Since $\overline{\cal K}\subseteq{\cal H}_{p}$, we have $-\mathfrak{s}+{\cal
H}=d_{b}(\mathfrak{h})$ as well. Thus $\overline{d_{b}}(\mathfrak{h})\subseteq
d_{b}(\mathfrak{h})$ and $\overline{d_{b}}$ extends $d_{b}$.

We must now show that $\overline{d_{b}}$ is a derivation. Towards this, let
$\mathfrak{h}_{i}+\epsilon\mathfrak{s}_{i}$ and
$[\mathfrak{h}_{1},\mathfrak{h}_{2}]+\epsilon\mathfrak{s}$ be elements of
$\overline{\cal K}$. By the closure of $\overline{\cal K}$ under Lie bracket,
we have:

$[\mathfrak{h}_{1},\mathfrak{s}_{2}]+[\mathfrak{s}_{1},\mathfrak{h}_{2}]\in\mathfrak{s}+{\cal
K}$

Let us verify the derivation condition:

$\begin{array}[]{rcl}\mathfrak{h}_{1}\circ\overline{d_{b}}(\mathfrak{h}_{2})-\mathfrak{h}_{2}\circ\overline{d_{b}}(\mathfrak{h}_{1})&=&-[\mathfrak{h}_{1},\mathfrak{s}_{2}]+[\mathfrak{h}_{2},\mathfrak{s}_{1}]+{\cal
K}\\\
&=&-([\mathfrak{h}_{1},\mathfrak{s}_{2}]+[\mathfrak{s}_{1},\mathfrak{h}_{2}])+{\cal
K}\\\ &=&-\mathfrak{s}+{\cal K}\\\
&=&\overline{d_{b}}([\mathfrak{h}_{1},\mathfrak{h}_{2}])\end{array}$

This proves the proposition. $\Box$

###### Remark 6.13

We have the exact sequence of ${\cal K}$-modules [Wag10]:

$0\longrightarrow{\cal H}/{\cal K}\longrightarrow{\cal G}/{\cal
K}\longrightarrow{\cal G}/{\cal H}\longrightarrow 0$

and the corresponding long exact sequence of cohomology modules:

$\begin{array}[]{rcl}0&\longrightarrow&H^{0}({\cal K},{\cal H}/{\cal
K})\longrightarrow H^{0}({\cal K},{\cal G}/{\cal K})\longrightarrow
H^{0}({\cal K},{\cal G}/{\cal H})\\\ &\longrightarrow&H^{1}({\cal K},{\cal
H}/{\cal K})\longrightarrow H^{1}({\cal K},{\cal G}/{\cal K})\longrightarrow
H^{1}({\cal K},{\cal G}/{\cal H})\\\ &\longrightarrow&H^{2}({\cal K},{\cal
H}/{\cal K})\longrightarrow\ldots\end{array}$

Since both $d_{b}$ and $\overline{d_{b}}$ are derivations, they belong to the
spaces $H^{1}({\cal K},{\cal G}/{\cal H})$ and $H^{1}({\cal K},{\cal G}/{\cal
K})$ respectively (but they may be $0$). Whence, parts of the long exact
sequence which are relevant to the extension of $d_{b}$ to $\overline{d_{b}}$
are:

$\longrightarrow H^{1}({\cal K},{\cal H}/{\cal K})\longrightarrow H^{1}({\cal
K},{\cal G}/{\cal K})\longrightarrow H^{1}({\cal K},{\cal G}/{\cal H})\\\
\longrightarrow H^{2}({\cal K},{\cal H}/{\cal K})\longrightarrow$

###### Remark 6.14

Our construction of the $\epsilon$-extension $\overline{\cal K}$ through a
module leads to derivations as the direction of infinitesimal extension. This
is a variation of the ideas of Nijenhuis and Richardson, [NR67], where the
2-cocyles are the infinitesimal directions of deformations. All the same, it
is likely that ${\cal K}_{0}$ is not rigid (again, [NR67]) while ${\cal
K}(t_{0})$ are, for generic $t_{0}\in\mathbb{C}$.

We now come to the main proposition of this section:

###### Definition 6.15

Let $A(t),f$ and $g$ be such that:

$A(t)\cdot f=f(t)=t^{a}g+t^{b}f_{b}+\mbox{ higher terms}$

We say that $g$ is a regular limit of $f$ via $A(t)$ if (i) in the above
expression $f_{c}\neq 0$ only when $c-a$ is a multiple of $b-a$, and (ii) if
${\cal H}$ is the stabilizer of $g$, then the stabilizer ${\cal K}(1)$ of $f$
is not contained inside ${\cal H}$.

###### Proposition 6.16

Let $g$ be a regular limit of $f$ via $A(t)$ and $f(t)$ and $f_{b}$ be as
above. Let ${\cal K}(t)$ be the stabilizing Lie algebra of $f(t)$ and ${\cal
H}$ that of $g$. Moreover, let ${\cal K}_{0}\subseteq{\cal H}_{b}$ be the
limit of ${\cal K}(t)$, as $t\rightarrow 0$. Let $d_{b}:{\cal
K}_{0}\rightarrow{\cal G}/{\cal H}$ be the derivation as above. Then there is
a derivation $\overline{d_{b}}:{\cal K}_{0}\rightarrow{\cal G}/{\cal K}_{0}$
which extends $d_{b}$.

Proof: Let $\delta=b-a$. Condition (i) of regularity implies that that after
removal of $t^{a}$, all non-zero terms in the expression for $f(t)$ have
powers of $t$ as $t^{k\delta}$ for some $k\geq 0$. We may absorb $t^{a}$ into
$A(t)$ and write:

$A(t)\cdot f=f(t)=g+t^{\delta}f_{b}+\mbox{ higher terms in powers of
$t^{\delta}$}$

This, in turn, implies that the matrices $M_{N}(t)$ and $M_{\cal S}(t)$ have
elements in $\mathbb{C}[t^{\delta}]$. Thus the basis for ${\cal K}(t)$ may be
written in the form $\\{\mathfrak{k}_{i}(t^{\delta})\\}_{i=1}^{k}$. Now
condition (ii) that ${\cal K}\not\subseteq{\cal H}$ implies that ${\cal
H}\cdot N\not\subseteq N$ and that there is a $\mathfrak{k}\in{\cal K}$ with
$\mathfrak{k}=\mathfrak{h}+\mathfrak{s}$, with $\mathfrak{h}\in{\cal H}$ and
$\mathfrak{s}\in{\cal S}$. This implies that $\lambda_{\cal
S}(\mathfrak{h}\cdot f_{b})\neq 0$ and thus $\lambda_{\cal
S}(\mathfrak{h}\cdot f^{+}(t))$ must be of the form
$t^{\delta}\mathfrak{s}_{1}+\mbox{higher terms}$. Thus some element
$\mathfrak{k}_{i}(t)=\mathfrak{h}_{0}+\mathfrak{g}_{1}t^{\delta}+\mbox{ higher
terms}$ is such that $\mathfrak{g}_{1}\neq 0$. This implies that if
$(\alpha_{ij}^{k}(T))$ are the structure constants of ${\cal K}(t)$, then
there are some constants $i^{\prime},j^{\prime},k^{\prime}$ such that
$\alpha_{i^{\prime}j^{\prime}}^{k^{\prime}}=c_{1}t^{\delta}+c_{2}t^{2\delta}+\ldots$,
where $c_{1}\neq 0$. Whence, there is a non-trivial $\epsilon$-extension
${\cal K}_{1}$ of ${\cal K}_{0}$. By Prop. 6.12, this is equivalent to the
assertion that there is a derivation $\overline{d_{b}}:{\cal
K}_{0}\rightarrow{\cal G}/{\cal K}_{0}$ which extends $d_{b}$.

###### Remark 6.17

The Lie alegbra $\overline{{\cal K}_{0}}$ may not yield the “additional
semisimplcity” which may be present in ${\cal K}(t_{0})$ (and absent in ${\cal
K}_{0}$), but its structure coefficients do approximate those of ${\cal K}(t)$
better as polynomials in $t$. See Example 6.1, where ${\cal K}_{0}$ is the
$1$-dimensional algebra generated by $A$ below, while, $\overline{{\cal
K}_{0}}$ is $2$-dimensional (over $\mathbb{C}$) and generated by
$A(\epsilon)$. On the other hand ${\cal K}(t_{0})$ is $1$-dimensional and
generated by $A(t_{0})$.

$A=\left[\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right]\mbox{
}A(\epsilon)=\left[\begin{array}[]{cc}0&1\\\
-\epsilon&0\end{array}\right]\mbox{ }A(t_{0})=\left[\begin{array}[]{cc}0&1\\\
-t_{0}^{2}&0\end{array}\right]$

###### Remark 6.18

The existence of the limit $\lim_{t\rightarrow 0}f(t)=g$ allows the
construction of ${\cal K}(t)$, the limit ${\cal K}_{0}$ and the tangent of
approach $f_{b}$. We also have the additional $\star$-action of ${\cal H}$ and
that ${\cal K}_{0}\subseteq{\cal H}_{b}$, the stabilizer of $\overline{f_{b}}$
under this $\star$-action. Thus, we have the picture below:

$\begin{array}[]{ccc}{\cal K}(t)&\stackrel{{\scriptstyle
lim}}{{\rightarrow}}&{\cal K}_{0}\\\ \downarrow&&\downarrow\\\
?&\stackrel{{\scriptstyle lim}}{{\rightarrow}}&{\cal H}_{b}\\\ \end{array}$

In other words, we already know that there is a local ${\cal G}$-action in the
vicinity of the orbit $O(g)$ of $g$. Does this action allow us to compute the
stabilizer ${\cal G}_{x,n}$ ( possibly over $\mathbb{C}(t)$), of the tangent
vector $n\in TV_{g}$? This may then be applied to the vector $f_{b}$ to obtain
a possible localization of ${\cal K}(t)$. The current recipe of
$\overline{{\cal K}_{0}}$ over $\mathbb{C}[\epsilon]$ is not entirely
satisfactory.

###### Remark 6.19

The codimension $1$ case This case yields two important simplifications.
Firstly, the equality of ${\cal H}_{b}$ and ${\cal K}_{0}$ simplifies the
extension problem as below:

${{{\cal G}/{\cal H}_{b}}}$${{{\cal G}/{\cal H}}}$${{\cal
H}_{b}}$$\scriptstyle{\overline{d_{b}}}$$\scriptstyle{d_{b}}$

The second simplification comes from the theorem of Hoffman[Hof65], stated
below:

###### Theorem 6.20

Let ${\cal K}\subseteq{\cal H}$ be Lie alegbras over $\mathbb{R}$ such that
${\cal K}$ is of codimension $1$. Then exactly one of the following three
cases must be true:

  1. 1.

Let $P$ be the $2$-dimensional parabolic subgroup of upper triangular matrices
in $sl_{2}$. Then there is an ideal ${\cal I}$ such that ${\cal K}/{\cal
I}\cong P$ and ${\cal H}/{\cal I}\cong sl_{2}$.

  2. 2.

Let $D\subseteq P$ the subalgebra of diagonal matrices in $P$. Then there is
an ideal ${\cal I}$ such that ${\cal K}/{\cal I}\cong D$ and ${\cal H}/{\cal
I}\cong P$

  3. 3.

${\cal K}$ is an ideal of ${\cal H}$.

Assuming that our Lie algebras are complex extensions of real Lie algebras, we
see that ${\cal H}_{b}\subset{\cal H}$ must fall into one of these. Whence,
the $1$-dimensional ${\cal H}_{b}$-module ${\cal H}/{\cal H}_{b}$ has a simple
structure in the exact sequence below:

$\longrightarrow H^{1}({\cal H}_{b},{\cal H}/{\cal H}_{b})\longrightarrow
H^{1}({\cal H}_{b},{\cal G}/{\cal H}_{b})\longrightarrow H^{1}({\cal
H}_{b},{\cal G}/{\cal H})\longrightarrow H^{2}({\cal H}_{b},{\cal H}/{\cal
H}_{b})\longrightarrow$

##  7 The single matrix under conjugation

In this section we use the local model to solve a different problem -
computing the projective limits of stable and polystable points of the vector
space of $n\times n$ matrices under conjugation. This problem has been studied
extensively in the (affine) setting of semisimple Lie groups under the adjoint
action. The proof of the main theorem in this section, Theorem 7.22,
illustrates the use of the local parametrization and the map $\theta$, to
obtain projective closures. The statement of the main theorem was inspired by
calculations performed using the local model of the neighbourhood of a matrix
with a single Jordon block resulting in Theorem 7.7.

###### Remark 7.1

Experts in the area inform us that the statement of Theorem 7.22 does not
surprise them and that it probably follows from earlier work. However we are
not aware of any literature where this question has been discussed.

Before proceeding to describe our results we set up some notation and recall
some well known results about closures of orbits in affine space, in type A.

Let $V$ be the $GL(X)$-module $End(X)$ where $GL(X)$ acts by conjugation.
Thus, given $A\in GL(X)$ and $Y\in V$, we have $A\cdot Y=AYA^{-1}$. Polystable
points in $V$ are the diagonalizable matrices, and a generic stable point is a
diagonalizable matrix with distinct eigenvalues, also called a regular
semisimple element.

It is well known that the null-cone ${\cal N}$ for this action is the set of
all nilpotent matrices,see [New78]. The closure of nilpotent matrices in
affine space is well understood, see [Ger61],[Hes76]. The main theorem of this
section has a description similar to that of closures of nilpotent matrices in
affine space, which we review in Remark 7.17.

If $N\in V$ is a nilpotent matrix, in its $G$-orbit is the Jordan canonical
form of $N$. Let us quickly recall this. The $b\times b$-matrix $J_{b}$ is
given by the conditions: $J_{b}(i,j)=0$ for all $(i,j)$ except for the tuple
$(i,i+1)$, for $i=1,\ldots,b-1$, where it is $1$. $J_{4}$ is shown below:

$J_{4}=\left[\begin{array}[]{cccc}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
0&0&0&0\end{array}\right]$

$J_{b}$ is determined by the properties that (i) it is a $b\times b$ matrix,
(ii) $J_{b}^{a}\neq 0$ for $a<b$, and (iii) $J_{b}^{b}=0$. Given $N$, any
$n\times n$ nilpotent matrix, its Jordan canonical form is given as the block
diagonal matrix with blocks $J_{\lambda_{1}},\ldots,J_{\lambda_{k}}$, with
$\lambda_{1}\geq\ldots\geq\lambda_{k}>0$ and $\sum_{i}\lambda_{i}=n$, or in
other words, a partition $\lambda$ of $n$. This partition $\lambda$ will be
called the nilpotent signature of $N$.

Let us assume now that $X$ is an $n$-dimensional complex vector space and that
we have chosen a basis of $X$. With respect to this basis every element in $V$
can be identified with an $n\times n$ matrix.

Let $J_{n}$ be the nilpotent matrix given by $J_{n}(i,i+1)=1$ for
$i=1,\ldots,n-1$, and zero otherwise. This is the single largest orbit in
${\cal N}$, and ${\cal N}$ is the closure of the orbit of $J_{n}$.

In the next section we study the neighbourhood of $J_{n}$ in a direction
normal to the orbit of $J_{n}$ under conjugation. We use the local model to
show that a polystable point whose projective orbit closure contains $J_{n}$
must be necessarily stable.

In Section 7.2 we consider the case $J_{a,b}$ and describe the semisimple and
nilpotent matrices in its neighborhood.

In Section 7.3, motivated by the above two computations, we state our main
theorem which describes the nilpotent matrices which are in the projective
closure of polystable points. This is described by the combinatorics of
multiplicities of the eigenvalues of the matrix.

To simplify notation we don’t use fraktur symbols to denote Lie algebra
elements in the next two sections. Instead we use letters $A,B,C\ldots$ to
denote matrices.

###  7.1 The neighbourhood of $J_{n}$

Recall that

$J_{n}=\left[\begin{array}[]{ccccc}0&1&0&\ldots&\\\ 0&0&1&0&\ldots\\\
&\vdots&&\vdots&\\\ 0&0&\ldots&0&1\\\ 0&0&\ldots&0&0\end{array}\right].$

The action $\rho$ of the Lie algebra $gl(X)$ on $V$ is given by
$\rho(A)(Y)=A\cdot Y-Y\cdot A$, where $A\in gl(X)$ and $Y\in V$. For
$k=-(n-1),\ldots,-1,0,1,\ldots,n-1$, let $Z_{k}$ be the $n\times n$-matrix
where $Z_{k}(i,j)=0$ unless $j=i+k$, and then $Z_{i,i+k}=1$. Thus
$J_{n}=Z_{1}$. We see that $Z_{i}Z_{j}=Z_{i+j}$ if $i,j\leq 0$ or $i,j\geq 0$.

###### Lemma 7.2

Let $J_{n}$ be as above. Then:

  1. 1.

The tangent space of the orbit $O$ of $p=J_{n}$ under the $GL(X)$ action is

$TO_{p}=\\{B\in End(X)|\sum_{i=-k+1}^{n}b_{i,i+k}=0\mbox{ for $-(n-1)\leq
k\leq 0$}\\}$

  2. 2.

A complement to $TO_{p}$ within $TV_{p}=End(X)$ is the space of matrices
${\cal C}_{n}$ is

${\cal C}_{n}(c)=\left[\begin{array}[]{lcccc}-c_{n-1}&0&0&\ldots&\\\
-c_{n-2}&0&0&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -c_{1}&0&\ldots&0&0\\\
-c_{0}&0&\ldots&0&0\end{array}\right]$

where $c=[c_{0},\ldots,c_{n-1}]^{T}\in\mathbb{C}^{n}$ is a column vector. Note
that the affine space $J_{n}+{\cal C}_{n}$ is the space of all companion
matrices:

$J_{n}+{\cal C}_{n}=\left[\begin{array}[]{lcccc}-c_{n-1}&1&0&\ldots&\\\
-c_{n-2}&0&1&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -c_{1}&0&\ldots&0&1\\\
-c_{0}&0&\ldots&0&0\end{array}\right]$

where $c_{0},\ldots,c_{n-1}\in\mathbb{C}$.

  3. 3.

The stabilizer Lie algebra ${\cal H}$ of $J_{n}$ is the collection of all
matrices $\sum_{i=0}^{n-1}a_{i}Z_{i}$, with $a_{i}\in\mathbb{C}$

  4. 4.

A complement ${\cal S}$ to ${\cal H}$ within $gl(X)$ is the collection of all
matrices $B\in gl(X)$ such that $B_{1,i}=0$ for $i=1,\ldots,n$ and for every
element $v\in TO_{p}$, there is a unique element $a\in{\cal S}$ such that
$\rho(a)(J_{n})=v$.

Proof: The action of the Lie algebra $gl(X)$ on $p$ is given by $A\rightarrow
AJ_{n}-J_{n}A=B$. But $B=(b_{ij})=(a_{i,j-1}-a_{i+1,j})$. We thus see that

$\begin{array}[]{lrcll}(I)&\sum_{i=1}^{n-k}b_{i,i+k}&=&a_{1,k}-a_{n-k+1,n}&\mbox{for
$1\leq k\leq n-1$}\\\ (II)&\sum_{i=-k+1}^{n}b_{i,i+k}&=&0&\mbox{for
$-(n-1)\leq k\leq 0$}\\\ \end{array}$

In fact, it is easily shown that for any matrix $B$ such that the $n$
equations constituting $(II)$ hold, we may find an $A\in gl(X)$ such that
$B=AJ_{n}-J_{n}A$. This proves the first assertion. The second assertion
follows from the fact that ${\cal C}_{n}\cap TO_{p}=\overline{0}$, and that
$dim({\cal C}_{n})=n$. Assertion (3) is classical. Assertion (4) follows from
(1) and (2). $\Box$

###### Remark 7.3

The minimal polynomial of $J_{n}+{\cal C}_{n}(c)$ is
$p(X)=X^{n}+c_{n-1}X^{n-1}+\ldots+c_{0}X^{0}$.

Proof: This is classical. Let us use $T$ for the matrix ${\cal C}_{n}(c)$. If
$e_{n}=[0,\ldots,0,1]^{T}\in\mathbb{C}^{n}$ is the $n$-th column vector, then
it is easily seen that $T^{i}\cdot e_{n}=e_{n-i}$ and that

$\begin{array}[]{rcl}T^{n}\cdot e_{n}&=&-c_{n-1}e_{1}-c_{n-2}e_{2}+\ldots-
c_{0}e_{n}\\\ &=&-c_{n-1}T^{n-1}\cdot e_{n}-c_{n-2}T^{n-2}\cdot e_{n}+\ldots-
c_{0}T^{0}\cdot e_{n}\\\ \end{array}$

Thus, we have $(T^{n}+c_{n-1}T^{n-1}+\ldots+c_{0}T^{0})\cdot e_{n}=0$. This,
and the fact that $\\{T^{i}\cdot e_{n}$ are linearly independent vectors show
that the polynomial $p$ is the minimial polynomial of $T$. $\Box$

We now prove the following lemma.

###### Lemma 7.4

$\theta^{i}(n)$ is identically zero for $i\geq 2$.

Proof: Since $\theta(n)$ is zero on $N\subset V$, it suffices to calculate it
on the orbit which we have identified with ${\cal S}$. Given a matrix
$B\in{\cal S}$ such that $B(1,i)=0$ for all $i$ and an element $C\in{\cal
C}_{n}$ shown below, we see that $BC-CB$ is precisely $BC$ since $CB=0$.

$C=\left[\begin{array}[]{lcccc}-c_{n-1}&0&0&\ldots&\\\
-c_{n-2}&0&0&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -c_{1}&0&\ldots&0&0\\\
-c_{0}&0&\ldots&0&0\end{array}\right]$

We may thus write $B=[0,b_{2}^{T},\ldots,b_{n}^{T}]^{T}$, where $b_{i}$ are
row matrices and $C=[c,0,\ldots,0]$, where $c$ is a column matrix, to get

$BC-CB=\left[\begin{array}[]{cccc}0&0&\ldots&0\\\ b_{2}\cdot c&0&\ldots&0\\\
\vdots&&\vdots&\\\ b_{n}\cdot c&0&\ldots&0\\\ \end{array}\right]$

Whence $BC-CB\in N$ Since $\theta^{i}(n)$, $i\geq 2$, is the composition of
projection of $\theta^{i-1}(n)$ onto ${\cal S}$ followed by $\theta$, it
follows that $\theta^{i}(n)$ is zero for every $i>1$.$\Box$

To carry out the program of the local model outlined in Theorem 2.7 we need to
understand how ${\cal H}$ acts on the space $N$, as given in the sequence
given below. Note from the description of ${\cal H}$ in Lemma 7.2, ${\cal R}$,
the reductive part of ${\cal H}$ is zero, so in fact ${\cal H}={\cal Q}$.

We show

${\cal H}\stackrel{{\scriptstyle\cdot
n}}{{\longrightarrow}}V\stackrel{{\scriptstyle\lambda_{\cal
S}}}{{\longrightarrow}}{\cal S}\stackrel{{\scriptstyle\cdot
n}}{{\longrightarrow}}0$

For $A\in{\cal H}$ and $C\in N$ we have the first arrow:

$CA-
AC=\left[\begin{array}[]{cccc}c_{n-1}a_{0}&c_{n-1}a_{1}&\ldots&c_{n-1}a_{n-1}\\\
c_{n-2}a_{0}&c_{n-2}a_{1}&\ldots&c_{n-2}a_{n-1}\\\ \vdots&&\vdots&\\\
c_{0}a_{0}&c_{0}a_{1}&\ldots&c_{0}a_{n-1}\\\
\end{array}\right]-\left[\begin{array}[]{cccc}\sum_{i=0}^{n-1}c_{n-1-i}a_{i}&0&\ldots&0\\\
\sum_{i=0}^{n-2}c_{n-2-i}a_{i}&0&\ldots&0\\\ \vdots&&\vdots&\\\
c_{0}a_{0}&0&\ldots&0\\\ \end{array}\right]$

Since the expression is linear for ${\cal H}$, let us assume that $a_{i}=1$
for a particular $i$, and $a_{j}=0$ for $j\neq i$. Whence, we have:

$CA_{i}-A_{i}C=\left[\begin{array}[]{ccccccccc}-c_{n-i}&0&\ldots&0\ldots&&0&0&0&0\\\
-c_{n-i-1}&0&\ldots&0\ldots&&0&0&0&0\\\
\vdots&\vdots&&\vdots\ldots&&\vdots&0\\\
-c_{0}&\vdots&\vdots&&&\vdots&0&0&0\\\ \vdots&&\vdots&&&c_{n-i}&0&0&0\\\
0&0&\ldots&0\ldots&&c_{n-i-1}&0&0&0\\\ \vdots&&\vdots&&&\vdots&0&0&0\\\
0&0&\ldots&&&c_{0}&0&0&0\\\ \end{array}\right]$

The sum of the entries in the diagonals $j,-(n-1)\leq j\leq 0$ are all zero,
so from Lemma 7.2 this is in the orbit $TO_{J_{n}}$.

Thus there is an $S\in{\cal S}$ such that $S\cdot J_{n}-J_{n}\cdot S=O_{i}$.
Since $S\cdot C-C\cdot S=0$ we have

${\cal Q}\stackrel{{\scriptstyle\cdot
n}}{{\longrightarrow}}V\stackrel{{\scriptstyle\lambda_{\cal
S}}}{{\longrightarrow}}{\cal S}\stackrel{{\scriptstyle\cdot
n}}{{\longrightarrow}}V=0$

$\Box$

Our final computation is the dimension of the stabilizer of an arbitrary point
sitting normally over $J_{n}$.

###### Lemma 7.5

The dimension of the stabilizer of any point $J_{n}+{\cal C}_{n}(c)$ is $n$.

Proof: Let $T={\cal C}_{n}(c)$. By Theorem 2.7, the stabilizer of the point
$J_{n}+T$ is governed by the equation:

$\Delta r\cdot T+\lambda_{N}\circ(1+\theta(n))^{-1}(\Delta q\cdot T)=0$

where ${\mathfrak{r}}\in{\cal R},\mathfrak{q}\in{\cal Q}$, the reductive and
nilpotent Lie subalgebras of ${\cal H}$. As mentioned earlier ${\cal R}=0$.
From the calculations in the preceeding paragraph, the image of ${\cal Q}\cdot
n$ is in ${\cal S}$ and so $\lambda_{N}\circ(1+\theta(n))^{-1}(\Delta q\cdot
T)=0$ for all ${\mathfrak{q}}$. So every elements of ${\cal Q}$ may be
supplemented by an element of ${\cal S}$ to given an element of the stabilier
of $T+J_{n}$. $\Box$

###### Remark 7.6

That the matrix $T+J_{n}$ is in companion form actually tells us that its
minimal polynomial is of degree $n$

We now come to the main theorem:

###### Theorem 7.7

Let $A\in End(X)$ be polystable under the action of $SL(X)$ by conjugation. If
$J_{n}$ is a projective limit of a matrix $A$ under the adjoint action, then
$A$ has $n$ distinct eigenvalues and is stable. Moreover, for any matrix $A$
such that $A$ has distinct eigenvalues $\lambda_{1},\ldots,\lambda_{n}$, there
is a family ${\cal A}(t)$ such that (i) the eigenvalues of ${\cal A}(t)$ are
$t\lambda_{1},\ldots,t\lambda_{n}$ and (ii) $\lim_{t\rightarrow 0}{\cal
A}(t)=J_{n}$.

Proof: Under the hypothesis, we know that there is a $GL(X)$-conjugate
$A^{\prime}$ such that $\|A^{\prime}-J_{n}\|<\epsilon$. By the local model
construction, we may further assume that $A^{\prime}$ is of the form
$J_{n}+T$, with $T\in N$. Since $A$ is polystable, so is $A^{\prime}$ and so
it is diagonalizable. Since $A^{\prime}=J_{n}+T$ as above, its minimal
polynomial is of degree $n$. Thus, $A^{\prime}$ and therefore $A$ has $n$
distinct eigenvalues. This proves the first part.

For the second part, let ${\cal C}_{n}(c)$ be the companion form of $A$. Thus,
the vector $c=(c_{n-1},\ldots,c_{0})$ are the coefficients of the
characteristic polynomial of $A$. Define
$c(t)=(tc_{n-1},t^{2}c_{n-2}t^{2},\ldots,t^{n}c_{0})$ and let ${\cal
A}(t)=J_{n}+{\cal C}(c(t))$, i.e., the companion matrix with column vector
$c(t)$. Then ${\cal A}(t)$ has eigenvalues
$(t\lambda_{1},\ldots,t\lambda_{n})$ and $\lim_{t\rightarrow 0}{\cal
A}(t)=J_{n}$. $\Box$

###### Example 7.8

Let us compute explicitly through ${\cal S}$-completion, the stabilizer of the
matrix $Z_{4}=J_{4}+C_{4}$ sitting ”normally” above $J_{4}$:

$Z_{4}=\left[\begin{array}[]{cccc}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
\end{array}\right]\>\>\>C_{4}=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 1&0&0&0\\\ \end{array}\right]$

The matrix $Z_{4}$ has the minimal polynomial $X^{4}-1$. The stabilizing Lie
algebra of $Z_{4}$ is given by the basis elements $Z_{4}^{i}$, for
$i=0,\ldots,3$.

The complementary space ${\cal S}$ is the 12-dimensional space of matrices
shown below:

${\cal S}=\left[\begin{array}[]{cccc}0&0&0&0\\\ &*&*&*\\\ &*&*&*\\\ &*&*&*\\\
\end{array}\right]$

The elementary matrices $E_{ij}$, for $i=2,3,4$ and $j=1,2,3,4$ form a basis
for ${\cal S}$. The tangent space to the orbit of $J_{4}$ is given by the
basis of 12 matrices $E_{ij}\cdot J_{4}-J_{4}\cdot E_{ij}$. The stabilizer
${\cal H}$ of $J_{4}$ is the space of matrices generated by $J_{4}^{i}$,
$i=0,\dots,3$. The stabilizer space of $Z_{4}$ is again $4$-dimensional and
given by the elements $h+s$ where $h\in{\cal H}$ and $s\in{\cal S}$ such that
$[s,J_{4}]+[h,C_{4}]=0$. Let us select the element $h=J_{4}^{3}$ to obtain
$[h,C_{4}]$ as shown below:

$h=\left[\begin{array}[]{cccc}0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{array}\right]\>\>\>[h,C_{4}]=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&-1\\\ \end{array}\right]$

One may check that $[s,J_{4}]$ below and verify that $[s,J_{4}]=-[h,C_{4}]$:

$s=E_{21}+E_{32}+E_{43}=\left[\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
\end{array}\right]\>\>\>[s,J_{4}]=\left[\begin{array}[]{cccc}-1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\ \end{array}\right]$

Thus $s+h$ shown below, stabilizes $Z_{4}$.

$s+h=\left[\begin{array}[]{cccc}0&0&0&1\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
\end{array}\right]$

###  7.2 The $J_{a,b}$ case

We now consider the partition $(a,b)$ where $a+b=n$ and $a\geq b$. The matrix
$J_{a,b}$ may be written in the block-diagonal form as below:

$J_{a,b}=\left[\begin{array}[]{cc}J_{a}&0\\\ 0&J_{b}\end{array}\right]$

###### Lemma 7.9

The stabilizer ${\cal H}$ of $J_{a,b}$ is of the form:

$\left[\begin{array}[]{cc}Z_{a}&Y_{ab}\\\ Y_{ba}&Z_{b}\end{array}\right]$

Where $Z_{m}$ is $m\times m$ such that (i) $Z_{m}(i,j)=0$ for all $i>j$, and
(ii) $Z_{m}(i+1,j+1)=Z_{m}(i,j)$. The forms for $Y_{ab}$ and $Y_{ba}$ are
given below:

$Y_{ab}=\left[\begin{array}[]{c}Z^{\prime}_{b}\\\
0\end{array}\right]\>\>\>\>Y_{ba}=\left[\begin{array}[]{cc}0&Z^{\prime\prime}_{b}\end{array}\right]$

The dimension of ${\cal H}$ is $a+3b$, $a$ coming from $Z_{a}$ and $b$ each
from the three matrices $Z_{b},Z^{\prime}_{b},Z^{\prime\prime}_{b}$.

###### Lemma 7.10

A normal space to the orbit of $J_{a,b}$ is ${\cal C}$ below:

${\cal C}=\left[\begin{array}[]{cc}{\cal C}_{a}&W_{ab}\\\ W_{ba}&{\cal
C}_{b}\end{array}\right]$

where ${\cal C}_{a}$ and ${\cal C}_{b}$ are as shown below:

${\cal C}_{a}(c)=\left[\begin{array}[]{lcccc}-c_{a-1}&0&0&\ldots&\\\
-c_{a-2}&0&0&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -c_{1}&0&\ldots&0&0\\\
-c_{0}&0&\ldots&0&0\end{array}\right]\>\>\>{\cal
C}_{b}(d)=\left[\begin{array}[]{lcccc}-d_{b-1}&0&0&\ldots&\\\
-d_{b-2}&0&0&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -d_{1}&0&\ldots&0&0\\\
-d_{0}&0&\ldots&0&0\end{array}\right]$

and $W_{ab}$ and $W_{ba}$ are of the following form:

$W_{ab}=\left[\begin{array}[]{ccc}&&\\\ &0_{(a-1)\times b}&\\\ &&\\\
\hline\cr\alpha_{1}&\ldots&\alpha_{b}\end{array}\right]\>\>\>W_{ba}=\left[\begin{array}[]{c|c}\beta_{1}&\\\
\vdots&0_{b\times{(a-1)}}\\\ \beta_{b}&\end{array}\right]$

where $c_{i},d_{j},\alpha_{r},\beta_{s}\in\mathbb{C}$ and $O_{r\times s}$ are
zero matrices. The dimension of ${\cal C}$ is also $a+3b$.

The proofs of these lemmas are straightforward. We now come to two main
lemmas.

###### Lemma 7.11

For any $C\in{\cal C}$, let $p_{C}$ be the minimal polynomial of
$T=J_{a,b}+C$. Then the degree of $p_{C}$ is at least $a$.

Proof: We may write $T$ in the jumbo-size matrix below:

$\left[\begin{array}[]{lcccc|lcccc}-c_{a-1}&1&0&\ldots&&&\\\
-c_{a-2}&0&1&0&\ldots&&&&\\\ \vdots&\vdots&&\vdots&&&\lx@intercol\hfil
0_{(a-1)\times b}\hfil\lx@intercol&\\\ -c_{1}&0&\ldots&0&1&&&&\\\
-c_{0}&0&\ldots&0&0&\alpha_{1}&\alpha_{2}&\ldots&&\alpha_{b}\\\
\hline\cr\beta_{1}&&&&&-d_{b-1}&1&0&\ldots&\\\
\beta_{2}&&&&&-d_{b-2}&0&1&0&\ldots\\\ \vdots&\lx@intercol\hfil
0_{b\times(a-1)}\hfil\lx@intercol&&\vdots&\vdots&&\vdots&\\\
&&&&&-d_{1}&0&\ldots&0&1\\\
\beta_{b}&&&&&-d_{0}&0&\ldots&0&0\end{array}\right]$

Let $e_{1},\ldots,e_{a},e_{a+1},\ldots,e_{a+b}$ be a basis of $\mathbb{C}^{n}$
(as column vectors). From the matrix structure of $T$, it is clear that for
$i=2,\ldots,a$ we have $T(e_{i})=e_{i-1}$. Thus if
$Q=T^{m}+\gamma_{m{-1}}T^{m-1}+\ldots+\gamma_{0}T^{0}$ is any operator, where
$m<a$, then $Q(e_{a})=e_{a_{m}}+\gamma_{m-1}e_{a-m-1}+\ldots+\gamma_{0}e_{a}$.
When $Q(e_{a})\neq 0$. Thus, $T$ cannot have a minimal polynomial of degree
less than $a$. $\Box$

###### Lemma 7.12

Let $T$ be as above and $\lambda\in C$, be any number. Let $ker(T-\lambda I)$
be the space of all $v$ such that $T(v)=\lambda v$, then $dim(ker(T-\lambda
I))\leq 2$. Moreover, any $v$ in this space is determined by $v_{1}$ and
$v_{a+1}$.

Proof: The condition that $(T-\lambda I)v=0$ gives us the following
conditions:

$\begin{array}[]{rl}-c_{a-k+1}v_{1}-\lambda v_{k-1}+v_{k}=0&\mbox{for
$k=2,\ldots,a$}\\\ \beta_{i-1}v_{1}-d_{b-i+1}v_{a+1}-\lambda
v_{a+i-1}+v_{a+i}=0&\mbox{for $i=2,\ldots,b$}\\\ \end{array}$

This proves the claim. $\Box$

For $T$ as above, we write

$T_{a}=\left[\begin{array}[]{lcccc}-c_{a-1}&1&0&\ldots&\\\
-c_{a-2}&0&1&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -c_{1}&0&\ldots&0&1\\\
-c_{0}&0&\ldots&0&0\end{array}\right]\>\>\>{T}_{b}=\left[\begin{array}[]{lcccc}-d_{b-1}&1&0&\ldots&\\\
-d_{b-2}&0&1&0&\ldots\\\ \vdots&\vdots&&\vdots&\\\ -d_{1}&0&\ldots&0&1\\\
-d_{0}&0&\ldots&0&0\end{array}\right]$

Note that, by Remark 7.3, the minimal polynomial of $T_{a}$ (respectively
$T_{b}$) is $T^{a}+c_{a-1}T^{a-1}+\ldots+c_{0}T^{0}$ (respectively
$T^{b}+d_{b-1}T^{a-1}+\ldots+d_{0}T^{0}$).

###### Lemma 7.13

Suppose that the minimal polynomial $p$ of $T$ has degree $a$. Then $p$ is the
minimal polynomial of $T_{a}$ and the minimal polynomial of $T_{b}$ divides
$p$.

Proof: Let $p(T)=T^{a}+\gamma_{a{-1}}T^{a-1}+\ldots+\gamma_{0}T^{0}$ be the
minimal polynomial of $T$ of degree $a$. Using the notation as in the proof of
Lemma 7.11, $p(T)(e_{a})=0$ implies that $\gamma_{i}=c_{i}$ for
$i=0,\ldots,a{-1}$ and $\beta_{j}=0$ for $j=1,\ldots,b$. Thus,
$p(T)=T^{a}+c_{a{-1}}T^{a-1}+\ldots+c_{0}T^{0}$ is also the minimal polynomial
of $T_{a}$.

We now write $T$ in block upper triangular form

$T=\left[\begin{array}[]{c|c}T_{a}&*\\\ \hline\cr 0_{b\times
a}&T_{b}\end{array}\right]$

It is easy to see that for any polynomial $Q$, $Q(T)$ is of the form

$Q(T)=\left[\begin{array}[]{c|c}Q(T_{a})&*\\\ \hline\cr 0_{b\times
a}&Q(T_{b})\end{array}\right]$

As $p(T)=0$, this shows that $p(T_{b})=0$. Therefore, the minimal polynomial
of $T_{b}$ divides $p$. $\Box$

###### Proposition 7.14

The nilpotent matrices in the space $J_{a,b}+{\cal C}$ have signature
$(a^{\prime},b^{\prime})$ with $a^{\prime}\geq a$. Moreover, $J_{a,b}$ is in
the projective orbit closure of $J_{a^{\prime},b^{\prime}}$.

Proof: Let $J$ be a nilpotent matrix of the above form and let $a^{\prime}$ be
the largest component of its signature. The conditions on the minimum degree
of the minimal polynomial implies that $a^{\prime}\geq a$. The dimension of
the kernel of $J$, with $\lambda=0$ in lemma 7.12, is only $2$. That forces
$J$ to have only two components. Thus, $J=J_{a^{\prime},b^{\prime}}$ with
$a^{\prime}\geq a$. That every such $J_{a^{\prime},b^{\prime}}$ is obtained is
verified by putting all $c$’s, $d$’s, $\beta$’s and $\alpha$’s to zero except
that $\alpha_{i}=t$, where $t\neq 0$. One may verify that this matrix
$J_{i}(t)$ is nilpotent with signature $a^{\prime}=a+b-i+1$ and
$b^{\prime}=i-1$. We also see that $\lim_{t\rightarrow 0}J_{i}(t)=J_{a,b}$,
and thus $J_{a,b}$ is in the projective orbit closure of
$J_{a^{\prime},b^{\prime}}$. $\Box$

###### Proposition 7.15

The diagonalizable matrices in the space $J_{a,b}+{\cal C}$ have at least $a$
distinct eigenvalues. Moreover, each eigenvalue has multiplicity at most $2$.
Moreover, for any such matrix $A$, the matrix $J_{a,b}$ is in its projective
closure.

Proof: For a diagonalizable matrix $T$, the number of distinct eigenvalues is
equal to the degree of the minimal polynomial. Further, for an eigenvalue
$\lambda$ of $T$, the (algebraic) multiplicity of $\lambda$ is equal to its
geometric multiplicity, namely $dim(ker(T-\lambda I))$. The claim now follows
from Lemmas 7.11 and 7.12.

Coming to the second assertion, let
$\\{\lambda_{1},\ldots,\lambda_{b^{\prime}},\lambda_{b^{\prime}+1},\ldots,\lambda_{a^{\prime}}\\}$
be an ordering of the eigenvalues of $T$ such that
$\lambda_{1},\ldots,\lambda_{b^{\prime}}$ have multiplicity $2$ and thus
$a^{\prime}+b^{\prime}=n$. By extending the argument of Theorem 7.7,
$J_{a^{\prime},b^{\prime}}$ is in the projective orbit closure of $T$. By
prop. 7.14, we have $a^{\prime}\geq a$, and thus $J_{a,b}$ is in its
projective orbit closure of $J_{a^{\prime},b^{\prime}}$, and therefore of $T$.
$\Box$

###  7.3 Orbit closure of polystable points

We end the section with a complete characterization of the orbit closure of
polystable points in the vicinity of a single nilpotent matrix. The closure is
completely determined by the combinatorics of the spectrum of the polystable
point under consideration. We give an algebraic proof which is motivated by
the constructions of the previous section.

Before we begin, let us recall some definitions. A partition $\alpha$ of $n$
is a sequence $(\alpha_{1}\geq\cdots\geq\alpha_{r}>0)$ such that
$\sum_{i=1}^{r}\alpha_{i}=n$.

###### Definition 7.16

Let $\alpha$ and $\beta$ be partitions of $n$. The transpose $\gamma$ of
$\alpha$ is given by the numbers $\gamma_{1}\geq\ldots\geq\gamma_{s}>0$ where
$\gamma_{i}=|\\{j|\alpha_{j}\geq i\\}|$, and is denoted by $\alpha^{T}$. It is
easy to check that $\gamma$ too is a partition of $n$. We say that $\alpha$
dominates $\beta$ iff for all $i>0$,
$\sum_{j=1}^{i}\alpha_{j}\geq\sum_{j=1}^{i}\beta_{j}$. This is denoted by
$\alpha\unrhd\beta$ or, equivalently, $\beta\unlhd\alpha$. It is easy to see
that $\alpha\unrhd\beta$ iff $\beta^{T}\unrhd\alpha^{T}$.

The order described above on partitions of $n$ is a partial order called the
dominance order.

###### Remark 7.17

Let $X$ be a nilpotent matrix with signature $\lambda$. The orbit closure of
$X$ under conjugation contains all nilpotent matrices with partition signature
$\lambda^{\prime}$ with $\lambda\unrhd\lambda^{\prime}$. This is a celebrated
result proved by Gerstenhaber, [Ger61].

If an $n\times n$-diagonalizable matrix has eigenvalues
$\mu_{1},\mu_{2},\ldots,\mu_{s}$ with multiplicities
$\lambda:=\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{s}$, we call
$\lambda$, the spectrum partition of that matrix.

###### Proposition 7.18

Let ${\cal X}_{k}^{r}$ denote the set of all $n\times n$ matrices $x$ for
which there exist $\lambda_{1},\ldots,\lambda_{k}$ such that
$\mathrm{rank}((x-\lambda_{1}I)(x-\lambda_{2}I)\ldots(x-\lambda_{k}I))\leq r$
and $\mathrm{det}(x-\lambda_{i}I)=0$ for all $i=1,\ldots,k$. Then ${\cal
X}_{k}^{r}$ is a projective variety. Moreover, ${\cal X}_{k}^{r}$ is invariant
under the action of $GL_{n}$ on $x$ by conjugation.

Proof: Let $X$ be an $n\times n$ matrix of variables and let
$Y=\\{y_{1},\ldots,y_{k}\\}$ be another set of indeterminates.

Let the affine variety ${\cal V}\subseteq{\mathbb{C}}^{n^{2}+k}$ be defined by
the homogeneous ideal $J$ in ${\mathbb{C}}[X,Y]$ generated by
$\mathrm{det}(X-y_{i}I)=0$ for $i=1,\ldots,k$. We define the projection
morphism $\phi:{\cal V}\to{\mathbb{C}}^{n^{2}}$ as follows:
$\phi(A,b_{1},\ldots,b_{k})=A$. The induced map of the co-ordinate rings is
simply the inclusion of ${\mathbb{C}}[X]\hookrightarrow{\mathbb{C}}[{\cal
V}]={\mathbb{C}}[X,Y]/J$. As $\mathrm{det}(X-y_{i}I)=0$, each
$y_{i}\in{\mathbb{C}}[{\cal V}]$ is integral over ${\mathbb{C}}[X]$. This
shows that $\phi$ is a finite map and hence maps closed sets to closed sets.

Consider the matrix $Z=(X-y_{1}I)(X-y_{2}I)\ldots(X-y_{k}I))$. Let $P,Q$ be
subsets of $[n]$ such that $|P|=|Q|=r+1$ and let $f_{P,Q}$ be the homogeneous
form $\mathrm{det}(Z_{P,Q})$, i.e., the determinant of the
$(P,Q)^{\mathrm{th}}$ minor of $Z$. Consider the ideal
$I\subseteq{\mathbb{C}}[{\cal Z}]$ generated by all such forms. This is a
homogeneous ideal in the variables of $X\cup Y$. Its variety, say ${\cal
U}\subseteq{\cal V}$ is indeed of points
$(A,\lambda_{1},\ldots,\lambda_{k})\in{\cal V}$ such that
$\mathrm{rank}((A-\lambda_{1}I)(A-\lambda_{2}I)\ldots(A-\lambda_{k}I))\leq r$.
The set ${\cal X}_{k}^{r}$ is precisely the image of the closed set ${\cal U}$
under the finite map $\phi$. As a result, ${\cal X}_{k}^{r}$ is also a closed
set. The fact that ${\cal X}_{k}^{r}$ is closed under homothety and invariant
under the conjugation action of $GL_{n}$ is obvious. $\Box$

###### Proposition 7.19

Let $x$ be an $n\times n$ diagonalizable matrix with spectrum partition
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\lambda_{k}\geq\cdots\geq\lambda_{s})$.
The $x\in{\cal X}_{k}^{r}$ iff $\lambda_{1}+\ldots+\lambda_{k}\geq n-r$.

The proof is obvious.

###### Proposition 7.20

Let $x=J_{\theta_{1}}\oplus\ldots\oplus J_{\theta_{s}}$ be a nilpotent matrix
with partition signature $(\theta_{1}\geq\ldots\geq\theta_{s})$. Then
$x\in{\cal X}_{k}^{r}$ iff $rank(x^{k})\leq r$. This is given by the condition
$\sum_{i:\theta_{i}>k}(\theta_{i}-k)\leq r$.

It will be useful to restate this condition in terms of the transpose
partition.

###### Proposition 7.21

Let $x=J_{\theta}$ be a nilpotent matrix with partition signature
$\theta=(\theta_{1}\geq\ldots\geq\theta_{s})$ whose transpose partition is
$\theta^{T}=(\theta^{\prime}_{1}\geq\cdots\geq\theta^{\prime}_{t})$. Then
$x\in{\cal X}_{k}^{r}$ iff $\theta^{\prime}_{1}+\ldots+\theta^{\prime}_{k}\geq
n-r$.

We state our main theorem.

###### Theorem 7.22

Let $x$ be a diagonalizable matrix in $End(X)$ with spectrum partition
$\lambda$. The projective orbit closure of $x$ under the conjugation action by
$GL(X)$ in ${\mathbb{P}}(End(X))$ contains nilpotent matrices whose partition
signature is $\theta$ iff $\theta\unlhd\lambda^{T}$.

Proof Let $y$ be a nilpotent matrix with partition signature $\theta$ such
that $\theta\not\\!\\!\unlhd~{}\lambda^{T}$. This implies that
$\lambda\not\\!\\!\unlhd~{}\theta^{T}=(\theta^{\prime}_{1}\geq\cdots\geq\theta^{\prime}_{s})$.
As a result, there is an index $k$ such that

$\sum_{j=1}^{k}\lambda_{j}>\sum_{j=1}^{k}\theta^{\prime}_{j}$

Now we choose $r$ such that $\sum_{j=1}^{k}\lambda_{j}=n-r$ and apply
Propositions 7.19 and 7.21 to conclude that $x\in{\cal X}_{k}^{r}$ and
$y\not\in{\cal X}_{k}^{r}$. This allows us to conclude that $y$ is not in the
projective orbit closure of $x$.

In view of Remark 7.17, it only remains to show that a nilpotent matrix $J$
with partition signature
$\lambda^{T}=(\lambda^{\prime}_{1}\geq\cdots\geq\lambda^{\prime}_{s})$ belongs
to the projective orbit closure of $x$. Towards this, we can easily adapt the
argument in the proof of Theorem 7.7. by writing $x$ as a block diagonal
matrix $x_{1}\oplus\cdots\oplus x_{s}$ where diagonal matrix $x_{i}$ is of
size $\lambda^{\prime}_{i}$ and has distinct eigenvalues, namely those
eigenvalues of $x$ which have multiplicity at least $i$. We skip the
straightforward details. $\Box$

Now we extend the above theorem to all matrices. Towards this, fix a matrix
$x$ in a Jordan canonical form with $s$ distinct eigenvalues
$\mu_{1},\mu_{2},\ldots,\mu_{s}$. For eigenvalue $\mu_{i}$, we associate a
partition $\lambda_{i}=(\lambda_{i1}\geq\lambda_{i2}\geq\ldots)$ which records
the sizes of Jordan blocks with diagonal entries $\mu_{i}$, in non-increasing
order. We now define the transpose block-spectrum partition of $x$ to be the
partition $\chi=(\chi_{1}\geq\chi_{2}\geq\ldots)$ where
$\chi_{j}=\lambda_{1j}+\lambda_{2j}+\ldots\lambda_{sj}$. Thus, $\chi$ is
simply the “sum” of the partitions $\lambda_{1},\ldots,\lambda_{s}$. Note that
when $x$ is a diagonal matrix, its transpose block-spectrum partition is
nothing but the transpose of its spectrum partition.

###### Theorem 7.23

Let $x$ be a matrix in $End(X)$ with transpose block-spectrum partition
$\chi$. The projective orbit closure of $x$ under the conjugation action by
$GL(X)$ in ${\mathbb{P}}(End(X))$ contains nilpotent matrices whose partition
signature is $\theta$ iff $\theta\unlhd\chi$.

Proof Let $y$ be a nilpotent matrix with partition signature
$\theta=(\theta_{1}\geq\theta_{2}\geq\ldots)$ such that
$\theta\not\\!\\!\unlhd~{}\chi$. This means, there is an index $\ell$ with

$\sum_{j=1}^{\ell}\theta_{j}>\sum_{j=1}^{\ell}\chi_{j}$

Henceforth we fix the least index $\ell$ with this property. This further
implies that $\theta_{\ell}>\chi_{\ell}$.

Now we set $k=\chi_{\ell+1}$, $r=\sum_{i=1}^{\ell}(\chi_{i}-\chi_{\ell+1})$
and claim that $x\in{\cal X}_{k}^{r}$ and $y\not\in{\cal X}_{k}^{r}$. We first
focus on showing that $y\not\in{\cal X}_{k}^{r}$. By Proposition 7.20, it
suffices to show that

$\sum_{i:\theta_{i}>k}(\theta_{i}-k)>r$

As $k=\chi_{\ell+1}$ and $\theta_{\ell}>\chi_{\ell}\geq\chi_{\ell+1}$, we have

$\sum_{i:\theta_{i}>k}(\theta_{i}-k)\geq\sum_{i=1}^{\ell}(\theta_{i}-\chi_{\ell+1})>\sum_{i=1}^{\ell}(\chi_{i}-\chi_{\ell+1})=r$

Now we turn our focus on showing that $x\in{\cal X}_{k}^{r}$. We make use of
the notation developed for defining $\chi$ just before the proof of the
theorem. In this notation, we show that

$\mathrm{rank}\left((x-\mu_{1}I)^{\lambda_{1,{\ell+1}}}(x-\mu_{2}I)^{\lambda_{2,{\ell+1}}}\ldots(x-\mu_{s}I)^{\lambda_{s,{\ell+1}}}\right)=r$

Note that, this would imply that $x\in{\cal X}_{k}^{r}$ as
$k=\chi_{\ell+1}=\lambda_{1,{\ell+1}}+\ldots+\lambda_{s,{\ell+1}}$. As the
generalized eigenspaces corresponding to different eigenvalues are independent
and $(x-\mu_{i}I)$ is nilpotent on the generalized eigenspace corresponding to
eigenvalue $\mu_{i}$ and invertible on other generalized eigenspaces, we can
use Proposition 7.20 to conclude that

$\mathrm{rank}\left((x-\mu_{i}I)^{\lambda_{i,{\ell+1}}}\right)=(\lambda_{i,1}-\lambda_{i,\ell+1})+\ldots+(\lambda_{i,\ell}-\lambda_{i,\ell+1})=\sum_{j=1}^{\ell}(\lambda_{i,j}-\lambda_{i,\ell+1})$

and hence,

$\begin{array}[]{ccl}\mathrm{rank}\left((x-\mu_{1}I)^{\lambda_{1,{\ell+1}}}(x-\mu_{2}I)^{\lambda_{2,{\ell+1}}}\ldots(x-\mu_{s}I)^{\lambda_{s,{\ell+1}}}\right)&=&\sum_{i=1}^{s}\sum_{j=1}^{\ell}(\lambda_{i,j}-\lambda_{i,\ell+1})\\\
&=&\sum_{j=1}^{\ell}\sum_{i=1}^{s}(\lambda_{i,j}-\lambda_{i,\ell+1})\\\
&=&\sum_{j=1}^{\ell}\left((\sum_{i=1}^{s}\lambda_{i,j})-(\sum_{i=1}^{s}\lambda_{i,\ell+1})\right)\\\
&=&\sum_{j=1}^{\ell}(\chi_{j}-\chi_{\ell+1})\\\ &=&r\end{array}$

We have thus proved that $x\in{\cal X}_{k}^{r}$ and $y\not\in{\cal
X}_{k}^{r}$. This shows that $y$ is not in the projective orbit closure of
$x$.

In order to complete the proof, we finally show that a nilpotent matrix of
partition signature $\chi=(\chi_{1},\ldots,\chi_{s})$ belongs to the
projective orbit closure of $x$. We first write $x$ as a block diagonal matrix
$x_{1}\oplus\cdots\oplus x_{s}$ where matrix $x_{j}$ is of size $\chi_{j}$ and
$x_{j}=y_{1,j}\oplus y_{2,j}\cdots\oplus y_{s,j}$ where $y_{i,j}$ is a Jordan
block of size $\lambda_{i,j}$ with diagonal entry $\mu_{i}$. The crucial
property of each $x_{j}$ is that different Jordan blocks in it correspond to
different eigenvalues and hence the minimal polynomial of $x_{j}$ equals its
characteristic polynomial. As a result, each $x_{j}$ can be conjugated to
obtain $x^{\prime}_{j}$ which is a companion matrix. We now consider
$x^{\prime}=x^{\prime}_{1}\oplus\cdots\oplus x^{\prime}_{s}$ to which $x$ can
be block-conjugated.

Now we use the 1-PS family $A(t)=\mathrm{diagonal}(t,t^{2},\ldots,t^{n})$ and
observe that the leading term of $A(t).x^{\prime}$ is the matrix
$t^{-1}J_{\chi}$ where $J_{\chi}=J_{\chi_{1}}\oplus\ldots\oplus J_{\chi_{s}}$
is a nilpotent matrix whose partition signature is $\chi$. This shows that
$J_{\chi}$ belongs to the projective orbit closure of $x$ and completes the
proof. $\Box$.

We now conclude this section with an example.

###### Example 7.24

Consider the following matrices in Jordan canonical forms

$x_{1}=\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{array}\right]\>\>\>x_{2}=\left[\begin{array}[]{ccc}1&1&0\\\ 0&1&0\\\
0&0&-1\\\ \end{array}\right]\>\>\>y_{1}=\left[\begin{array}[]{ccc}0&1&0\\\
0&0&1\\\ 0&0&0\\\
\end{array}\right]\>\>\>y_{2}=\left[\begin{array}[]{ccc}0&1&0\\\ 0&0&0\\\
0&0&0\\\ \end{array}\right]$

Clearly, $x_{1}$ is diagonal and its spectrum partition as well as transpose
spectrum partition is $(2,1)$. The transpose block-spectrum partition of
$x_{2}$ is $(3)$. Note that $x_{1}$ is in the affine orbit closure of $x_{2}$.
Further, $y_{1}$ and $y_{2}$ are nilpotent matrices with partition signatures
$(3)$ and $(2,1)$ respectively.

The above results allow us to conclude that the projective orbit closure of
$x_{1}$ contains $y_{2}$ but not $y_{1}$, and the projective orbit closure of
$x_{2}$ contains both $y_{1}$ and $y_{2}$.

##  8 Local Differential Geometry of Orbits

So far we have explored algebraic techniques which relate properties of the
point $y$, the tangent vector $\ell\cdot y\in T_{y}O(y)$ and the ultimate
limit $x$ for paths which arise from 1-PS. A natural question is whether these
paths satisfy additional differential geometric properties, or if there are
other special paths which take us to specific limit points $x$ and if the
tangent vectors of exit can be determined using the local geometry at $y$. We
formulate this as the following path problem.

###### Definition 8.1

Given a point $y\in V$ and an $x\in\overline{O(y)}$, the path problem is to
determine a path $\pi$ from $y$ to $x$ satisfying specific properties derived
from those of $y$ and $x$.

This question is intriguing even when the orbit of $y$ is affine and we wish
to reach representative points of codimension $1$ components, i.e., the
boundary of the orbit $O(y)\subseteq\mathbb{P}V$. Specifically, is there an
optimization problem formulation to arrive at the codimension $1$ components
and the tangents of exit from $y$ leading to these components.

An important step in this direction would be to develop the local manifold
geometry of the orbit. We have already seen in Remark 2.11, that the function
$\Phi:{\cal S}\times N\to{\cal S}$ of the local model is intimately connected
to the 2-form $\Pi$. However, one technical issue is that our interest lies in
the closure in projective space. In Lemma 8.10 we extend Remark 2.11 to
compute the Riemannian curvature tensor on a chart of the projective space
containing $y$. This is developed further in Section 8.3, where we combine the
optimization approach and the computation of the curvatures for a specific
problem.

Let us begin the discussion with the familiar 1-PS situation with a
$GL(X)$-module $V$, and $x,y\in V$ with the action of a 1-PS $\lambda(t)$ as
shown below:

$\lambda(t)\cdot y=x_{a}\cdot t^{a}+x_{b}t^{b}+\ldots+x_{d}t^{d}$

Here $x=x_{a}$ is the projective limit of $y$ via the 1-PS $\lambda(t)$. We
will call $a$ as the degree of the limit $x$ under $\lambda$. If
$\ell=\log_{t}(\lambda(t))$ then $\ell\cdot y$ is the tangent of exit of the
curve $\lambda(t)\cdot y$ as it exits $y$ and approaches its limit $x=x_{a}$.
We see how the two may be related by a smooth optimization problem.

###  8.1 Degree and the structure of the orbit closure

In this subsection, $O(y)$ will denote the $SL(X)$-orbit of $y$,
$\overline{O(y)}$ its closure in $V$, and
$\mathbb{P}\overline{O(y)}\subset\mathbb{P}V$ will denote the projective
closure of the orbit $O(y)$. The structure of $\mathbb{P}\overline{O(y)}$
consists of closures $\mathbb{P}\overline{O(x)}$ of various lower dimensional
orbits $O(x)$ of limit points $x$ of the orbit $O(y)$. If $O(y)$ is closed in
$V$ and the stabilizer $K$ of $y$ is reductive, then in fact, the set
$\mathbb{P}\overline{O(y)}-\mathbb{P}O(y)$ is the union of a finite number of
co-dimension 1 projective varieties $\mathbb{P}\overline{O(x_{i})}$. This
offers the first decomposition of the limit points of $O(y)$.

And yet, while the boundary of $\mathbb{P}O(y)$ may be of small co-dimension,
the limit $x$ of a typical 1-PS $\lambda(t)$ acting on $y$ may be embedded
deep within this boundary as the following examples show.

###### Example 8.2

Let $A$ be an $n\times n$-matrix such that $A(1,n)\neq 0$. Let $\lambda(t)$ be
the 1-PS consisting of diagonal matrices $diag([1,t,\ldots,t^{n-1}])$. Then
the projective limit of $\lambda(t)A\lambda(t)^{-1}$ is the matrix $E_{1n}$
with degree $-(n-1)$. On the other hand, we know that if $A$ has distinct
eigenvalues then the orbit $\mathbb{P}\overline{O(J_{n})}$ of $J_{n}$ is of
co-dimension 1 within $\mathbb{P}\overline{O(A))}$. The point $J_{n}$ is
approached by a special 1-PS which is conjugate to $\lambda(t)$ and where the
degree of $J_{n}$ is $-1$.

###### Example 8.3

Consider the variables $X=\\{x_{1},\ldots,x_{9}\\}$ and the form
$det_{3}(X)\in Sym^{3}(X)$. The dimension of the stabilizer $K\subseteq GL(X)$
of $det_{3}$ is $16$. There is another basis ${\cal
X}=\\{X_{1},\ldots,X_{9}\\}$ of $\mathbb{C}X$ and a 1-PS $\lambda_{2}(t)$ with
the following properties ( see 5.2 for details). Let
$Y=\\{X_{1},X_{2},X_{3}\\}$ and $Z=\\{X_{4},\ldots,X_{9}\\}$. Let
$\lambda_{2}(t)$ be such that $\lambda_{2}(t)(y)=y$ for $y\in Y$, while
$\lambda_{2}(t)(z)=tz$ for all $t\in Z$. Then $\lambda_{2}\cdot det_{3}({\cal
X})=tQ_{2}({\cal X})+t^{3}Q_{3}({\cal X})$, where $Q_{2}({\cal
X})=X_{1}^{2}X_{4}+X_{2}^{2}X_{5}+X_{3}^{2}X_{6}+X_{1}X_{2}X_{7}+X_{2}X_{3}X_{8}+X_{1}X_{3}X_{9}$.
Moreover, the degree of $Q_{2}$ is $1$ and its orbit is one of the co-
dimension 1 forms in the projective closure of the orbit of $det_{3}$, and the
dimension of its stabilizer is 17.

On the other hand, for $\lambda^{\prime}_{2}(t)$ defined as
$\lambda^{\prime}_{2}(t)x_{i}=x_{i}$ for $i=1,2,3$ and
$\lambda^{\prime}_{2}(t)(x_{j})=tx_{j}$, for $j=4,\ldots,9$, we have the limit
$Q^{\prime}_{2}=x_{1}x_{2}x_{3}$ of degree $0$. The stabilizer of
$Q^{\prime}_{2}$ is of dimension $19$. Indeed, for most $\mu(t)$ conjugate to
$\lambda_{2}$, we will have
$\mu(t)(det_{3}(X))=t^{0}P_{0}+t^{1}P_{1}+t^{2}P_{2}+t^{3}P_{3}$, i.e., with
the degree of the leading term $P_{0}$ as zero, which is lower than that of
$Q_{2}$. Moreover, the dimension of the orbit of this generic limit is lower
than that of $Q_{2}$.

Thus, an important problem is to locate directions, which in the algebraic
case, are 1-PS, to specific limit points. A foundation for such analysis was
provided by Kempf, [Kem78], in the case when $\lambda(t)\subseteq SL(X)$ and
the degree $a$ at which the limit is found is non-negative. The case of $a>0$
corresponds to $y$ being in the nullcone with the limit $x=0$. The case $a=0$
corresponds to a limit $x\neq 0$, with the orbit of $O(x)$ lying in the affine
closure of the $SL(X)$-orbit of $y$.

A key construction in Kempf’s analysis is the computation of an optimal 1-PS
$\lambda(t)$ within a fixed torus $T$. The uniqueness of this $\lambda$ within
one torus allows its comparsion with optimal points in other tori, and the
identification of a canonical parabolic subgroup $P(\lambda)\subseteq GL(X)$
to be associated with $y$ and $x$. The first computation has a generalization
to projective limits, which we outline below.

Let us fix a maximal torus $T\subseteq GL(X)$. For simplicity, let us assume
that it is the space of invertible diagonal matrices $D_{n}$. Let ${\cal
D}_{n}$ be the Lie algebra of $D_{n}$. Let

$O_{n-2}=\\{p\in\mathbb{R}^{n}\mbox{ such that }|\sum_{i}p_{i}^{2}=1\mbox{ and
}\sum_{i}p_{i}=0\\}$

For any $p\in\mathbb{R}^{n}$, let us denote by $\overline{p}$ the $n\times n$
diagonal matrix with $p$ as its diagonal. We regard the action of
$\overline{p}$ as an element of $gl(x)$, the Lie algebra of $GL(X)$. Let
$t^{p}$ denote the diagonal matrix with diagonal
$[t^{p_{1}},\ldots,t^{p_{n}}]$. Note that for $p\in O_{n-2}$, we have
$\overline{p}\in{\cal D}_{n}\subseteq sl(X)$ and $t^{p}\in D_{n}\subseteq
SL(X)$.

Let $\Xi\subseteq{\cal D}_{n}^{*}$ be the weight-space of the module $V$ under
$T$. For any $v\in V$ let $v=\sum_{\chi\in\Xi}v_{\chi}$ be the decomposition
of $v$ by this weight-space. Let $\Xi(v)$ be those $\chi\in\Xi$ for which
$v_{\chi}\neq 0$. Thus $\Xi(v)$ is the support of $v$. To simplify notation we
omit the subscripts in ${\cal D}_{n},D_{n}$.

###### Lemma 8.4

For any element $\ell\in{\cal D}$, we have:

$\begin{array}[]{rcll}\ell\cdot v&=&\sum_{\chi}\langle\ell,\chi\rangle
v_{\chi}&\mbox{ via the Lie algebra action}\\\ t^{\ell}\cdot
v&=&\sum_{\chi}t^{\langle\ell,\chi\rangle}v_{\chi}&\mbox{ via the group
action}\\\ \end{array}$

We now state Kempf’s basic optimization result.

###### Proposition 8.5

(Kempf) We say that $v\in V$ is the the null-cone (with respect to ${\cal
D}$), if there is an $\ell\in{\cal D}$ such that for all $\chi\in\Xi(v)$ we
have $\langle\ell,\chi\rangle>0$. Let
$\mu(\ell,v)=\min\\{\langle\ell,\chi\rangle|\chi\in\Xi(v)\\}$. If $v$ is
unstable then there is a unique $\ell_{0}\in sl(X)$ such that
$\mu(\ell_{0},v)>0$ and $\mu(\ell_{0},v)>\mu(\ell,v)$ for all
$\ell\neq\ell_{0}$.

We have the following alternate optimization formulation.

###### Proposition 8.6

Let $v\in V$ be arbitrary. Let ${\cal L}_{\alpha}(v)=\\{\ell\in{\cal
D}|\mu(\ell,v)>\alpha\\}$. Suppose that ${\cal L}_{\alpha}(v)$ is non-empty.
Consider the function

$f(t,\ell)=\sum_{\chi\in\Xi}\|v_{\chi}\|^{2}t^{-\langle\ell,\chi\rangle}$

For fixed $v\in V$ and $t\in\mathbb{R}$ the above function is a smooth
function on the compact set $O_{n-2}$. Let $\ell_{f}(t)$ minimize $f$ on
$O_{n-2}$. Then there is a constant $t_{0}>1$ depending on $v$, such that for
all $t>t_{0}$ we have $\ell_{f}(t)\in{\cal L}_{\alpha}(v)$.

Proof: Let $\ell_{K}\in{\cal L}_{\alpha}(v)$ be any element and let
$\mu_{K}=\mu(\ell_{K},v)$. Note that $\mu_{K}>\alpha$. Whence, for $t>1$, we
have $f(t,\ell_{K})<\|v\|^{2}t^{-\mu_{K}}$.

Let $\Xi_{-}(t)\subseteq\Xi(v)$ be those $\chi\in\Xi(v)$ such that
$\langle\ell_{f}(t),\chi\rangle\leq\alpha$. Similarly, let $\Xi_{+}(t)$ be
those $\chi\in\Xi(v)$ such that $\langle\ell_{f}(t),\chi\rangle>\alpha$. The
claim is that $\Xi_{-}(t)$ is empty for $t$ sufficiently large.

Suppose not and there is a $\Xi_{-}\subseteq\Xi(v)$ such that
$\Xi_{-}=\Xi_{-}(t)$ for a divergent sequence $(t_{i})_{i=1}^{\infty}$ with
$t_{i}>1$ for all $i$. Then there is a $\nu\in\Xi_{-}\subseteq\Xi(v)$ such
that $\langle\ell_{f}(t_{i}),\nu\rangle\leq\alpha$ for all $i$. Whence
$f(t_{i},\ell_{i})\geq\|v_{\nu}\|^{2}t^{-\alpha}>0$ for all $t_{i}$. But
$f(t,\ell_{K})<\|v\|^{2}t^{-\mu_{K}}$ for all $t$. Thus we have:

$\|v_{\nu}\|^{2}t^{-\alpha}\leq f(t_{i},\ell_{i})\leq
f(t_{i},\ell_{K})<\|v\|^{2}t_{i}^{-\mu_{K}}$

This gives us $t_{i}^{\mu_{k}-\alpha}<\|v\|^{2}/\|v_{\nu}\|^{2}$, with
$\mu_{K}>\alpha$.

This is a contradiction since we can choose a $t_{i}$ large enough for this to
not hold. This proves the proposition. $\Box$

###### Remark 8.7

For any $\ell\in{\cal L}_{\alpha}(v)$, let $\alpha_{0}=\mu(\ell,v)$ and
$\Xi_{\ell}=\\{\chi\in\Xi(v)|\langle\ell,\chi\rangle=\alpha_{0}\\}$. Let
$v_{\ell}=\sum_{\chi\in\Xi_{\ell}}v_{\chi}$. Then $v_{\ell}$ is a projective
limit of $v$. We do see that larger values of $\alpha$ do lead to more
fundamental limit points of $v$.

Thus, if $x$ is the minimum degree limit for the point $y$, and is
approachable by a 1-PS within $T$, then the optimization function $f(t,\ell)$
on ${\cal D}\subseteq sl(X)$ does indeed offer an algorithm to discover this.
However, in the general case, this must be done across all tori within
$sl(X)={\cal G}$ in a unified manner. An example where this has been achieved
is in designing efficient algorithms to decide intersections of orbit closures
in the space $(Mat_{n})^{k}$, i.e., $k$-tuples of matrices with $GL_{n}\times
GL_{n}$ acting diagonally on the the left and right. [AZGL+18] formulate this
problem as a geodesic convex optimization problem on the space of positive
definite matrices with their natural hyperbolic metric.

Thus, what is needed are global optimization functions
$\tilde{f}(\mathfrak{g},y)$, where $\mathfrak{g}\in{\cal G}$. or more
specifically on ${\cal G}/{\cal K}$ where ${\cal K}$ is the stabilizer of $y$.
Since ${\cal G}/{\cal Y}$ is the tangent space of the $GL(X)$-orbit of $y$,
with a ${\cal K}$-structure, the differential geometric structure of this
space needs closer examination. The Riemannian approach, and its associated
tensors and the geometry of geodesic paths offers an alternate analysis of
this algebraic object. We begin with curvature of the orbit $O(y)$ in the next
subsection.

###  8.2 The Curvature Tensors

We now look at the differential geometry of $y$ as a point in $\mathbb{P}V$.

Remark 2.11 already connects the algebraic gadgets $\theta$ and $\Phi$ with
the local 2-form $\Pi$ of the orbit $O(y)\subset V$. Thus, it is worthwhile to
investigate if there is a differential geometric equivalent of the triple
stabilizer condition. The hope is that the local geometry may guide us to
identify potential tangent vectors and standard paths (e.g., along 1-PS or
geodesics) which lead us to various limit points.

However, one difficulty in pursuing this further is that while the Lie algebra
acts on $V$, the topology and the computation of the limit happens in
$\mathbb{P}V$. Whence, we must develop a local chart ${\cal C}$ for the point
$y$ as a member of $\mathbb{P}V$ and transfer the action of ${\cal G}$ to this
chart.

The approach we adopt here is to restrict this computation to the real case.
Whence, we assume that $V$ is equipped with a positive definite metric
$\langle\cdot,\cdot\rangle$. Moreover, let us assume that a basis for $V$ is
chosen such that $\langle v,v^{\prime}\rangle=v^{T}v^{\prime}$, for any
$v,v^{\prime}\in V$ (treated as column vectors).

The chart ${\cal C}$ is constructed from the ball $B_{\epsilon}(y)$ as
follows:

${\cal C}=\\{v\in B_{\epsilon}(y)\mbox{ such that }\|v\|=\|y\|\\}$

We define the projection $\pi_{\cal C}:B_{\epsilon}(y)\rightarrow{\cal C}$ as

$\pi_{\cal C}(v)=\frac{\|y\|}{\|v\|}v$

###### Definition 8.8

We say that the representation $\rho:GL(X)\rightarrow GL(V)$ is of (A) degree
zero if $\mathbb{R}^{*}I_{X}$ is in the kernel of this map. Otherwise, (B) we
say that $\rho$ has a non-zero degree. In this case, and if $V$ is
irreducible, then there is an integer $d\neq 0$ such
$\rho(tI_{X})=t^{d}I_{V}$.

We say that a point $v$ is simple if in the case (A) above, the orbit $O(y)$
does not contain any other multiple of $y$, or in the case (B) above, the
condition $\mathfrak{g}\in{\cal G}$ and $\mathfrak{g}\cdot v\in\mathbb{R}v$
implies that $\mathfrak{g}\in\mathbb{R}I_{X}$. In this section, in case (B),
we restrict ${\cal G}$ to $sl(X)$ and the orbit $O(v)$ to be the $SL(X)$-orbit
of $v$.

Let us define $\tilde{O}(y)$ as the image of the projection of $O(y)\cap
B_{\epsilon}(y)$ onto ${\cal C}$. We have the following simple lemma:

###### Lemma 8.9

Suppose that $y$ is a simple point. Then we may choose $\epsilon>0$ such that
all points $v\in B_{\epsilon}(y)$ are also simple. Then the projection
$\pi_{\cal C}:O(y)\rightarrow\tilde{O}(y)$ is a diffeomorphism at all points.

Proof: The condition of simplicity is equivalent to $v\not\in{\cal G}\cdot v$
for (A) ${\cal G}=gl(X)$ or (B) ${\cal G}=sl(X)$. This is clearly an open
condition. The second condition is immediate. $\Box$

The above lemma allows us to define $\rho_{\cal C}:{\cal G}\rightarrow
Vec({\cal C})$ given by $\rho_{\cal C}(\mathfrak{g})(v)=\pi_{\cal
C}^{*}(\rho(\mathfrak{g})(v))$. Our next task is to compute the second
fundamental form $\Pi_{\cal C}$ on ${\cal C}$. As before, we will use the
vector fields $\rho_{\cal C}(\mathfrak{s})$ for $\mathfrak{s}\in{\cal S}$, to
compute this.

###### Lemma 8.10

Let $y\in\tilde{O}(y)\subset{\cal C}\subset V$ be a simple point with
stabilizer ${\cal K}$ and complement ${\cal S}$ of dimension $K$. Let
$X=\rho(\mathfrak{s})(v)=\nabla Sv$, be the linear vector field, in the
notation of Remark 2.11. For $i=1,\ldots,K$, let $\mathfrak{s}_{i}$ be a basis
of ${\cal S}$, $X_{i}=\rho(\mathfrak{s}_{i})=\nabla S_{i}v$ and
$\tilde{X_{i}}=\pi^{*}_{\cal C}(X_{i})=\rho_{\cal C}(\mathfrak{s}_{i})$. Then:

  1. 1.

The normal space $\tilde{N}$ to $\tilde{O}(y)$ within ${\cal C}$ is
$\pi^{*}_{\cal C}(y)(N)$.

  2. 2.

Let $\pi^{*}_{\cal C}(y)$ be the projection at point $y$ and
$\tilde{S}_{i}=\pi^{*}_{\cal C}(y)S_{i}$, then:

$(D_{\tilde{X}_{j}}\tilde{X}_{i})(y)=-\frac{y^{T}S_{i}y}{y^{T}y}\tilde{S}_{j}y+\tilde{S}_{i}\tilde{S}_{j}y$

  3. 3.

Then the second fundamental form $\Pi_{\cal C}$ of $\tilde{O}(y)$ as a
submanifold of ${\cal C}$ is given as follows:

$\Pi_{\cal
C}(\tilde{X}_{j},\tilde{X}_{i})(y)=\lambda_{\tilde{N}}(D_{\tilde{X}_{j}}\tilde{X}_{i})(y)=\lambda_{\tilde{N}}\tilde{S}_{i}\tilde{S}_{j}y$

  4. 4.

Finally, if $TO(y)=T\tilde{O}(y)$, i.e., the orbit and its projection osculate
at $y$, and $\langle D_{X_{i}}y,y\rangle=0$, i.e., $y^{T}S_{i}y=0$ for all
$i$, then:

$\Pi_{\cal C}(\tilde{X}_{j},\tilde{X}_{i})(y)=\pi_{\cal
C}^{*}(y)(\Pi(X_{j},X_{i})(y))$

Proof: The first part is easily proved by taking up cases (A) and (B)
separately.

Next, the field $\tilde{X}_{i}=\rho_{\cal C}(\mathfrak{s}_{i})$ is given by:

$\tilde{X}_{i}=\pi_{\cal C}^{*}(v)(\nabla
S_{i}v)=\nabla(I-\frac{1}{v^{T}v}vv^{T})S_{i}v$

Since $\pi^{*}_{\cal C}(y)=(I-\frac{1}{y^{T}y}yy^{T})$, we have:

$\tilde{X}_{j}(y)=\nabla\pi^{*}_{\cal C}(y)S_{i}y=\nabla\tilde{S_{i}}y$

We then have:

$\begin{array}[]{rcl}(D_{\tilde{X}_{j}}\tilde{X}_{i})(y)&=&\pi^{*}_{\cal
C}(y)\circ[[{\tilde{X}_{j}}(y)(I-\frac{1}{v^{T}v}vv^{T})(y)]S_{i}y+(I-\frac{1}{y^{T}y}yy^{T})[{\tilde{X}_{j}}(y)(S_{i}v)(y)]]\\\
&=&\pi^{*}_{\cal
C}(y)\circ[[2(\tilde{S}_{j}y^{T}y)yy^{T}S_{i}y-\frac{1}{y^{T}y}[\tilde{X}_{j}(y)(vv^{T})(y)]S_{i}y+\pi^{*}_{\cal
C}(y)S_{i}\pi^{*}_{\cal C}(y)S_{j}y]]\\\ &=&\pi^{*}_{\cal
C}(y)[-\frac{1}{y^{T}y}\tilde{S}_{j}yy^{T}S_{i}y-\frac{1}{yTy}yy^{T}\tilde{S}_{j}^{T}S_{i}y+\tilde{S}_{i}\tilde{S}_{j}y]\\\
&=&\pi^{*}_{\cal
C}(y)[-\frac{y^{T}S_{i}y}{y^{T}y}\tilde{S}_{j}y-\frac{y^{T}\tilde{S}_{j}S_{i}y}{yTy}y+\tilde{S}_{i}\tilde{S}_{j}y]\\\
&=&-\frac{y^{T}S_{i}y}{y^{T}y}\tilde{S}_{j}y+\tilde{S}_{i}\tilde{S}_{j}y\\\
\end{array}$

This proves (2). Now, since $\tilde{N}=\pi^{*}_{\cal C}(y)(N)$, we have
$\tilde{S}_{j}y\in T_{y}\tilde{O(y)}$, we have
$\lambda_{\tilde{N}}(\tilde{S}_{j}y)=0$. Whence we have:

$\Pi_{\cal
C}(\tilde{X}_{j},\tilde{X}_{i})(y)=\lambda_{\tilde{N}}(D_{\tilde{X}_{j}}\tilde{X}_{i})(y)=\lambda_{\tilde{N}}\tilde{S}_{i}\tilde{S}_{j}y$

This proves (3). In the case when $O(y)$ and $\tilde{O}(y)$ osculate, i.e.,
$TO_{y}=T\tilde{O}_{y}$, i.e., the tangent planes match, we have
$\tilde{S}_{i}=\pi^{*}_{\cal C}(y)S_{i}=S_{i}$, whence

$\Pi_{\cal
C}(\tilde{X}_{j},\tilde{X}_{i})(y)=\lambda_{\tilde{N}}S_{i}S_{j}y=\pi^{*}_{\cal
C}(y)\lambda_{N}S_{i}S_{j}y=\pi_{\cal C}^{*}(\Pi(X_{j},X_{i})(y))$

This proves (4). $\Box$

We now recall the fundamental result on Riemannian curvature tensor for smooth
manifolds.

###### Proposition 8.11

Let $M$ be a submanifold of a Riemannian manifold $P$. Let $X_{i},X_{j}$ and
$X_{k}$ be vector fields on $M$. Then the Riemannian curvature tensor of $M$
is given by:

$R(X_{i},X_{j})X_{k}=\overline{D}_{X_{i}}\overline{D}_{X_{j}}X_{k}-\overline{D}_{X_{j}}\overline{D}_{X_{i}}X_{k}-\overline{D}_{[X_{i},X_{j}]}X_{k}$

Moreover, if $X_{l}$ is another vector field on $M$ then The quantity
$r_{ijkl}=\langle R(X_{i},X_{j})(X_{k})(x),X_{l}(x))\rangle$ equals the
quantity
$\langle\Pi(X_{i},X_{l})(x),\Pi(X_{j},X_{k})(x)\rangle-\langle\Pi(X_{j},X_{l})(x),\Pi(X_{i},X_{k})(x)\rangle$.

The proof of this is standard.

###### Example 8.12

Let us consider $\mathbb{R}^{n}$ with the standard basis $e_{1},\ldots,e_{n}$
and under the action of $SO(n)$ and Lie algebra $so(n)$. Consider the vector
$x=(r,0,\ldots,0)=r\cdot e_{1}$, whose orbit is $S=rS^{n-1}$, the sphere of
radius $r$ in $\mathbb{R}^{n}$. Note that in this case, $O(x)=\tilde{O}(x)$.
The tangent space $T_{x}S$ is the span of $e_{2},\ldots,e_{n}$. We have
$\mathbb{R}^{n}=\mathbb{R}e_{1}\oplus T_{x}S$ and thus the normal space is
only $1$-dimensional. Let $\mathfrak{s}_{i}\in so(n)$ be the element with
matrix $S^{i}$ with the only non-zero entries as
$S^{i}(1,i)=-1/r,S^{i}(i,1)=1/r$. Thus $X_{i}=\nabla S^{i}\overline{v}$ is the
corresponding vector field such that $X_{i}(x)=e_{i}$.

For $i=2,\ldots,n$ and $j=2,\ldots,n$, we may build that data $c^{i}_{j}$,
i.e., the “normal” part of $j$-th column of the matrix $S_{i}$. Since the
normal part is only $1$-dimensional, i.e., the first row, we have:

$\Pi(X_{j},X_{i})=c^{i}_{j}=\left\\{\begin{array}[]{lr}0&\mbox{if $i\neq
j$}\\\ -1/r&\mbox{if $i=j$}\end{array}\right.$

The Ricci curvature $R^{c}_{jk}$ is the sum below:

$R^{c}_{jk}=\sum_{i}r_{ijki}=\sum_{i}c^{i}_{i}c^{k}_{j}-c^{i}_{j}c^{k}_{i}$

This gives us:

$R^{c}_{jk}=\left\\{\begin{array}[]{lr}0&\mbox{if $j\neq k$}\\\
(n-1)/r^{2}&\mbox{if $j=k$}\end{array}\right.$

###### Example 8.13

Let us consider the adjoint action of $GL_{n}$ on $Matn_{n}$, i.e.,
$\rho(A)(X)=AXA^{-1}$, where $A\in GL_{n}$ and $X\in Matn_{n}$, and the
corresponding Lie algebra action of $gl_{n}$, given by $\rho(a)(X)=aX-Xa$,
where $a\in gl_{n}$.

Let $x=diag(\overline{\lambda})$, the diagonal matrix with distinct
eigenvalues $\overline{\lambda}=(\lambda_{1},\ldots,\lambda_{n})$. Then ${\cal
H}={\cal D}$, the Lie algebra of all diagonal matrices and ${\cal S}$ is the
linear span of $\\{E_{ij}|i\neq j\\}$, the non-diagonal elementary matrices.
Note that $[E_{ij},x]=(\lambda_{j}-\lambda_{i})x$ and thus ${\cal S}$ is both
the complement to ${\cal D}$ within $gl_{n}$ and to complement in $Matn_{n}$
to $T_{x}O$ the tangent space of the orbit. Note that in the standard inner
product on matrices, we have $Tr([E_{ij},x]x^{T})=0$ and thus the point $x$
satisfies the osculation conditions of Lemma 8.10 (4).

Let $e_{ij}=E_{ij}/(\lambda_{j}-\lambda_{i})$ and note that
$\rho(e_{ij})(x)=e_{ij}$. We now compute
$\Pi(e_{rs},e_{pq})=c^{pq}_{rs}(x)=\lambda_{N}D_{e_{rs}}\rho(e_{pq})(x)$. This
is given by $\lambda_{N}$ of the column vector corresponding to $e_{rs}$ in
the matrix $\rho(e_{pq})$. Since $N$ consists of diagonal matrices, this
equals the diagonal entries in the action of $e_{pq}$ on $Mat_{n}$, in other
words, the diagonal matrix in $[e_{pq},e_{rs}]$. Thus, the only $(r,s)$ for
which this happens is $(q,p)$ and thus we see that:

$c^{pq}_{rs}=\left\\{\begin{array}[]{lr}0&\mbox{if $(rs)\neq(qp)$}\\\
d_{pq}&\mbox{if $(rs)=(qp)$}\end{array}\right.$

where $d_{pq}$ is the diagonal matrix below:

$\left\\{\begin{array}[]{ll}d_{pq}(p,p)&=-(\lambda_{q}-\lambda_{p})^{2}\\\
d_{pq}(q,q)&=(\lambda_{q}-\lambda_{p})^{2}\\\ d_{pq}(i,j)&=0\mbox{ for all
other tuples $(i,j)$}\\\ \end{array}\right.$

Finally, note that $\langle d_{pq},x\rangle\neq 0$ unles
$\lambda_{p}\neq\lambda_{q}$, $\Pi_{\cal C}(c_{rs}^{pq})$ will need a final
projection which is skipped here.

###### Example 8.14

Let us consider the case when $n=2m$ in the example above and

$x=\left[\begin{array}[]{cc}\lambda I_{m}&0\\\ 0&\mu I_{m}\\\
\end{array}\right]\mbox{ with $\lambda,\mu\neq 0$}$

Clearly, the stabilizer ${\cal H}$ of $x$ is the Lie algebra $gl_{m}\times
gl_{m}$ and ${\cal S}$ is the space below:

${\cal S}=\left\\{=\left[\begin{array}[]{cc}0&X\\\ Y&0\\\
\end{array}\right]|X,Y\in gl_{m}\right\\}$

Note that ${\cal S}=T_{x}O$ and that $N$ is the space of all block-diagonal
matrices:

$N=\left\\{=\left[\begin{array}[]{cc}Z&0\\\ 0&W\\\ \end{array}\right]|Z,W\in
gl_{m}\right\\}$

We use the ${\cal H}$-invariant inner product $\langle A,B\rangle=Tr(AB)$ on
the space $gl_{n}$. We decompose ${\cal S}={\cal S}^{+}\oplus{\cal S}^{-}$, as
given below:

${\cal S}^{+}=\left\\{=\left[\begin{array}[]{cc}0&X\\\ 0&0\\\
\end{array}\right]|X\in gl_{m}\right\\}\mbox{ and }{\cal
S}^{-}=\left\\{=\left[\begin{array}[]{cc}0&0\\\ Y&0\\\ \end{array}\right]|Y\in
gl_{m}\right\\}$

For matrices $X,Y\in{\cal S}^{\pm}$, it is easy to see that
$\Pi(X,Y)(x)=\lambda_{N}D_{X}Y(x)$ is precisely the block diagonal component
of $[X,Y]$. Whence we have:

$\Pi(X,Y)(x)=\left\\{\begin{array}[]{rl}\left[\begin{array}[]{ll}XY&0\\\
0&-YX\end{array}\right]&\mbox{when $X\in{\cal S}^{+},Y\in{\cal S}^{-}$}\\\
\vskip 8.53581pt\cr\left[\begin{array}[]{cc}-YX&0\\\
0&XY\end{array}\right]&\mbox{when $X\in{\cal S}^{-},Y\in{\cal S}^{+}$}\\\
\vskip 8.53581pt\cr 0&\mbox{otherwise}\end{array}\right.$

###  8.3 The Cyclic Shift Matrix

We now illustrate an optimization formulation of the path problem and its
relationship with local curvatures.

Let $X$ be the space $Mat_{n}$ as before acted upon by $gl_{n}$ my
conjugation. Let us examine the case of the approach of a special matrix to
its projective limit point expressed in a basis in which the required 1-PS is
easily written.

Let $\mathbb{Z}_{n}=\\{0,\ldots,n-1\\}$ be the set of integers under modulo
$n$ addition. This will also be an index set for defining the matrix
$\mathfrak{c}_{n}$ (or simply $\mathfrak{c})$ as the cyclic shift matrix:

$\mathfrak{c}(i,j)=\left\\{\begin{array}[]{rl}1&\mbox{ if $j-i=1$}\\\ 0&\mbox{
otherwise}\end{array}\right.$

Thus $\mathfrak{c}$ is the matrix below:

$\mathfrak{c}=\left[\begin{array}[]{ccccc}0&1&\ldots&&0\\\ 0&0&1&\ldots&0\\\
\vdots&&\vdots&&\\\ 0&\ldots&0&0&1\\\ 1&\ldots&0&0&0\\\ \end{array}\right]$

Let $\lambda(t)$ be the $1$-PS below:

$\lambda(t)=\left[\begin{array}[]{ccccc}t&0&\ldots&&0\\\ 0&t^{2}&0&\ldots&0\\\
\vdots&&\vdots&&\\\ 0&\ldots&0&0&t^{n}\\\ \end{array}\right]$

We the see that $\log(\lambda(t))$ is the matrix $\ell$ below:

$\ell=\left[\begin{array}[]{ccccc}1&0&\ldots&&0\\\ 0&2&0&\ldots&0\\\
\vdots&&\vdots&&\\\ 0&\ldots&0&0&n\\\ \end{array}\right]$

The action of a generic element $\mathfrak{a}=(a_{ij})$ of $gl_{n}$ on
$\mathfrak{c}$ is given by:

$t_{n}=\mathfrak{a}\cdot\mathfrak{c}=a\cdot\mathfrak{c}-\mathfrak{c}\cdot
a=\left[\begin{array}[]{ccccc}a_{0,n-1}-a_{10}&a_{00}-a_{11}&\ldots&&a_{0,n-2}-a_{1,n-1}\\\
\vdots&&\vdots&&\\\
a_{n-1,n-1}-a_{0,0}&a_{n-1,0}-a_{0,1}&\ldots&&a_{n-1,n-1}-a_{0,n-1}\\\
\end{array}\right]$

Thus the matrix $t_{n}$ is given by:

$t_{n}(i,j)=a_{i,j-1}-a_{i+1,j}$

We use the standard inner product on both $Mat_{n}$ as well as $gl_{n}$, viz.,
$\langle x,y\rangle=Tr(x\overline{y}^{T})$, for any matrices $x,y$. Since
$(\mathfrak{c}^{i})^{T}=\mathfrak{c}^{n-i}$, we have the following
straightforward lemma:

###### Lemma 8.15

The tangent space $T_{n}X$ to $\mathfrak{c}$ is perpendicular to
$\mathfrak{c}$. In other words $Tr(t_{n}\mathfrak{c}^{T})=0$ for all
$\mathfrak{a}\in gl_{n}$.

The stabilizer ${\cal H}$ of $\mathfrak{c}$ is the space
$\oplus_{i=0}^{n-1}\mathfrak{c}^{i}$. Let us compute the complement ${\cal S}$
to ${\cal H}$ under the above inner product. For $k=0,\ldots,n-1$, let $D_{k}$
be the indices

$D_{k}=\\{(i,i+k)|i=1,\ldots,n\\}$

Thus $D_{k}$ is $k$-th shifted diagonal, i.e., the support $c_{n}^{k}$. Thus

${\cal S}=\\{\mathfrak{s}=(s_{ij})\mbox{ such that }\sum_{(i,j)\in
D_{k}}s_{ij}=0\mbox{ for all $k$}\\}$

Let us look at
$\gamma(\mathfrak{s})=\|\mathfrak{s}\cdot\mathfrak{c}\|/\|\mathfrak{s}\|$, the
compression achieved by $\mathfrak{s}$. We show that $\ell$ above is one of
the elements of ${\cal S}$ which minimize $\gamma$. For convenience, let us
denote by ${\cal S}_{k}$ as the elements $(s=(s_{ij})\in{\cal S}$ where
$j-i=k$, i.e., matrices in ${\cal S}$ with support in $D_{k}$. Thus, ${\cal
S}=\oplus_{k=0}^{n-1}{\cal S}_{i}$.

Let us assume that $\gamma$ achieves a minima at $\mathfrak{x}\in{\cal S}$ and
that $\mathfrak{t}=\mathfrak{x}\cdot\mathfrak{c}$. We may decompose these
elements as above, by their support and express $\mathfrak{x}$ and
$\mathfrak{t}$ as a sum $\mathfrak{x}=\sum_{i=0}^{n-1}\mathfrak{x}_{i}$ and
$\mathfrak{t}=\sum_{i=0}^{n-1}\mathfrak{t}_{i}$. We observe that:

$\gamma(\mathfrak{x})=\frac{\sqrt{\|\mathfrak{t}_{0}\|^{2}+\ldots+\|\mathfrak{t}_{n-1}\|^{2}}}{\sqrt{\|\mathfrak{x}_{0}\|^{2}+\ldots+\|\mathfrak{x}_{n-1}\|^{2}}}$

Since $\mathfrak{x}_{i}\cdot\mathfrak{c}=\mathfrak{t}_{i+1}$, the minima is
also achieved where $\mathfrak{x}=\mathfrak{x}_{i}$ for some $i$. We choose
$i=0$; the other cases are similar. Thus $\mathfrak{x}\in{\cal S}_{0}$ is a
diagonal matrix $(x_{ii})$ such that $\sum_{i}x_{ii}=0$. We see then that:

$\gamma(\mathfrak{x})=\frac{\sum_{i=1}^{n}(x_{ii}-x_{i+1,i+1})^{2}}{\sum_{i=1}^{n}x_{ii}^{2}}$

The condition that $\sum_{i}x_{ii}=0$ requires that some $x_{ii}\neq
x_{i+1,i+1}$ for some $i$. Suppose then that $x_{nn}-x_{11}=1$. It is then
easily seen that the optimal distribution of the intermediate $x_{ii}$ is
obtained by evenly spacing them, i.e., $x_{i+1,i+1}-x_{ii}=1/(n-1)$, for
$i=1,\ldots,n-2$.

Recall $\ell$ as before and let $\overline{\ell}$ be obtained by subtracting
constant $\frac{(n+1)}{2}$ from each entry of $\ell$. This makes
$\overline{\ell}$ an element of ${\cal S}$. We have the following lemma, which
is easily proved.

###### Lemma 8.16

Let $\ell_{ij}=\mathfrak{c}^{i}\overline{\ell}\mathfrak{c}^{-j}$.

  * •

Then $\ell_{ij}\in{\cal S}_{i-j}$ and the set $L_{k}=\\{\ell_{ij}|i-j=k\\}$ is
a basis for ${\cal S}_{i}$. $L=\cup_{k}L_{k}$ is a basis for ${\cal S}$.

  * •

The elements $\mathfrak{x}\in{\cal S}_{k}$, for some $k$, optimizing $\gamma$
above are scalar multiples of some element of $L$.

The above lemma is not a completely happy situation. For $\mathfrak{r}\in{\cal
S}_{0}$ with $\ell_{ii}=\mathfrak{c}^{i}\overline{\ell}\mathfrak{c}^{-i}$, we
see that the leading term of
$\ell_{ii}(\mathfrak{c})=[\ell_{ii},\mathfrak{c}]=\ell_{ii}\mathfrak{c}-\mathfrak{c}\ell_{ii}$
is $\mathfrak{c}^{i}J_{n}\mathfrak{c}^{-i}$, which is a cyclic renumbering of
the nilpotent flag of $J_{n}$ and thus $\ell_{ii}$ is not only an optimal
tangent of exit, but it also leads to the desired highest dimension limit
point. This is the good part.

However, for $i\neq j$,
$\ell_{ij}=\mathfrak{c}^{i}\cdot\overline{\ell}\cdot\mathfrak{c}^{-j}\in{\cal
S}_{j-i}$ does minimize $\gamma$, but we have the following:

$LT([\ell_{ij},\mathfrak{c}])=\ell_{ij}\mathfrak{c}-\mathfrak{c}\ell_{ij}=(\mathfrak{c}^{i}J_{n}\mathfrak{c}^{-1})c^{i-j}$

This limit is not conjugate to $J_{n}$, and need not even be a nilpotent
matrix. Thus the direction $\ell_{ij}\in{\cal S}_{i-j}$ is a false start from
the point $\mathfrak{c}$. The resolution of this situation lies in the
analysis of the curvature form at the point $\mathfrak{c}$. Note that thus
form $\Pi:{\cal S}\times{\cal S}\rightarrow N$, where we have the basis
$\cup_{k}L_{i}$ for ${\cal S}$ and $\\{\mathfrak{c}^{i}|i=0,\ldots,n-1\\}$ for
$N$.

Our next task will be to find an expression for
$\langle\Pi(\ell_{i,j},\ell_{k,l}),\mathfrak{c}^{r}\rangle$ for general
$i,j,k,l,r$ and to evaluate these for a suitable subset. For this, we will use
the lemma:

###### Lemma 8.17

With the above notation, we have:

$\langle\Pi(\ell_{i,j},\ell_{k,l}),\mathfrak{c}^{r}\rangle=\langle\lambda_{N}(\rho(\ell_{k,l})(\rho(\ell_{i,j}),(\mathfrak{c}^{r}))\rangle=\frac{1}{n}Tr([\ell_{kl},[\ell_{ij},\mathfrak{c}^{1}]]\cdot(\mathfrak{c}^{r})^{T})$

Proof: The first equality follows from Lemma 8.10. For the second equality,
note that
$\rho(\ell_{k,l})(\rho(\ell_{i,j})(\mathfrak{c}^{r}))=[\ell_{kl},[\ell_{ij},\mathfrak{c}^{1}]]$.
Next, since $\\{\mathfrak{c}^{i}|i=0,\ldots,n-1\\}$ is an orthogonal basis in
the norm $\langle X,Y\rangle=Tr(XY^{T})$, we have that the component along
$\mathfrak{c}^{r}$ is precisely
$\langle\cdot,\mathfrak{c}^{r}\rangle/\langle\mathfrak{c}^{r},\mathfrak{c}^{r}\rangle$.
The result follows since
$\langle\mathfrak{c}^{r},\mathfrak{c}^{r}\rangle=Tr(\mathfrak{c}^{r}(\mathfrak{c}^{r})^{T})=n$.
$\Box$

###### Lemma 8.18

Let $L_{i}=\ell\cdot\mathfrak{c}^{i}=\ell_{0,-i}$ and let:

$\Pi(L_{i},L_{j})=\sum_{k=0}^{n-1}p_{ij}^{k}\mathfrak{c}^{k}$

Then $p_{ij}^{k}$ are given by:

$p_{ij}^{k}=\left\\{\begin{array}[]{rl}(n-1)-(i+j)&\mbox{ when $k\neq 0$ and
$k=i+j+1$ in $\mathbb{Z}_{n}$}\\\ 0&\mbox{otherwise}\end{array}\right.$

Proof: Let is use the index set $\mathbb{Z}_{n}=\\{0,\ldots,n-1\\}$. Let us
first note that for any $i,j$, there is at most one $k$ for which $p_{ijk}\neq
0$. This is simply because $\Pi(L_{i},L_{j})=[L_{j},[L_{i},\mathfrak{c}^{1}]]$
has support only in $D_{i+j+1}$. Thus, $k=i+j+1$. Let us denote the inner term
$[\ell\mathfrak{c}^{i},\mathfrak{c}^{1}]$ as $\Delta_{i}$ which is given by:

$\Delta_{i}(r,s)=\left\\{\begin{array}[]{rl}-1&\mbox{if
$r\in\\{0,\ldots,n-2\\}$ and $s=r+i+1$ in $\mathbb{Z}_{n}$}\\\
n-1&\mbox{$r=n-1$ and $s=r+i+1$ in $\mathbb{Z}_{n}$}\\\ 0&\mbox{otherwise}\\\
\end{array}\right.$

$L_{j}$ itself is given by:

$L_{j}(r,s)=\left\\{\begin{array}[]{rl}r&\mbox{if $r\in\\{0,\ldots,n-1\\}$ and
$s=r+j$ in $\mathbb{Z}_{n}$}\\\ 0&\mbox{otherwise}\\\ \end{array}\right.$

Then the matrix of interest $L_{j}\Delta_{i}-\Delta_{i}L_{j}$. We see that:

$\begin{array}[]{rcl}Tr(L_{j}\Delta_{i}(\mathfrak{c}^{k})^{T})&=&\sum_{r=0}^{n-1}(L_{j})_{r,r+j}(\Delta_{i})_{r+j,r+k}\\\
&=&\sum_{r=0,r+j\neq n-1}^{n-1}(-r)+r\cdot(n-1)\delta_{r+j,n-1}\\\
&=&\sum_{r=0}^{n-1}(-r)+r\cdot(n)\delta_{r+j,n-1}\\\ &=&-n(n-1)/2+n(n-1-j)\\\
&=&n(n-1)/2-j\cdot n\\\ \end{array}$

In the same manner, we have:

$\begin{array}[]{rcl}Tr(\Delta_{i}L_{j}(\mathfrak{c}^{k})^{T})&=&\sum_{r=0}^{n-1}(\Delta_{i})_{r,r+i+1}(L_{j})_{r+i+1,r+k}\\\
&=&\sum_{r=0}^{n-2}(\Delta_{i})_{r,r+i+1}(L_{j})_{r+i+1,r+k}+(n-1)(L_{j})_{n+i,n-1+k}\\\
&=&-\sum_{r=0}^{n-2-i}r-\sum_{n-2-i+1}^{n-2}(r-n)+(n-1)i\\\
&=&-\sum_{r=0}^{n-2}(r+i+1)+\sum_{r=n-2-i+1}^{n-2}(r+i+1-n)+(n-1)i\\\
&=&-(i+1+n+i-1)(n-1)/2+in+i(n-1)\\\ &=&-(n-1)n/2+in\\\ \end{array}$

This gives us:

$p_{ij}^{k}=\frac{1}{n}Tr((L_{j}\Delta_{i}-\Delta_{i}L_{j})(\mathfrak{c}^{k})^{T})=(n-1)-(i+j)$

where $k=i+j+1$. This proves the lemma. $\Box$

###### Example 8.19

Let $\Pi^{k}$ be the matrices $(p^{k}_{ij})$, then for $n=5$ we have the
following.

$P^{1}=\left[\begin{array}[]{rrrrr}4&0&0&0&0\\\ 0&0&0&0&-1\\\ 0&0&0&-1&0\\\
0&0&-1&0&0\\\ 0&-1&0&0&0\\\
\end{array}\right]P^{2}=\left[\begin{array}[]{rrrrr}0&3&0&0&0\\\ 3&0&0&0&0\\\
0&0&0&0&-2\\\ 0&0&0&-2&0\\\ 0&0&-2&0&0\\\ \end{array}\right]$
$P^{3}=\left[\begin{array}[]{rrrrr}0&0&2&0&0\\\ 0&2&0&0&0\\\ 2&0&0&0&0\\\
0&0&0&0&-3\\\ 0&0&0&-3&0\\\
\end{array}\right]P^{4}=\left[\begin{array}[]{rrrrr}0&0&0&1&0\\\ 0&0&1&0&0\\\
0&1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&-4\\\ \end{array}\right]$

Note that $\Pi(L_{i},L_{j})$ will have no component along $\mathfrak{c}^{0}$,
the identity matrix, simply because the matrix
$[L_{i}[L_{j},\mathfrak{c}^{1}]]$ will have zero trace. Next, the normal
$\mathfrak{c}^{1}$ is indeed special since, the in any local model of the
projective orbit $\mathbb{P}O(\mathfrak{c})$, the normal $\mathfrak{c}$ to
$O(\mathfrak{c})$ will collapse. We then have the following important lemma:

###### Lemma 8.20

For the element $\mathfrak{c}\in Mat_{n}$, we have $\Pi(L_{0},L_{0})=0$ for
the submanifold $\mathbb{P}O(\mathfrak{c})\subseteq\mathbb{P}V$. Moreover, if
$n$ is odd then of all $L_{i}$, it is unique in having this property.

This analysis does show that the limiting 1-PS is indeed special in the local
differential geometry of the starting point $y$. However, the exact connection
between this and the algebraic conditions which lead us to the limit $x$, and
its properties, is not clear.

##  9 Conclusion

Our motivating question has been to understand the conditions under which a
point $x\in V$ is in the projective orbit closure of $y\in V$, when both $x$
and $y$ have distinctive stabilizers, resp. $H$ and $K$, with Lie algebras
${\cal H}$ and ${\cal K}$. Our primary objective has been to provide some
initial Lie algebraic tools to analyse this important problem.

Towards this, we develop a local model for $x$, i.e., an open neighborhood of
$x$ and a Lie algebra action of ${\cal G}$ with certain useful properties. We
show that this open neighborhood has a convenient product structure $M\times
N$, where $M$ corresponds to a neighborhood of the orbit $O(x)$ identified
with ${\cal S}$, a complement to ${\cal H}$ in ${\cal G}$, and $N$, a
complementary space to $T_{x}O(x)\subseteq V$, the tangent space of the orbit.
We show that for points $y=x+n\in M\times N$, the Lie algebra ${\cal K}$ of
the stabilizer $K$ of $y$ is determined by a subspace of ${\cal H}$. Indeed,
${\cal K}$ is the ${\cal S}$-completion of this subspace of ${\cal H}$, where
${\cal S}$ above is a complement to ${\cal H}$ in ${\cal G}$. Moreover, we
show that the computation of ${\cal K}$ is made effective by a map $\theta$,
which measures the “shear” due to ${\cal Q}$, the unipotent radical of ${\cal
H}$, and its close relative $\Phi$, which is intimately connected with the
curvature of the orbit at $x$. This alignment between ${\cal H}$ and ${\cal
K}$ is an extension of Luna’s results on reductive stabilizers, but at the Lie
algebra level.

Our main application is on forms, i.e., on the space $V=Sym^{d}(X)$, and
points $x,y\in V$, which have distinctive stabilizers, and where $x$ is
obtained from $y$ by taking limits through 1-parameter subgroups $\lambda(t)$
of $GL(X)$. We show that that there is a leading term subalgebra ${\cal
K}_{0}$ of ${\cal K}$, the stabilizer of $y$, which appears as a subalgebra of
${\cal H}$, the stabilizer of $x$. We also show that a generic $\lambda$ leads
to solvable ${\cal K}_{0}$, while non-generic $\lambda$ connect the form $x$,
the tangent of approach, the tangent of exit and the form $y$ through triple-
stabilizers.

The connection between ${\cal K}_{0}$ and ${\cal K}$ is explored further by
considering ${\cal K}$ as an extension of ${\cal K}_{0}$. Under a
transversality assumption, there is an extension ${\cal K}(\epsilon)$ of
${\cal K}_{0}$ whose structure constants agree with that of ${\cal K}$ up to
the first order. This leads us to derivations of Lie algebras and certain Lie
cohomology conditions. These tools are applied to the co-dimension 1
subvarieties within orbit closures.

The local model is next applied to the celebrated problem of computing
projective closures of orbits of a single matrix under conjugation. Here we
reprove the classical result of [Ger61, Hes79] on the orbit closure of a
nilpotent matrix, as well as prove, perhaps for the first time, a result on
the projective closures of a general semisimple matrix.

Finally, we consider the path problem, of the choice of paths which lead out
of $y$ and their limit points $x$. We show how Kempf’s algebraic construction
for a torus does have an unconstrained optimization formulation, and thus
points to local gradient descent schemes at the local tangent space of $y$. We
then connect this with the map $\Phi$ above and compute curvatures at $y$ and
show, through an example, that these may indicate good choices of paths out of
$y$.

Coming back to GCT, our motivation for the problems considered in this paper,
identifying linkages between forms and their stabilizer groups has been an
important theme in the GCT approach. The expectation is that such linkages
would lead to obstructions, or witnesses of why a form $g$ with stabilizer
$H$, cannot be in the orbit closure of another form, $f$, with stabilizer $K$,
where both $H$ and $K$ are distinctive. Indeed, in [MS01] and [MS08] the
authors proposed representation theoretic obstructions which arose from a
generalization of the Peter-Weyl theorem to this situation. These were
$G$-modules in coordinate rings of the orbit closure of $g$ and $f$, and the
obstruction arose from the multiplicities of these modules in the respective
rings. Whether these multiplicities obstructions will be enough to separate
$g$ from $f$ has been intensely analysed.

Perhaps the local model allows for another source of obstructions connecting
${\cal H}$ and ${\cal K}$, the corresponding Lie algebra stabilizers. If
indeed $g$ is obtained as a limit of $f$, then the local model gives us two
intermediate structures which link ${\cal H}$ and ${\cal K}$. The first is a
“tangent of approach”, a form $f_{b}$, along which $f$ approaches $g$, and the
tangent of exit $\ell f$. The second is the Lie algebra ${\cal K}_{0}$, which
is obtained from ${\cal K}$ through a limiting process. ${\cal K}_{0}$ is also
a sublagebra of ${\cal H}$, which stabilizes the vector $\overline{f_{b}}$ in
the “quotient” representation $V/TO_{g}$ of ${\cal H}$. This also requires an
alignment in certain Lie algebra cohomology modules, which may be the place
for potential obstructions. We believe that these obstructions may be
especially computable when the orbit of $g$ is expected to be within the orbit
of $f$ as a sub-variety of codimension one.

## References

  * [AM95] DV Alekseevsky and Peter W Michor.  Differential geometry of g-manifolds.  Differential Geometry and its Applications, 5(4):371–403, 1995\.
  * [AZGL+18] Zeyuan Allen-Zhu, Ankit Garg, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson.  Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing.  In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 172–181, 2018.
  * [BGO+18] Peter Bürgisser, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson.  Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory.  In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018.
  * [BIP19] Peter Bürgisser, Christian Ikenmeyer, and Greta Panova.  No occurrence obstructions in Geometric Complexity Theory.  Journal of the American Mathematical Society, 32(1):163–193, 2019\.
  * [BLMW11] Peter Bürgisser, Joseph M Landsberg, Laurent Manivel, and Jerzy Weyman.  An overview of mathematical issues arising in the Geometric Complexity Theory approach to VP $\neq$ VNP.  SIAM Journal on Computing, 40(4):1179–1209, 2011.
  * [DM17] Harm Derksen and Visu Makam.  Polynomial degree bounds for matrix semi-invariants.  Advances in Mathematics, 310:44–63, 2017.
  * [Dol03] Igor Dolgachev.  Lectures on invariant theory.  Number 296. Cambridge University Press, 2003.
  * [Ger61] Murray Gerstenhaber.  Dominance over the classical groups.  Annals of Mathematics, pages 532–569, 1961.
  * [GGOW19] Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson.  Operator scaling: theory and applications.  Foundations of Computational Mathematics, pages 1–68, 2019.
  * [Gro15] Joshua A Grochow.  Unifying known lower bounds via Geometric Complexity Theory.  computational complexity, 24(2):393–475, 2015.
  * [Hes76] Wim Hesselink.  Singularities in the nilpotent scheme of a classical group.  Transactions of the american mathematical society, 222:1–32, 1976\.
  * [Hes79] Wim H Hesselink.  Desingularizations of varieties of nullforms.  Inventiones mathematicae, 55(2):141–163, 1979.
  * [Hil93] David Hilbert.  Über die vollen Invariantensysteme.  Mathematische annalen, 42(3):313–373, 1893.
  * [HL16] Jesko Hüttenhain and Pierre Lairez.  The boundary of the orbit of the 3-by-3 determinant polynomial.  Comptes rendus Mathematique, 354(9):931–935, 2016.
  * [Hof65] Karl Heinrich Hofmann.  Lie algebras with subalgebras of co-dimension one.  Illinois Journal of Mathematics, 9(4):636–643, 1965.
  * [Hüt17] Jesko Hüttenhain.  Geometric Complexity Theory and orbit closures of homogeneous forms.  Ph.D Thesis,Technichen Universit$\ddot{a}$t, Berlin, 2017.
  * [IP17] Christian Ikenmeyer and Greta Panova.  Rectangular Kronecker coefficients and plethysms in Geometric Complexity Theory.  Advances in Mathematics, 319:40–66, 2017.
  * [IQS17] Gábor Ivanyos, Youming Qiao, and KV Subrahmanyam.  Non-commutative Edmonds’ problem and matrix semi-invariants.  computational complexity, 26(3):717–763, 2017.
  * [Kem78] George R Kempf.  Instability in invariant theory.  Annals of Mathematics, 108(2):299–316, 1978.
  * [Lan15] Joseph M Landsberg.  Geometric Complexity Theory: an introduction for geometers.  Annali dell’universita’di Ferrara, 61(1):65–117, 2015.
  * [Lee01] Dong Hoon Lee.  The structure of complex Lie groups.  CRC Press, 2001.
  * [LL89] Thomas Lehmkuhl and Thomas Lickteig.  On the order of approximation in approximative triadic decompositions of tensors.  Theoretical computer science, 66(1):1–14, 1989.
  * [Lun73] Domingo Luna.  Slices étales.  Bull. Soc. Math. France, 33:81–105, 1973.
  * [Mat60] Yozô Matsushima.  Espaces homogenes de Stein des groupes de Lie complexes.  Nagoya Mathematical Journal, 16:205–218, 1960.
  * [MFK94] David Mumford, John Fogarty, and Frances Kirwan.  Geometric Invariant Theory, volume 34.  Springer Science & Business Media, 1994.
  * [Mic08] Peter W Michor.  Topics in differential geometry, volume 93.  American Mathematical Soc., 2008.
  * [MLR13] Laurent Manivel, Joseph M Landsberg, and Nicolas Ressayre.  Hypersurfaces with degenerate duals and the Geometric Complexity Theory program.  Commentarii Mathematici Helvetici, 88(2):469–484, 2013.
  * [MR04] Thierry Mignon and Nicolas Ressayre.  A quadratic bound for the determinant and permanent problem.  International Mathematics Research Notices, 2004(79):4241–4253, 2004.
  * [MS01] Ketan D Mulmuley and Milind Sohoni.  Geometric Complexity Theory i: An approach to the P vs. NP and related problems.  SIAM Journal on Computing, 31(2):496–526, 2001.
  * [MS08] Ketan D Mulmuley and Milind Sohoni.  Geometric Complexity Theory ii: Towards explicit obstructions for embeddings among class varieties.  SIAM Journal on Computing, 38(3):1175–1206, 2008.
  * [New78] Peter E Newstead.  Introduction to moduli problems.  Tata Inst. Lecture Notes, 1978.
  * [NR67] Albert Nijenhuis and R.W. Richardson.  Deformations of lie algebra structures.  Journal of Mathematics and Mechanics, 17(1):89–105, 1967.
  * [Pop03] Vladimir Leonidovich Popov.  The cone of Hilbert nullforms.  Trudy Matematicheskogo Instituta imeni VA Steklova, 241:192–209, 2003.
  * [Pop09] Vladimir L Popov.  Two orbits: When is one in the closure of the other?  Proceedings of the Steklov Institute of Mathematics, 264(1):146–158, 2009.
  * [Val79] L. G. Valiant.  Completeness classes in algebra.  In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC ’79, page 249–261, New York, NY, USA, 1979. Association for Computing Machinery.
  * [Wag10] Fredrich Wagemann.  Introduction to Lie algebra cohomology with a view towards BRST cohomology.  Lecture notes, Université Nantes, 2010.

##  Appendix A Algebraic complexity and the geometric complexity theory - a
brief introduction

Let $Y$ denote a vector of indeterminates suitably arranged, (e.g., as an
$m\times m$-matrix) and let $Sym^{d}(Y)$ be the space of forms of degree $d$.
A natural model to compute forms are straight-line programs (also known as
arithmetic circuits). Such a program consists of an ordered sequence of
arithmetic operations involving $+,-,*$ (called lines), starting with
variables $y_{i}\in Y$ and constants from ${\mathbb{C}}$. A form $f\in
Sym^{d}(Y)$ is said to have an efficient computation if there is straight-line
program computing $f$ in which the number of lines is bounded by a polynomial
in $|Y|$ and $d$. The determinant form $det_{m}(Y)\in Sym^{m}(Y)$ (where $Y$
is an $m\times m$-matrix) is known be be efficiently computable. On the other
hand, for the permanent form, $perm_{m}(Y)$, no such efficient computation is
known. The analysis of this divergence has been a central problem of the
theory.

In a remarkable construction, Valiant[Val79] showed that for a form $f(Y)\in
Sym^{d}(Y)$, if there is a program111In this construction the program
corresponds to an arithmetic formula. of $L$ lines then there is an
$O(L)\times O(L)$-matrix with entries as linear forms $A_{ij}(Y^{\prime})$ in
the variables $Y^{\prime}=Y\cup\\{z\\}$ (where $z$ is merely a degree
homogenizing variable) such that $det_{L}(A(Y^{\prime}))=z^{(L-d)}f(Y)$.

We may thus define the determinantal complexity of a function $f\in
Sym^{d}(Y)$ as the smallest integer $n$ such that $f$ can be written as the
determinant of an $n\times n$ matrix whose entries are linear forms in the
variables $Y^{\prime}=Y\cup\\{z\\}$. Valiant’s construction shows that the
determinant is universal, and that every form $f$ has a finite determinantal
complexity.

Let us now connect determinantal complexity with the projective orbit closure
of the form $det_{n}(X)\in Sym^{n}(X)$. Suppose that the determinantal
complexity of $f\in Sym^{d}(Y)$ is $n$ (naturally, with $n>|Y|$). Then (i)
define $X=\\{x_{ij},1\leq i,j\leq n\\}$ as a set of new $n^{2}$ variables,
(ii) identify $y_{ij}$ with $x_{ij}$ for $1\leq i,j\leq m$ and $z$ with
$x_{nn}$, and finally (iii) define the form $f_{d,n}(X):=x_{nn}^{n-d}f(Y)\in
Sym^{n}(X)$. The condition that $f_{d,n}(X)=z^{n-d}det(A(Y^{\prime}))$ tell us
that there is a (possibly singular) matrix $A\in End(\mathbb{C}\cdot X)$ such
that $det(A\cdot X)=f_{d,n}(X)$. Whence this homogenized version of $f$ is in
the $GL(X)$ projective orbit closure of $det_{n}$. Thus, computability of
forms in various models of computation is intimately connected with the
algebraic closures of orbits of universal functions associated with these
models. This theme was first proposed as the Geometric Complexity Theoretic
(GCT) approach in the study of computational complexity[MS01].

A central question in computational complexity has been to determine the
determinantal complexity of $perm_{m}$, and thus to determine those $n$ for
which $perm_{m,n}\in\overline{O(det_{n})}$.

It is well known that invariant properties of these functions play an
important part in the design of algorithms. For example, the invariance of the
determinant to elementary row operations, has paved the way for its efficient
evaluation. The group of symmetries of the permanent too is large, and also
well known. In fact, both $perm_{m}(Y)$ and $det_{m}(Y)$ as forms in
$Sym^{m}(Y)$ are determined by their stabilizers within $GL(Y)$ \- and yet
this divergence in their computational complexity.

One approach to understanding this divergence is by a careful examination of
their symmetries. More concretely, it is to ask if the question of
$[perm_{m,n}]\not\in\overline{O([det_{n}])}$ can be cast in terms of the group
theory, of linkages, or their absence, between the stabilizers of $perm_{m,n}$
and $det_{n}$. This broad strategy of finding “obstructions” was also proposed
as a method in the GCT approach through a sequence of papers, which we review
briefly later.

###  A.1 Obstructions for membership in orbit closures

Let us now review a key approach of Geometric Complexity Theory, that of the
hunt for obstructions for $[x]$ to be in the orbit closure of $[y]$. This was
formulated in [MS01], in the context of the permanent and determinant forms,
both of which have large and distinctive stabilizers. We outline this briefly.
Recall that $X$ denotes linear dual of the complex $n^{2}$-dimensional vector
space of $n\times n$-matrices over ${\mathbb{C}}$.

Let $\Delta[det_{n}]$ be the projective orbit closure of $[det_{n}]$ in
${\mathbb{P}}(Sym^{n}(X))$. If $[perm_{m,n}]$ is in $\Delta[det_{n}]$ then
there would be a $GL(X)$ equivariant embedding of the projective orbit closure
of $[perm_{m,n}]$, , $Z_{m,n}$, in $\Delta[det_{n}]$. This would give rise to
a surjective $GL(X)$-morphism from the coordinate ring
${\mathbb{C}}[\Delta[det_{n}]]$ of $\Delta[det_{n}]$, to the coordinate ring
${\mathbb{C}}[Z_{m,n}]$ of $Z_{m,n}$.

The distinctive nature of the stabilizers $K$ of $[det_{n}]$ and $H$ of
$[perm_{m,n}]$ leads to a complete classification of the $GL(X)$-modules which
may appear in the above coordinate rings. Whence, the surjection implies that
every irreducible $GL(X)$-representation which occurs in
${\mathbb{C}}[Z_{m,n}]$ must occur in ${\mathbb{C}}[\Delta[det_{n}]]$. An
irreducible representation $\lambda$ of $GL(X)$ which occurs in
${\mathbb{C}}[Z_{m,n}]$ and does not occur in ${\mathbb{C}}[\Delta[det_{n}]]$
is called an occurrence obstruction. Mulmuley and Sohoni conjectured that for
every non negative integer $c$ and for infinitely many $m$, there exist
irreducible representations $\lambda$ of $GL(X)$ (with $n=m^{c}$) which occur
in ${\mathbb{C}}[Z_{m,m^{c}}]$ but not in ${\mathbb{C}}[\Delta[det_{m^{c}}]]$.
This conjecture was shown to be false for irreducible representations of
$GL(X)$ of shape $n\times d$ in [IP17]. In [BIP19] the authors showed that the
conjecture is false for all $\lambda$, when $n\geq m^{25}$. Note that these
results do not rule out the GCT approach. If one were to show that the
multiplicity of some irreducible $GL(X)$-module $\lambda$ is more in
${\mathbb{C}}[Z_{m,m^{c}}]$ than in ${\mathbb{C}}[\Delta[det_{m^{c}}]]$ then
an obstruction has been found. If one were to show this for infinitely many
$m$ and every $c$ that would separate the complexity class $VP$ from $VNP$,
the flagship problem of GCT.

Mignon and Ressayre [MR04] showed that the determinantal complexity of
$perm_{m}$ is $\Omega(m^{2})$. They consider the zero locus of the determinant
hypersurface $det_{n}=0$ in ${\mathbb{C}}^{n^{2}}$ and the zero locus of the
permanent hypersurface $perm_{m}=0$ in ${\mathbb{C}}^{m^{2}}$. They show that
the Hessian of the determinant at any point on $det_{n}=0$ has rank at most
$2n+1$ and this rank is $m^{2}$ at a generic point on $perm_{m}=0$. Their
proof follows from this beautiful calculation. Although their proof does not
explicitly use obstructions, Growchow [Gro15] showed, using a construction of
[MLR13], that their proof can also be seen through the lens of representation
theoretic obstructions.

##  Appendix B Stabilizer conditions in special cases

Let’s analyse the stabilization conditions of Theorem 3.13 for an element
$\mathfrak{h}\in{\cal K}_{0}$ in a special case. We have:

$\mathfrak{s}\cdot g=\mathfrak{h}\cdot f_{b}$

We may split $\mathfrak{s}$ further as
$\mathfrak{s}=\mathfrak{s}_{-1}+\mathfrak{s}_{0}+\mathfrak{s}_{1}$, and
$\mathfrak{h}$ as
$\mathfrak{h}=\mathfrak{h}_{-1}+\mathfrak{h}_{0}+\mathfrak{h}_{1}$ to give us:

$\mathfrak{s}_{-1}\cdot g+\mathfrak{s}_{0}\cdot g+\mathfrak{s}_{1}\cdot
g=\mathfrak{h}_{-1}\cdot f_{b}+\mathfrak{h}_{0}\cdot
f_{b}+\mathfrak{h}_{1}f_{b}$

Now note that $\mathfrak{s}_{0},\mathfrak{h}_{0}$ are elements of
$Hom(Y,Y)\oplus Hom(Z,Z)$. Typically, the dimension of $Hom(Y,Y)$ may be much
smaller that the dimension of ${\cal K}_{0}$. Thus, there is a subspace ${\cal
K}^{\prime}\subseteq{\cal K}_{0}$ of dimension $k-|Y|^{2}$ where we have that
for $\mathfrak{h}\in{\cal K}^{\prime}$, we have
$\mathfrak{h}=\mathfrak{h}_{-1}+\mathfrak{h}_{Z}+\mathfrak{h}_{1}$, where
$\mathfrak{h}_{Z}\in Hom(Z,Z)$, and that the stabilizer condition at degree
$b$ and $b-1$ is satisfied as follows:

$\mathfrak{s}_{0}\cdot g+\mathfrak{s}_{1}\cdot g=\mathfrak{h}_{-1}\cdot
f_{b}+\mathfrak{h}_{Z}f_{b}$

Noting that $g\in V_{a}$ while $f_{b}\in V_{b}$ gives us the following
possibilities, (i) $b=a+1$, (ii) $b=a+2$, and (iii) $b>a+2$. Let us take each
of them one by one.

  1. 1.

Case $b=a+1$. We have $\mathfrak{h}_{-1}f_{b}=\mathfrak{s}_{0}\cdot g$ and
$\mathfrak{s}_{1}g=\mathfrak{h}_{Z}f_{b}$.

  2. 2.

Case $b=a+2$. We have $\mathfrak{s}_{1}g=\mathfrak{h}_{-1}f_{b}$ and
$\mathfrak{h}_{Z}f_{b}=0$.

  3. 3.

Case $b>a+2$. We have $\mathfrak{s}_{1}g=0,\mathfrak{h}_{-1}f_{b}=0$ and
$\mathfrak{h}_{Z}f_{b}=0$.

Each condition gives us certain stringent conditions on the relationship
between $g$ and $f_{b}$. As an example, let us take the case $b=a+1$ and an
$\mathfrak{h}\in{\cal K}^{\prime}$. Let us assume that $|Z|=r$ and $|Y|=s$. We
may find a basis for $\mathbb{C}\cdot Z$ such that $\mathfrak{h}_{Z}$ is in
Jordan canonical form. Assume for the moment that $\mathfrak{h}_{Z}$ is
diagonal with eigenvalues $\mu_{1},\ldots,\mu_{r}$ where the first
$r^{\prime}$ are non-zero. We may express $f_{b}$ in this new basis as
$f_{b}=\sum_{\alpha}z_{\alpha}f_{b}^{\alpha}$, where $z_{\alpha}$ are distinct
monomials of degree $a+1$ and $f^{\alpha}_{b}\in Sym^{d-a-1}(Y)$. By the same
token, we may express $g$ as $\sum_{\beta}z_{\beta}g^{\beta}$, where
$z^{\beta}\in Sym^{a}(Z)$ and $g^{\beta}\in Sym^{d-a}(Y)$. The condition
$\mathfrak{s}_{1}g=\mathfrak{r}_{Z}b$ gives :

$\sum_{\beta}\sum_{i=1}^{s}L_{i}(z)\cdot\partial g/\partial
y_{i}=\sum_{\alpha}\mu^{\alpha}z_{\alpha}f^{\alpha}_{b}$

This tells us that those ${\alpha}$ such that $\mu^{\alpha}\neq 0$, we have
that $f_{b}^{\alpha}$ are $\mathbb{C}$-linear combinations of $\partial
g/\partial y_{i}$, in other words, the “minors” of $g$. The first condition,
viz., $\mathfrak{q}f_{b}=0$ gives us that there are linear relations between
the $f^{\alpha}_{b}$, i.e., the minors of $g$, with coefficients in
$Sym^{1}(Y)$. These are akin to the Laplace’s conditions on the expansion of
the determinant.