[subfigure]style=plain,subcapbesideposition=top

# On semi-simplicity of KMY algebras

N. M. Alraddadi and A. E. Parker  Mathematics Department
Al-Lith University College
Umm Al-Qura University
Al-Lith
Saudi Arabia  Department of Mathematics
University of Leeds
Leeds, LS2 9JT
UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the algebra, $J_{l,n}(\delta)$, introduced by Kadar-Martin-Yu in [4].
We show that these algebras are iterated inflation algebras. We give a set of
generators for $J_{l,n}(\delta)$. We show that this algebra satisifies the
CMPX axiomatic framework in [1] when the base field is the complex numbers and
$\delta\neq 0$. We show that $J_{l,n}(\delta)$ is semisimple if $\delta$ is
complex and not real.

## Introduction

This paper continues the study of a certain family of algebras that live
inside the Brauer algebra, thought of diagrammatically. These algebras were
introduced by Kadar, Martin and Yu in [4]. We will introduce these algebras
with less formality than that used in [4] as once certain crucial properties
are established (e.g. that heights cannot increase upon multiplication) the
strict formulism of [4] is more of a hindrance to understanding. We give a
rather more seminar style introduction to these algebras with the knowledge
that should more formalism be needed, it is all there in [4] for the
interested reader.

The bulk of this paper is taken from the first author’s PhD thesis, the second
author was the first author’s main supervisor. We have added one result not in
the thesis, that KMY algebras are iterated inflation algebras, and generalised
another, extending the result about semi-simplicity to all $l$.

There are four new main results in this paper. The first, Theorem 2.1, is that
the KMY algebras are iterated inflation algebras, as is true for Brauer
algebras.

The second, section 3, is that when the ring (field) of coefficients is
$\mathbb{C}$ then the KMY algebras satisfy the axiomatic framework for a tower
of recollement introduced in [1]. Much of the groundwork for this result was
laid down in the KMY paper, [4].

The third, Theorem 4.2, gives a generating set for the KMY algebra.

The fourth, Theorem 5.3, gives a criterion for the KMY algebra to be semi-
simple, when the ring (field) of coeefficients is $\mathbb{C}$, namely that
the parameter $\delta$ is not real.

##  1\. Review

A _pair partition_ of a set is a set partition of the set where each part has
size two (i.e. each element is paired with another). A _Brauer diagram_ is a
picture/diagram of a pair partition of

$\\{1,2,\ldots,n\\}\cup\\{1^{\prime},2^{\prime},\ldots,n^{\prime}\\}$

which is a rectangle (also called a _frame_) with $n$ (labelled) vertices on
the top and $n$ (labelled) vertices on the bottom that are joined by a line,
(which is often smooth and we allow to be isotopically deformed), if the
labels on the end vertices form a pair in the pair partition. The labels on
the vertices are usually omitted if it is clear. The important information
here is the pairs in the pair partition, represented visually in the diagram,
rather than the details of the diagram itself. E.g. for $n=5$, the pair
partition
$\\{\\{1,2\\},\\{2^{\prime},3\\},\\{1,4\\},\\{4^{\prime},5\\},\\{3^{\prime},5^{\prime}\\}\\}$
can be represented by the diagram:

.

We fix $n$ and use all possible diagrams as a basis for a vector space over a
field (or ring) $K$. To make an algebra we stack diagrams on top of each other
to multiply: E.g.

$\raisebox{-0.45pt}{\includegraphics{prodeg1left.eps}}\quad=\quad\raisebox{-0.4pt}{\includegraphics{prodeg1right.eps}}$

We might get loops:

$\raisebox{-0.45pt}{\includegraphics{prodeg2left.eps}}\quad=\quad\delta\quad\raisebox{-0.4pt}{\includegraphics{prodeg2right.eps}}$

The power of the parameter $\delta\in K$ records the number of loops removed.

$\raisebox{-10.0pt}{\includegraphics{loopsnested.eps}}\rightsquigarrow\delta^{2}\qquad\raisebox{-5.0pt}{\includegraphics{loopslink.eps}}\rightsquigarrow\delta^{2}$

A twist gets untwisted — we don’t record untangling of twists.

The Brauer algebra has the symmetric group algebra as a sub-algebra via
permutation diagrams. E.g. for $n=5$:

$\raisebox{-15.0pt}{\includegraphics{perm.eps}}\leftrightarrow\begin{pmatrix}1&2&3&4&5\\\
2&4&1&5&3\end{pmatrix}$

We call a line _propagating_ if it goes from top to bottom. Otherwise it’s
called an _arc_ , or sometimes a cap $\cap$ or a cup $\cup$.

With this diagram realisation we have

$K\mathfrak{S}_{n}\hookrightarrow\mathrm{Br}_{n}$

where $\mathfrak{S}_{n}$ is the symmetric group on $\\{1,2,\ldots,n\\}$ and
$\mathrm{Br}_{n}$ is the Brauer algebra on $n$.

Note:  is the identity permutation and is the same as the identity diagram in
$\mathrm{Br}_{n}$.

We note that the representation theory of $\mathrm{Br}_{n}$ is at least as
complicated as that for $K\mathfrak{S}_{n}$ and indeed it’s unknown for
$\operatorname{char}K=p<n$.

There is another algebra inside $\operatorname{Br}_{n}$, namely the
_Temperley-Lieb algebra_ , $\mathrm{TL}_{n}$.

Take a _planar_ representation of a diagram. If this can be deformed in the
plane to have _no_ crossings then it’s a _Temperley-Lieb diagram_. E.g.

or

The representation theory of $\mathrm{TL}_{n}$ is _very_ well understood.
(Analogous to $q$-$\mathrm{GL}_{2}$.)

Are there algebras in between that might be easier to understand? Well yes.
Take a Brauer diagram. Define _height_ as follows. Take the following diagram:

We first consider the interior connected regions (alcoves) in the diagram and
number from $0$ from the left, increasing the number by one if we cross a
line.

$\to$ $\to$ $\to$ $\to$

The number in each alcove therefore records the minimum number of lines that
are crossed by any path joining that alcove to the left most side of the
rectangle (frame) for the diagram.

We circle all the _crossing points_ , that is the points where two lines
intersect. (Remembering that we do not allow diagrams where more than 2 lines
intersect at a point.) We then label each crossing point with the smallest
number of the alcove that the circle around the crossing point intersects.

$\to$ $\to$

The _height_ of this particular diagram is then the maximum of all the heights
of the crossing points. The _height_ of a diagram (as a pair partition) is the
minimum of the heights of all possible ways of representing the pair partition
with a planar diagram.

Temperley-Lieb diagrams which have no crossings, are defined to have height
$-1$.

That this definition is possible is covered in [4].

Let $h(d)$ be the height of a diagram $d$. In [4] they prove:

$h(d_{1}d_{2})\leqslant\max\\{h(d_{1}),h(d_{2})\\}$

where $d_{1}$ and $d_{2}$ are Brauer diagrams.

That is the height can go _down_ but never go _up_ when we multiply.

Thus diagrams of height at most $l$ give a basis for a subalgebra,
$J_{l,n}(\delta)$, of $\operatorname{Br}_{n}$.

This algebra was first defined in [4] and we will henceforth call it _the KMY-
algebra_.

We have

$J_{-1,n}(\delta)=\mathrm{TL}_{n}$ $J_{n-2,n}(\delta)=\operatorname{Br}_{n}$

as Brauer diagrams have maximum height $n-2$.

Many of the properties of $\operatorname{Br}_{n}$ are also true for
$J_{l,n}(\delta)$.

Let $\\#(d)$ be the number of propagating lines in $d$. Then:

$\\#(d_{1}d_{2})\leqslant\min\\{\\#(d_{1}),\\#(d_{2})\\}.$

We define ideals, $I_{m}=\langle d\in J_{l,n}(\delta)\mid\\#(d)\leqslant
m\rangle$, and these ideals filter the algebra. For $n$ even:

$I_{0}\leqslant I_{2}\leqslant\cdots\leqslant I_{n-2}\leqslant
I_{n}=J_{l,n}(\delta).$

And for $n$ odd:

$I_{1}\leqslant I_{3}\leqslant\cdots\leqslant I_{n-2}\leqslant
I_{n}=J_{l,n}(\delta).$

One result of [4] is that the KMY algebra, $J_{l,n}(\delta)$, is _cellular_
with a similar cell basis as for $\operatorname{Br}_{n}$. This uses half
diagrams and bases for Specht modules for $K\mathfrak{S}_{r}$, where
$r\leqslant n$. Our labelling poset for the cell modules consists of pairs
$(r,\lambda)$ where $r$ is the number of propagating lines in the half diagram
basis for the cell module, and $\lambda$ is a partition of either $l+2$ or $r$
depending on which is smaller.

Half diagrams are formed from cutting the propagating lines.

We ignore crossings in the propagating lines:

Let $S^{\lambda}$ be a Specht module for $K\mathfrak{S}_{m}$, with $\lambda$ a
partition of $m$, $\lambda\vdash m$.

We form a basis for the cell module $\Delta_{l,n}(r,\lambda)$ by taking half
diagrams with $r$ propagating lines, height at most $l$,
$\lambda\vdash\min\\{r,l+2\\}$ and a (fixed) basis for $S^{\lambda}$.

NB: We are agnostic about what basis to take for the Specht module. We treat
it rather like a black box. In the applications in this paper, the Specht
modules will all be simple.

Let $v$ be in a basis for $S^{\lambda}$. Elements of the half diagram basis
look like:

We get an action of $J_{l,n}(\delta)$ like so:

where $s_{1}s_{2}=(1,2)(2,3)$ is a permutation. That is any crossings in the
propagating lines (which mean the written diagram is not in the basis) get
converted into a permutiation on $\min\\{r,l+2\\}$ lines which is then pushed
into the box representing the element of the Specht module. We then act with
this permutation in the Specht module, getting ultimately a linear combination
of basis elements.

An arc acts as zero: $\raisebox{-5.0pt}{\includegraphics{arczero.eps}}=0$.
(Kills off propagating lines.)

E.g. for $n=4$, $l=0$ we get

$\Delta_{0,4}(0,\emptyset)=\left\langle\raisebox{3.0pt}{\includegraphics{twoarcs.eps}},\raisebox{1.5pt}{\includegraphics{nestedarcs.eps}},\raisebox{3.0pt}{\includegraphics{crossarcs.eps}}\right\rangle$
$\Delta_{0,4}(2,\lambda)=\left\langle\raisebox{-5.0pt}{\includegraphics{042one.eps}},\raisebox{-5.0pt}{\includegraphics{042four.eps}},\raisebox{-5.0pt}{\includegraphics{042two.eps}},\raisebox{-5.0pt}{\includegraphics{042three.eps}}\right\rangle$

where $\lambda\vdash 2$ and $S^{\lambda}=\operatorname{Span}\\{v\\}$.

$\Delta_{0,4}(4,\lambda)=\left\langle\raisebox{-5.0pt}{\includegraphics{044one.eps}},\right\rangle$

where $\lambda\vdash 2$ and $S^{\lambda}=\operatorname{Span}\\{v\\}$.

We may use these bases to define Gram matrices and determinants as usual for
cellular algebras.

##  2\. KMY algebras are iterated inflation algebras

######  Theorem 2.1.

The algebra $J_{l,n}(\delta)$ is an iterated inflation of group algebras of
symmetric groups.

I.e., the algebra $J_{l,n}(\delta)$ has a filtration by ideals: n even:

$I_{0}\leqslant I_{2}\leqslant\cdots\leqslant I_{n-2}\leqslant
I_{n}=J_{l,n}(\delta).$

n odd:

$I_{1}\leqslant I_{3}\leqslant\cdots\leqslant I_{n-2}\leqslant
I_{n}=J_{l,n}(\delta).$

where

$I_{m}/I_{m-2}\cong V_{m}\otimes k\mathfrak{S}_{m_{l}}\otimes V_{m}^{\circ}$

as a $K$-module, $m_{l}=\min\\{l+2,m\\}$, $V_{m}$ is the free $K$-module with
basis the upper half diagrams on $m$ propagating lines, and $V_{m}^{\circ}$ is
the free $K$-module with basis the lower half diagrams on $m$ propagating
lines.

We may take $I_{m}=\langle d\in J_{l,n}(\delta)\mid\\#(d)\leqslant m\rangle$

The construction of the cell module basis essentially gives us this result and
it follows exactly as for the Brauer algebra.

##  3\. KMY algebras satisfy the CMPX axioms

For this section and much of the remainder of the paper we will set
$K=\mathbb{C}$, the complex numbers. The main sticking point for generalising
to other $K$ is axiom (A2). This could be generalised if we can ensure that
the symmetric group algebra $K\mathfrak{S}_{n}$ was semisimple for that $K$.
I.e. we have to assume that $\operatorname{ch}K>l$, and this is indeed
possible, in a way that it’s not for the Brauer algebra, where we would need
$\operatorname{ch}K>n$ with $n$ increasing up the chain. We leave the details
to the interested reader.

In 2006, Cox, Martin, Parker and Xi [1], introduce an axiomatic framework for
studying the representation theory of tower of algebras. Each family of
algebras that satisfy these axioms is called a _tower of recollement_.

###  3.1. Axioms for towers of recollement

Let $e$ be an idempotent of an algebra $A.$

######  Theorem 3.1 ([2]).

Let $\\{L(\lambda)\mid\lambda\in\Lambda\\}$ be a full set of simple
$A$-modules, and set $\Lambda^{e}=\\{\lambda\in\Lambda\mid eL(\lambda)\neq
0\\}.$ Then $\\{eL(\lambda)\mid\lambda\in\Lambda^{e}\\}$ is a full set of
simple $eAe$-modules. Further, the simple modules $L(\lambda)$ with
$\lambda\in\Lambda\setminus\Lambda^{e}$ are a full set of simple
$A/AeA$-modules.

Let $A_{n}$ with $n\geq 0$ be a family of finite-dimensional algebras with
idempotents $e_{n}$ in $A_{n}$.

######  Definition 3.2 ([1, Section 1]).

Let $\Lambda_{n}$ be the indexing set for the simple $A_{n}$-modules and
$\Lambda^{n}$ be the indexing set for the simple
$A_{n}/A_{n}e_{n}A_{n}$-modules.

######  Definition 3.3 ([1, Section 1]).

For $m,n\in\mathbb{N}$ with $m-n$ even we set $\Lambda_{m}^{n}=\Lambda^{n}$
regarded as a subset of $\Lambda_{m}$ if $m\geq n$, and
$\Lambda_{m}^{n}=\emptyset$ otherwise.

######  Definition 3.4 ([1, Section 1]).

A $\Delta_{n}$-filtration is defined to be a filtration of a module such that
successive quotients are isomorphic to $\Delta_{n}(\mu)$ for some

######  Definition 3.5 ([1, Section 1]).

If a module $M$ in $A_{n}$-$\operatorname{mod}$ has a $\Delta_{n}$-filtration
we define the support of $M$, denoted $\operatorname{supp}_{n}(M)$, to be the
set of labels $\lambda$ for which $\Delta_{n}(\lambda)$ occurs in this
filtration.

Now we present the axioms of a tower of recollement as in [1].

  1. (A1)

For each $n\geq 2$ we have an isomorphism

$\Phi_{n}:A_{n-2}\longrightarrow e_{n}A_{n}e_{n}.$

We have

$\Lambda_{n}=\Lambda^{n}\sqcup\Lambda_{n-2}.$

Set $e_{n,0}=1$ in $A_{n}$ and $1\leq i\leq\frac{n}{2}$ define new idempotents
in $A_{n}$ by setting $e_{n,i}=\Phi_{n}(e_{n-2,i-1}).$

  1. (A2)

For each $n\geq 0$ the algebra $A_{n}$ is quasi-hereditary with heredity chain
of the form

$0\subset\cdots\subset A_{n}e_{n,i}A_{n}\subset\cdots\subset
A_{n}e_{n,0}A_{n}=A_{n}.$

As $A_{n}$ is quasi-hereditary, there is for each $\lambda\in\Lambda_{n}$ a
standard module $\Delta_{n}(\lambda)$ with simple head $L_{n}(\lambda)$. If we
set $\lambda\leq\mu$ if either $\lambda=\mu$ or $\lambda\in\Lambda_{n}^{r}$
and $\mu\in\Lambda_{n}^{s}$ with $r>s$, then all other composition factors of
$\Delta_{n}(\lambda)$ are labelled by weights $\mu$ with $\mu<\lambda$. For
$\lambda\in\Lambda_{n}^{n}$, we have that $\Delta_{n}(\lambda)\cong
L_{n}(\lambda),$ and that this is just the lift of a simple module for the
quotient algebra $A_{n}/A_{n}e_{n}A_{n}$.

We recall the localisation functor, $\mathcal{F}_{n}$,

$\displaystyle\mathcal{F}_{n}:A\text{-}\operatorname{mod}$
$\displaystyle\rightarrow eAe\text{-}\operatorname{mod}$ $\displaystyle M$
$\displaystyle\rightarrow eM$

and the globalisation functor, $\mathcal{G}_{n}$,

$\displaystyle\mathcal{G}_{n}:eAe\text{-}\operatorname{mod}$
$\displaystyle\rightarrow A\text{-}\operatorname{mod}$ $\displaystyle N$
$\displaystyle\rightarrow Ae\otimes_{eAe}N.$

From [6, Proposition 3] we have

$\mathcal{G}_{n}(\Delta_{n}(\lambda))\cong\Delta_{n+2}(\lambda).$
$\mathcal{F}_{n}(\Delta_{n}(\lambda))\cong\begin{cases}\Delta_{n-2}(\lambda),&\mbox{if
}\lambda\in\Lambda_{n-2}\\\ 0,&\mbox{if }\lambda\in\Lambda^{n}.\end{cases}$

  1. (A3)

For each $n\geq 0$ the algebra $A_{n}$ can be identified with a subalgebra of
$A_{n+1}$.

  2. (A4)

For all $n\geq 1$ we have $A_{n}e_{n}\cong A_{n-1}$ as a left $A_{n-1}$-,
right $A_{n-2}$-bimodule. where we view $A_{n}e_{n}$ as a left $A_{n-1}$
module via the embedding in (A3), and as a right $A_{n-2}$ via the isomorphism
from (A1) and we view $A_{n-1}$ as a right $A_{n-2}$-module via the embedding
from (A3).

  3. (A5)

For each $\lambda\in\Lambda_{n}^{m}$ we have that the restriction from $A_{n}$
to $A_{n-1}$ (via the embedding in (A3)), $\Delta_{n}(\lambda)\downarrow$, has
a $\Delta$-filtration and

$\operatorname{supp}(\Delta_{n}(\lambda)\downarrow)\subseteq\Lambda_{n-1}^{m-1}\sqcup\Lambda_{n-1}^{m+1}.$

  4. (A6)

For each $\lambda\in\Lambda_{n}^{n}$ there exist $\mu\in\Lambda_{n+1}^{n-1}$
such that $\lambda\in\operatorname{supp}(\Delta_{n+1}(\mu)\downarrow).$

If axioms (A1) to (A6) are satisfied then we have the following theorem for
finding homomorphisms between two cell modules and a criterion for semi-
simplicity.

######  Theorem 3.6 ([1, theorem 1.1]).

Let $\lambda\in\Lambda_{n}^{m}$ and $\mu\in\Lambda_{n}^{t}$.

  1. (i)

For all pairs of weights $\lambda\in\Lambda_{n}^{m}$ and
$\mu\in\Lambda_{n}^{t}$ we have

$\operatorname{Hom}(\Delta_{n}(\lambda),\Delta_{n}(\mu))\cong\begin{cases}\operatorname{Hom}(\Delta_{m}(\lambda),\Delta_{m}(\mu)),&\mbox{if
}t\leq m\\\ 0,&\mbox{otherwise}.\end{cases}$

  2. (ii)

Suppose that for all $n\geq 0$ and pairs of weights
$\lambda\in\Lambda_{n}^{n}$ and $\mu\in\Lambda_{n}^{n-2}$ we have

$\operatorname{Hom}\bigl{(}\Delta_{n}(\lambda),\Delta_{n}(\mu)\bigr{)}=0.$

Then each of the algebras $A_{n}$ is semisimple.

###  3.2. Algebra $J_{l,n}(\delta)$ satisfies the axioms (A1) to (A4)

In the following we can see that $J_{l,n}(\delta)$ satisfies the axioms (A1)
to (A4). Firstly we define the idempotent $e_{n}$ in algebra
$J_{l,n}(\delta).$

######  Definition 3.7.

For $n\geq 2$ and $\delta\neq 0,$ we define the idempotent element $e_{n}\in
J_{l,n}(\delta)$ to be the diagram

$e_{n}={\frac{1}{\delta}}(1_{n-2}\otimes U)={\frac{1}{\delta}}$$\cdots$$n-2$

where there are $n-2$ propagating lines in the diagram for $e_{n}$,
$U=\leavevmode\hbox to14.63pt{\vbox
to23.16pt{\pgfpicture\makeatletter\hbox{\hskip-99.38466pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{22.76228pt}\pgfsys@curveto{99.58466pt}{11.38092pt}{113.81104pt}{11.38092pt}{113.81104pt}{22.76228pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{99.58466pt}{0.0pt}\pgfsys@curveto{99.58466pt}{11.38092pt}{113.81104pt}{11.38092pt}{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ and $\otimes$
denotes side-by-side concatenation.

Now we prove that the algebra $J_{l,n}(\delta)$ satisfies all the axioms in
the CMPX framework when our base field is the complex numbers and $\delta\neq
0$.

######  Proposition 3.8 ([4, Proposition 4.1]).

For $n>2$, we have

$J_{l,n-2}\cong e_{n}J_{l,n}e_{n}$

so $J_{l,n}(\delta)$ satisfies axiom (A1).

For $n\geq 2t$, where $t\in\mathbb{N}$, we define $e_{n,t}=1_{n-2t}\otimes
U^{\otimes t}$ and when $\delta\in\mathbb{C}^{\times}$ we define
$e_{n,t}^{\prime}=\delta^{-t}e_{n,t}.$ We can see that $e_{n,t}\in
J_{-1}(n,n-2t,n),$ i.e. $e_{n,t}$ has no crossings and is a Temperley-Lieb
diagram.

######  Theorem 3.9 ([4, Theorem 4.12]).

If $\delta$ invertible in $\mathbb{C}$ then $J_{l,n}$ is quasi-hereditary,
with heredity chain given by $(1,e_{n,1}^{\prime},e_{n,2}^{\prime},\ldots)$.
So $J_{l,n}(\delta)$ satisfies axiom (A2).

######  Theorem 3.10.

For all $n\geq 1$, the algebra $J_{l,n}(\delta)$ is a subalgebra of
$J_{l,n+1}(\delta)$. So $J_{l,n}(\delta)$ satisfies axiom (A3).

###### Proof.

We define an inclusion map
$i_{n}:J_{l,n}(\delta)\xhookrightarrow{}J_{l,n+1}(\delta)$ by adding a
vertical line to the right of the diagram $d\in J_{l,n}(\delta)$ and then
extended by linearity to all of $J_{l,n}(\delta)$. This inclusion sends the
identity diagram in $J_{l,n}(\delta)$ to the identity in $J_{l,n+1}(\delta)$.

Assume that $d_{1}$, $d_{2}\in J_{l,n}(\delta)$, we have

$i(d_{1}d_{2})=i(d_{1})i(d_{2}).$

This means if we multiply two diagrams $d_{1}$ and $d_{2}$, then add a
vertical line on the right of the multiplication $d_{1}d_{2}$ is the same when
we add a vertical line on the right of $d_{1}$ and $d_{2}$ then multiply them.
Therefore the inclusion map $i$ is a homomorphism. Since different diagrams in
$J_{l,n}(\delta)$ have different images in $J_{l,n+1}(\delta)$, the map $i$ is
injective. ∎

######  Proposition 3.11 ([4, Lemma 6.2]).

For all $n\geq 1$ we have $J_{l,n}e_{n}\cong J_{l,n-1}$ as a left
$J_{l.n-1}$-, right $J_{l,n-2}$-bimodule. So $J_{l,n}(\delta)$ satisfies axiom
(A4).

###  3.3. The localisation and globalisation functors

We use the previously definied idempotent $e_{n}$ to define functors. The
_localisation_ functor is given by

$\displaystyle\mathcal{F}_{n}:J_{l,n}\text{-}\operatorname{mod}$
$\displaystyle\rightarrow J_{l,n-2}\text{-}\operatorname{mod}$ $\displaystyle
M$ $\displaystyle\mapsto e_{n}M.$

The _globalisation_ functor is given by

$\displaystyle\mathcal{G}_{n}:J_{l,n}\text{-}\operatorname{mod}$
$\displaystyle\rightarrow J_{l,n+2}\text{-}\operatorname{mod}$ $\displaystyle
M$ $\displaystyle\mapsto J_{l,n}e_{n}\otimes_{e_{n}J_{l,n}e_{n}}M.$

###  3.4. Restriction of the cell module $\Delta_{l,n}(p,\lambda)$

The restriction of $\Delta_{l,n}(p,\lambda)$ to $J_{l,n-1}(\delta)$ via the
embedding $i_{n-1}$ in (A3) is denoted by $\Delta_{l,n}(p,\lambda)\downarrow$.
In the following we present restriction rules for the cell modules in the
algebra $J_{l,n}(\delta)$.

######  Theorem 3.12 ([4, Proposition 5.3]).

Consider $i_{n-1}:J_{l,n-1}(\delta)\hookrightarrow J_{l,n}(\delta)$, denote
the corresponding restriction of $J_{l,n}(\delta)$ to a $J_{l,n-1}(\delta)$ by
$\Delta_{l,n}(p,\lambda)\downarrow$. Then we have for fixed $l$ the following
exact sequences for the restriction

  1. (i)

For $p<l+2$ we have a short exact sequence

$0\rightarrow\bigoplus_{i}\Delta_{l,n-1}(p-1,\lambda-
e_{i})\rightarrow\Delta_{l,n}(p,\lambda)\downarrow\rightarrow\bigoplus_{i}\Delta_{l,n-1}{(p+1,\lambda+e_{i})}\rightarrow
0$

where the sums are over addable/removable boxes of the Young diagram and
$\lambda-e_{i}$ denotes $\lambda$ with the $i$-th removable box removed and so
on.

  2. (ii)

For $p=l+2$

$0\rightarrow\bigoplus_{i}\Delta_{l,n-1}(p-1,\lambda-
e_{i})\rightarrow\Delta_{l,n}(p,\lambda)\downarrow\rightarrow\Delta_{l,n-1}{(p+1,\lambda)}\rightarrow
0$

where the sums are over removable boxes of the Young diagram and $\lambda-
e_{i}$ denotes $\lambda$ with the $i$-th removable box removed and so on.

  3. (iii)

For $p>l+2$

$0\rightarrow\Delta_{l,n-1}(p-1,\lambda)\rightarrow\Delta_{l,n}(p,\lambda)\downarrow\rightarrow\Delta_{l,n-1}{(p+1,\lambda)}\rightarrow
0$

###  3.5. Proof of the axioms (A5) and (A6)

######  Proposition 3.13.

For each $(p,\lambda)\in\Lambda_{n,p_{l}}^{p}$ we have that
$\Delta_{l,n}(p,\lambda)\downarrow$ has a $\Delta$-filtration and

$\operatorname{supp}(\Delta_{l,n}(p,\lambda)\downarrow)\subseteq\Lambda_{{n-1,p_{l}}}^{p-1}\sqcup\Lambda_{{n-1,p_{l}}}^{p+1}.$

So the algebra $J_{l,n}(\delta)$ satisfies axiom (A5).

###### Proof.

Let $(p,\lambda)\in\Lambda_{n,p_{l}}^{p}.$ By applying Theorem 3.12 we have
Case 1: If $p<l+2,$ we have

$\displaystyle\begin{split}\operatorname{supp}(\Delta_{l,n}(p,\lambda)\downarrow)&=\bigcup_{i}\\{(p-1,\lambda-
e_{i})\\}\cup\bigcup_{i}\\{(p+1,\lambda+e_{i})\\}\\\
&\subseteq\Lambda_{{n-1,p_{l}}}^{p-1}\sqcup\Lambda_{{n-1,p_{l}}}^{p+1}\end{split}$

Case 2: If $p=l+2,$ we have

$\displaystyle\begin{split}\operatorname{supp}(\Delta_{l,n}(p,\lambda)\downarrow)&=\bigcup_{i}\\{(p-1,\lambda-
e_{i})\\}\cup\\{(p+1,\lambda)\\}\\\
&\subseteq\Lambda_{{n-1,p_{l}}}^{p-1}\sqcup\Lambda_{{n-1,p_{l}}}^{p+1}\end{split}$

Case 3: If $p>l+2,$ we have

$\displaystyle\begin{split}\operatorname{supp}(\Delta_{l,n}(p,\lambda)\downarrow)&=\\{(p-1,\lambda)\\}\cup\\{(p+1,\lambda)\\}\\\
&\subseteq\Lambda_{{n-1,p_{l}}}^{p-1}\sqcup\Lambda_{{n-1,p_{l}}}^{p+1}\qed\end{split}$

######  Proposition 3.14.

For each $(p,\lambda)\in\Lambda_{n,p_{l}}^{p}$ there exists
$(p^{\prime},\mu)\in\Lambda_{n+1,p_{l}^{\prime}}^{p^{\prime}-1}$ such that

$(p,\lambda)\in\operatorname{supp}(\Delta_{l,n+1}(p^{\prime},\mu)\downarrow).$

I.e., the algebra $J_{l,n}(\delta)$ satisfies axiom (A6).

###### Proof.

Let $(p,\lambda)\in\Lambda_{n}^{p}$ and let $p^{\prime}>p$, we take
$p^{\prime}=p+1.$ We have the following cases:
Case 1: If $p>l+2,$ we have $p^{\prime}=p+1>l+2.$ So $\lambda\vdash l+2$ and
$\mu\vdash l+2.$
We have $\lambda=\mu$ and $p=p^{\prime}-1.$

$(p,\lambda)=(p^{\prime}-1,\mu).$

Then we have
$(p,\lambda)\in\operatorname{supp}(\Delta_{l,n+1}(p^{\prime},\mu)\downarrow).$

Case 2: If $p=l+2,$ we have $\lambda\vdash p=l+2$ and $\mu\vdash l+2=p.$ So

$(p,\lambda)=(p^{\prime}-1,\mu).$

Therefore we have
$(p,\lambda)\in\operatorname{supp}(\Delta_{l,n+1}(p^{\prime},\mu)\downarrow).$

Case 3: If $p<l+2,$ we have $\lambda\vdash p.$ $\lambda$ has an addable node
$e_{i}$, either $\mu\vdash p^{\prime},p^{\prime}<l+2$ or $\mu\vdash
p^{\prime}=l+2.$
$\mu$ has a removable node, let $\mu=\lambda+e_{i}.$
We have $(p,\lambda)=(p^{\prime}-1,\mu-e_{i}).$ Then
$(p,\lambda)\in\operatorname{supp}(\Delta_{l,n+1}(p^{\prime},\mu)\downarrow).$
∎

The algebra $J_{l,n}(\delta)$ satisfies axioms (A1) to (A6). We restate
Theorem 1.1 (theorem 3.6 in this paper) from [1] for the context of our
algebra $J_{l,n}(\delta)$. It is this theorem that will allow us to prove
Theorem 5.3.

######  Theorem 3.15.

  1. (i)

For $l=\\{-1,\ldots,n-2\\}$ and all pairs of weight
$(p,\lambda)\in\Lambda_{n,p_{l}}^{p}$ and
$(p^{\prime},\mu)\in\Lambda_{n,p^{\prime}_{l}}^{p^{\prime}}$ we have

$\operatorname{Hom}(\Delta_{l,n}(p,\lambda),\Delta_{l,n}(p^{\prime},\mu))\cong\begin{cases}\operatorname{Hom}(\Delta_{l,p}(p,\lambda),\Delta_{l,p}(p^{\prime},\mu)),&\mbox{if
}p^{\prime}\leq p\\\ 0,&\mbox{otherwise}.\end{cases}$

  2. (ii)

Suppose that for all $n\geq 0$ and pairs of weights
$(n,\lambda)\in\Lambda_{n,p_{l}}^{n}$ and
$(n-2,\mu)\in\Lambda_{n,p_{l}}^{n-2}$ we have

$\operatorname{Hom}(\Delta_{l,n}(n,\lambda),\Delta_{l,n}(n-2,\mu))=0.$

Then each of the algebras $J_{l,n}(\delta)$ is semisimple.

##  4\. A generating set for the algebra $J_{l,n}(\delta)$

In this section we give a generating set for the algebra $J_{l,n}(\delta)$. We
work over the ring (field) of coefficients $\mathbb{C}$ with $\delta$ as a
parameter. However, in this section the ring of coefficients does not play a
strong role, and the result holds more generally.

The first type of generator $u_{i}$, $1\leq i\leq n-1$ are all TemperleyLieb
diagrams (height $-1$) and generate the Temperley-Lieb algebra as a subalgebra
of $J_{l,n}(\delta)$. The second type are the crossings $s_{m}$, where $1\leq
m\leq l+1$, and generate the symmetric group algebra
$\mathbb{C}\mathfrak{S}_{m}$ as a subalgebra of $J_{l,n}(\delta)$.

$\displaystyle u_{1}=\raisebox{-0.4pt}{ \leavevmode\hbox to71.53pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-6.9132pt\lower-0.2pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
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}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{78.24509pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{78.24509pt}{28.45276pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{14.22638pt}{28.45276pt}\pgfsys@curveto{14.22638pt}{17.07182pt}{28.45276pt}{17.07182pt}{28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@curveto{14.22638pt}{11.38092pt}{28.45276pt}{11.38092pt}{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{71.1319pt}{28.45276pt}\pgfsys@lineto{71.1319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,u_{2}=\raisebox{-0.4pt}{
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{}{}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{92.47147pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{92.47147pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{14.22638pt}{28.45276pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@curveto{28.45276pt}{11.38092pt}{42.67914pt}{11.38092pt}{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{85.35828pt}{28.45276pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,\ldots,u_{n-1}=\raisebox{-0.5pt}{
\leavevmode\hbox to78.65pt{\vbox
to48.15pt{\pgfpicture\makeatletter\hbox{\hskip-6.9132pt\lower-11.91235pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{14.22638pt}{0.0pt}\pgfsys@lineto{14.22638pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{{}}{}{{}}{}{{}}{}{}{}{{}}{{}}{}\pgfsys@moveto{56.90552pt}{28.45276pt}\pgfsys@curveto{56.90552pt}{17.07182pt}{78.24509pt}{17.07182pt}{78.24509pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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s_{1}=\raisebox{-0.4pt}{ \leavevmode\hbox to71.53pt{\vbox
to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-6.9132pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{78.24509pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{78.24509pt}{28.45276pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{14.22638pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{71.1319pt}{28.45276pt}\pgfsys@lineto{71.1319pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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to28.85pt{\pgfpicture\makeatletter\hbox{\hskip-6.9132pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{7.11319pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{28.45276pt}\pgfsys@lineto{92.47147pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{92.47147pt}{28.45276pt}\pgfsys@lineto{92.47147pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{92.47147pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{14.22638pt}{28.45276pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{28.45276pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{28.45276pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{56.90552pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{85.35828pt}{28.45276pt}\pgfsys@lineto{85.35828pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,\ldots,s_{l+1}=\raisebox{-0.5pt}{
\leavevmode\hbox to121.32pt{\vbox
to49.91pt{\pgfpicture\makeatletter\hbox{\hskip-6.9132pt\lower-11.91235pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
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} {}{{}}{}
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} {}{{}}{}
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} {}{{}}{}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{42.67914pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{{}}{{}}{}\pgfsys@moveto{56.90552pt}{28.45276pt}\pgfsys@lineto{78.24509pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\tiny$l+1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\tiny$(l+2)^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{{}}{{}}{}\pgfsys@moveto{78.24509pt}{28.45276pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\tiny$(l+1)^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{73.17123pt}{31.19272pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{92.47147pt}{28.45276pt}\pgfsys@lineto{92.47147pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{102.94785pt}{11.72638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{120.92422pt}{28.45276pt}\pgfsys@lineto{120.92422pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$ Figure 1. The
generators for $J_{l,n}(\delta)$

We introduce some terminology for the diagrams and also a “standard” picture
for a diagram $d$ with $m$ propagating lines.

######  Definition 4.1.

We draw a diagram with the least number of total crossings. An arc is
_covering_ a crossing if the crossing is contained by the arc. I.e. when we
draw the diagram, the end points of the arc are either side of the crossing
and the crossing is above (respectively below) the arc if it its a top
(bottom) arc respectively. We will also say an arc _covers_ another arc if the
end points of the second arc lie between the end points of the first arc.

By “turning point” in the following we mean that when the line is drawn on the
plane, a turning point is a point on that line where is there a horizontal
tangent to that line.

We “standardise” the drawing for the diagram $d$ with $m$ propagating lines
and height $l$ as follows:

Step 1: draw all arcs as “simpler” arcs, i.e. the lines only have one
stationary point.

Step 2: draw all top arcs so they do not intersect any bottom arcs.

Step 3: If an arc covers another arc, then we draw $d$ so that these arcs do
not intersect.

Step 4: pull propagating lines “taut”, i.e. All propagating lines now have no
turning points in them.

Step 5: pull any crossings in propagating lines to the middle of the diagram
so that they are not covered by any arcs.

The diagram $d$ then has a top half-diagram, a permutation on the $m$
propagating lines ($\pi\in\mathfrak{S}_{m}$) in the middle and a bottom half-
diagram.

Step 2 reduces crossings in the arcs, and leaves crossings with propagating
lines the same, so this does not produce any new crossings. This also leaves a
region in the middle of the diagram that touches both sides of the diagram and
contains no points from arcs, although it may contain propagating lines (if
$m>0$). (If $m=0$ then this region is empty.)

In Step 5, the crossing number of this crossing either stays the same or goes
down with this procedure. The total number of crossings in the propagating
lines stays the same.

This procedure either reduces the number of crossings of left-height $l$ in
the picture for the diagram $d$ or leaves it the same, but it cannot increase
it. In particular, it does not create a crossing with a higher height than
$l$.

NB: if standardisation removes all crossings of height $l$ in a diagram, then
that diagram did not have height $l$ to start with, as we now have a picture
for $d$ where all crossings are of height smaller than $l$.

######  Theorem 4.2.

The algebra $J_{l,n}(\delta)$ with left height $l$ is generated by the
diagrams $u_{i}$, $1\leq i\leq n-1$ and $s_{m}$, $1\leq m\leq l+1$ with
$l=\\{0,1,2,\ldots,n-1\\}$, as in Figure 1.

###### Proof.

We want to prove that the algebra $J_{l,n}(\delta)$ is generated by the set
$A$

$A:=\\{u_{i}\\}_{i=1}^{i=n-1}\cup\\{s_{m}\\}_{m=1}^{m=l+1}.$

It is clear that the set of diagrams which is generated by $A$ is contained in
$J_{l,n}(\delta)$ since the concatenation of these diagrams can never increase
the left height $l$. So we need to prove that every element in
$J_{l,n}(\delta)$ can be written as a product of elements belonging to $A$.
Now the strategy of the proof is as follows:

  * •

Use the induction on the left-height $l$ of any diagram $d\in J_{l,n}(\delta)$
and on the number of features $r$ of left-height $l$

  * •

“standardise” the drawing for the diagram $d$

  * •

Divide the diagram $d$ into $d_{1}$, $s_{l+1}$ and $d_{2}$ such that $d_{1}$
and $d_{2}$ have less than $r$ features of left-height $l$

$d=$$\leadsto$

$d_{1}$$d_{2}$$s_{\tiny{l+1}}$

If $l=-1$, then the result follows from that for the Temperley-Lieb algebra
$\mathrm{TL}_{n}(\delta)$[5, Section 6.1 and 6.3].

Now suppose $d$ is any diagram in $J_{l,n}(\delta)$ with fixed left height
$l\in\\{0,1,2,\ldots,n-2\\}$. Suppose $d$ is a diagram with $r$ features of
left height $l$ and $m$ propagating lines. First, we “standardise” drawing for
the diagram $d$ as above. Let $w_{1}$ be the top half-diagram with $c_{1}$
features of left height $l$, let $w_{2}$ be the bottom half-diagram with
$c_{2}$ features of left height $l$ and $\pi$ is the middle of diagram $d$
with left height $l$. This standardisation either reduces the number of
crossings of left-height $l$ in the picture for the diagram $d$ or leaves it
the same, but it cannot increase it. In particular, it does not create a
crossing with a higher height than $l$.

$w_{1}$$w_{2}$$\pi$

If $r=0$, then we are done by induction. For any $r\neq 0$, we have the
following cases:
(1) There is a feature of height $l$ in $\pi$. I.e. a crossing of two
propagating lines.
(2) There is no crossing of height $l$ in $\pi$, but the crossing of left
height $l$ is in $w_{1}$ or $w_{2}$.

$l$$k$$l$$k$$\cdots$$\cdots$$\pi_{1}$$w_{2}$$w_{1}$$\pi_{2}$$\pi_{3}$ (a)

$l$$k$$l$$k$$\cdots$$\cdots$$w_{2}$$w_{1}$$\tilde{\pi_{1}}$$\tilde{\pi_{2}}$$\tilde{\pi_{3}}$$b_{1}\text{``wiggles''}$
(b)

Figure 2. Case 1 the feature of height $l$ in $\pi$

Case 1: There is a feature of height $l$ in $\pi.$ I.e. a crossing of two
propagating lines.
We pull $\pi$ apart in the following way: Firstly we may draw $\pi$ so that no
two crossings lie on the same (invisible) horizontal line. Then consider the
crossing of height $l.$ We may take a small horizontal strip containing this
crossing and no other crossing. Then we draw two dotted lines above and below
the crossing of height $l$ giving this strip. This splits $\pi$ into a product
of three permutations $\pi=\pi_{1}\pi_{2}\pi_{3}\in S_{m}.$ We have $\pi_{2}$
is the permutation $(l+1,l+2)$ in $S_{m}$ as it is of height $l.$ Since we did
not create any new crossings, we have $w_{1}\pi_{1}$ has one less crossing
than $w_{1}\pi_{1}\pi_{2}.$ Also $\pi_{3}w_{2}$ has one less crossing than
$\pi_{2}\pi_{3}w_{2}$ and the total number of crossing in $w_{1}\pi_{1}$ and
$\pi_{3}w_{2}$ is less than $r.$

Now to make $\pi_{2}$ into $s_{l+1}$ between the two dotted lines, we need to
get the right numbers of vertices and lines on the two dotted lines. We need
$n=l+k+2+2b_{1},$ where $b_{1}$ is the number of wiggles in Figure 2(b). So we
need $n$ to be the same parity as $m$ which is true. The middle region
$\tilde{\pi_{2}}$ that is shown in Figure 2(b) is then $s_{l+1}.$ Then $d_{1}$
is essentially $w_{1}\tilde{\pi_{1}}$ and has the same number of features of
height $l$ as in $w_{1}\pi_{1}$, $d_{2}$ is $\pi_{3}\tilde{w_{2}}$ and has the
same number of features of height $l$ as in $\pi_{3}w_{2}.$ Thus
$d_{1}\tilde{\pi_{2}}d_{2}$ gives us required decomposition for the induction.

$\vdots$$\cdots$$C$$B$$A$$l$$\pi$$w_{2}$$w_{1}$ (a)

$\vdots$$h_{1}$$h_{2}$$h_{3}$$\cdots$$b_{2}\text{``wiggles''}$$C$$l$$B$$A$$\pi$$w_{2}$$m-k-1$$x_{1}$
(b)

Figure 3. Case 2a the feature of height $l$ in $w_{1}$ is the intersection an
arc and a propagating line

Case 2: There is no crossing in $\pi,$ but the crossing of left height $l$ is
in $w_{1}$ or $w_{2}.$

Case 2a: The feature of left height $l$ is the intersection of an arc and a
propagating line.
We assume without loss of generality that the arc is in the top half-diagram,
$w_{1}$, the bottom case is similar. The region labelled $A$ in the Figure
3(a) can have arcs or propagating lines with crossing because the crossings in
this region have left height less than or equal $l.$ The regions labelled $B$
in Figure 3(a) have only arcs without any crossing because any crossing this
would create a crossing with left height $l+1.$ The region labelled $C$ in
Figure 3(a) could have crossings of propagating lines or arcs such that the
left height less than or equal $l.$

We push all covering arcs down without creating new crossings. Then we pull
the crossing of left height $l$ down and draw two dotted horizontal lines
above and below this crossing to get a single crossing, it has left height $l$
so it looks like $s_{l+1}.$ Let $k\geq 0$ be the number of propagating lines
on the right of the crossing of left height $l$ and suppose that $j$ is the
number of covering arcs.

We need to get the right number of vertices and lines on the two dotted lines.
There are $(m-k-1)+3+2j+2b_{2}+k$ vertices on each of the dotted lines as
shown in Figure 3(b), where $b_{2}$ is the number of wiggles. We have
$m+2j+2+2b_{2}$ has the same parity of $n.$ Now we need to add
$2b_{2}=n-(m+2j+2)$ vertices to each of the dotted lines. We wiggle the arc at
position $l+1$ for $b_{2}$ times to get the right numbers of vertices and
vertical lines on each of the dotted lines. The middle region $h_{2}$ that is
shown in Figure 3(b) looks like $s_{l+1}.$ The diagram $h_{2}$ has one feature
of left height $l.$ We can decompose the diagram $w_{1}$ into a product of
three smaller diagrams $w_{1}=h_{1}h_{2}h_{3}$ with $c_{1}$ features of left-
height $l.$ The top diagram $h_{1}$ has strictly less features of height $l$
than $w_{1}.$ Any crossing in region labelled $C$ stay in $C$ and these
crossings are in $h_{1}$. The bottom diagram $h_{3}$ has all covering arcs and
any crossings of propagating lines with these covering arcs have left height
at most $l,$ the number of features of height $l$ in $h_{3}$ is the same as
$\pi w_{2}<r.$ The total number of crossings of left height $l$ in $h_{1}$ and
$h_{3}$ is less than $r.$ Then $d_{1}$ is essentially $h_{1}$ has number of
features of left-height $l$ less than $r,$ $d_{2}=h_{3}\pi w_{2}$ and has the
same number of features in $w_{2}$ that is less than $r$ and the diagram
$s_{l+1}=h_{2}$ has one feature of height $l.$ Thus $d_{1}s_{l+1}d_{2}$ gives
us required decomposition for the induction.

$A$$D$$D$$B$$C$$\vdots$$\cdots$$\pi$$w_{2}$ (a)

$A$$\vdots$$D$$B$$C$$h_{1}$$h_{2}$$h_{3}$$\cdots$$b_{2}\text{``wiggles''}$$\pi$$w_{2}$$m-k$
(b)

Figure 4. Case 2b the feature of height $l$ in $w_{1}$ is the intersection of
two arcs

Case 2b: The feature of left height $l$ in $w_{1}$ is the intersection of two
arcs.
We assume without loss of generality that the arc is in the top half-diagram,
$w_{1}$, the bottom case is similar. We can deform the arcs to look like
Figure 4(b) by pulling the two arcs down (into the middle between two dotted
lines) without creating new crossings. We may assume the two arcs themselves
only cross each other once. The region labelled $A$ in the Figure 4(a) can
have arcs or propagating lines because they have left height less than or
equal $l.$ The region labelled $B$ has only arcs without any crossings because
any crossing this would create a crossing with left height $l+1.$ The region
labelled $C$ could have arcs or propagating lines with crossings such that
these crossings in the alcove of left height at most $l.$ The region labelled
$D$ may contain arcs or propagating lines. Also the region $D$ could have
crossings of height at most $l.$ Let $k$ be the number of propagating lines on
the right of crossing of height $l.$

Now we take a horizonal strip as in case 1 this gives two horizonal dotted
lines above and below the crossing it has left height $l$ so it looks like
$s_{l+1}.$ We push all the covering arcs down under the dotted lines.

Now to create $h_{2}$ (see figure 5) between the two dotted lines, we need to
get the right number of vertices and vertical lines on two dotted lines. Let
$j$ be the number of covering arcs. There are $m-k$ propagating lines on the
left of the crossing. There are $(m-k)+k+2j+2b_{3}+4$ vertices on each of the
dotted lines as shown in Figure 4(b), where $b_{3}$ is the number of wiggles.
We have $m+2j+2b_{3}+4$ has the same parity of $n.$ Now we need to add
$2b_{3}=n-(m+2j+4)$ vertices to each of the dotted lines. We wiggle the arc at
position $l+1$ for $b_{3}$ times to get the right numbers of vertices and
vertical lines on each of the dotted lines. The middle region $h_{2}$ that is
shown in Figure 4(b) looks like $s_{l+1}.$

We have $h_{2}$ has one feature of left height $l.$ Any crossing in region
labelled $D$ stay in $h_{1}.$ Then $w_{1}$ decompose to three small diagrams
$w_{1}=h_{1}h_{2}h_{3}.$ The diagram $h_{1}$ has number of features of height
$l$ less than $w_{1},$ this include the crossing in region labelled $D.$ The
diagram $h_{3}$ has all crossing of propagating lines with covering arcs. This
bottom diagram has the same number of features of height as $\pi w_{2}<r.$
Then $d_{1}$ is essentially $h_{1}$ and the number of features of height
$l<r,$ $s_{l+1}=h_{2}$ and the diagram $d_{2}$ is $h_{3}\pi w_{2}$ and has
number of features of height $l$ as $\pi w_{2}<r.$ The total number of
features of height $l$ in $d_{1}$ and $d_{3}$ is less than $r.$ So we can
decompose the diagram $d$ into a product of three diagrams with
$d_{1}s_{l+1}d_{2}$ with the desired properties. For example,
The diagram $d$ has $2$ features of left-height $l=2.$ Now to make the diagram
$d$ in standardisation, we push all arcs down under the dotted lines and pull
the two arcs in the region $h_{2}$. Suppose $w_{1}=h_{1}h_{2}h_{3}$, then
$w_{1}$ has $2$ features of left-height equal $2.$ The region $h_{2}$ should
has one crossing of left-height $2$ and it likes $s_{3}.$ The diagram $h_{3}$
has the same left-height equal $1$ as in $\pi w_{2}$. The diagram $h_{1}$ has
$1$ feature of left-height $2.$ Then the diagram $d_{1}=h_{1}$, $h_{2}=s_{3}$
and $d_{2}=h_{3}\pi w_{2}$. We can decompose the diagram $d$ into product of
three diagrams.

$d=$$=$$h_{1}$$h_{2}$$h_{3}$$\pi$$w_{1}$ Figure 5. creating $h_{2}$ in $d$.

∎

######  Example 4.3.

Let $n=8$ and $l=3$. We can decompose the diagram $d$ below which has height
$3$, into a product of generators.

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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
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{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{128.03741pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{128.03741pt}{28.45276pt}\pgfsys@lineto{128.03741pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
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}
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}
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}
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}
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}
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{}{{}}{}{{}}{}{{}}{}{}{}{}\pgfsys@moveto{71.1319pt}{0.0pt}\pgfsys@curveto{71.1319pt}{8.5359pt}{113.81104pt}{8.5359pt}{113.81104pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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$\displaystyle=s_{3}s_{4}u_{5}s_{1}s_{2}u_{6}s_{4}u_{5}u_{7}s_{3}u_{6}s_{4}u_{5}s_{4}s_{3}$

##  5\. KMY algebras are semi-simple if $\delta$ not real

######  Theorem 5.1 ([3, Theorem 48]).

All eigenvalues of a real symmetric matrix are real.

Let $\\{v_{i}\\}$ be a basis for $S^{\lambda}$. Then we may choose the inner
product on $S^{\lambda}$ so that it has real values on the basis elements and
so that $\langle v_{i}\mid v_{j}\rangle\in\mathbb{R}$ for $i\neq j$. Indeed,
the Specht modules are defined over $\mathbb{Z}$ and thus we can choose to
have integral values for the inner product between basis elements.

######  Lemma 5.2.

If $\delta$ is not real then the determinant of Gram matrix for the cell
module $\Delta_{l,n}(n-2,\lambda)$, $\lambda\vdash n$ is not zero.

###### Proof.

Let $G_{l,n}(n-2,\lambda)$ be the Gram matrix of $\Delta_{l,n}(n-2,\lambda)$.
In computing the Gram matrix, we note that it is only possible to get loops
when the basis elements are multiplied by themselves. “Mixed” products have
either a real value (coming from the inner product on the Specht module) or
are zero.

A basis element of $\Delta_{l,n}(n-1,\lambda)$ has one of two typical types.
Either it has one arc that can cross any of the first l-1 propagating lines,
or is has an arc that does not cross any lines. Let $v$ and $v^{\prime}$ be
basis elements in the Specht module $S^{\lambda}$. Elements of the half
diagram basis for $\Delta_{l,n}(n-1,\lambda)$ look like:

or

A typical inner product then looks like:

or  or  or

and other variations. Thus we can write the matrix $G_{l,n}(n-2,\lambda)$ as
$M+\delta I_{n}$ where and $I_{n}$ is the unit matrix and $M$ is a real
matrix.

Now $\xi+\delta$ is an eigenvalue of $M+\delta I_{n}$ if and only if $\xi$ is
an eigenvalue of $M$. Since the matrix $M$ is real and symmetric the
eigenvalues of $M$ are real from Theorem 5.1. Thus if $\delta$ is complex,
$M+\delta I_{n}$ can have no real eigenvalues. So the determinant cannot be
zero as no eigenvalue is zero. ∎

######  Theorem 5.3.

For $n\geq 0$, if $\delta$ is not real then the algebra $J_{l,n}(\delta)$ is
semisimple.

###### Proof.

Take $(n,\lambda)\in\Lambda_{n,2}^{n}$ and $(n-2,\mu)\in\Lambda_{n,2}^{n-2}.$
The module $\Delta_{l,n}(n,\lambda)$ is isomorphic to the Specht module
$S^{\lambda}$, $\lambda\vdash l+2$ which is simple in characteristic zero. We
consider $\Delta_{l,n}(n-2,\mu),\mu\vdash\min\\{n,l+2\\}$. We have from Lemma
5.2 that $\det G_{l,n}(n-2,\mu)\neq 0$ if $\delta$ is not real. So
$\Delta_{l,n}(n-2,\mu)$ is simple. As $J_{l,n}(\delta)$ is quasi-hereditary
these two modules are not isomorphic. So we have

$\operatorname{Hom}\bigl{(}\Delta_{l,n}(n,\lambda),\Delta_{l,n}(n-2,\mu)\bigr{)}=0$

therefore by Theorem 3.15(ii) the algebra $J_{l,n}(\delta)$ is semisimple. ∎

## References

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