# Topological Noetherianity of the infinite half-spin representations

Christopher Chiu  Department of Mathematics, KU Leuven, Celestijnenlaan 200b,
box 02400, 3001 Heverlee, Belgium<EMAIL_ADDRESS>,  Jan
Draisma  Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012
Bern, Switzerland<EMAIL_ADDRESS>,  Rob Eggermont  Department of
Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box
513, 5600MB, Eindhoven, the Netherlands<EMAIL_ADDRESS>,  Tim Seynnaeve
Department of Computer Science, KU Leuven, Celestijnenlaan 200A, Room 02.183,
3001 Leuven, Belgium<EMAIL_ADDRESS>and  Nafie Tairi  Mathematical
Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

We prove that the infinite half-spin representations are topologically
Noetherian with respect to the infinite spin group. As a consequence we obtain
that half-spin varieties, which we introduce, are defined by the pullback of
equations at a finite level. The main example for such varieties is the
infinite isotropic Grassmannian in its spinor embedding, for which we
explicitly determine its defining equations.

##  1\. Introduction

###  1.1. Purpose of this paper and main theorem

The purpose of this paper is to study certain varieties $X_{n}$ that live in
the half-spin representations of the even spin groups
$\operatorname{Spin}(2n)$ with $n$ varying. In particular, we will show that
these varieties are defined, for all $n$, by pulling back the equations for a
single $X_{n_{0}}$ along suitable contraction maps. The simplest instance of
such a variety is the Grassmannian of $n$-dimensional isotropic spaces in a
$2n$-dimensional orthogonal space. In this case, we use earlier work [ST24] by
the last two authors to show that $n_{0}$ can be taken equal to $4$; see
Theorem 6.1.

But the half-spin varieties that we introduce go far beyond the maximal
isotropic Grassmannian. Indeed, this class of varieties is preserved under
linear operations such as joins and tangential varieties, and under finite
unions and arbitrary intersections. Consequently, any variety obtained from
several copies of the maximal isotropic Grassmannian by such operations is
defined by equations of some degree bounded independently of $n$. We stress,
though, that these results are of a purely topological/set-theoretic nature.
It is not true, for instance, that one gets the entire ideal of the maximal
isotropic Grassmannian of $n$-spaces in a $2n$-space by pulling back equations
for $X_{4}$ along the maps that we define.

Our main results about half-spin varieties are Theorem 5.6, which establishes
a descending chain condition for these, and Corollary 5.8, which implies the
results mentioned above. These results follow from a companion result in
infinite dimensions, which is a little easier to state here. We will construct
a direct limit $\operatorname{Spin}(V_{\infty})$ of all spin groups; here
$V_{\infty}=\bigcup_{n}V_{n}$ is a countable-dimensional vector space with
basis $e_{1},f_{1},e_{2},f_{2},e_{3},f_{3},\ldots$ and a bilinear form
determined by $(e_{i}|e_{j})=(f_{i}|f_{j})=0$ and $(e_{i}|f_{j})=\delta_{ij}$.
Furthermore, we will construct a direct limit
$\bigwedge\nolimits_{\infty}^{+}E_{\infty}$ of all even half-spin
representations. This space has as basis all formal infinite products

$e_{i_{1}}\wedge e_{i_{2}}\wedge e_{i_{3}}\wedge\cdots$

where $\\{i_{1}<i_{2}<\ldots\\}$ is a cofinite subset of the positive
integers. The group $\operatorname{Spin}(V_{\infty})$ acts naturally on this
space, and hence on its dual
$(\bigwedge\nolimits_{\infty}^{+}E_{\infty})^{*}$, which we regard as the
spectrum of the symmetric algebra on
$\bigwedge\nolimits_{\infty}^{+}E_{\infty}$. Our main theorem is as follows.

######  Theorem 1.1.

The scheme $(\bigwedge\nolimits_{\infty}^{+}E_{\infty})^{*}$ is topologically
$\operatorname{Spin}(V_{\infty})$-Noetherian. That is, every chain

$X_{1}\supseteq X_{2}\supseteq X_{3}\supseteq\ldots$

of $\operatorname{Spin}(V_{\infty})$-stable reduced closed subschemes
stabilises.

###  1.2. Relations to the literature

Our work is primarily motivated by earlier work by the second and third author
on Plücker varieties, which live in exterior powers
$\bigwedge\nolimits^{n}K^{p+n}$ with both $p$ and $n$ varying. The results in
[DE18] on Plücker varieties are analoguous to the results we establish here
for half-spin varieties, and the main result in [Nek20] is an exact analogue
of Theorem 1.1 for the dual infinite wedge, acted upon by the infinite general
linear group.

On the one hand, we now have much better tools available to study these kind
of questions than we had at the time of [DE18]—notably the topological
Noetherianity of polynomial functors [Dra19] and their generalisation to
algebraic representations [ES22]. But on the other hand, spin representations
are much more intricate than polynomial functors, and a part of the current
paper will be devoted to establishing the precise relationship between the
infinite half-spin representation and algebraic representations of the
infinite general linear group, so as to use those tools.

This paper fits in a general programme that asks for which sequences of
representations of increasing groups one can expect Noetherianity results.
This seems to be an extremely delicate question. Indeed, while Theorem 1.1
establishes Noetherianity of the dual infinite half-spin representation, we do
not know whether the dual infinite spin representation is
$\operatorname{Spin}(V_{\infty})$-Noetherian; see Remark 4.9. Similarly, we do
not know whether a suitable inverse limit of exterior powers
$\bigwedge\nolimits^{n}V_{n}$ is
$\operatorname{SO}(V_{\infty})$-Noetherian—and there are many more natural
sequences of representations for which we do not yet have satisfactory
results.

In the context of secant varieties, we point out the work by Sam on Veronese
varieties: the $k$-th secant variety of the $d$-th Veronese embedding of
$\mathbb{P}(K^{n})$ is defined ideal-theoretically by finitely many types of
equations, independently of $n$—and in particular in bounded degree [Sam17a].
Furthermore, a similar statement holds for the $p$-th syzygies for any fixed
$p$ [Sam17b]. Similar results for ordinary Grassmannians were established by
Laudone in [Lau18]. It would be very interesting to know whether their
techniques apply to secant varieties of the maximal isotropic Grassmannian in
its spinor embedding. Our results here give a weaker set-theoretic statement,
but for a more general class of varieties.

After establishing Noetherianity, it would be natural to try and study
additional geometric properties of $\operatorname{Spin}(V_{\infty})$-stable
subvarieties of the dual infinite half-spin representation. Perhaps there is a
theory there analogous to the theory of $\operatorname{GL}$-varietes [BDES23a,
BDES23b]. However, we are currently quite far from any such deeper
understanding!

###  1.3. Organisation of this paper

In §2, we recall the construction of the (finite-dimensional) half-spin
representations. We mostly do this in a coordinate-free manner, only
choosing—as one must—a maximal isotropic subspace of an orthogonal space for
the construction. But for the construction of the infinite half-spin
representation, we will need explicit formulas, and these are derived in §2,
as well.

In §3, we first describe the embedding of the maximal isotropic Grassmannian
in the projectivised half-spin representation. Then, we define suitable
contraction and multiplication maps, which we show preserve the cones over
these isotropic Grassmannians. Finally, we use these maps to construct the
infinite-dimensional half-spin representations.

In §4, we prove Theorem 1.1 (see Theorem 4.1); and in §5, we state and prove
the main results about half-spin varieties discussed above. Finally, in §6 we
prove the universality of the isotropic Grassmannian of $4$-spaces in an
$8$-dimensional space. We do so by relating the half-spin representations via
the Cartan map to the exterior power representations and using results from
[ST24].

##  2\. Finite spin representations and the spin group

In this section we collect some preliminaries on spin groups and their
defining representations. Throughout we will assume that $K$ is an
algebraically closed field of characteristic $0$. We follow [Man09] in our
set-up; for more general references on spin groups and their representations
see [LM89, Pro07].

###  2.1. The Clifford algebra

Let $V$ be a finite-dimensional vector space over $K$ endowed with a quadratic
form $q$. The Clifford algebra $\operatorname{Cl}(V,q)$ of $V$ is the quotient
of the tensor algebra $T(V)=\bigoplus_{d\geq 0}V^{\otimes d}$ by the two-sided
ideal generated by all elements

$v\otimes v-q(v)\cdot 1,\>v\in V.$ (2.1)

This is also the two-sided ideal generated by

$v\otimes w+w\otimes v-2(v|w)\cdot 1,\>v,w\in V,$ (2.2)

where $(\cdot|\cdot)$ denotes the bilinear form associated to $q$ defined by
$(v|w):=\frac{1}{2}(q(v+w)-q(v)-q(w))$.

The Clifford algebra is a functor from the category of vector spaces equipped
with a quadratic form to the category of (unital) associative algebras. That
is, any linear map $\varphi\colon(V,q)\to(V^{\prime},q^{\prime})$ with
$q^{\prime}(\varphi(v))=q(v)$ for all $v\in V$ induces a homomorphism of
associative algebras
$\operatorname{Cl}(\varphi)\colon\operatorname{Cl}(V,q)\to\operatorname{Cl}(V^{\prime},q^{\prime})$.
If $\varphi$ is an inclusion $V\subseteq V^{\prime}$, then
$\operatorname{Cl}(\varphi)$ is injective, and hence $\operatorname{Cl}(V,q)$
is a subalgebra of $\operatorname{Cl}(V^{\prime},q^{\prime})$.

The decomposition of $T(V)$ into the even part $T^{+}(V):=\bigoplus_{d\text{
even}}V^{\otimes d}$ and the odd part $T^{-}(V):=\bigoplus_{d\text{
odd}}V^{\otimes d}$ induces a decomposition
$\operatorname{Cl}(V,q)=\operatorname{Cl}^{+}(V,q)\oplus\operatorname{Cl}^{-}(V,q)$,
turning $\operatorname{Cl}(V,q)$ into a ${\mathbb{Z}}/2{\mathbb{Z}}$-graded
associative algebra. Note that, via the commutator on
$\operatorname{Cl}(V,q)$, the even Clifford algebra
$\operatorname{Cl}^{+}(V,q)$ is a Lie subalgebra of $\operatorname{Cl}(V,q)$.

The anti-automorphism of $T(V)$ determined by $v_{1}\otimes\cdots\otimes
v_{d}\mapsto v_{d}\otimes\cdots\otimes v_{1}$ preserves the ideal in the
definition of $\operatorname{Cl}(V,q)$ and therefore induces an anti-
automorphism $x\mapsto x^{*}$ of $\operatorname{Cl}(V,q)$.

###  2.2. The Grassmann algebra as a $\operatorname{Cl}(V)$-module

From now on, we will write $\operatorname{Cl}(V)$ for $\operatorname{Cl}(V,q)$
when $q$ is clear from the context. If $q=0$, then
$\operatorname{Cl}(V)=\bigwedge\nolimits V$, the Grassmann algebra of $V$. If
$E\subseteq V$ is an isotropic subspace, that is, a subspace for which
$q|_{E}=0$, then this fact allows us to identify $\bigwedge\nolimits E$ with
the subalgebra $\operatorname{Cl}(E)$ of $\operatorname{Cl}(V)$.

For general $q$, $\operatorname{Cl}(V)$ is not isomorphic as an algebra to
$\bigwedge\nolimits V$, but $\bigwedge\nolimits V$ is naturally a
$\operatorname{Cl}(V)$-module as follows. For $v\in V$ define
$o(v):\bigwedge\nolimits V\to\bigwedge\nolimits V$ (the “outer product”) as
the linear map

$o(v)\omega:=v\wedge\omega$

and $\iota(v):\bigwedge\nolimits V\to\bigwedge\nolimits V$ (the “inner
product”) as the linear map determined by

$\iota(v)w_{1}\wedge\cdots\wedge w_{k}:=\sum_{i=1}^{k}(-1)^{i-1}(w_{i}\mid
v)w_{1}\wedge\cdots\wedge\widehat{w}_{i}\wedge\cdots\wedge w_{k}.$

Here, and elsewhere in the paper, $\widehat{\cdot}$ indicates a factor that is
left out. Now $v\mapsto\iota(v)+o(v)$ extends to an algebra homomorphism
$\operatorname{Cl}(V)\to\operatorname{End}(\bigwedge\nolimits V)$. To see
this, it suffices to consider $v,w_{1},\ldots,w_{k}\in V$ and verify

$(\iota(v)+o(v))^{2}w_{1}\wedge\cdots\wedge w_{k}=(v|v)w_{1}\wedge\cdots\wedge
w_{k}.$

We write $a\bullet\omega$ for the outcome of $a\in\operatorname{Cl}(V)$ acting
on $\omega\in\bigwedge\nolimits V$. Using induction on the degree of a
product, the linear map $\operatorname{Cl}(V)\to\bigwedge\nolimits V,a\mapsto
a\bullet 1$ is easily seen to be an isomorphism of vector spaces. In
particular, $\operatorname{Cl}(V)$ has dimension $2^{\dim V}$.

###  2.3. Embedding $\mathfrak{so}(V)$ into the Clifford algebra

From now on, we assume that $q$ is nondegenerate and write
$\operatorname{SO}(V)=\operatorname{SO}(V,q)$ for the special orthogonal group
of $q$. Its Lie algebra $\mathfrak{so}(V)$ consists of linear maps $V\to V$
that are skew-symmetric with respect to $(\cdot|\cdot)$, that is, those
$A\in\operatorname{End}(V)$ such that $(Av|w)=-(v|Aw)$ for all $v,w\in V$. We
have a unique linear map
$\psi:\bigwedge\nolimits^{2}V\to\operatorname{Cl}^{+}(V)$ with $\psi(u\wedge
v)=uv-vu$, and $\psi$ is injective. A straightforward computation shows that
the image $L$ of $\psi$ is closed under the commutator in
$\operatorname{Cl}(V)$, hence a Lie subalgebra. We claim that $L$ is
isomorphic to $\mathfrak{so}(V)$. Indeed, for $u,v,w\in V$ we have

$[\psi(u\wedge v),w]=[[u,v],w]=4(v|w)u-4(u|w)v.$

We see, first, that $V\subseteq\operatorname{Cl}(V)$ is preserved under the
adjoint action of $L$; and second, that $L$ acts on $V$ via skew-symmetric
linear maps, so that $L$ maps into $\mathfrak{so}(V)$. Since every map in
$\mathfrak{so}(V)$ is a linear combination of the linear maps above, and since
$\dim(L)=\dim(\mathfrak{so}(V))$, the map $L\to\mathfrak{so}(V)$ is an
isomorphism. We will identify $\mathfrak{so}(V)$ with the Lie subalgebra
$L\subseteq\operatorname{Cl}(V)$ via the inverse of this isomorphism, and we
will identify $\bigwedge\nolimits^{2}V$ with $\mathfrak{so}(V)$ via the map
$u\wedge v\mapsto(w\mapsto(v|w)u-(u|w)v)$. The concatenation of these
identifications is the linear map $\frac{1}{4}\psi$.

###  2.4. The half-spin representations

From now on, we assume that $\dim(V)=2n$. We believe that all our results hold
mutatis mutandis also in the odd-dimensional case, but we have not checked the
details. A maximal isotropic subspace $U$ of $V$ is an isotropic subspace
which is maximal with respect to inclusion. Since $K$ is algebraically closed,
$q$ has maximal Witt index, so that every maximal isotropic subspace of $V$
has dimension $n$.

The spin representation of $\mathfrak{so}(V)$ is constructed as follows. Let
$F$ be a maximal isotropic subspace of $V$ and let $f_{1},\ldots,f_{n}$ be a
basis of $F$. Define $f:=f_{1}\cdots f_{n}\in\operatorname{Cl}(F)$; this
element in $\operatorname{Cl}(F)=\bigwedge\nolimits F$ is well-defined up to a
scalar. Then the left ideal $\operatorname{Cl}(V)\cdot f$ is a left module for
the associative algebra $\operatorname{Cl}(V)$, and hence for its Lie
subalgebra $\mathfrak{so}(V)$. This ideal is called the spin representation of
$\mathfrak{so}(V)$. As $\operatorname{Cl}(V)$ is
${\mathbb{Z}}/2{\mathbb{Z}}$-graded, the spin representation splits into a
direct sum of two subrepresentations for $\operatorname{Cl}^{+}(V)$, and hence
for $\mathfrak{so}(V)\subseteq\operatorname{Cl}^{+}(V)$, namely,
$\operatorname{Cl}^{+}(V)\cdot f$ and $\operatorname{Cl}^{-}(V)\cdot f$. These
representations are called the half-spin representations of
$\mathfrak{so}(V)$.

###  2.5. Explicit formulas

We will need more explicit formulas for the action of $\mathfrak{so}(V)$ on
the half-spin representations. To this end, let $E$ be another isotropic
$n$-dimensional subspace of $V$ such that $V=E\oplus F$. Then the map

$\bigwedge\nolimits
E=\operatorname{Cl}(E)\to\operatorname{Cl}(V)f,\quad\omega\mapsto\omega f$

is a linear isomorphism, and we use it to identify $\bigwedge\nolimits E$ with
the spin representation. We write
$\rho:\mathfrak{so}(V)\to\operatorname{End}(\bigwedge\nolimits E)$ for the
corresponding representation. It splits as a direct sum of the half-spin
representations
$\rho_{+}:\mathfrak{so}(V)\to\operatorname{End}(\bigwedge\nolimits^{+}E)$ and
$\rho_{-}:\mathfrak{so}(V)\to\operatorname{End}(\bigwedge\nolimits^{-}E)$
where $\bigwedge\nolimits^{+}E=\bigoplus_{d\text{
even}}\bigwedge\nolimits^{d}E$ and $\bigwedge\nolimits^{-}E=\bigoplus_{d\text{
odd}}\bigwedge\nolimits^{d}E$.

In this model of the spin representation, the action of $v\in
E\subseteq\operatorname{Cl}(V)$ on the spin representation $\bigwedge\nolimits
E$ is just the outer product on $\bigwedge E:o(v):\bigwedge\nolimits
E\to\bigwedge\nolimits E,\ \omega\mapsto v\wedge\omega$, while the action of
$v\in F\subseteq\operatorname{Cl}(V)$ is twice the inner product on $\bigwedge
E$:

$2\iota(v)w_{1}\wedge\cdots\wedge
w_{k}=2\sum_{i=1}^{k}(-1)^{i-1}(v|w_{i})w_{1}\wedge\cdots\wedge\widehat{w_{i}}\wedge\cdots\wedge
w_{k}.$

The factor $2$ and the alternating signs come from the following identity in
$\operatorname{Cl}(V)$:

$vv_{i}=2(v|v_{i})-v_{i}v\text{ for }v\in F\text{ and }v_{i}\in E.$

For a general $v\in V$ we write $v=v^{\prime}+v^{\prime\prime}$ with
$v^{\prime}\in E$, $v^{\prime\prime}\in F$. Then the action of $V$ on
$\bigwedge\nolimits E$ is given by

$v\mapsto o(v^{\prime})+2\iota(v^{\prime\prime}).$

We now compute the linear maps by means of which $\mathfrak{so}(V)$ acts on
$\bigwedge\nolimits E$. To this end, recall that a pair $e,f\in V$ is called
hyperbolic if $e$, $f$ are isotropic and $(e|f)=1$. Given the basis
$f_{1},\ldots,f_{n}$ of $F$, there is a unique basis $e_{1},\ldots,e_{n}$ of
$E$ so that $(e_{i}|f_{j})=\delta_{ij}$; then
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ is called a hyperbolic basis of $V$.
Now the element $e_{i}\wedge e_{j}\in\mathfrak{so}(V)$ acts on
$\bigwedge\nolimits E\simeq\operatorname{Cl}(V)f$ via the linear map

$\frac{1}{4}(o(e_{i})o(e_{j})-o(e_{j})o(e_{i}))=\frac{1}{2}o(e_{i})o(e_{j});$

the element $f_{i}\wedge f_{j}$ acts via the linear map

$\frac{1}{4}(4\iota(f_{i})\iota(f_{j})-4\iota(f_{j})\iota(f_{i}))=2\iota(f_{i})\iota(f_{j});$

and the element $e_{i}\wedge f_{j}$ acts via the linear map

$\frac{1}{4}(o(e_{i})2\iota(f_{j})-2\iota(f_{j})o(e_{i}))=\frac{1}{2}(o(e_{i})\iota(f_{j})-\iota(f_{j})o(e_{i})).$

In particular, $\omega_{0}:=e_{1}\wedge\cdots\wedge e_{n}\in\bigwedge\nolimits
E$ is mapped to $0$ by all elements $e_{i}\wedge e_{j}$ and all elements
$e_{i}\wedge f_{j}$ with $i\neq j$, and it is mapped to
$\frac{1}{2}\omega_{0}$ by all $e_{i}\wedge f_{i}$.

###  2.6. Highest weights of the half-spin representations

Recall, e.g. from [Jac62, Chapter IV, pages 140–141], that in the basis
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$, matrices in $\mathfrak{so}(V)$ have
the form

$\begin{bmatrix}A&B\\\ C&-A^{T}\end{bmatrix}\text{ with }B^{T}=-B,\text{ and
}C^{T}=-C.$

Here the $(e_{i},e_{j})$-entry of $A$ is the coefficient of $e_{i}\wedge
f_{j}$, the $(e_{i},f_{j})$-entry of $B$ is the coefficient of $e_{i}\wedge
e_{j}$, and the $(f_{i},e_{j})$-entry of $C$ is the coefficient of
$f_{i}\wedge f_{j}$.

The diagonal matrices $e_{i}\wedge f_{i}$ span a Cartan subalgebra of
$\mathfrak{so}(V)$ with standard (Chevalley) basis consisting of
$h_{i}:=e_{i}\wedge f_{i}-e_{i+1}\wedge f_{i+1}$ for $i=1,\ldots,n-1$ and
$h_{n}:=e_{n-1}\wedge f_{n-1}+e_{n}\wedge f_{n}$ (this last element is
forgotten in the basis of the Cartan algebra on [Jac62, page 140]).

Now $(e_{i}\wedge e_{j})\omega_{0}=(e_{i}\wedge f_{j})\omega_{0}=0$ for all
$i\neq j$. Furthermore, the elements $h_{1},\ldots,h_{n-1}$ map $\omega_{0}$
to $0$, while $h_{n}$ maps $\omega_{0}$ to $\omega_{0}$. Thus the Borel
subalgebra maps the line $K\omega_{0}$ into itself and $\omega_{0}$ is a
highest weight vector of the fundamental weight $\lambda_{0}:=(0,\ldots,0,1)$
in the standard basis. Summarising, $\omega_{0}\in\bigwedge\nolimits E$
generates a copy of the irreducible $\mathfrak{so}(V)$-module
$V_{\lambda_{0}}$ with highest weight $\lambda_{0}$. Clearly, the
$\mathfrak{so}(V)$-module generated by $\omega_{0}$ is contained in
$\bigwedge\nolimits^{+}E$ if $n$ is even, and contained in
$\bigwedge\nolimits^{-}E$ when $n$ is odd. One can also show that both half-
spin representations are irreducible, hence one of them is a copy of
$V_{\lambda_{0}}$. For the other half-spin representation, consider the
element

$\omega_{1}:=e_{1}\wedge\cdots\wedge e_{n-1}\in\bigwedge\nolimits E.$

This element is mapped to zero by $e_{i}\wedge e_{j}$ for all $i\neq j$ and by
$e_{i}\wedge f_{j}$ for all $i<j$. It is further mapped to $0$ by
$h_{1},\ldots,h_{n-2},h_{n}$, and to $\omega_{1}$ by $h_{n-1}$. For example,
we have

$\displaystyle h_{n}\omega_{1}$
$\displaystyle=\frac{1}{2}\left(o(e_{n-1})\iota(f_{n-1})-\iota(f_{n-1})o(e_{n-1})+o(e_{n})\iota(f_{n})-\iota(f_{n})o(e_{n})\right)e_{1}\wedge\cdots\wedge
e_{n-1}$ $\displaystyle=\frac{1}{2}(1-0+0-1)\omega_{1}=0,\text{ and
similarly}$ $\displaystyle h_{n-1}\omega_{1}$
$\displaystyle=\frac{1}{2}(1-0-0+1)\omega_{1}=\omega_{1}.$

Hence $\omega_{1}$ generates a copy of $V_{\lambda_{1}}$, the irreducible
$\mathfrak{so}(V)$-module of highest weight $\lambda_{1}:=(0,\ldots,0,1,0)$;
this is the other half-spin representation.

###  2.7. The spin group

Let $\rho:\mathfrak{so}(V)\to\operatorname{End}(\bigwedge\nolimits E)$ be the
spin representation. The spin group $\operatorname{Spin}(V)$ can be defined as
the subgroup of $\operatorname{GL}(\bigwedge\nolimits E)$ generated by the
one-parameter subgroups $t\mapsto\exp(t\rho(X))$ where $X$ runs over the root
vectors $e_{i}\wedge e_{j},f_{i}\wedge f_{j}$ and $e_{i}\wedge f_{j}$ with
$i\neq j$. Note that $\rho(X)$ is nilpotent for each of these root vectors, so
that $t\mapsto\exp(t\rho(X))$ is an algebraic group homomorphism
$K\to\operatorname{GL}(\bigwedge\nolimits E)$. It is a standard fact that the
subgroup generated by irreducible curves through the identity in an algebraic
group is itself a connected algebraic group; see [Bor91, Proposition 2.2]. So
$\operatorname{Spin}(V)$ is a connected algebraic group, and one verifies that
its Lie algebra is isomorphic to the Lie algebra generated by the root vectors
$X$, i.e., to $\mathfrak{so}(V)$.

By construction, the (half-)spin representations $\bigwedge\nolimits E$,
$\bigwedge\nolimits^{+}E$ and $\bigwedge\nolimits^{-}E$ are representations of
$\operatorname{Spin}(V)$. We use the same notation
$\rho:\operatorname{Spin}(V)\to\operatorname{GL}(\bigwedge\nolimits E),\
\rho_{+}\colon\operatorname{Spin}(V)\to\operatorname{GL}(\bigwedge\nolimits^{+}E),\text{
and
}\rho_{-}\colon\operatorname{Spin}(V)\to\operatorname{GL}(\bigwedge\nolimits^{-}E)$
for these as for the corresponding Lie algebra representations.

######  Remark 2.1.

The algebraic group $\operatorname{Spin}(V)$ is usually constructed as a
subgroup of the unit group $\operatorname{Cl}^{\times}(V)$ as follows:
consider first

$\Gamma(V)=\\{x\in\operatorname{Cl}^{\times}(V)\mid xVx^{-1}=V\\},$

sometimes called the Clifford group. Then $\operatorname{Spin}(V)$ is the
subgroup of $\Gamma(V)$ of elements of _spinor norm_ $1$; that is, $xx^{*}=1$,
where $x^{*}$ denotes the involution defined in Section 2.1. In this model of
the spin group it is easy to see that it admits a $2:1$ covering
$\operatorname{Spin}(V)\to\operatorname{SO}(V)$, namely, the restriction of
the homomorphism $\Gamma(V)\to\mathrm{O}(V)$ given by associating to
$x\in\Gamma(V)$ the orthogonal transformation $w\mapsto xwx^{-1}$. For more
details see [Pro07]. Since our later computations involve the Lie algebra
$\mathfrak{so}(V)$ only, the definition of $\operatorname{Spin}(V)$ above
suffices for our purposes.

The half-spin representations are not representations of the group
$\operatorname{SO}(V)$; this can be checked, e.g., by showing that the highest
weights $\lambda_{0}$ and $\lambda_{1}$ are not in the weight lattice of
$\operatorname{SO}(V)$.

###  2.8. Two actions of $\mathfrak{gl}(E)$ on $\bigwedge\nolimits E$

The definition of the (half-)spin representation(s) of $\mathfrak{so}(V)$ and
$\operatorname{Spin}(V)$ as $\operatorname{Cl}^{(\pm)}(V)f$ involves only the
quadratic form $q$ and the choice of a maximal isotropic space $F\subseteq V$.
Consequently, any linear automorphism of $V$ that preserves $q$ and maps $F$
into itself also acts on $\operatorname{Cl}^{(\pm)}(V)f$. These linear
automorphisms form the stabiliser of $F$ in $\operatorname{SO}(V)$, which is
the parabolic subgroup whose Lie algebra consists of the matrices in
$\operatorname{SO}(V)$ that are block lower triangular in the basis
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$. So, while $\operatorname{SO}(V)$ does
not act naturally on the (half-)spin representation(s), this stabiliser does.

In particular, in our model $\bigwedge\nolimits^{(\pm)}E$ of the (half-)spin
representation(s), the group $\operatorname{GL}(E)$, embedded into
$\operatorname{SO}(V)$ as the subgroup of block diagonal matrices

$\begin{bmatrix}a&0\\\ 0&-a^{T}\end{bmatrix}$

acts on $\bigwedge\nolimits E$ in the natural manner. We stress that this is
not the action obtained by integrating the action of
$\mathfrak{gl}(E)\subseteq\mathfrak{so}(V)$ on $\bigwedge\nolimits E$ regarded
as the spin representation. Indeed, the standard action of $e_{i}\wedge
f_{j}\in\mathfrak{gl}(E)$ on $\omega:=e_{i_{1}}\wedge\cdots\wedge
e_{i_{k}}\in\bigwedge\nolimits^{k}E$ yields

$\sum_{l=1}^{k}e_{i_{1}}\wedge\cdots\wedge
e_{i}(f_{j}|e_{i_{l}})\wedge\cdots\wedge e_{i_{k}}=\begin{cases}0\text{ if
$j\not\in\\{i_{1},\ldots,i_{k}\\}$}\\\ (-1)^{l-1}e_{i}\wedge
e_{i_{1}}\wedge\cdots\wedge\widehat{e_{i_{l}}}\wedge\cdots\wedge
e_{i_{k}}\text{ if $j=i_{l}$.}\end{cases}$

On the other hand, in the spin representation the action is given by the
linear map $\frac{1}{2}(o(e_{i})\iota(f_{j})-\iota(f_{j})o(e_{i}))$. If $j\neq
i$ and $j\not\in\\{i_{1},\ldots,i_{k}\\}$, then

$o(e_{i})\iota(f_{j})\omega=\iota(f_{j})o(e_{i})\omega=0.$

If $j\neq i$ and $j=i_{l}$, then

$o(e_{i})\iota(f_{j})\omega=(-1)^{l-1}e_{i}\wedge
e_{i_{1}}\wedge\cdots\wedge\widehat{e_{i_{l}}}\wedge\cdots\wedge
e_{i_{k}}=-\iota(f_{j})o(e_{i})\omega.$

We conclude that for $i\neq j$, the action of $e_{i}\wedge f_{j}$ is the same
in both representations. However, if $i=j$, then

$\frac{1}{2}(o(e_{i})\iota(f_{i})-\iota(f_{i})o(e_{i}))\omega=\begin{cases}-\frac{1}{2}\omega\text{
if $i\not\in\\{i_{1},\ldots,i_{k}\\}$, and}\\\
\frac{1}{2}(-1)^{l-1}e_{i}\wedge
e_{i_{1}}\wedge\cdots\wedge\widehat{e_{i_{l}}}\wedge\cdots\wedge
e_{i_{k}}=\frac{1}{2}\omega\text{ if $i=i_{l}$}.\end{cases}$

We conclude that if
$\tilde{\rho}:\mathfrak{gl}(E)\to\operatorname{End}(\bigwedge\nolimits E)$ is
the standard representation of $\mathfrak{gl}(E)$, then the restriction of the
spin representation
$\rho:\mathfrak{so}(V)\to\operatorname{End}(\bigwedge\nolimits E)$ to
$\mathfrak{gl}(E)$ as a subalgebra of $\mathfrak{so}(V)$ satisfies

$\rho(A)=\tilde{\rho}(A)-\frac{1}{2}\operatorname{tr}(A)\operatorname{Id}_{\bigwedge\nolimits
E}.$ (2.3)

At the group level, this is to be understood as follows. The pre-image of
$\operatorname{GL}(E)\subseteq\operatorname{SO}(V)$ in
$\operatorname{Spin}(V)$ is isomorphic to the connected algebraic group

$H:=\big{\\{}(g,t)\in\operatorname{GL}(E)\times
K^{*}\mid\det(g)=t^{2}\big{\\}}$

for which $(g,t)\mapsto g$ is a $2:1$ cover of $\operatorname{GL}(E)$, and the
restriction of $\rho$ to $H$ satisfies $\rho(g,t)=\tilde{\rho}(g)\cdot
t^{-1}$—a “twist of the standard representation by the inverse square root of
the determinant”.

##  3\. The isotropic Grassmannian and infinite spin representations

###  3.1. The isotropic Grassmannian in its spinor embedding

As before, let $V$ be a $2n$-dimensional vector space over $K$ endowed with a
nondegenerate quadratic form. The (maximal) _isotropic Grassmannian_
$\operatorname{Gr}_{\operatorname{iso}}(V,q)$ parametrizes all maximal
isotropic subspaces of $V$. It has two connected components, denoted
$\operatorname{Gr}^{+}_{\operatorname{iso}}(V)$ and
$\operatorname{Gr}^{-}_{\operatorname{iso}}(V)$. The goal of this subsection
is to introduce the isotropic Grassmann cone, which is an affine cone over
$\operatorname{Gr}_{\operatorname{iso}}(V,q)$ in the spin representation.

Fix a maximal isotropic subspace $F\subseteq V$ and as before set $f\coloneqq
f_{1}\cdots f_{n}\in\operatorname{Cl}(V)$, where $f_{1},\dots,f_{n}$ is any
basis of $F$. Now let $H\subseteq V$ be another maximal isotropic space. Then
we claim that the space

$S_{H}:=\\{\omega\in\operatorname{Cl}(V)f\mid v\cdot\omega=0\text{ for all
}v\in H\\}\subseteq\operatorname{Cl}(V)f$ (3.1)

is $1$-dimensional. Indeed, we may find a hyperbolic basis
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ of $V$ such that $f_{1},\ldots,f_{k}$
span $H\cap F$, $f_{1},\ldots,f_{n}$ span $F$, and
$e_{k+1},\ldots,e_{n},f_{1},\ldots,f_{k}$ span $H$. We call this hyperbolic
basis adapted to $H$ and $F$. Then the element

$\omega_{H}:=e_{k+1}\cdots e_{n}f_{1}\cdots f_{k}f_{k+1}\cdots
f_{n}\in\operatorname{Cl}(V)f$

lies in $S_{H}$ since $e_{i}\omega_{H}=f_{j}\omega_{H}=0$ for all $i>k$ and
$j\leq k$. Conversely, if $\mu\in S_{H}$, then write

$\mu=\sum_{l=0}^{n}\sum_{i_{1}<\ldots<i_{l}}c_{\\{i_{1},\ldots,i_{l}\\}}e_{i_{1}}\cdots
e_{i_{l}}f.$

If $c_{I}\neq 0$ for some $I$ with $I\not\supseteq\\{k+1,\ldots,n\\}$, then
for any $j\in\\{k+1,\ldots,n\\}\setminus I$ we find that $e_{j}\mu\neq 0$. So
all $I$ with $c_{I}\neq 0$ contain $\\{k+1,\ldots,n\\}$. If some $I$ with
$c_{I}\neq 0$ further contains an $i\leq k$, then $f_{i}\mu$ is nonzero. Hence
$S_{H}$ is spanned by $\omega_{H}$, as claimed. In what follows, by slight
abuse of notation, we will write $\omega_{H}$ for any nonzero vector in
$S_{H}$.

The space $H$ can be uniquely recovered from $\omega_{H}$ via

$H=\\{v\in V\mid v\cdot\omega_{H}=0\\}.$

Indeed, we have already seen $\subseteq$. For the converse, observe that the
vectors $e_{i}\omega_{H},f_{j}\omega_{H}$ with $i\leq k$ and $j>k$ are
linearly independent.

The map that sends $H\in\operatorname{Gr}_{\operatorname{iso}}(V,q)$ to the
projective point representing it, i.e.,

$H\mapsto[\omega_{H}]\in{\mathbb{P}}(\operatorname{Cl}(V)f),$

is therefore injective, and it is called the spinor embedding of the isotropic
Grassmannian (see [Man09]). The isotropic Grassmann cone is defined as

$\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V,q)\coloneqq\bigcup_{H}\langle\omega_{H}\rangle\subseteq\operatorname{Cl}(V)f,$

where the union is the taken over all maximal isotropic subspaces $H\subseteq
V$. We denote by
$\widehat{\operatorname{Gr}}^{\pm}_{\operatorname{iso}}(V,q)\coloneqq\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V,q)\cap\operatorname{Cl}^{\pm}(V)f$
the cones over the connected components of the isotropic Grassmannian in its
spinor embedding.

###  3.2. Contraction with an isotropic vector

Let $e\in V$ be a nonzero isotropic vector. Then $V_{e}:=e^{\perp}/\langle
e\rangle$ is equipped with a natural nondegenerate quadratic form, and there
is a rational map
$\operatorname{Gr}_{\operatorname{iso}}(V)\to\operatorname{Gr}_{\operatorname{iso}}(V_{e})$
that maps an $n$-dimensional isotropic space $H$ to the image in $V_{e}$ of
the $(n-1)$-dimensional isotropic space $H\cap e^{\perp}$ (this is defined if
$e\not\in H$, which by maximality of $H$ is equivalent to $H\not\subseteq
e^{\perp}$). This map is the restriction to
$\operatorname{Gr}_{\operatorname{iso}}(V)$ of the rational map
$\mathbb{P}(\bigwedge\nolimits^{n}V)\to\mathbb{P}(\bigwedge\nolimits^{n-1}V_{e})$
induced by the linear map (“contraction with $e$”):

$c_{e}:\bigwedge\nolimits^{n}V\to\bigwedge\nolimits^{n-1}V_{e},\quad
v_{1}\wedge\cdots\wedge
v_{n}\mapsto\sum_{i=1}^{n}(-1)^{i-1}(e|v_{i})\overline{v_{1}}\wedge\cdots\wedge\widehat{v_{i}}\wedge\cdots\wedge\overline{v_{n}}$

where $\overline{v_{i}}$ is the image of $v_{i}$ in $V/\langle e\rangle$. Note
first that this map is the inner product $\iota(e)$ followed by a projection.
Furthermore, a priori, the codomain of this map is the larger space
$\bigwedge\nolimits^{n-1}(V/\langle e\rangle)$, but one may choose
$v_{1},\ldots,v_{n}$ such that $(e|v_{i})=0$ for $i>1$, and then it is evident
that the image is indeed in $\bigwedge\nolimits^{n-1}V_{e}$.

We want to construct a similar contraction map at the level of the spin
representation. For reasons that will become clear in a moment, we restrict
our attention first to a map between two half-spin representations, as
follows. Assume that $e\notin F$, and choose a basis $f_{1},\ldots,f_{n}$ of
$F$ such that $(e|f_{i})=\delta_{in}$. As usual, write $f:=f_{1}\cdots f_{n}$,
and write $\overline{f}:=\overline{f}_{1}\cdots\overline{f}_{n-1}$, so that
$\operatorname{Cl}^{+}(V_{e})\overline{f}$ is a half-spin representation of
$\mathfrak{so}(V_{e})$.

Then we define the map

$\pi_{e}:\operatorname{Cl}^{+}(V)f\to\operatorname{Cl}^{+}(V_{e})\overline{f},\quad\pi_{e}(af):=\text{
the image of }\frac{1}{2}((-1)^{n-1}eaf+afe)\text{ in
}\operatorname{Cl}(V_{e})\overline{f},$

where the implicit claim is that the expression on the right lies in
$\operatorname{Cl}(e^{\perp})f_{1}\cdots f_{n-1}$, so that its image in
$\operatorname{Cl}(V_{e})\overline{f}$ is well-defined (note that the
projection $e^{\perp}\to V_{e}$ induces a homomorphism of Clifford algebras),
and that this image lies in the left ideal generated by $\overline{f}$. To
verify this claim, and to derive a more explicit formula for the map above,
let $e_{1},\ldots,e_{n}=e$ be a basis of an isotropic space $E$ complementary
to $F$. Then it suffices to consider the case where $a=e_{i_{1}}\cdots
e_{i_{k}}$ for some $i_{1}<\ldots<i_{k}$. We then have

$\displaystyle eaf$ $\displaystyle=ee_{i_{1}}\cdots e_{i_{k}}f_{1}\cdots
f_{n}$ $\displaystyle=\begin{cases}0\text{ if $i_{k}=n$, and}\\\
2(-1)^{k+n-1}e_{i_{1}}\cdots e_{i_{k}}f_{1}\cdots
f_{n-1}+(-1)^{k+n}e_{i_{1}}\cdots e_{i_{k}}f_{1}\cdots f_{n}e\text{
otherwise.}\end{cases}$

Multiplying by $(-1)^{n-1}$ and using that $k$ is even, the latter expression
becomes

$2e_{i_{1}}\cdots e_{i_{k}}f_{1}\cdots f_{n-1}-afe.$

Hence we conclude that

$\pi_{e}(e_{i_{1}}\cdots e_{i_{k}}f)=\begin{cases}0&\text{if $i_{k}=n$,
and}\\\
\overline{e}_{i_{1}}\cdots\overline{e}_{i_{k}}\overline{f}&\text{otherwise.}\end{cases}$

In short, in our models $\bigwedge\nolimits^{+}E$ and
$\bigwedge\nolimits^{+}(E/\langle e\rangle)$ for the half-spin representations
of $\mathfrak{so}(V)$ and $\mathfrak{so}(V_{e})$, $\pi_{e}$ is just the
reduction-mod-$e$ map. We leave it to the reader to check that the reduction-
mod-$e$ map $\bigwedge\nolimits^{-}E\to\bigwedge\nolimits^{-}(E/\langle
e\rangle)$ arises in a similar fashion from the map

$\pi_{e}:\operatorname{Cl}(V)^{-}f\to\operatorname{Cl}(V_{e})^{-}\overline{f},\quad\pi_{e}(af):=\text{
the image of }\frac{1}{2}((-1)^{n}eaf+afe)\text{ in
}\operatorname{Cl}(V_{e})\overline{f}.$

We will informally call the maps $\pi_{e}$ “contraction with $e$”. Together
they define a map on $\operatorname{Cl}(V)f$ which we also denote by
$\pi_{e}$.

######  Proposition 3.1.

The contraction map
$\pi_{e}:\operatorname{Cl}(V)f\to\operatorname{Cl}(V)\overline{f}$ is a
homomorphism of $\operatorname{Cl}(e^{\perp})$-representations.

###### Proof.

Let $v\in e^{\perp}$ and consider $a\in\operatorname{Cl}^{-}(V)$. Then
$va\in\operatorname{Cl}^{+}(V)$ and hence $\pi_{e}(vaf)$ is the image in
$\operatorname{Cl}(V_{e})\overline{f}$ of

$\frac{1}{2}((-1)^{n-1}evaf+vafe)=\frac{1}{2}((-1)^{n}veaf+vafe)=v\frac{1}{2}((-1)^{n}eaf+afe),$

where we have used $(v|e)=0$ in the first equality. The right-hand side
clearly equals $\overline{v}$ times the image of $\pi_{e}(af)$ in
$\operatorname{Cl}(V_{e})\overline{f}$. ∎

###  3.3. Multiplying with an isotropic vector

In a sense dual to the contraction maps
$c_{e}:\bigwedge\nolimits^{n}V\to\bigwedge\nolimits^{n-1}V_{e}$ are
multiplication maps defined as follows. Let $e,h\in V$ be isotropic with
$(e|h)=1$; such a pair is called a hyperbolic pair. We then have $V=\langle
e,h\rangle\oplus\langle e,h\rangle^{\perp}$, and the map from the second
summand to $V_{e}=e^{\perp}/\langle e\rangle$ is an isometry. We use this
isometry to identify $V_{e}$ with the subspace $\langle e,h\rangle^{\perp}$ of
$V$ and write $s_{e}$ for the corresponding inclusion map. Then we define

$m_{h}:\bigwedge\nolimits^{n-1}V_{e}\to\bigwedge\nolimits^{n}V,\quad\overline{v}_{1}\wedge\cdots\wedge\overline{v}_{n-1}\mapsto
h\wedge\overline{v}_{1}\wedge\cdots\wedge\overline{v}_{n-1},$

which is just the outer product $o(h)$. The projectivisation of this map sends
$\operatorname{Gr}_{\operatorname{iso}}(V_{e})$ isomorphically to the closed
subset of $\operatorname{Gr}_{\operatorname{iso}}(V)$ consisting of all $H$
containing $h$. We further observe that

$c_{e}\circ m_{h}=\operatorname{id}_{\bigwedge\nolimits^{n-1}V_{e}}.$

We define a corresponding multiplication map at the level of spin
representations as follows: first, we assume that $h\in F$, and choose a basis
$f_{1},\ldots,f_{n}=h$ of $F$ such that $(e|f_{i})=\delta_{in}$. As usual, we
set $f=f_{1}\cdots f_{n}$ and
$\overline{f}=\overline{f}_{1}\cdots\overline{f}_{n-1}$. Then we define

$\tau_{h}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f,\quad\tau_{h}(a\overline{f}):=a\overline{f}f_{n}=af.$

Note that, for $a\in\operatorname{Cl}(V_{e})$, we have

$\pi_{e}(\tau_{h}(a\overline{f}))=\pi_{e}(af)=a\overline{f},$

where the last identity can be seen verified in the model $\bigwedge\nolimits
E$ for the spin representation, where $\pi_{e}$ is the reduction-mod-$e$ map,
and $\tau_{h}$ is just the inclusion $\bigwedge\nolimits E/\langle
e\rangle\to\bigwedge\nolimits E$ corresponding to the inclusion $V_{e}\to V$.
So
$\pi_{e}\circ\tau_{h}=\operatorname{id}_{\operatorname{Cl}(V_{e})\overline{f}}$.
We will informally call $\tau_{h}$ the multiplication map with $h$.

######  Proposition 3.2.

The multiplication map
$\tau_{h}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f$ is a
homomorphism of $\operatorname{Cl}(V_{e})$-representations, where
$\operatorname{Cl}(V_{e})$ is regarded a subalgebra of $\operatorname{Cl}(V)$
via the section $s_{e}:V_{e}\to V$.

###### Proof.

Let $v\in V_{e}$ and let $a\in\operatorname{Cl}(V_{e})$. Then

$\tau_{h}(va\overline{f})=va\overline{f}f_{n}=vaf,$

as desired. ∎

######  Corollary 3.3.

Both the map
$\pi_{e}:\operatorname{Cl}(V)f\to\operatorname{Cl}(V_{e})\overline{f}$ and the
map $\tau_{h}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f$
are $\operatorname{Spin}(V_{e})$-equivariant, where
$\operatorname{Spin}(V_{e})$ is regarded as a subgroup of
$\operatorname{Spin}(V)$ via the orthogonal decomposition
$V=V_{e}\oplus\langle e,h\rangle$.

###### Proof.

Propositions 3.1 and 3.2 imply that both maps are homomorphisms of
$\mathfrak{so}(V_{e})$-representations. Since $\operatorname{Spin}(V_{e})$ is
generated by one-parameter subgroups corresponding to nilpotent elements of
$\mathfrak{so}(V_{e})$, $\pi_{e}$ and $\tau_{h}$ are
$\operatorname{Spin}(V_{e})$-equivariant. ∎

###  3.4. Properties of the isotropic Grassmannian

The goal of this subsection is to collect properties of the isotropic
Grassmann cone that will later motivate the definition of a (half-)spin
variety (see Section 5). We fix a maximal isotropic subspace $F\subseteq V$
and a hyperbolic pair $(e,h)$ with $h\in F$ and $e\not\in F$ and identify
$V_{e}=e^{\perp}/\langle e\rangle$ with the subspace $\langle
e,h\rangle^{\perp}$ of $V$. We choose any basis $f_{1},\ldots,f_{n}$ of $F$
with $f_{n}=h$ and $(e|f_{i})=0$ for $i<n$ and write $f:=f_{1}\cdots
f_{n}\in\operatorname{Cl}(V)$ and
$\overline{f}:=\overline{f_{1}}\cdots\overline{f_{n-1}}\in\operatorname{Cl}(V_{e})$.

######  Proposition 3.4.

The isotropic Grassmann cone in $\operatorname{Cl}(V)f$ has the following
properties:

  1. (1)

$\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V)\subseteq\operatorname{Cl}(V)f$
is Zariski-closed and $\operatorname{Spin}(V)$-stable.

  2. (2)

Let $\pi_{e}:\operatorname{Cl}(V)f\to\operatorname{Cl}(V_{e})\overline{f}$ be
the contraction defined in §3.2. Then for every maximal isotropic subspace
$H\subseteq V$ we have

$\pi_{e}(S_{H})\subseteq S_{H_{e}},$

where $H_{e}\subseteq V_{e}$ is the image of $e^{\perp}\cap H$ in $V_{e}$.

  3. (3)

Let $\tau_{h}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f$ be
the map defined in §3.3. Then for every maximal isotropic $H^{\prime}\subseteq
V_{e}$ we have

$\tau_{h}(S_{H^{\prime}})=S_{{H^{\prime}}\oplus\langle h\rangle}.$

In particular, the contraction and multiplication map $\pi_{e}$ and $\tau_{h}$
preserve the isotropic Grassmann cones, i.e.,

$\pi_{e}\big{(}\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V)\big{)}\subseteq\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V_{e})\quad\text{and}\quad\tau_{h}\big{(}\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V_{e})\big{)}\subseteq\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V).$

###### Proof of Proposition 3.4.

  1. (1)

This is well known. Indeed, the isotropic Grassmann cone is the union of the
cones over the two connected components, and these cones are the union of
$\\{0\\}$ with the orbits of the highest weight vectors $\omega_{0}$ and
$\omega_{1}$. These minimal orbits are always Zariski closed. For more detail
see [Pro07, Theorem 1, p.428].

  2. (2)

Let $\omega_{H}$ be a spanning element of $S_{H}$. Then for all $v\in
e^{\perp}\cap H$ we have

$\overline{v}\cdot\pi_{e}(\omega_{H})=\pi_{e}(v\cdot\omega_{H})=\pi_{e}(0)=0,$

where the first equality follows from Proposition 3.1. Hence
$\pi_{e}(\omega_{H})$ lies in $S_{H_{e}}$.

  3. (3)

Let $\omega_{H^{\prime}}$ be a spanning element of $S_{H^{\prime}}$. Then for
all $v\in H^{\prime}$ we have

$v\cdot\tau_{h}(\omega_{H^{\prime}})=\tau_{h}(v\cdot\omega_{H^{\prime}})=\tau_{h}(0)=0$

where the first equality holds by Proposition 3.2. Furthermore, we have

$h\cdot\tau_{h}(\omega_{H^{\prime}})=h\cdot\omega_{H^{\prime}}f_{n}=0,$

where we used the definition of $\tau_{h}$ and $h=f_{n}$. Thus
$\tau_{h}(\omega_{H^{\prime}})$ lies in $S_{{H^{\prime}}\oplus\langle
h\rangle}$. The equality now follows from the fact that $\tau_{h}$ is
injective. ∎

######  Remark 3.5.

If $h\in H$, then $H=H_{e}\oplus\langle h\rangle$ and since
$\pi_{e}\circ\tau_{h}$ is the identity on
$\operatorname{Cl}(V_{e})\overline{f}$ we find that

$\pi_{e}(S_{H})=\pi_{e}(\tau_{h}(S_{H_{e}}))=S_{H_{e}},$

i.e., equality holds in (2) of Proposition 3.4. Later we will see that
equality holds under the weaker condition that $e\not\in H$, while
$\pi_{e}(S_{H})=\\{0\\}$ when $e\in H$. These statements can also be checked
by direct computations, but some care is needed since for $e,H,F$ in general
position one cannot construct a hyperbolic basis adapted to $H$ and $F$ that
moreover contains $e$.

###  3.5. The dual of contraction

Let $e\not\in F\subseteq V$ be an isotropic vector. We want to compute the
dual of the contraction map
$\pi_{e}:\operatorname{Cl}(V)f\to\operatorname{Cl}(V_{e})\overline{f}$;
indeed, we claim that this is essentially the map

$\psi_{e}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f$

defined by its restriction
$\operatorname{Cl}^{\pm}(V_{e})\overline{f}\to\operatorname{Cl}^{\mp}(V)f$ as

$\psi_{e}(\overline{b}\cdot\overline{f}_{1}\cdots\overline{f_{n-1}}):=\pm
ebf_{1}\cdots f_{n},$

where the sign is $+$ on $\operatorname{Cl}^{+}(V_{e})\overline{f}$ and $-$ on
$\operatorname{Cl}^{-}(V_{e})\overline{f}$. The reason for the “flip” of the
choice of half-spin representation in the dual will become obvious below.
Observe that $\psi_{e}$ is well-defined and, given a basis
$e_{1},\ldots,e_{n}=e$ of an isotropic space complementary to $F$ such that
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ is a hyperbolic basis, maps
$\overline{e_{J}}\overline{f}$ to $e_{J\cup\\{n\\}}f$.

######  Proposition 3.6.

The following diagram:

$\textstyle{(\operatorname{Cl}(V_{e})\overline{f})^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\pi_{e}^{*}}$$\textstyle{(\operatorname{Cl}(V)f)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{Cl}(V_{e})\overline{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{e}}$$\textstyle{\operatorname{Cl}(V)f}$

can be made commuting via a $\operatorname{Spin}(V_{e})$-module isomorphism on
the left vertical arrow and a $\operatorname{Spin}(V)$-module isomorphism on
the right vertical arrow.

######  Remark 3.7.

The statement of Proposition 3.6 holds true when replacing
$\operatorname{Cl}(V)f$ by either one of the two half-spin representations by
considering the correct “flip”. For example, if $n=\dim F$ is even, and
$e_{1},\dots,e_{n},f_{1},\dots,f_{n}$ is a hyperbolic basis as above, then in
the $\bigwedge\nolimits E$-model the correct grading is

$\textstyle{\left(\bigwedge\nolimits^{+}E_{n-1}\right)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\pi_{e_{n}}^{*}}$$\textstyle{\left(\bigwedge\nolimits^{+}E_{n}\right)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigwedge\nolimits^{-}E_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{e_{n}}}$$\textstyle{\bigwedge\nolimits^{+}E_{n}.}$

To prove Proposition 3.6 we consider the bilinear form $\beta$ on the spin
representation $\operatorname{Cl}(V)f$ defined as in [Pro07] as follows: for
$af,bf\in\operatorname{Cl}(V)f$ it turns out that $(af)^{*}bf=f^{*}a^{*}bf$,
where $*$ denotes the anti-automorphism from §2.1, is a scalar multiple of
$f$. The scalar is denoted $\beta(af,bf)$. We have the following properties:

######  Lemma 3.8 ([Pro07, p. 430]).

Let $\beta$ be the bilinear form defined as above.

  1. (1)

The form $\beta$ is nondegenerate and $\operatorname{Spin}(V)$-invariant.

  2. (2)

$\beta$ is symmetric if $n\equiv 0,1\mod 4$, and it is skew-symmetric if
$n\equiv 2,3\mod 4$.

  3. (3)

The two half-spin representations are self-dual via $\beta$ if $n$ is even,
and each is the dual of the other if $n$ is odd.

In the proof of Proposition 3.6 we will use a hyperbolic basis
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ with $e_{n}=e$. For a subset
$I=\\{i_{1}<\ldots<i_{k}\\}\subseteq[n]$ set $e_{I}:=e_{i_{1}}\cdots
e_{i_{k}}\in\operatorname{Cl}(E)\simeq\bigwedge\nolimits E$, where $E$ is the
span of the $e_{i}$. We have seen in §2.5 that the spin representation has as
a basis the elements $e_{I}f$ with $I$ running through all subsets of $[n]$.

###### Proof of Proposition 3.6.

Consider the bilinear forms $\beta$ on $\operatorname{Cl}(V)f$ and $\beta_{e}$
on $\operatorname{Cl}(V_{e})\overline{f}$ as defined above. By Lemma 3.8 the
spin representations $\operatorname{Cl}(V)f$ and
$\operatorname{Cl}(V_{e})\overline{f}$ are self-dual via $\beta$ and
$\beta_{e}$, respectively. Thus it suffices to prove, for
$a\in\operatorname{Cl}(V)$ and $b\in\operatorname{Cl}(e^{\perp})$, that

$\beta_{e}(\pi_{e}(af),\overline{b}\overline{f})=\frac{(-1)^{n-1}}{2}\beta(af,\psi_{e}(\overline{b}\overline{f})).$

We may assume that $a=e_{I}$, $b=e_{J}$ with $I\subseteq[n]$,
$J\subseteq[n-1]$.

In the $\bigwedge\nolimits E$-model $\pi_{e}$ is the mod-$e$ map, and hence
the left-hand side is zero if $n\in I$. If $n\not\in I$, then the left-hand
side equals the coefficient of $\overline{f}$ in
$\overline{f}^{*}\overline{e_{I}}^{*}\overline{e_{J}}\overline{f}$. This is
nonzero if and only if $[n-1]$ is the disjoint union of $I$ and $J$, and then
it is $2^{n-1}$ times a sign corresponding to the number of swaps needed to
move the factors $\overline{f_{i}}$ of $\overline{f}^{*}$ to just before the
corresponding factor $\overline{e_{i}}$ in either $\overline{e_{I}}^{*}$ or
$\overline{e_{J}}$.

Apart from the factor $\frac{(-1)^{n-1}}{2}$, the right-hand side is the
coefficient of $f$ in $f^{*}e_{I}e_{J}e_{n}f$. This is nonzero if and only if
$[n]$ is the disjoint union of the sets $\\{n\\},J,I$, and in that case it is
$2^{n}$ times a sign corresponding to the number of swaps needed to move the
factors $f_{i}$ of $f^{*}$ to the corresponding factor $e_{i}$ in either
$e_{I}$ or $e_{J}$ or (in the case of $f_{n}$) to just before the factor
$e_{n}$. The latter contributes $(-1)^{n-1}$, and apart from this factor the
sign is the same as on the left-hand side. ∎

###  3.6. Two infinite spin representations

Let $V_{\infty}$ be the countable-dimensional vector space with basis
$e_{1},f_{1},e_{2},f_{2},\dots$, and equip $V_{\infty}$ with the quadratic
form for which this is a hyperbolic basis, i.e.,
$(e_{i}|e_{j})=(f_{i}|f_{j})=0$ and $(e_{i}|f_{j})=\delta_{ij}$ for all $i,j$.
We write $E_{\infty}$ and $F_{\infty}$ for the subspaces of $V_{\infty}$
spanned by the $e_{i}$ and the $f_{i}$, respectively.

Let $V_{n}$ be the subspace of $V_{\infty}$ spanned by
$e_{1},f_{1},e_{2},f_{2},\dots,e_{n},f_{n}$, with the restricted quadratic
form. We further set $E_{n}:=V_{n}\cap E_{\infty}$ and $F_{n}:=V_{n}\cap
F_{\infty}$. We define the infinite spin group as
$\operatorname{Spin}(V_{\infty}):=\varinjlim_{n}\operatorname{Spin}(V_{n})$
where $\operatorname{Spin}(V_{n-1})$ is embedded into
$\operatorname{Spin}(V_{n})$ as the subgroup that fixes $\langle
e_{n},f_{n}\rangle$ element-wise. Similarly, we write
$\operatorname{GL}(E_{\infty}):=\varinjlim_{n}\operatorname{GL}(E_{n})$ and
$H$ for the preimage of $\operatorname{GL}(E_{\infty})$ in
$\operatorname{Spin}(V_{\infty})$. We use the notation
$\mathfrak{so}(V_{\infty})$ and $\mathfrak{gl}(E_{\infty})$ for the
corresponding direct limits of the Lie algebras $\mathfrak{so}(V_{n})$ and
$\mathfrak{gl}(E_{n})$. Here the direct limits are taken in the categories of
abstract groups, and Lie algebras, respectively.

The previous paragraphs give rise to various
$\operatorname{Spin}(V_{n-1})$-equivariant maps between the spin
representations of $\operatorname{Spin}(V_{n-1})$ and
$\operatorname{Spin}(V_{n})$. First, contraction with $e_{n}$,

$\pi_{e_{n}}:\operatorname{Cl}(V_{n})f_{1}\cdots
f_{n}\to\operatorname{Cl}(V_{n-1})f_{1}\cdots f_{n-1},$

and second, multiplication with $f_{n}$,

$\tau_{f_{n}}:\operatorname{Cl}(V_{n-1})f_{1}\cdots
f_{n-1}\to\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}.$

We have that these satisfy $\pi_{e_{n}}\circ\tau_{f_{n}}=\operatorname{id}$.
Third, the map

$\psi_{e_{n}}:\operatorname{Cl}(V_{n-1})f_{1}\cdots
f_{n-1}\to\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}$

that is dual to $\pi_{e_{n}}$ in the sense of Proposition 3.6.

######  Definition 3.9.

The direct (infinite) spin representation is the direct limit of all spaces
$\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}$ along the maps $\psi_{e_{n}}$. The
inverse (infinite) spin representation is the inverse limit of all spaces
$\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}$ along the maps $\pi_{e_{n}}$.

Since the maps $\psi_{e_{n}},\pi_{e_{n}}$ are
$\operatorname{Spin}(V_{n-1})$-equivariant, both of these spaces are
$\operatorname{Spin}(V_{\infty})$-modules. As the dual of a direct limit is
the inverse limit of the duals, and since the maps $\psi_{e_{n}}$ and
$\pi_{e_{n}}$ are dual to each other by Proposition 3.6, the inverse spin
representation is the dual space of the direct spin representation.

In our model $\bigwedge\nolimits E_{n}$ of
$\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}$, the map $\psi_{e_{n}}$ is just
the right multiplication

$\bigwedge\nolimits E_{n-1}\to\bigwedge\nolimits
E_{n},\;\omega\mapsto\omega\wedge e_{n}.$

Hence the direct spin representation has as a basis formal infinite products

$e_{i_{1}}\wedge e_{i_{2}}\wedge\ldots=:e_{I}$

where $I=\\{i_{1}<i_{2}<\ldots\\}$ is a cofinite subset of ${\mathbb{N}}$. We
will write $\bigwedge\nolimits_{\infty}E_{\infty}$ for this countable-
dimensional vector space. The action of the Lie algebra
$\mathfrak{so}(V_{\infty})$ of $\operatorname{Spin}(V_{\infty})$ on this space
is given via the explicit formulas from §2.5. In particular, the span of the
$e_{I}$ with $|{\mathbb{N}}\setminus I|$ even (respectively, odd) is a
$\operatorname{Spin}(V_{\infty})$-submodule, and
$\bigwedge\nolimits_{\infty}E_{\infty}$ is the direct sum of these
(irreducible) modules.

######  Remark 3.10.

The reader may wonder why we do not introduce the direct spin representation
as the direct limit of all $\operatorname{Cl}(V)f_{1}\cdots f_{n}$ along the
maps $\tau_{f_{n}}$. This would make the ordinary Grassmann algebra
$\bigwedge\nolimits E_{\infty}$ a model for the direct spin representation,
instead of the slightly more complicated-looking space
$\bigwedge\nolimits_{\infty}E_{\infty}$. However, the maps dual to the
$\tau_{f_{n}}$ correspond to contraction maps with $f_{n}\in F$, which we have
not discussed and which interchange even and odd half-spin representations. We
believe that our theorem below goes through for this different setting, as
well, but we have not checked the details.

###  3.7. Four infinite half-spin representations

Keeping in mind that the maps $\psi_{e_{n}}$ interchange the even and odd
subrepresentations, we define the direct (infinite) half-spin representations
$\bigwedge\nolimits_{\infty}^{\pm}E_{\infty}$ to be the direct limit

$\bigwedge\nolimits_{\infty}^{\pm}E_{\infty}=\varinjlim\left(\bigwedge\nolimits^{\pm}E_{0}\to\bigwedge\nolimits^{\mp}E_{1}\to\bigwedge\nolimits^{\pm}E_{2}\to\bigwedge\nolimits^{\mp}E_{3}\to\bigwedge\nolimits^{\pm}E_{4}\to\cdots\right)$

along the maps $\psi_{e_{n}}$. For the sake of readability we will abbreviate
this by

$\bigwedge\nolimits_{\infty}^{\pm}E_{\infty}=\varinjlim_{n}\bigwedge\nolimits^{\pm(-1)^{n}}E_{n},$
(3.2)

where $\pm(-1)^{n}$ denotes $\pm$ if $n$ is even and $\mp$ if $n$ is odd. In
terms of the basis $e_{I}$ introduced in §3.6, the half-spin representation
$\bigwedge\nolimits_{\infty}^{+}E_{\infty}$ is spanned by all $e_{I}$ with
$|{\mathbb{N}}\setminus I|$ even, and
$\bigwedge\nolimits_{\infty}^{-}E_{\infty}$ by those with
$|{\mathbb{N}}\setminus I|$ odd. The inverse (infinite) half-spin
representations are defined as the duals of the direct (infinite) half-spin
representations. Using the isomorphisms from Remark 3.7 we observe

$\left(\bigwedge\nolimits^{\pm}_{\infty}E_{\infty}\right)^{*}=\varprojlim_{n}\left(\bigwedge\nolimits^{\pm(-1)^{n}}E_{n}\right)^{\ast}\simeq\varprojlim_{n}\bigwedge\nolimits^{\pm}E_{n}.$
(3.3)

So the inverse (infinite) half-spin representations can be identified with the
inverse limits of the half-spin representations
$\bigwedge\nolimits^{\pm}E_{n}$ along the projections $\pi_{e_{n}}$.

We can enrich the inverse spin representation to an affine scheme whose
coordinate ring is the symmetric algebra on
$\bigwedge\nolimits_{\infty}E_{\infty}$, recalling the following remark.

######  Remark 3.11.

Let $K$ be any field (not necessarily algebraically closed) and $W$ any
$K$-vector space (not necessarily finite dimensional). Then there are
canonical identifications

$W^{\ast}={\rm Spec}\big{(}{\rm Sym}(W)\big{)}(K)\subseteq\left\\{\text{closed
points in }{\rm Spec}\big{(}{\rm Sym}(W)\big{)}\right\\}.$

So ${\rm Spec}\big{(}{\rm Sym}(W)\big{)}$ can be seen as an enrichment of
$W^{\ast}$ to an affine scheme. If $W$ is a linear representation for a group
$G$, then $G$ acts via $K$-algebra automorphisms on $\operatorname{Sym}W$ and
hence via $K$-automorphisms on the affine scheme corresponding to $W^{*}$. For
$W=\bigwedge\nolimits^{\pm}_{\infty}E_{\infty}$, this construction extends the
natural $\operatorname{Spin}(V_{\infty})$-action on the vector space
$\varprojlim_{n}\bigwedge\nolimits^{\pm}E_{n}\simeq W^{*}$ to the
corresponding affine scheme.

By abuse of notation, we will write
$\left(\bigwedge\nolimits_{\infty}E_{\infty}\right)^{*}$ also for the scheme
itself, and similarly for the inverse half-spin representations
$\left(\bigwedge\nolimits^{\pm}_{\infty}E_{\infty}\right)^{*}$. Later we will
also write $\bigwedge\nolimits^{\pm}E_{n}$ for the affine scheme ${\rm
Spec}\left({\rm Sym}\left(\bigwedge\nolimits^{\pm(-1)^{n}}E_{n}\right)\right)$
by identifying
$\bigwedge\nolimits^{\pm}E_{n}\cong\left(\bigwedge\nolimits^{\pm(-1)^{n}}E_{n}\right)^{*}$
as in Equation 3.3.

##  4\. Noetherianity of the inverse half-spin representations

In this section we prove our main theorem.

######  Theorem 4.1.

The inverse half-spin representation
$(\bigwedge\nolimits_{\infty}^{+}E_{\infty})^{*}$ is topologically Noetherian
with respect to the action of $\operatorname{Spin}(V_{\infty})$. That is,
every descending chain

$\left(\bigwedge\nolimits_{\infty}^{+}E_{\infty}\right)^{*}\supseteq
X_{1}\supseteq X_{2}\supseteq\ldots$

of closed, reduced $\operatorname{Spin}(V_{\infty})$-stable subschemes
stabilises, and the same holds for the other inverse half-spin representation.

Recall that the action of $\operatorname{Spin}(V_{\infty})$ on the inverse
half-spin representation (as an affine scheme) is given by $K$-automorphisms,
as described in Remark 3.11. We write $R$ for the symmetric algebra on the
direct spin representation $\bigwedge\nolimits_{\infty}E_{\infty}$, so the
inverse spin representation is $\operatorname{Spec}(R)$. Similarly, we write
$R^{\pm}$ for the symmetric algebras on the direct half-spin representations,
so $R^{\pm}$ is the coordinate ring of
$\varprojlim_{n}\bigwedge\nolimits^{\pm}E_{n}$, respectively.

Let us briefly outline the proof strategy. We will proceed by induction on the
minimal degree of an equation defining a closed subset $X$. Starting with such
an equation $p$, we show that there exists a partial derivative
$q:=\frac{\partial p}{\partial e_{I}}$ such that the principal open $X[1/q]$
is topologically $H_{n}$-Noetherian, where $H_{n}$ is the subgroup of
$\operatorname{Spin}(V_{\infty})$ defined below. For that we use that the
$H_{n}$-action corresponds to a “twist” of the usual
$\operatorname{GL}(E_{\infty})$-action, as observed in Section 2.8 (for the
exact formula see (2.3)); this allows us to apply the main result of [ES22].
Finally, for those points which are contained in the vanishing set of the
$\operatorname{Spin}(V_{\infty})$-orbit of $q$ we can apply induction, as the
minimal degree of a defining equation has been lowered by $1$.

###  4.1. Shifting

Let $G_{n}$ be the subgroup of $G$ that fixes
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ element-wise. Note that $G_{n}$ is
isomorphic to $G$; at the level of the Lie algebras the isomorphism from $G$
to $G_{n}$ is given by the map

$\begin{bmatrix}A&B\\\ C&-A^{T}\end{bmatrix}\mapsto\begin{bmatrix}0&0&0&0\\\
0&A&0&B\\\ 0&0&0&0\\\ 0&C&0&-A^{T}\end{bmatrix}$

where the widths of the blocks are $n,\infty,n,\infty$, respectively. We write
$H_{n}$ for $H\cap G_{n}$, where $H\subseteq\operatorname{Spin}(V_{\infty})$
is the subgroup corresponding to the subalgebra
$\mathfrak{gl}(E_{\infty})\subseteq\mathfrak{so}(V_{\infty})$. Then $H_{n}$ is
the pre-image in $\operatorname{Spin}(V_{\infty})$ of the subgroup
$\operatorname{GL}(E_{\infty})_{n}\subseteq\operatorname{GL}(E_{\infty})$ of
all $g$ that fix $e_{1},\ldots,e_{n}$ element-wise and maps the span of the
$e_{i}$ with $i>n$ into itself. The Lie algebra of $H_{n}$ and of
$\operatorname{GL}(E_{\infty})_{n}$ consists of the matrices above on the
right with $B=C=0$.

###  4.2. Acting with the general linear group on $E$

For every fixed $k\in{\mathbb{Z}}_{\geq 0}$, the Lie algebra
$\mathfrak{gl}(E_{\infty})\subseteq\mathfrak{so}(V_{\infty})$ preserves the
linear space

$\left(\bigwedge\nolimits_{\infty}E_{\infty}\right)_{k}:=\big{\langle}\\{e_{I}:|{\mathbb{N}}\setminus
I|=k\\}\big{\rangle},$

and hence so does the corresponding subgroup
$H\subseteq\operatorname{Spin}(V_{\infty})$. We let $R_{\leq\ell}\subseteq R$
be the subalgebra generated by the spaces
$(\bigwedge\nolimits_{\infty}E_{\infty})_{k}$ with $k\leq\ell$. Crucial in the
proof of Theorem 4.1 is the following result.

######  Proposition 4.2.

For every choice of nonnegative integers $\ell$ and $n$,
$\operatorname{Spec}(R_{\leq\ell})$ is topologically $H_{n}$-Noetherian, i.e.,
every descending chain

$\operatorname{Spec}(R_{\leq\ell})\supseteq X_{1}\supseteq
X_{2}\supseteq\ldots$

of $H_{n}$-stable closed and reduced subschemes stabilizes.

The key ingredient in the proof of Proposition 4.2 is the main result of
[ES22]. In order to apply their result we need to do some preparatory work. We
will start with the following lemma.

######  Lemma 4.3.

Every $H_{n}$-stable closed subscheme of $\operatorname{Spec}(R_{\leq\ell})$
is also stable under the group $\operatorname{GL}(E_{\infty})_{n}$ acting in
the natural manner on $\bigwedge\nolimits_{\infty}E_{\infty}$ and its dual,
and vice versa.

###### Proof.

Equation 2.3 implies that
$\mathfrak{gl}(E_{\infty})\subseteq\mathfrak{so}(V_{\infty})$ acts on
$\bigwedge\nolimits_{\infty}E_{\infty}$ via

$\rho(A)=\tilde{\rho}(A)-\frac{1}{2}\operatorname{tr}(A)\operatorname{id}_{\bigwedge\nolimits_{\infty}E_{\infty}}$

where $\tilde{\rho}$ is the standard representation of
$\mathfrak{gl}(E_{\infty})$ on $\bigwedge\nolimits_{\infty}E_{\infty}$. An
$H_{n}$-stable closed subscheme $X$ of $\operatorname{Spec}(R_{\leq\ell})$ is
given by an $H_{n}$-stable ideal $I$ in the symmetric algebra $R_{\leq\ell}$.
Such an $I$ is then also stable under the action of the Lie algebra
$\mathfrak{gl}(E_{\infty})_{n}$ of $H_{n}$ by derivations that act on
variables in
$\bigoplus_{k=0}^{\ell}(\bigwedge\nolimits_{\infty}E_{\infty})_{k}$ via
$\rho$.

We claim that $I$ is a homogeneous ideal. Indeed, for $f\in I$, choose $m>n$
such that all variables in $f$ (which are basis elements $e_{I}$) contain the
basis element $e_{m}$ of $E_{\infty}$. Let $A\in\mathfrak{gl}(E_{\infty})_{n}$
be the diagonal matrix with $0$’s everywhere except a $1$ on position $(m,m)$.
Then $\rho(A)$ maps each variable in $f$ to $\frac{1}{2}$ times itself. Hence,
by the Leibniz rule, $\rho(A)$ scales the homogeneous part of degree $d$ in
$f$ by $\frac{d}{2}$. Since $I$ is preserved by $\rho(A)$, it follows that $I$
contains all homogeneous components of $f$, and hence $I$ is a homogeneous
ideal.

Now let $B\in\mathfrak{gl}(E_{\infty})_{n}$ and $f\in I$ be arbitrary. By the
previous paragraph we can assume $f$ to be homogeneous of degree $d$, and we
then have

$\rho(B)f=\tilde{\rho}(B)f-\frac{d}{2}\operatorname{tr}(B)f,$

and since $I$ is $\rho(B)$-stable, we deduce $\tilde{\rho}(B)f\in I$. This
completes the proof in one direction. The proof in the opposite direction is
identical. ∎

######  Remark 4.4.

By the proof above, any $\operatorname{Spin}(V_{\infty})$-stable closed
subscheme $X$ of $(\bigwedge\nolimits_{\infty}E_{\infty})^{*}$ is an affine
cone.

Following [ES22] the restricted dual $(E_{\infty})_{*}$ of $E_{\infty}$ is
defined as the union $\bigcup_{n\geq 1}(E_{n})^{*}.$ We will denote by
$\varepsilon^{1},\varepsilon^{2},\dots$ the basis of $(E_{\infty})_{*}$ that
is dual to the canonical basis $e_{1},e_{2},\dots$ of $E_{\infty}$ given by
$\varepsilon^{i}(e_{j})=\delta_{ij}$.

######  Lemma 4.5.

There is an $\operatorname{SL}(E_{\infty})$-equivariant isomorphism

$\bigwedge\nolimits_{\infty}E_{\infty}\longrightarrow\bigwedge(E_{\infty})_{*},$

which restricts to an isomorphism

$\left(\bigwedge\nolimits_{\infty}E_{\infty}\right)_{k}\longrightarrow\bigwedge\nolimits^{k}(E_{\infty})_{*}.$

We will use this isomorphism to regard $\bigwedge\nolimits_{\infty}E_{\infty}$
as the restricted dual of the Grassmann algebra $\bigwedge\nolimits
E_{\infty}$. We stress, though, that this isomorphism is not
$GL(E_{\infty})$-equivariant.

###### Proof.

We have a natural bilinear map

$\bigwedge\nolimits
E_{\infty}\times\bigwedge\nolimits_{\infty}E_{\infty}\to\bigwedge\nolimits_{\infty}E_{\infty},\quad(\omega,\omega^{\prime})\mapsto\omega\wedge\omega^{\prime}.$

If $I\subseteq{\mathbb{N}}$ is finite and $J\subseteq{\mathbb{N}}$ is
cofinite, then $e_{I}\wedge e_{J}$ is $0$ if $I\cap J\neq\emptyset$ and $\pm
e_{I\cup J}$ otherwise, where the sign is determined by the permutation
required to order the sequence $I,J$. We then define a perfect pairing
$\gamma$ between the two spaces by

$\gamma(\omega,\omega^{\prime}):=\text{the coefficient of $e_{\mathbb{N}}$ in
}\omega\wedge\omega^{\prime}.$

The map
$\Phi_{\gamma}:\bigwedge\nolimits_{\infty}E_{\infty}\to\bigwedge\nolimits(E_{\infty})_{*},\;\omega^{\prime}\mapsto\gamma(\cdot,\omega^{\prime})$
induced by $\gamma$ is the isomorphism given by
$e_{I}\mapsto\pm\varepsilon^{I^{c}}$, where $I^{c}\subseteq{\mathbb{N}}$ is
the complement of $I$ and
$\varepsilon^{J}\coloneqq\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{k}}$
for a finite set $J=\\{j_{1},\dots,j_{k}\\}$. Note that
$\gamma(A\cdot\omega,A\cdot\omega^{\prime})=\det(A)\gamma(\omega,\omega^{\prime})$
for all $A\in\operatorname{GL}(E_{\infty})$, and hence $\gamma$ is
$\operatorname{SL}(E_{\infty})$-invariant. Therefore, the isomorphism
$\Phi_{\gamma}$ is $\operatorname{SL}(E_{\infty})$-equivariant. ∎

######  Lemma 4.6.

An ideal $I\subseteq\operatorname{Sym}(\bigwedge(E_{\infty})_{*})$ is
$\operatorname{SL}(E_{\infty})$-stable if and only if it is
$\operatorname{GL}(E_{\infty})$-stable. The same holds for
$\operatorname{SL}(E_{\infty})_{n}$ and $\operatorname{GL}(E_{\infty})_{n}$.

###### Proof.

Assume that $I$ is $\operatorname{SL}(E_{\infty})$-stable. Let $f\in I$ and
$A\in\operatorname{GL}(E_{\infty})$ be arbitrary. Choose
$m=m(f,A)\in{\mathbb{N}}$ large enough so that
$f\in\operatorname{Sym}(\bigwedge\nolimits(E_{m})^{*})$ and $A$ is the image
of some $A_{m}\in\operatorname{GL}(E_{m})$. Define
$A_{m+1}\in\operatorname{GL}(E_{m+1})$ as the map given by
$A_{m+1}(e_{i})=A_{m}(e_{i})$ for $i\leq m$ and
$A_{m+1}(e_{m+1})=(\det(A_{m}))^{-1}(e_{m+1})$, and let $A^{\prime}$ be the
image of $A_{m+1}$ in $\operatorname{SL}(E_{\infty})$. Then the action of
$A_{m}$ and $A_{m+1}$ agree on $(E_{m})^{*}$. Hence they also agree on
$\operatorname{Sym}(\bigwedge\nolimits(E_{m})^{*})$. So $A\cdot
f=A^{\prime}\cdot f\in I$ since $I$ was assumed to be
$\operatorname{SL}(E_{\infty})$-stable and
$A^{\prime}\in\operatorname{SL}(E_{\infty})$. As $f\in I$ and
$A\in\operatorname{GL}(E_{\infty})$ were arbitrary, this shows that $I$ is
$GL(E_{\infty})$-stable. ∎

###### Proof of Proposition 4.2..

First, we claim that
$\operatorname{Spec}\big{(}\operatorname{Sym}\big{(}\bigoplus_{k=0}^{\ell}\bigwedge\nolimits^{k}(E_{\infty})_{*}\big{)}\big{)}$
is topologically $\operatorname{GL}(E_{\infty})_{n}$-Noetherian. Indeed, the
standard $\operatorname{GL}(E_{\infty})$-representation of the space
$\bigoplus_{k=0}^{\ell}\bigwedge\nolimits^{k}(E_{\infty})_{*}$ is an algebraic
representation and this also remains true when we act with
$\operatorname{GL}(E_{\infty})$ via its isomorphism into
$\operatorname{GL}(E_{\infty})_{n}$. Hence, the claim follows from [ES22,
Theorem 2]. Let
$(X_{i})_{i\in{\mathbb{N}}}\subseteq\operatorname{Spec}(R_{\leq\ell})$ be a
descending chain of $H_{n}$-stable, closed, reduced subschemes. By Lemma 4.3
every $X_{i}$ is also $\operatorname{GL}(E_{\infty})_{n}$-stable. By Lemma 4.5
there is an $\operatorname{SL}(E_{\infty})_{n}$-equivariant isomorphism
$\operatorname{Spec}(R_{\leq\ell})\cong\operatorname{Spec}\big{(}\operatorname{Sym}\big{(}\bigoplus_{k=0}^{\ell}\bigwedge\nolimits^{k}(E_{\infty})_{*}\big{)}\big{)}$.
Let
$X_{i}^{\prime}\subseteq\operatorname{Spec}\big{(}\operatorname{Sym}\big{(}\bigoplus_{k=0}^{\ell}\bigwedge\nolimits^{k}(E_{\infty})_{*}\big{)}\big{)}$
be the closed, reduced, $\operatorname{SL}(E_{\infty})$-stable subscheme
corresponding to $X_{i}$ under this isomorphism. Using Lemma 4.6 we see that
the subschemes $X_{i}^{\prime}$ are also
$\operatorname{GL}(E_{\infty})_{n}$-stable. Therefore, the chain
$(X_{i}^{\prime})_{i\in{\mathbb{N}}}$ stabilizes by our first claim.
Consequently, also the chain $(X_{i})_{i\in{\mathbb{N}}}$ stabilizes. ∎

Before we come to the proof of Theorem 4.1, let us recall the action of
$f_{i}\wedge f_{j}\in\mathfrak{so}(V_{\infty})$ on
$\bigwedge\nolimits^{+}_{\infty}E_{\infty}$ and its symmetric algebra $R^{+}$
in explicit terms. Recall from Section 3.6 that a basis for
$\bigwedge\nolimits^{+}_{\infty}E_{\infty}$ is given by $e_{I}=e_{i_{1}}\wedge
e_{i_{2}}\wedge\cdots$, where
$I=\\{i_{1}<i_{2}<\cdots\\}\subseteq{\mathbb{N}}$ is cofinite and
$|{\mathbb{N}}\setminus I|$ even. Then we have

$(f_{i}\wedge
f_{j})e_{I}=\begin{cases}(-1)^{c_{i,j}(I)}e_{I\setminus\\{i,j\\}}&\text{if
}i,j\in I,\text{ and}\\\ 0&\text{otherwise,}\end{cases}$

where $c_{i,j}(I)$ depends on the position of $i,j$ in $I$. (Note that there
is no factor $4$, since in our identification of $\bigwedge\nolimits^{2}V$ to
the Lie subalgebra $L$ of $\operatorname{Cl}(V)$ we had a factor
$\frac{1}{4}$.) The corresponding action of $f_{i}\wedge f_{j}$ on polynomials
in $R^{+}$ is as a derivation.

###  4.3. Proof of Theorem 4.1

Let $R^{+}\subseteq R$ be the symmetric algebra on the direct half-spin
representation $\bigwedge\nolimits^{+}_{\infty}E_{\infty}$, so that
$\operatorname{Spec}(R^{+})$ is the inverse half-spin representation
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$. We prove topological
$\operatorname{Spin}(V_{\infty})$-Noetherianity of
$\operatorname{Spec}(R^{+})$; the corresponding statement for
$\operatorname{Spec}(R^{-})$ is proved in exactly the same manner.

For a closed, reduced $\operatorname{Spin}(V_{\infty})$-stable subscheme $X$
of $\operatorname{Spec}(R^{+})$, we denote by
$\delta_{X}\in\\{0,1,2,\ldots,\infty\\}$ the lowest degree of a nonzero
polynomial in the ideal $I(X)\subseteq R^{+}$ of $X$. Here we consider the
natural grading on
$R^{+}=\operatorname{Sym}(\bigwedge\nolimits^{+}_{\infty}E_{\infty})$, where
the elements of $\bigwedge\nolimits^{+}_{\infty}E_{\infty}$ all have degree
$1$.

We proceed by induction on $\delta_{X}$ to show that $X$ is topologically
Noetherian; we may therefore assume that this is true for all $Y$ with
$\delta_{Y}<\delta_{X}$. We have $\delta_{X}=\infty$ if and only if
$X=\operatorname{Spec}(R^{+})$. Then a chain

$\operatorname{Spec}(R^{+})=X\supseteq X_{1}\supseteq X_{2}\supseteq\ldots$

of $\operatorname{Spin}(V_{\infty})$-closed subsets is either constant or else
there exists an $i$ with $\delta_{X_{i}}<\infty$. Hence it suffices to prove
that $X$ is Noetherian under the additional assumption that
$\delta_{X}<\infty$. At the other extreme, if $\delta_{X}=0$, then $X$ is
empty and there is nothing to prove. So we assume that $0<\delta_{X}<\infty$
and that all $Y$ with $\delta_{Y}<\delta_{X}$ are
$\operatorname{Spin}(V_{\infty})$-Noetherian.

Let $p\in R^{+}$ be a nonzero polynomial in the ideal of $X$ of degree
$\delta_{X}$. By Remark 4.4, since $X$ is a cone, $p$ is in fact homogeneous
of degree $\delta_{X}$. Let $e_{I}$ be a variable appearing in $p$ such that
$k:=|I^{c}|$ is maximal among all variables in $p$; note that $k$ is even.
Then choose $n\geq k+2$ even such that all variables of $p$ are contained in
$\bigwedge\nolimits^{+}E_{n}$, i.e., they are of the form $e_{J}$ with
$J\supseteq\\{n+1,n+2,\ldots\\}$.

Now act on $p$ with the element $f_{i_{1}}\wedge
f_{i_{2}}\in\mathfrak{so}(V_{\infty})$ with $i_{1}<i_{2}$ the two smallest
elements in $I$. Since $X$ is $\operatorname{Spin}(V_{\infty})$-stable, the
result $p_{1}$ is again in the ideal of $X$. Furthermore, $p_{1}$ has the form

$p_{1}=\pm e_{I\setminus\\{i_{1},i_{2}\\}}\cdot q+r_{1}$

where $q=\frac{\partial p}{\partial e_{I}}$ contains only variables $e_{J}$
with $|J^{c}|\leq k$ and where $r_{1}$ does not contain
$e_{I\setminus\\{i_{1},i_{2}\\}}$ but may contain other variables $e_{J}$ with
$|J^{c}|=k+2$ (namely, those with $i_{1},i_{2}\not\in J$ for which
$e_{J\cup\\{i_{1},i_{2}\\}}$ appears in $p$).

If $n=k+2$, then $I\setminus\\{i_{1},i_{2}\\}=\\{n+1,n+2,\ldots\\}$ and, since
all variables $e_{J}$ in $p_{1}$ satisfy $J\supseteq\\{n+1,n+2,\ldots\\}$,
$e_{I\setminus\\{i_{1},i_{2}\\}}$ is the only variable $e_{J}$ in $p_{1}$ with
$|J^{c}|=k+2$. If $n>k+2$, then we continue in the same manner, now acting
with $f_{i_{3}}\wedge f_{i_{4}}$ on $p_{1}$, where $i_{3}<i_{4}$ are the two
smallest elements in $I\setminus\\{i_{1},i_{2}\\}$. We write $p_{2}$ for the
result, which is now of the form

$p_{2}=\pm e_{I\setminus\\{i_{1},i_{2},i_{3},i_{4}\\}}\cdot q+r_{2}$

where $q$ is the same polynomial as before and $r_{2}$ does not contain the
variable $e_{I\setminus\\{i_{1},i_{2},i_{3},i_{4}\\}}$ but may contain other
variables $e_{J}$ with $|J^{c}|=k+4$.

Iterating this construction we find the polynomial

$p_{\ell}=\pm e_{\\{n+1,n+2,\ldots\\}}\cdot q+r_{\ell}$

in the ideal of $X$, where $\ell=(n-k)/2$, $q$ is the same polynomial as
before and $r_{\ell}$ only contains variables $e_{J}$ with $|J^{c}|<n$. Let
$Z:=X[1/q]$ be the open subset of $X$ where $q$ is nonzero.

######  Lemma 4.7.

For every variable $e_{J}$ with $|J^{c}|\geq n$, the ideal of $Z$ in the
localisation $R^{+}[1/q]$ contains a polynomial of the form $e_{J}-s/q^{d}$
for some $d\in{\mathbb{Z}}_{\geq 0}$ and some $s\in R^{+}_{\leq n-2}$.

###### Proof.

We proceed by induction on $|J^{c}|=:m$. By successively acting on $p_{\ell}$
with the elements $f_{n}\wedge f_{n+1},f_{n+2}\wedge
f_{n+3},\ldots,f_{m-1}\wedge f_{m}$, we find the polynomial

$\pm e_{\\{m+1,m+2,\ldots\\}}\cdot q+r$

in the ideal of $X$, where $r$ contains only variables $e_{L}$ with
$|L^{c}|<m$. Now act with elements of $\mathfrak{gl}(E_{\infty})$ to obtain an
element

$\pm e_{J}\cdot q+\tilde{r}$

where $\tilde{r}$ still contains only variables $e_{L}$ with $|L^{c}|<m$.
Inverting $q$, this can be used to express $e_{J}$ in such variables $e_{L}$.
By the induction hypothesis, all those $e_{L}$ admit an expression, on $Z$, as
a polynomial in $R^{+}_{\leq n-2}$ times a negative power of $q$. Then the
same holds for $e_{J}$. ∎

######  Lemma 4.8.

The open subscheme $Z=X[1/q]$ is stable under the group $H_{n}$ and
$H_{n}$-Noetherian.

###### Proof.

By Lemma 4.3, $X$ is stable under $\operatorname{GL}(E_{\infty})_{n}$. The
polynomial $q$ is homogeneous and contains only variables $e_{J}$ with
$J\supseteq\\{n+1,n+2,\ldots\\}$. Every
$g\in\operatorname{GL}(E_{\infty})_{n}$ scales each such variable with
$\det(g)$, and hence maps $q$ to a scalar multiple of itself. We conclude that
$Z$ is stable under $\operatorname{GL}(E_{\infty})_{n}$, hence by (a slight
variant of) Lemma 4.3 also under $H_{n}$.

By Lemma 4.7, the projection dual to the inclusion $R^{+}_{\leq
n-2}[1/q]\subseteq R^{+}[1/q]$ restricts on $Z$ to a closed embedding, and
this embedding is $H_{n}$-equivariant. By Proposition 4.2, the image of $Z$ is
$H_{n}$-Noetherian, hence so is $Z$ itself. ∎

###### Proof of Theorem 4.1.

Let

$X\supseteq X_{1}\supseteq\ldots$

be a chain of reduced, $\operatorname{Spin}(V_{\infty})$-stable closed
subschemes. Let $Y\subseteq X$ be the reduced closed subscheme defined by the
orbit $\operatorname{Spin}(V_{\infty})\cdot q$. Since $q$ has degree
$\delta_{X}-1$, we have $\delta_{Y}<\delta_{X}$ and hence $Y$ is
$\operatorname{Spin}(V_{\infty})$-Noetherian by the induction hypothesis. It
follows that the chain

$Y\supseteq(Y\cap X_{1})^{\mathrm{red}}\supseteq\ldots$

is eventually stable. On the other hand, the chain

$Z\supseteq(Z\cap X_{1})^{\mathrm{red}}\supseteq\ldots$

consists of reduced, $H_{n}$-stable closed subschemes of $Z$, hence it is
eventually stable by Lemma 4.8.

Now pick a (not necessarily closed) point $P\in X_{i}$ for $i\gg 0$. If $P\in
Y\cap X_{i}$, then $P\in Y\cap X_{i-1}$ by the first stabilisation. On the
other hand, if $P\not\in Y\cap X_{i}$, then there exists a
$g\in\operatorname{Spin}(V_{\infty})$ such that $gP\in Z$. Then $gP$ lies in
$X_{i}\cap Z$, which by the second stabilisation equals $X_{i-1}\cap Z$, hence
$P=g^{-1}(gP)$ lies in $X_{i-1}$, as well. We conclude that the chain
$(X_{i})_{i}$ of closed, reduced subschemes of $X$ stabilises. Hence the
inverse half-spin representation
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$ is topologically
$\operatorname{Spin}(V_{\infty})$-Noetherian. ∎

######  Remark 4.9.

While the proof of Theorem 4.1 for the even half-spin case is easily adapted
to a proof for the odd half-spin case, we do not know whether the spin
representation $(\bigwedge\nolimits_{\infty}E_{\infty})^{*}$ itself is
topologically $\operatorname{Spin}(V_{\infty})$-Noetherian! Also, despite much
effort, we have not succeeded in proving that the inverse limit
$\varprojlim_{n}\bigwedge\nolimits^{n}V_{n}$ along the contraction maps
$c_{e_{n}}$ is topologically $\operatorname{SO}(V_{\infty})$-Noetherian.
Indeed, the situation is worse for this question: like the inverse spin
representation, this limit is the dual of a countable-dimensional module that
splits as a direct sum of two $\operatorname{SO}(V_{\infty})$-modules—and here
we do not even know whether the dual of one of these modules is topologically
Noetherian!

##  5\. Half-spin varieties and applications

In this section we introduce the notion of half-spin varieties and reformulate
our main result Theorem 4.1 in this language. We start by fixing the necessary
data determining the half-spin representations of $\operatorname{Spin}(V)$.

######  Notation 5.1.

As shorthand, we write $\mathcal{V}=(V,q,F)\in\mathcal{Q}$ to refer to a
triple where

  1. (1)

$V$ is an even-dimensional vector space over $K$,

  2. (2)

$q$ is a nondegenerate symmetric quadratic form on $V$, and

  3. (3)

$F$ is a maximal isotropic subspace of $V$.

An isomorphism
$\mathcal{V}\to\mathcal{V}^{\prime}=(V^{\prime},q^{\prime},F^{\prime})$ of
such triples is a linear bijection $\varphi:V\to V^{\prime}$ with
$q^{\prime}(\varphi(v))=q(v)$ and $\varphi(F)=F^{\prime}$.

Given a triple $\mathcal{V}$, we have half-spin representations
$\operatorname{Cl}^{\pm}(V)f$, where $f=f_{1}\cdots f_{n}$ with
$f_{1},\ldots,f_{n}$ a basis of $F$ (recall that the left ideal
$\operatorname{Cl}(V)f$ does not depend on this basis). Half-spin varieties
are $\operatorname{Spin}(V)$-invariant subvarieties of these half-spin
representations that are preserved by the contraction maps $\pi_{e}$ from §3.2
and the multiplication maps $\tau_{h}$ from §3.3. The precise definition below
is inspired by the that of a Plücker variety in [DE18]. It involves a uniform
choice of either even or odd half-spin representations. For convenience of
notation, we will only explicitly work with the even half-spin
representations, but all further results are valid for the odd counterparts as
well.

######  Definition 5.2 (Half-spin variety).

A half-spin variety is a rule $X$ that assigns to each triple
$\mathcal{V}=(V,q,F)\in\mathcal{Q}$ a closed, reduced subscheme
$X(\mathcal{V})\subseteq\operatorname{Cl}^{+}(V)f$ such that

  1. (1)

$X(\mathcal{V})$ is $\operatorname{Spin}(V)$-stable;

  2. (2)

for any isomorphism $\varphi:\mathcal{V}\to\mathcal{V}^{\prime}$, the map
$\operatorname{Cl}^{+}(\varphi)$ maps $X(\mathcal{V})$ into
$X(\mathcal{V}^{\prime})$;

  3. (3)

for any isotropic $e\in V$ with $e\not\in F$, if we set
$V^{\prime}:=e^{\perp}/\langle e\rangle$, $q^{\prime}$ the induced form on
$V^{\prime}$, $F^{\prime}$ the image of $F\cap e^{\perp}$ in $V^{\prime}$, and
$\mathcal{V}^{\prime}:=(V^{\prime},q^{\prime},F^{\prime})$, then the
contraction map
$\pi_{e}:\operatorname{Cl}^{+}(V)f\to\operatorname{Cl}^{+}(V^{\prime})f^{\prime}$
maps $X(\mathcal{V})$ into $X(\mathcal{V}^{\prime})$; and

  4. (4)

for any $\mathcal{V}=(V,q,F)$, if we construct a triple $\mathcal{V}^{\prime}$
by setting $V^{\prime}:=V\oplus\langle e,h\rangle$, $q^{\prime}$ the quadratic
form that restricts to $q$ on $V$, makes the direct sum orthogonal, and makes
$e,h$ a hyperbolic basis, if we set $f^{\prime}:=f\cdot h$, then the map
$\tau_{h}:\operatorname{Cl}^{+}(V)f\to\operatorname{Cl}^{+}(V^{\prime})f^{\prime}$
maps $X(\mathcal{V})$ into $X(\mathcal{V}^{\prime})$.

######  Example 5.3.

The following are examples of half-spin varieties.

  1. (1)

Trivially, $X(\mathcal{V}):=\operatorname{Cl}^{+}(V)f$,
$X(\mathcal{V}):=\\{0\\}$ and $X(\mathcal{V}):=\emptyset$ define half-spin
varieties.

  2. (2)

The even component of the cone over the isotropic Grassmannian,
$X(\mathcal{V}):=\widehat{\operatorname{Gr}}_{\operatorname{iso}}^{+}(V,q)$ is
a half-spin variety by Proposition 3.4.

  3. (3)

For two half-spin varieties $X$ and $X^{\prime}$ their join $X+X^{\prime}$
defined by

$(X+X^{\prime})(\mathcal{V}):=\overline{\\{x+x^{\prime}\mid x\in
X(\mathcal{V}),x^{\prime}\in X^{\prime}(\mathcal{V})\\}}$

is a half-spin variety.

  4. (4)

The intersection of two half-spin varieties $X$ and $X^{\prime}$ is a half-
spin variety, which is defined by $(X\cap
X^{\prime})(\mathcal{V}):=X(\mathcal{V})\cap X^{\prime}(\mathcal{V})$.

Similar as in §3.6 we will use the following notation: for every
$n\in{\mathbb{N}}$, we consider the vector space $V_{n}=\langle
e_{1},\dots,e_{n},f_{1},\dots,f_{n}\rangle$ together with the quadratic form
$q_{n}$ whose corresponding bilinear form $(\cdot|\cdot)$ satisfies

$(e_{i}|e_{j})=0,\quad(f_{i}|f_{j})=0\quad\text{and}\quad(e_{i}|f_{j})=\delta_{ij}.$

Furthermore, let $E_{n}=\langle e_{1},\dots,e_{n}\rangle$ and $F_{n}=\langle
f_{1},\dots,f_{n}\rangle$; these are maximal isotropic subspaces of $V_{n}$.
We will denote the associated tuple by $\mathcal{V}_{n}=(V_{n},q_{n},F_{n})$.

######  Remark 5.4.

A half-spin variety $X$ is completely determined by the values
$X(\mathcal{V}_{n})$, that is, if $X$ and $X^{\prime}$ are half-spin varieties
such that $X(\mathcal{V}_{n})=X^{\prime}(\mathcal{V}_{n})$ for all
$n\in{\mathbb{N}}$, then $X(\mathcal{V})=X^{\prime}(\mathcal{V})$ for all
tuples $\mathcal{V}$.

We now want to associate to each half-spin variety $X$ an infinite-dimensional
scheme $X_{\infty}$ embedded inside the inverse half-spin representation
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$ as follows. Since
$V_{n}=E_{n}\oplus F_{n}$, we can use the isomorphism from §2.5 to embed
$X(\mathcal{V}_{n})$ as a reduced subscheme of $\bigwedge^{+}E_{n}$ (recall
from §3.7 that we view $\bigwedge^{+}E_{n}$ as the affine scheme with
coordinate ring ${\rm Sym}(\bigwedge\nolimits^{+(-1)^{n}}E_{n})$). We
abbreviate $X_{n}\coloneqq X(\mathcal{V}_{n})\subseteq\bigwedge^{+}E_{n}$.

For $N\geq n$ let
$\pi_{N,n}:\bigwedge\nolimits^{+}E_{N}\to\bigwedge\nolimits^{+}E_{n}$, resp.
$\tau_{n,N}:\bigwedge\nolimits^{+}E_{n}\to\bigwedge\nolimits^{+}E_{N}$ be the
maps induced by by canonical projection $E_{N}\to E_{n}$, resp. by the
injection $E_{n}\hookrightarrow E_{N}$. Note that $\tau_{n,N}$ is a section of
$\pi_{N,n}$. Recall that
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}=\varprojlim_{n}\bigwedge^{+}E_{n}$.
We denote the structure maps by
$\pi_{\infty,n}:(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}\to\bigwedge\nolimits^{+}E_{n}$
and by
$\tau_{n,\infty}:\bigwedge\nolimits^{+}E_{n}\to(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$
the inclusion maps induced by $\tau_{n,N}$.

From the definition of a half-spin variety it follows that

$\pi_{N,n}(X_{N})\subseteq X_{n}\quad\text{and}\quad\tau_{n,N}(X_{n})\subseteq
X_{N}.$ (5.1)

Hence the inverse limit

$X_{\infty}\coloneqq\underset{n}{\varprojlim}\ X_{n}$

is well-defined, and a closed, reduced,
$\operatorname{Spin}(V_{\infty})$-stable subscheme of
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$. In order to see this, write
$R_{n}\coloneqq\operatorname{Sym}(\bigwedge^{+(-1)^{n}}E_{n})$ and
$R_{\infty}\coloneqq\operatorname{Sym}(\bigwedge\nolimits^{+}_{\infty}E_{\infty})$.
Let $I_{n}\subseteq R_{n}$ be the radical ideal associated to
$X_{n}\subseteq\operatorname{Spec}(R_{n})$, i.e.
$X_{n}=V(I_{n})=\operatorname{Spec}(R_{n}/I_{n})$. As
$\operatorname{Spec}(\cdot)$ is a contravariant equivalence of categories, it
holds that

$X_{\infty}\coloneqq\varprojlim_{n}X_{n}=\varprojlim_{n}\operatorname{Spec}(R_{n}/I_{n})=\operatorname{Spec}(\varinjlim_{n}(R_{n}/I_{n})).$

So $X_{\infty}$ corresponds to the ideal
$I_{\infty}\coloneqq\varinjlim_{n}I_{n}\subseteq R_{\infty}$. As all
$I_{n}\subseteq R_{n}$ are radical, so is $I_{\infty}\subseteq R_{\infty}$ and
therefore $X_{\infty}$ is a reduced subscheme.

It follows from Equation 5.1 that

$\pi_{\infty,n}(X_{\infty})\subseteq
X_{n}\quad\text{and}\quad\tau_{n,\infty}(X_{n})\subseteq X_{\infty}.$ (5.2)

######  Lemma 5.5.

The mapping

$X\mapsto X_{\infty}$

is injective. That is, if $X$ and $X^{\prime}$ are half-spin varieties such
that $X_{\infty}=X^{\prime}_{\infty}$, then $X=X^{\prime}$, i.e.
$X(\mathcal{V})=X^{\prime}(\mathcal{V})$ for all tuples $\mathcal{V}$.

###### Proof.

Note that, for all $n\in{\mathbb{N}}$, it holds that

$X_{n}=\pi_{\infty,n}(X_{\infty}).$

Indeed, the inclusion $\supseteq$ is contained in Equation 5.2, and the other
direction $\subseteq$ follows from the fact that $\tau_{n,\infty}:X_{n}\to
X_{\infty}$ is a section of $\pi_{\infty,n}$. Hence, if
$X_{\infty}=X^{\prime}_{\infty}$, then

$X_{n}=\pi_{\infty,n}(X_{\infty})=\pi_{\infty,n}(X^{\prime}_{\infty})=X^{\prime}_{n}.$

By Remark 5.4 this shows that $X=X^{\prime}$. ∎

For two half-spin varieties $X$ and $X^{\prime}$, we will write $X\subseteq
X^{\prime}$ if $X(\mathcal{V})\subseteq X^{\prime}(\mathcal{V})$ for all
$\mathcal{V}=(V,q,F)$. Theorem 4.1 then implies the following.

######  Theorem 5.6 (Noetherianity of half-spin varieties).

Every descending chain of half-spin varieties

$X^{(0)}\supseteq X^{(1)}\supseteq X^{(2)}\supseteq X^{(3)}\supseteq\ldots$

stabilizes, that is, there exists $m_{0}\in{\mathbb{N}}$ such that
$X^{(m)}=X^{(m_{0})}$ for all $m\geq m_{0}$.

###### Proof.

Note that the mapping $X\mapsto X_{\infty}$ is order preserving, that is, if
$X\subseteq X^{\prime}$, then $X_{\infty}\subseteq X^{\prime}_{\infty}$.
Hence, a chain

$X^{(0)}\supseteq X^{(1)}\supseteq X^{(2)}\supseteq X^{(3)}\supseteq\ldots$

of half-spin varieties induces a chain

$X^{(0)}_{\infty}\supseteq X^{(1)}_{\infty}\supseteq X^{(2)}_{\infty}\supseteq
X^{(3)}_{\infty}\supseteq\ldots$

of closed, reduced, $\operatorname{Spin}(V_{\infty})$-stable subschemes in
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$. By Theorem 4.1 we know that
$(\bigwedge\nolimits^{+}_{\infty}E_{\infty})^{*}$ is topologically
$\operatorname{Spin}(V_{\infty})$-Noetherian. Hence, the chain
$X^{(m)}_{\infty}$ stabilizes. But then, by Lemma 5.5 also the chain of half-
spin varieties $X^{(m)}$ stabilizes. This completes the proof. ∎

As a consequence we obtain the next results, which state how $X_{\infty}$ is
determined by the data coming from some finite level of $X$.

######  Theorem 5.7.

Let $X$ be a half-spin variety. Then there exists $n_{0}\in{\mathbb{N}}$ such
that

$X_{\infty}=V\big{(}\operatorname{rad}(\operatorname{Spin}(V_{\infty})\cdot
I_{n_{0}})\big{)},$

where $\operatorname{rad}(\operatorname{Spin}(V_{\infty})\cdot
I_{n_{0}})\subseteq\operatorname{Sym}(\bigwedge\nolimits^{+}_{\infty}E_{\infty})$
is the radical ideal generated by the $\operatorname{Spin}(V_{\infty})$-orbits
of the ideal
$I_{n_{0}}\subseteq\operatorname{Sym}(\bigwedge^{+(-1)^{n_{0}}}E_{n_{0}})$
defining $X_{n_{0}}\subseteq\bigwedge^{+}E_{n_{0}}$.

###### Proof.

For each $n\in{\mathbb{N}}$ set
$J_{n}\coloneqq\operatorname{rad}(\operatorname{Spin}(V_{\infty})\cdot
I_{n})\subseteq\operatorname{Sym}(\bigwedge\nolimits^{+}_{\infty}E_{\infty})$.
We denote by
$I_{\infty}\subseteq\operatorname{Sym}(\bigwedge\nolimits^{+}_{\infty}E_{\infty})$
the ideal associated to $X_{\infty}$. This ideal is
$\operatorname{Spin}(V_{\infty})$-stable, radical and it holds that
$I_{\infty}=\varinjlim_{n}I_{n}$. Thus, $\bigcup_{n}J_{n}=I_{\infty}$. Since
$(J_{n})_{n\in{\mathbb{N}}}$ is an increasing chain of closed
$\operatorname{Spin}(V_{\infty})$-stable radical ideals, by Theorem 4.1 there
exists $n_{0}\in{\mathbb{N}}$ such that $J_{n}=J_{n_{0}}$ for all $n\geq
n_{0}$. Therefore, $I_{\infty}=\bigcup_{n}J_{n}=J_{n_{0}}$ and hence
$X_{\infty}=V(I_{\infty})=V(J_{n_{0}}).$ ∎

######  Corollary 5.8.

Let $X$ be a half-spin variety. There exists $n_{0}\in{\mathbb{N}}$ such that
for all $n\geq n_{0}$ it holds that

$X_{n}=V(\operatorname{rad}(\operatorname{Spin}(V_{n})\cdot I_{n_{0}})).$

###### Proof.

Take $n_{0}$ as in Theorem 5.7. Then the statement follows from that theorem
and [Dra10, Lemma 2.1]. To apply that lemma, we must check condition (*) in
that paper, namely, that for $q\geq n\geq n_{0}$ and
$g\in\operatorname{Spin}(V_{q})$ we can write

$\pi_{q,n_{0}}\circ
g\circ\tau_{n,q}=g^{\prime\prime}\circ\tau_{m,n_{0}}\circ\pi_{n,m}\circ
g^{\prime}$

for suitable $m\leq n_{0}$ and $g^{\prime}\in\operatorname{Spin}(V_{n})$ and
$g^{\prime\prime}\in\operatorname{Spin}(V_{n_{0}})$. In fact, since half-spin
varieties are affine cones, it suffices that this identity holds up to a
scalar factor. It also suffices to prove this for $g$ in an open dense subset
$U$ of $\operatorname{Spin}(V_{q})$, because the equations for $X_{n_{0}}$
pulled back along the map on the left for $g\in U$ imply the equations for all
$g$. We will prove this, with $m=n_{0}$, using the Cartan map in Lemma 6.6
below. ∎

##  6\. Universality of
$\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(4,8)$ and the Cartan map

###  6.1. Statement

In [ST24] the last two authors showed that in even dimension, the isotropic
Grassmannian in its Plücker embedding is set-theoretically defined by pulling
back equations coming from
$\widehat{\operatorname{Gr}}_{\operatorname{iso}}(4,8)$. Using the Cartan map
we can translate this into a statement about the isotropic Grassmannian in its
spinor embedding and prove the following result.

######  Theorem 6.1.

For all $n\geq 4$ we have

$\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V_{n})=V(\operatorname{rad}(\operatorname{Spin}(V_{n})\cdot
I_{4})),$

where $I_{4}$ is the ideal of polynomials defining
$\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V_{4})\subseteq\operatorname{Cl}^{+}(V_{4})f.$

In other words, the bound $n_{0}$ from Corollary 5.8 can be taken equal to $4$
for the cone over the isotropic Grassmannian. We give the proof of Theorem 6.1
in §6.5 using properties of the Cartan map that will be established in the
following sections.

###  6.2. Definition of the Cartan map

When we regard $e_{1}\wedge\cdots\wedge e_{n}$ as an element of the $n$-th
exterior power $\bigwedge\nolimits^{n}V$ of the standard representation $V$ of
$\mathfrak{so}(V)$, then it is a highest weight vector of weight
$(0,\ldots,0,2)=2\lambda_{0}$, where $\lambda_{0}$ is the fundamental weight
introduced in §2.6 and the highest weight of the half-spin representation
$\operatorname{Cl}^{(-1)^{n}}(V)f$. Similarly, the element $e_{1}\wedge
e_{2}\wedge\cdots\wedge e_{n-1}\wedge f_{n}\in\bigwedge\nolimits^{n}V$ is a
highest weight vector of weight $(0,\ldots,0,2,0)=2\lambda_{1}$, where
$\lambda_{1}$ is the highest weight of the other half-spin representation. So
$\bigwedge\nolimits^{n}V$ contains copies of the irreducible representations
$V_{2\lambda_{0}},V_{2\lambda_{1}}$ of $\mathfrak{so}(V)$; in fact, it is well
known to be the direct sum of these. To compare our results in this paper
about spin representations with earlier work by the last two authors about
exterior powers, we will need the following considerations.

Consider any connected, reductive algebraic group $G$, with maximal torus $T$
and Borel subgroup $B\supseteq T$. Let $\lambda$ be a dominant weight of $G$,
let $V_{\lambda}$ be the corresponding irreducible representation, and let
$v_{\lambda}\in V_{\lambda}$ be a nonzero highest-weight vector (which is
unique up to scalar multiples). Then the symmetric square $S^{2}V_{\lambda}$
contains a one-dimensional space of vectors of weight $2\lambda$, spanned by
$v_{2\lambda}:=v_{\lambda}^{2}$. This vector is itself a highest-weight
vector, and hence generates a copy of $V_{2\lambda}$; this is sometimes called
the Cartan component of $S^{2}V_{\lambda}$. By semisimplicity, there is a
$G$-equivariant linear projection $\pi:S^{2}V_{\lambda}\to V_{2\lambda}$ that
restricts to the identity on $V_{2\lambda}$. The map

$\widehat{\nu_{2}}:V_{\lambda}\to V_{2\lambda},\quad v\mapsto\pi(v^{2}).$

is a nonzero polynomial map, homogeneous of degree $2$, and hence induces a
rational map $\nu_{2}:\mathbb{P}V_{\lambda}\to\mathbb{P}V_{2\lambda}$. Note
that this is the composition of the quadratic Veronese embedding and the
projection $\pi$. We will refer to $\nu_{2}$ and to $\widehat{\nu_{2}}$ as the
Cartan map.

######  Lemma 6.2.

The rational map $\nu_{2}$ is a morphism and injective.

We thank J.M. Landsberg for help with the following proof.

###### Proof.

To show that $\nu_{2}$ is a morphism, we need to show that $\pi(v^{2})$ is
nonzero whenever $v$ is. Now the set $Q$ of all $[v]\in\mathbb{P}V_{\lambda}$
for which $\pi(v^{2})$ is zero is closed and $B$-stable. Hence, if
$Q\neq\emptyset$, then by Borel’s fixed point theorem, $Q$ contains a
$B$-fixed point. But the only $B$-fixed point in $\mathbb{P}V_{\lambda}$ is
$[v_{\lambda}]$, and $v_{\lambda}$ is mapped to the nonzero vector
$v_{2\lambda}$. Hence $Q=\emptyset$.

Injectivity is similar but slightly more subtle. Assume that there exist
distinct $[v],[w]$ with $\nu_{2}([v])=\nu_{2}([w])$. Then $\\{[v],[w]\\}$
represents a point in the Hilbert scheme of two points in
$\mathbb{P}V_{\lambda}$. Now the locus $Q$ of points $S$ in said Hilbert
scheme such that $\nu_{2}(S)$ is a single reduced point is a closed subset of
a projective scheme, hence $Q$ contains a $B$-stable point $S$. This scheme
$S$ cannot consist of two distinct reduced points, since there is only one
$B$-stable point. Therefore, the reduced subscheme of $S$ is
$\\{[v_{\lambda}]\\}$, and $S$ represents the point $[v_{\lambda}]$ together
with a nonzero tangent direction in
$T_{[v_{\lambda}]}\mathbb{P}V_{\lambda}=V_{\lambda}/Kv_{\lambda}$, represented
by $w\in V_{\lambda}$. Furthermore, $B$-stability of $S$ implies that the
$B$-module generated by $w$ equals $\langle w,v_{\lambda}\rangle_{K}$. That
$S$ lies in $Q$ means that

$\pi((v_{\lambda}+\epsilon w)^{2})=v_{2\lambda}\mod\epsilon^{2}.$

We find that $\pi(v_{\lambda}w)=0$, so that the $G$-module generated by
$v_{\lambda}w\in S^{2}V$ does not contain $V_{2\lambda}$. But since
$v_{\lambda}$ is (up to a scalar) fixed by $B$, the $B$-module generated by
$v_{\lambda}w$ equals $v_{\lambda}$ times the $B$-module gene rated by $w$,
and hence contains $v_{\lambda}^{2}=v_{2\lambda}$, a contradiction. ∎

Observe that $\nu_{2}$ maps the unique closed orbit $G\cdot[v_{\lambda}]$ in
$\mathbb{P}V_{\lambda}$ isomorphically to the unique closed orbit
$G\cdot[v_{2\lambda}]$—both are isomorphic to $G/P$, where $P\supseteq B$ is
the stabiliser of the line $Kv_{\lambda}$ and of the line $Kv_{2\lambda}$. In
our setting above, where $G=\operatorname{Spin}(V)$ and
$\lambda\in\\{\lambda_{0},\lambda_{1}\\}$, the closed orbit
$G\cdot[v_{2\lambda}]$ is one of the two connected components of the
Grassmannian $\operatorname{Gr}_{\operatorname{iso}}(V)$ of $n$-dimensional
isotropic subspaces of $V$, in its Plücker embedding; and the closed orbit in
the projectivised half-spin representation $\mathbb{P}V_{\lambda}$ is the same
component of the isotropic Grassmannian but now in its spinor embedding.

In what follows we will need a more explicit understanding both of the
embedding of the isotropic Grassmannian in the projectivised (half-)spin
representations and of the map $\widehat{\nu_{2}}$. These are treated in the
next two paragraphs.

###  6.3. The map $\widehat{\nu_{2}}$ from the spin representation to the
exterior power

In §6.2 we argued the existence of $\operatorname{Spin}(V)$-equivariant
quadratic maps from the half-spin representations to the two summands of
$\bigwedge\nolimits^{n}V$. In [Man09] these two maps are described jointly as

$\widehat{\nu_{2}}:\operatorname{Cl}(V)f\to\bigwedge\nolimits^{n}V,\quad
af\mapsto\text{ the component in $\bigwedge\nolimits^{n}V$ of
}(afa^{*})\bullet 1\in\bigwedge\nolimits V,$

where $\bullet$ stands for the $\operatorname{Cl}(V)$-module structure of
$\bigwedge\nolimits V$ from §2.2 and $a^{*}$ refers to the anti-automorphism
of the Clifford algebra from §2.1.

######  Lemma 6.3.

The map $\hat{\nu}_{2}$ maps the isotropic Grassmann cone in its spinor
embedding to the isotropic Grassmann cone in its Plücker embedding, i.e.,

$\hat{\nu}_{2}\big{(}\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V)\big{)}=\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(V),$

where
$\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(V)$ is
the isotropic Grassmann cone in its Plücker embedding (see [ST24, Definition
3.7]).

###### Proof.

Let $H\subseteq V$ be a maximal isotropic subspace that intersects $F$ in a
$k$-dimensional space. Choose a hyperbolic basis
$e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ adapted to $H$ and $F$, so that
$H=\langle e_{k+1},\ldots,e_{n},f_{1},\ldots,f_{k}\rangle$ is represented by
the vector $\omega_{H}:=e_{k+1}\cdots
e_{n}f\in\widehat{\operatorname{Gr}}_{\operatorname{iso}}(V)$ where
$f=f_{1}\cdots f_{n}$; see §3.1. Set $a:=e_{k+1}\cdots e_{n}$. Now

$\displaystyle afa^{*}$ $\displaystyle=e_{k+1}\cdots e_{n}f_{1}\cdots
f_{n}e_{n}\cdots e_{k+1}$ $\displaystyle=e_{k+1}\cdots e_{n}f_{1}\cdots
f_{n-1}(2-e_{n}f_{n})e_{n-1}\cdots e_{k+1}$ $\displaystyle=2e_{k+1}\cdots
e_{n}f_{1}\cdots f_{n-1}e_{n-1}\cdots e_{k+1}$ $\displaystyle=\ldots$
$\displaystyle=2^{n-k}e_{k+1}\cdots e_{n}f_{1}\cdots f_{k}$

where we have used the definition of $\operatorname{Cl}(V)$ (in the first
step), the fact that the second copy of $e_{n}$ is perpendicular to all
elements before it and multiplies to zero with the first copy of $e_{n}$ (in
the second step), and have repeated this another $n-k-1$ times in the last
step. We now find that

$(afa^{*})\bullet 1=2^{n-k}e_{k+1}\wedge\cdots\wedge e_{n}\wedge
f_{1}\wedge\cdots\wedge f_{k},$

so that $(afa^{*})\bullet 1$ lies in one of the two summands of
$\bigwedge\nolimits^{n}V$ and spans the line representing the space $H$ in the
Plücker embedding. This shows that $\hat{\nu_{2}}$ maps the isotropic
Grassmann cone in its spinor embedding to the isotropic Grassmann cone in its
Plücker embedding, as desired. ∎

######  Remark 6.4.

While the spin representation $\operatorname{Cl}(V)f$ depends only on the
space $F$—since $F$ determines $f$ up to a scalar, which doesn’t affect the
left ideal $\operatorname{Cl}(V)f$—the map $\widehat{\nu_{2}}$ actually
depends on $f$ itself: for $\tilde{f}:=tf$ with $t\in K^{*}$, the map
$\widehat{\nu_{2}}$ constructed from $\tilde{f}$ sends $af=(t^{-1}a)\tilde{f}$
to $t^{-1}a\tilde{f}t^{-1}a^{*}=t^{-1}afa^{*}$, so the new $\widehat{\nu_{2}}$
is $t^{-1}$ times the old map.

###  6.4. Contraction and the Cartan map commute

Recall from §6.2 that we have quadratic maps $\widehat{\nu_{2}}$ from the
half-spin representations to the two summands of $\bigwedge\nolimits^{n}V$;
together, these form a quadratic map $\widehat{\nu_{2}}$ which we discussed in
§6.3. By abuse of terminology, we call this, too, the Cartan map. Given an
isotropic vector $e\in V$ that is not in $F$, we write $\widehat{\nu_{2}}$
also for the Cartan map
$\operatorname{Cl}(V_{e})\overline{f}\to\bigwedge\nolimits^{n-1}V_{e}$
(notation as in §3.2). Recall from §3.2 the contraction map
$c_{e}:\bigwedge\nolimits^{n}V\to\bigwedge\nolimits^{n-1}V_{e}$ and its spin
analogue
$\pi_{e}:\operatorname{Cl}(V)f\to\operatorname{Cl}(V_{e})\overline{f}$. Also,
for a fixed $h=f_{n}\in F$ with $\langle e,h\rangle=1$, recall from §3.3 the
multiplication map
$m_{h}:\bigwedge\nolimits^{n-1}V_{e}\to\bigwedge\nolimits^{n}V$ and its spin
analogue
$\tau_{h}:\operatorname{Cl}(V_{e})\overline{f}\to\operatorname{Cl}(V)f$. The
relations between these maps are as follows.

######  Proposition 6.5.

The following diagrams essentially commute:

$\textstyle{\operatorname{Cl}(V)f\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nu_{2}}}$$\scriptstyle{\pi_{e}}$$\textstyle{\operatorname{Cl}(V_{e})\overline{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nu_{2}}}$$\textstyle{\bigwedge\nolimits^{n}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{e}}$$\textstyle{\bigwedge\nolimits^{n-1}V_{e}}$
and
$\textstyle{\operatorname{Cl}(V_{e})\overline{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nu_{2}}}$$\scriptstyle{\tau_{h}}$$\textstyle{\operatorname{Cl}(V)f\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\nu_{2}}}$$\textstyle{\bigwedge\nolimits^{n-1}V_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m_{h}}$$\textstyle{\bigwedge\nolimits^{n}V.}$
(6.1)

More precisely, one can rescale the restrictions of $c_{e}$ to the two
$\mathfrak{so}(V)$-submodules of $\bigwedge\nolimits^{n}V$ each by $\pm 1$ in
such a manner that the diagram commutes, and similarly for $m_{h}$.

Naturally, we could have chosen the scalars in the definition of $c_{e}$ (or,
using a square root of $-1$, in that of $\pi_{e}$) such that the diagram
literally commutes. However, we have chosen the scalars such that $c_{e}$ has
the most natural formula and $\pi_{e},\tau_{h}$ have the most natural formulas
in our model $\bigwedge\nolimits E$ for the spin representation.

###### Proof.

We may choose a hyperbolic basis $e_{1},\ldots,e_{n},f_{1},\ldots,f_{n}$ of
$V$ such that $e=e_{n}$ and $f_{1},\ldots,f_{n}$ is a basis of $F$. We write
$f:=f_{1}\cdots f_{n}$ and
$\overline{f}:=\overline{f}_{1}\cdots\overline{f}_{n-1}$.

Since the vertical maps are quadratic, it is not sufficient to show
commutativity on a spanning set. We therefore consider

$a:=\sum_{I\subseteq[n]}c_{I}e_{I}$

where, for $I=\\{i_{1}<\ldots<i_{k}\\}$ we write $e_{I}:=e_{i_{1}}\cdots
e_{i_{k}}$. We then have

$\pi_{e}(af)=\sum_{I:n\not\in
I}c_{I}\overline{e_{I}}\overline{f}=:\overline{a}\overline{f}$

and

$\widehat{\nu_{2}}(\overline{a}\overline{f})=\text{the component in
$\bigwedge\nolimits^{n-1}V_{e}$ of}\sum_{I,J:n\not\in I\cup
J}(c_{I}c_{J}\overline{e_{I}}\overline{f}\overline{e_{J}}^{*})\bullet
1\in\bigwedge\nolimits V_{e}.$

Now note that, since $\overline{f}$ has $n-1$ factors, if $I,J$ do not have
the same parity, then acting with
$\overline{e_{I}}\overline{f}\overline{e_{J}}^{*}$ on $1$ yields a zero
contribution in $\bigwedge\nolimits^{n-1}V_{e}$. Hence the sum above may be
split into two sums, one of which is

$\text{the component in $\bigwedge\nolimits^{n-1}V_{e}$
of}\sum_{I,J:|I|,|J|\text{ even, }n\not\in I\cup
J}(c_{I}c_{J}\overline{e_{I}}\overline{f}\overline{e_{J}}^{*})\bullet 1.$
(6.2)

On the other hand, consider

$\widehat{\nu_{2}}(af)=\text{the component in $\bigwedge\nolimits^{n}V$
of}\sum_{I,J}(c_{I}c_{J}e_{I}fe_{J}^{*})\bullet 1\in\bigwedge\nolimits V.$

For the same reason as above, this splits into two sums, and we want to
compare the following expression to (6.2):

$c_{e}(\text{the component in $\bigwedge\nolimits^{n}V$ of
}\sum_{I,J:|I|,|J|\text{ even}}(c_{I}c_{J}e_{I}fe_{J}^{*})\bullet 1).$ (6.3)

Now recall that the action of $e=e_{n}\in V\subseteq\operatorname{Cl}(V)$ on
$\bigwedge\nolimits V$ is via $o(e)+\iota(e)$, while $c_{e}$ is $\iota_{e}$
followed by projection to $\bigwedge\nolimits^{n-1}V_{e}$. Hence to compute
(6.3), we may as well compute the summands of

$\text{the component in $\bigwedge\nolimits^{n}V$ of }\sum_{I,J:|I|,|J|\text{
even}}(c_{I}c_{J}\cdot e\cdot e_{I}fe_{J}^{*})\bullet 1$

that do not contain a factor $e$. Terms with $n\in I$ do not contribute,
because then $ee_{I}=0$. Terms with $n\not\in I$ but $n\in J$ do not
contribute because when $e$ gets contracted with $f_{n}$ a factor $e$ in
$e_{J}^{*}$ survives, and when $e$ does not get contracted with $f_{n}$, we
use $ee_{J}^{*}=0$. So we may restrict attention to the terms with $n\not\in
I\cup J$. Let $I,J$ correspond to such a term, that is, $|I|,|J|$ are even and
$n\not\in I\cup J$. Write $I=\\{i_{1}<\ldots<i_{k}\\}$ and
$J=\\{j_{1}<\ldots<j_{l}\\}$. Then

$\displaystyle(ee_{I}fe_{J}^{*})\bullet 1$
$\displaystyle=((-1)^{n-1}e_{I}f_{1}\cdots f_{n-1}ef_{n}e_{J}^{*})\bullet 1$
$\displaystyle=((-1)^{n-1}e_{I}f_{1}\cdots f_{n-1}e)\bullet(f_{n}\wedge
e_{j_{l}}\wedge\cdots\wedge e_{j_{1}})$
$\displaystyle=((-1)^{n-1}e_{I}f_{1}\cdots
f_{n-1})\bullet(e_{j_{l}}\wedge\cdots\wedge e_{j_{1}}+e\wedge f_{n}\wedge
e_{j_{l}}\wedge\cdots\wedge e_{j_{1}}).$

The second term in the last expression will contribute only terms with a
factor $e$ to the final result, and the former term contributes

$\text{the component in $\bigwedge\nolimits^{n-1}V_{e}$ of
}(-1)^{n-1}(\overline{e_{I}}\overline{f}\overline{e_{J}}^{*})\bullet 1.$

Comparing this with (6.2), we see that the diagram commutes on terms in
$\operatorname{Cl}^{+}(V)f$ up to the factor $(-1)^{n-1}$. A similar
computation shows that it commutes on terms in $\operatorname{Cl}^{-}(V)f$ up
to a factor factor $(-1)^{n}$.

We now consider the second diagram, where $V$ is split as the orthogonal
direct sum $V_{e}\oplus\langle e,h\rangle$ with $e=e_{n},h=f_{n}$. Consider
$a\in\operatorname{Cl}(\langle e_{1},\ldots,e_{n-1}\rangle)$. By the same
argument as above, it suffices to consider the case where all summands of $a$
in the basis $e_{I}$ have indices $I$ with $|I|$ of the same parity, say even.
Then $\widehat{\nu}_{2}\circ\tau_{h}$ in the diagram sends $a\overline{f}$ to
the component in $\bigwedge\nolimits^{n}V$ of $afa^{*}\bullet 1$. Since the
summands $e_{I}$ in $a$ all have $n\not\in I$, in $afa^{*}\bullet 1$ all
summands have a factor $f_{n}$, and indeed

$(afa^{*})\bullet 1=f_{n}\wedge(a\overline{f}a^{*}\bullet 1)$

(when all terms in $a$ have $|I|$ odd, we get a minus sign). The component in
$\bigwedge\nolimits^{n}V$ of this expression is the same as the one obtained
via $m_{h}\circ\widehat{\nu}_{2}$. ∎

###  6.5. Proof of Theorem 6.1

In this section we use the Cartan map to prove Theorem 6.1, and finish the
proof of Corollary 5.8 via a similar argument.

###### Proof of Theorem 6.1.

For a quadratic space of dimension $2n$, denote by
$\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(V)\subseteq\bigwedge\nolimits^{n}V$
the isotropic Grassmann cone over the Plücker embedding. Given a maximal
isotropic subspace $F\subseteq V$ with basis $f_{1},\ldots,f_{n}$ and
$f:=f_{1}\cdots f_{n}$, let
$\hat{\nu}_{2}:\operatorname{Cl}^{+}(V)f\to\bigwedge\nolimits^{n}V$ be the
Cartan map defined in §6.3. For any isotropic $v\in V\setminus F$ the diagram

${\bigwedge\nolimits^{n}V}$${\bigwedge\nolimits^{n-1}V_{v}}$${\operatorname{Cl}(V)f}$${\operatorname{Cl}(V_{v})\overline{f}}$$\scriptstyle{c_{v}}$$\scriptstyle{\pi_{v}}$$\scriptstyle{\hat{\nu}_{2}}$$\scriptstyle{\hat{\nu}_{2}}$

commutes up to scalar factor at the bottom by Proposition 6.5, where
$V_{v}:=v^{\perp}/\langle v\rangle$ and where $\overline{f}$ is the image of a
product of a basis of $v^{\perp}\cap F_{n}$.

The proof of [ST24, Corollary 4.2] shows that for
$\omega\in\bigwedge\nolimits^{n}V$ the following are equivalent:

  1. (1)

$\omega\in\widehat{\operatorname{Gr}}^{\rm Pl}_{\operatorname{iso}}(V)$;

  2. (2)

For every sequence of isotropic vectors $v_{1}\in V$, $v_{2}\in V_{v_{1}}$,
$v_{3}\in(V_{v_{1}})_{v_{2}},\dots,v_{n-4}\in(\cdots((V_{v_{1}})_{v_{2}})_{v_{3}}\cdots)_{v_{n-3}}$
it holds

$C(\omega)\in\widehat{\operatorname{Gr}}^{\rm Pl}_{\operatorname{iso}}(W),$

where we abbreviate $W:=(\cdots((V_{v_{1}})_{v_{2}})_{v_{3}}\cdots)_{v_{n-4}}$
and $C:\bigwedge\nolimits^{n}V\to\bigwedge\nolimits^{4}W$ is the composition
$C:=c_{v_{n-4}}\circ\cdots\circ c_{v_{1}}$ of the contraction maps $c_{v_{i}}$
introduced in Section 3.2.

By slight abuse of notation, we also write $v_{1},\ldots,v_{n-4}$ for
preimages of these vectors in $V$. These span an $(n-4)$-dimensional isotropic
subspace $U$ of $V$ (provided that each $v_{i}$ chosen above in the successive
quotients is nonzero), and $W$ equals $U^{\perp}/U$. For any fixed $\omega$,
the condition that $C(\omega)$ lies in $\widehat{\operatorname{Gr}}^{\rm
Pl}_{\operatorname{iso}}(W)$ is a closed condition on $U$, and hence it
suffices to check that condition for $U$ in a dense subset of the Grassmannian
of isotropic $(n-4)$-dimensional subspaces of $V$. In particular, it suffices
to check this when $U\cap F_{n}=\\{0\\}$.

Fix $n\geq 4$ and $x\in\operatorname{Cl}(V_{n})f_{1}\cdots f_{n}$ such that
$p(g\cdot x)=0$ for all $g\in{\operatorname{Spin}}(V_{n})$ and all $p\in
I_{4}$. This means precisely that $\pi_{n,4}(g\cdot
x)\in\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V_{4})$ for all
$g\in{\operatorname{Spin}}(V_{n})$. We need to show that
$x\in\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V_{n})$. To this
end, consider $\omega:=\hat{\nu}_{2}(x)\in\bigwedge\nolimits^{n}V_{n}$. It
suffices to show that
$\omega\in\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(V_{n})$.
Indeed, this follows from the fact that
$\hat{\nu}_{2}\big{(}\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V)\big{)}$
is one of the two irreducible components of
$\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(V)$ (see
Lemma 6.3) and because $\nu_{2}$ is an injective morphism by Lemma 6.2. Let
$v_{1},v_{2},\ldots,v_{n-4}\in V_{n}$ as above: linearly independent, and such
that the span $U:=\langle v_{1},\ldots,v_{n-4}\rangle$ is an isotropic space
that intersects $F_{n}$ trivially. Let $C:=c_{v_{n-4}}\circ\cdots\circ
c_{v_{1}}$ be the composition of the associated contractions. We need to show
that
$C(\omega)\in\widehat{\operatorname{Gr}}^{\operatorname{Pl}}_{\operatorname{iso}}(W)$,
where $W:=U^{\perp}/U$.

Now
$\hat{\nu}_{2}\big{(}\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(W)\big{)}\subseteq\widehat{\operatorname{Gr}}^{\rm
Pl}_{\operatorname{iso}}(W)$ by Lemma 6.3, and the diagram

${\bigwedge\nolimits^{n}V_{n}}$${\bigwedge\nolimits^{4}W}$${\operatorname{Cl}(V_{n})f}$${\
\operatorname{Cl}(W)\overline{f},}$$\scriptstyle{C}$$\scriptstyle{\pi_{v_{n-4}}\circ\cdots\circ\pi_{v_{1}}}$$\scriptstyle{\hat{\nu}_{2}}$$\scriptstyle{\hat{\nu}_{2}}$

where $\overline{f}$ is the image of the product of a basis of $U^{\perp}\cap
F_{n}$, commutes up to a scalar factor in the bottom map due to Proposition
6.5. Hence it suffices to check that
$\pi_{v_{n-4}}\circ\cdots\circ\pi_{v_{1}}(x)\in\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(W)$.
Now there exists an element $g\in\operatorname{Spin}(V_{n})$ that maps $F_{n}$
into itself (not with the identity!) and sends $v_{i}$ to $e_{n+1-i}$ for
$i=1,\ldots,n-4$. This induces an isometry
$W:=U^{\perp}/U\to(U^{\prime})^{\perp}/U^{\prime}=V_{4}=\langle
e_{1},\ldots,e_{4},f_{1},\ldots,f_{4}\rangle$, where $U^{\prime}:=\langle
e_{5},\ldots,e_{n}\rangle$. This in turn induces a linear isomorphism (unique
up to a scalar)
$\operatorname{Cl}(W)\cdot\overline{f}\to\operatorname{Cl}(V_{4})\cdot
f_{1}\cdots f_{4}$ (where $f$ on the left is the product of a basis of
$F_{n}\cap U^{\perp}$) that maps
$\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(W)$ onto
$\widehat{\operatorname{Gr}}^{+}_{\operatorname{iso}}(V_{4})$. Since, by
assumption, $\pi_{n,4}(g\cdot x)=\pi_{e_{5}}\circ\cdots\circ\pi_{e_{n}}(g\cdot
x)$ lies in the latter isotropic Grassmann cone,
$\pi_{v_{n-4}}\circ\cdots\circ\pi_{v_{1}}(x)$ lies in the former. ∎

######  Lemma 6.6.

Let $q\geq n\geq n_{0}$. Then for all $g$ in some open dense subset of
$\operatorname{Spin}(V_{q})$ there exist
$g^{\prime}\in\operatorname{Spin}(V_{n})$ and
$g^{\prime\prime}\in\operatorname{Spin}(V_{n_{0}})$ such that

$\pi_{q,n_{0}}\circ g\circ\tau_{n,q}=g^{\prime\prime}\circ\pi_{n,n_{0}}\circ
g^{\prime}$

holds up to a scalar factor.

###### Proof.

The proof is similar to that above; we just give a sketch. Using the Cartan
map, which is equivariant for the relevant spin groups, this lemma follows
from a similar statement for the corresponding (halfs of) exterior power
representations. Specifically, define

$\displaystyle E$ $\displaystyle:=\langle
e_{n_{0}+1},\ldots,e_{q}\rangle\subseteq V_{q},$ $\displaystyle E^{\prime}$
$\displaystyle:=\langle e_{n_{0}+1},\ldots,e_{n}\rangle\subseteq V_{n},\text{
and}$ $\displaystyle F$ $\displaystyle:=\langle
f_{n+1},\ldots,f_{q}\rangle\subseteq V_{q}.$

Then the desired identity is

$c_{E}\circ g\circ m_{F}=g^{\prime\prime}\circ c_{E^{\prime}}\circ g^{\prime}$
(6.4)

(up to a scalar), where

$\displaystyle c_{E}$ $\displaystyle:=c_{e_{n_{0}+1}}\circ\cdots\circ
c_{e_{q}}:\bigwedge\nolimits^{q}V_{q}\to\bigwedge\nolimits^{n_{0}}V_{n_{0}},$
$\displaystyle c_{E^{\prime}}$ $\displaystyle:=c_{n_{0}+1}\circ\cdots\circ
c_{e_{n}}:\bigwedge\nolimits^{n}V_{n}\to\bigwedge\nolimits^{n_{0}}V_{n_{0}},\text{
and}$ $\displaystyle m_{F}$ $\displaystyle:=m_{f_{q}}\circ\cdots\circ
m_{f_{n+1}}:\bigwedge\nolimits^{n}V_{n}\to\bigwedge\nolimits^{q}V_{q}$

and the $c_{e_{i}}$ and $m_{f_{j}}$ are as defined in §3.2 and §3.3,
respectively. Furthermore, since the exterior powers are representations of
the special orthogonal groups, we may take $g,g^{\prime},g^{\prime\prime}$ to
be in
$\operatorname{SO}(V_{q}),\operatorname{SO}(V_{n}),\operatorname{SO}(V_{n_{0}})$,
respectively.

We investigate the effect of the map on the left on (a pure tensor in
$\bigwedge\nolimits^{n}V_{n}$ corresponding to) a maximal (i.e.,
$n$-dimensional) isotropic subspace $W$ of $V_{n}$. First, $W$ is extended to
$W^{\prime}:=W\oplus F$, then $g$ is applied to $W^{\prime}$, and the final
contraction map sends $gW^{\prime}$ to the image in $V_{q}/E$ of
$(gW^{\prime})\cap E^{\perp}$.

Instead of intersecting $gW^{\prime}$ with $E^{\perp}$, we may intersect
$W^{\prime}=W\oplus F$ with $(E^{\prime\prime})^{\perp}$ where
$E^{\prime\prime}:=g^{-1}E$, followed by the isometry
$\overline{g}:(E^{\prime\prime})^{\perp}/E^{\prime\prime}\to E^{\perp}/E$
induced by $g$. Accordingly, one can verify that the map on the left-hand side
of (6.4) becomes (a scalar multiple of)

$\overline{g}\circ c_{E^{\prime\prime}}\circ m_{F}$

where
$c_{E^{\prime\prime}}:\bigwedge\nolimits^{q}V_{q}\to\bigwedge\nolimits^{n_{0}}((E^{\prime\prime})^{\perp}/E^{\prime\prime})$
is the composition of contractions with a basis of $E^{\prime\prime}$, and
where we write $\overline{g}$ also for the map that $\overline{g}$ induces
from $\bigwedge\nolimits^{n_{0}}((E^{\prime\prime})^{\perp}/E^{\prime\prime})$
to $\bigwedge\nolimits^{n_{0}}(E^{\perp}/E)$.

Now consider the space $E^{\prime\prime}\cap(V_{n}\oplus F)\subseteq V_{q}$.
For $g$ in an open dense subset of $\operatorname{SO}(V_{q})$, this
intersection has the expected dimension $(q-n_{0})+(2n+q-n)-2q=n-n_{0}$, and
for $g$ in an open dense subset of $\operatorname{SO}(V_{q})$ we also have
$(E^{\prime\prime})^{\perp}\cap F=\\{0\\}$ (because
$(E^{\prime\prime})^{\perp}$ has codimension $q-n_{0}$, which is at least the
dimension $q-n$ of $F$). We restrict ourselves to such $g$. Then in particular
$E^{\prime\prime}\cap F=\\{0\\}$ and therefore the projection
$\widetilde{E}\subseteq V_{n}$ of $E^{\prime\prime}\cap(V_{n}\oplus F)$ along
$F$ has dimension $n-n_{0}$, as well. Note that $\widetilde{E}$ is isotropic
because $E^{\prime\prime}$ is and because $F$ is the radical of the bilinear
form on $V_{n}\oplus F$.

Furthermore, the projection $V_{n}\oplus F\to V_{n}$ restricts to a linear
isomorphism $(V_{n}\oplus
F)\cap(E^{\prime\prime})^{\perp}\to\widetilde{E}^{\perp}$, where the latter is
the orthogonal complement of $\widetilde{E}$ inside $V_{n}$. This linear
isomorphism induces an isometry

$h_{1}:((V_{n}\oplus F)\cap(E^{\prime\prime})^{\perp})/((V_{n}\oplus F)\cap
E^{\prime\prime})\to\widetilde{E}^{\perp}/\widetilde{E}$

between spaces of dimension $2n_{0}$ equipped with a nondegenerate bilinear
forms. On the other hand, the inclusion $V_{n}\oplus F\to V_{q}$ also induces
an isometry

$h_{2}:((V_{n}\oplus F)\cap(E^{\prime\prime})^{\perp})/((V_{n}\oplus F)\cap
E^{\prime\prime})\to(E^{\prime\prime})^{\perp}/E^{\prime\prime}.$

Now a computation shows that, up to a scalar, we have

$c_{E^{\prime\prime}}\circ m_{F}=h_{2}\circ h_{1}^{-1}\circ
c_{\widetilde{E}},$

where
$c_{\widetilde{E}}:\bigwedge\nolimits^{n}V_{n}\to\bigwedge\nolimits^{n_{0}}(\widetilde{E}^{\perp}/\widetilde{E})$
is a composition of contractions with a basis of $\widetilde{E}$. Now choose
$g^{\prime}\in\operatorname{SO}(V_{n})$ such that
$g^{\prime}\widetilde{E}=E^{\prime}$, so that we have

$c_{E^{\prime}}\circ g^{\prime}=\overline{g^{\prime}}\circ c_{\widetilde{E}},$

where $\overline{g^{\prime}}$ is the isometry
$\widetilde{E}^{\perp}/\widetilde{E}\to(E^{\prime})^{\perp}/E^{\prime}$
induced by $g^{\prime}$. We then conclude that

$c_{E}\circ g\circ m_{F}=\overline{g}\circ h_{2}\circ
h_{1}^{-1}\circ(\overline{g^{\prime}})^{-1}\circ c_{E^{\prime}}\circ
g^{\prime}$

and hence we are done if we set

$g^{\prime\prime}:=\overline{g}\circ h_{2}\circ
h_{1}^{-1}\circ(\overline{g^{\prime}})^{-1}\in\operatorname{SO}((E^{\prime})^{\perp}/E^{\prime})=\operatorname{SO}(V_{n_{0}}).\qed$

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