# Generalised Temperley-Lieb algebras of type $G(r,p,n)$

Gus Lehrer and Mengfan Lyu  School of Mathematics and Statistics, University
of Sydney, N.S.W. 2006, Australia
gustav.lehrer<EMAIL_ADDRESS>
###### Abstract.

In an earlier work, we defined a “generalised Temperley-Lieb algebra”
$TL_{r,1,n}$ corresponding to the imprimitive reflection group $G(r,1,n)$ as a
quotient of the cyclotomic Hecke algebra. In this work we introduce the
generalised Temperley-Lieb algebra $TL_{r,p,n}$ which corresponds to the
complex reflection group $G(r,p,n)$. Our definition identifies $TL_{r,p,n}$ as
the fixed-point subalgebra of $TL_{r,1,n}$ under a certain automorphism
$\sigma$. We prove the cellularity of $TL_{r,p,n}$ by proving that $\sigma$
induces a special shift automorphism with respect to the cellular structure of
$TL_{r,1,n}$. Finally, we give a description of the cell modules of
$TL_{r,p,n}$ and their decomposition numbers.

###### Key words and phrases:

Temperley-Lieb algebra, Hecke algebras, KLR algebras, cellular basis,
decomposition numbers

###### 2020 Mathematics Subject Classification:

Primary 16G20, 20C08; Secondary 20G42

###### Contents

  1. 1 Introduction
  2. 2 The cyclotomic Hecke algebra $H(r,p,n)$
  3. 3 The generalised Temperley-Lieb algebra $TL_{r,1,n}$
  4. 4 The definition of $TL_{r,p,n}$
  5. 5 3-dimensional multipartitions and the cellular structure of $TL_{r,1,n}$
  6. 6 Skew cellular algebras
  7. 7 The skew cellularity of $TL_{r,p,n}$
  8. 8 The representations of $TL_{r,p,n}$
    1. 8.1 The cell modules of $TL_{r,p,n}$ and irreducible cells
    2. 8.2 The reducible cells and their decomposition numbers

##  1\. Introduction

The Temperley-Lieb algebra was initially introduced in [13] in connection with
transition matrices in statistical mechanics. Since then, it has been linked
with many diverse areas of mathematics, such as operator algebras, quantum
groups, categorification and representation theory. It can be defined in
several ways, including as a quotient of the Hecke algebra and an associative
diagram algebra.

In [8], we defined a generalised Temperley-Lieb algebra $TL_{r,1,n}$
corresponding to the complex reflection group $G(r,1,n)$ as a generalisation
of the Temperley-Lieb algebras of types $A_{n-1}$ and $B_{n}$. In that work,
we give a cellular structure for $TL_{r,1,n}$ and find the decomposition
matrix of the corresponding cell modules using the technology of KLR algebras.

We generalise that work in this paper. Specifically, we introduce a
generalised Temperley-Lieb algebra $TL_{r,p,n}$ which corresponds to an
arbitrary imprimitive complex reflection group $G(r,p,n)$. It is defined as
the fixed-point subalgebra of a suitably specialised generalised Temperley-
Lieb algebra $TL_{r,1,n}$ under an automorphism $\sigma_{TL}$ defined in
(4.3).

Moreover, we introduce a modified multipartition which is called a
3-dimensional multipartition in Section 5, and construct a new cellular
structure of our specialised $TL_{r,1,n}$ in Theorem 7.5. Inspired by the skew
cellularity introduced by Hu, Mathas and Rostam in [7], we show that
$\sigma_{TL}$ induces a shift automorphism with respect to this cellular basis
in Theorem 7.7 and deduce the cellularity of $TL_{r,p,n}$ in Theorem 7.9.
Using this cellular structure of $TL_{r,p,n}$ inherited from $TL_{r,1,n}$, we
determine the decomposition matrix of the cell modules of $TL_{r,p,n}$ in
Theorem 8.10.

##  2\. The cyclotomic Hecke algebra $H(r,p,n)$

To define the generalised Temperley-Lieb algebra $TL_{r,p,n}$, we first recall
the definition of the cyclotomic Hecke algebra $H(r,p,n)$ corresponding to the
imprimitive reflection group $G(r,p,n)$. In this section, we also recall
Ariki’s construction in [1] showing that $H(r,p,n)$ is the fixed point
subalgebra under an automorphism $\sigma$ of $H_{n}^{\Lambda}(q,\zeta)$, a
specialization of $H(r,1,n)$. We will use this automorphism to define
$TL_{r,p,n}$.

The following lemma gives a presentation of the imprimitive reflection group
$G(r,p,n)$.

######  Lemma 2.1.

(cf. [2],Appendix 2) Let $r,p$ and $n$ be positive integers such that $p|r$
and let $d=\dfrac{r}{p}$. The complex reflection group $G(r,p,n)$ is the group
generated by $s_{0},s_{1},s_{1}^{\prime},s_{2},$ $s_{3},\dots,s_{n-1}$ subject
to the following relations:

$\displaystyle s_{0}^{d}=s_{1}^{2}=s_{1}^{\prime 2}=s_{2}^{2}$
$\displaystyle=\dots=s_{n-1}^{2}=1;$ $\displaystyle s_{0}s_{1}s_{0}s_{1}$
$\displaystyle=s_{1}s_{0}s_{1}s_{0};$ $\displaystyle
s_{0}s_{1}^{\prime}s_{0}s_{1}^{\prime}$
$\displaystyle=s_{1}^{\prime}s_{0}s_{1}^{\prime}s_{0};$ $\displaystyle
s_{0}s_{1}s_{1}^{\prime}$ $\displaystyle=s_{1}s_{1}^{\prime}s_{0};$
$\displaystyle s_{0}s_{i}=s_{i}s_{0}$ $\displaystyle\text{ for }i\geq 2;$
$\displaystyle s_{1}^{\prime}s_{2}s_{1}^{\prime}$
$\displaystyle=s_{2}s_{1}^{\prime}s_{2};$ $\displaystyle
s_{1}^{\prime}s_{i}=s_{i}s_{1}^{\prime}$ $\displaystyle\text{ for }i\geq 3;$
$\displaystyle s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$ $\displaystyle\text{ for
}i\geq 1;$ $\displaystyle s_{i}s_{j}=s_{j}s_{i}$ $\displaystyle\text{ for
}|i-j|\geq 2\text{ and }i,j\geq 1;$
$\displaystyle\underbrace{s_{0}s_{1}s_{1}^{\prime}s_{1}s_{1}^{\prime}\dots}_{p+1}$
$\displaystyle=\underbrace{s_{1}^{\prime}s_{0}s_{1}s_{1}^{\prime}s_{1}\dots}_{p+1}.$

$G(r,p,n)$ may be viewed as the group consisting of all $n\times n$ monomial
matrices such that each non-zero entry is an $r^{th}$ root of unity and the
product of the non-zero entries is a $d^{th}$ root of unity, w

$d=\frac{r}{p}$. More precisely, let $\zeta\in\mathbb{C}^{*}$ is a primitive
$r^{th}$ root of unity. The complex reflection group $G(r,p,n)$ is generated
by the following elements:

$\displaystyle s_{0}$ $\displaystyle=\zeta^{p}E_{1,1}+\sum_{k=2}^{n}E_{k,k},$
$\displaystyle s_{1}^{\prime}$ $\displaystyle=\zeta
E_{1,2}+\zeta^{-1}E_{2,1}+\sum_{k=3}^{n}E_{k,k}$ $\displaystyle s_{i}$
$\displaystyle=\sum_{1\leq k\leq i-1}E_{k,k}+E_{i+1,i}+E_{i,i+1}+\sum_{i+2\leq
k\leq n}E_{k,k}\text{ for all }1\leq i\leq n-1$

where $E_{i,j}$ is the elementary $n\times n$ matrix with $(i,j)$-entry equal
to $1$ and all other entries equal to zero. Since the complex reflection group
$G(r,1,n)$ is generated by $s_{i}(1\leq i\leq n-1)$ as well as the following
matrix:

$t_{0}=\zeta E_{1,1}+\sum_{k=2}^{n}E_{k,k},$

the complex reflection group $G(r,p,n)$ is a subgroup of $G(r,1,n)$.

The Hecke algebra of type $G(r,p,n)$ is a deformation of its complex group
ring. Following [2], the Hecke algebra $H(r,p,n)$ may be defined as follows:

######  Definition 2.2.

([2],Definition 4.21) Let $R$ be a commutative ring with 1 and
$q,u_{1},u_{2},\dots,u_{r}\in R^{*}$. The Hecke algebra $H(r,p,n)$ is the
unitary associative $R$-algebra generated by
$S,T_{1}^{\prime},T_{1},T_{2},\dots,T_{n-1}$ subject to the following
relations:

$\displaystyle(S-u_{1})(S-u_{2})\dots$ $\displaystyle(S-u_{d})=0;$
$\displaystyle(T_{1}^{\prime}-q)(T_{1}^{\prime}+1)=(T_{i}-q)$
$\displaystyle(T_{i}+1)=0\text{ for }1\leq i\leq n-1;$ $\displaystyle
ST_{1}ST_{1}$ $\displaystyle=T_{1}ST_{1}S;$ $\displaystyle
ST_{1}^{\prime}ST_{1}^{\prime}$
$\displaystyle=T_{1}^{\prime}ST_{1}^{\prime}S;$ $\displaystyle
ST_{1}T_{1}^{\prime}$ $\displaystyle=T_{1}T_{1}^{\prime}S;$ $\displaystyle
ST_{i}=T_{i}S$ $\displaystyle\text{ for }i\geq 2;$ $\displaystyle
T_{1}^{\prime}T_{2}T_{1}^{\prime}$ $\displaystyle=T_{2}T_{1}^{\prime}T_{2};$
$\displaystyle T_{1}^{\prime}T_{i}=T_{i}T_{1}^{\prime}$ $\displaystyle\text{
for }i\geq 3;$ $\displaystyle T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$
$\displaystyle\text{ for }i\geq 1;$ $\displaystyle T_{i}T_{j}=T_{j}T_{i}\text{
for }$ $\displaystyle|i-j|\geq 2\text{ and }i,j\geq 1;$
$\displaystyle\underbrace{ST_{1}T_{1}^{\prime}T_{1}T_{1}^{\prime}\dots}_{p+1}$
$\displaystyle=\underbrace{T_{1}^{\prime}ST_{1}T_{1}^{\prime}T_{1}\dots}_{p+1}.$

In analogy with the restrictions on the parameters in [8], we require that $R$
is a field of characteristic 0 with a $p^{th}$ primitive root of unity and
$q,u_{1},u_{2},\dots,u_{d}\in R^{*}$ such that

(2.1) $\frac{u_{i}}{u_{j}}\neq 1,q\textbf{ or }q^{2}$

for all $i\neq j$ and

(2.2) $(1+q)(1+q+q^{2})\neq 0.$

Moreover, we assume that there exists $w_{i}\in R$ such that $w_{i}^{p}=u_{i}$
for $i=1,2,\dots,d$.

We see from their matrix realisations that $G(r,p,n)$ is a subgroup of
$G(r,1,n)$. It is a natural question to ask whether there is a connection
between the two Hecke algebras. To review the construction of the Hecke
algebra of $G(r,p,n)$ by Ariki in [1] which indicates the connection between
$H(r,1,n)$ and $H(r,p,n)$, we recall the definition of $H(r,1,n)$.

######  Definition 2.3.

([2],Definition 4.21) Let $R$ be as defined above and
$q,v_{1},v_{2},\dots,v_{r}\in R^{*}$. The cyclotomic Hecke algebra $H(r,1,n)$
corresponding to $G(r,1,n)$ over $R$ is the associative algebra generated by
$T_{0},T_{1},\dots,T_{n-1}$ subject to the following relations:

(2.3) $\displaystyle(T_{0}-v_{1})(T_{0}-v_{2})\dots(T_{0}-v_{r})$
$\displaystyle=0;$ (2.4) $\displaystyle(T_{i}-q)(T_{i}+1)$
$\displaystyle=0\textit{ for }1\leq i\leq n-1;$ (2.5) $\displaystyle
T_{0}T_{1}T_{0}T_{1}$ $\displaystyle=T_{1}T_{0}T_{1}T_{0};$ (2.6)
$\displaystyle T_{i}T_{i+1}T_{i}$ $\displaystyle=T_{i+1}T_{i}T_{i+1}\textit{
for }1\leq i\leq n-2;$ (2.7) $\displaystyle T_{i}T_{j}$
$\displaystyle=T_{j}T_{i}\textit{ for }|i-j|\geq 2.$

We remind readers that in the original definition of $H(r,1,n)$, the base
field $R$ is only required as a commutative ring with unity. The restrictions
on $R$ are for the following specialisation:

Let $\zeta\in R$ be a primitive root of unity of order $p$. Choose the
parameters $v_{1},v_{2},\dots,v_{r}$ in $(\ref{ordering relation})$ such that
the equation is of the following form:

(2.8) $\prod_{k=1}^{d}\prod_{i=1}^{p}(T_{0}-\zeta^{i}w_{k})=0,$

where $w_{k}$ is an element in $R$ such that $w_{k}^{p}=u_{k}$. Then this
equation can be written as

(2.9) $\prod_{k=1}^{d}(T_{0}^{p}-u_{k})=0.$

Denote by $H_{n}(q,u_{k})$ the algebra $H(r,1,n)$ with the specialization
above. Then we have an automorphism $\sigma:H_{n}(q,u_{k})\mapsto
H_{n}(q,u_{k})$ such that:

(2.10) $\sigma(T_{0})=\zeta T_{0},\sigma(T_{i})=T_{i}\text{ for all }1\leq
i\leq n-1.$

Ariki shows in [1]:

######  Proposition 2.4.

([1], Proposition 1.6) The algebra homomorphism $\phi:H(r,p,n)\to
H_{n}(q,u_{i})$ given by:

$\displaystyle\phi(S)$ $\displaystyle=T_{0}^{p},$
$\displaystyle\phi(T_{1}^{\prime})$ $\displaystyle=T_{0}^{-1}T_{1}T_{0},$
$\displaystyle\phi(T_{i})$ $\displaystyle=T_{i}\text{ for }1\leq i\leq n-1$

is well-defined and one-to-one. And $H(r,p,n)\cong\phi(H(r,p,n))$.

Further, we have

######  Proposition 2.5.

([11],Corollary 1.18) The algebra $H(r,p,n)$ is isomorphic to
$H_{n}(q,u_{k})^{\sigma}$ via $\phi$, where the latter algebra is the fixed
point subalgebra of $H_{n}(q,u_{k})$ under $\sigma$.

This proposition implies that the cyclotomic Hecke algebra $H(r,p,n)$ can be
realised as the subalgebra of a specialisation of the cyclotomic Hecke algebra
$H(r,1,n)$, consisting of the points fixed by the automorphism $\sigma$
defined in (2.14). In analogy with this situation,, we will define the
generalised Temperley-Lieb algebra $TL_{r,p,n}$ as a fixed point subalgebra of
$TL_{r,1,n}$ under $\sigma$ in Section 4.

In the remaining part of this section, we interpret the automorphism $\sigma$
into the language of quivers which are used to define a KLR algebra, from
which we obtain the cellularity of $TL_{r,p,n}$. This interpretation was first
discussed by Rostam in Section 3.1 in [11].

Let $q\in R^{*}$ be the parameter in definition 2.2 and $e$ be the smallest
positive integer such that $1+q+\dots+q^{e-1}=0$, setting $e:=0$ if no such
integer exists. Let $\Gamma_{e,p}$ be the quiver with vertex set $K:=I\times
J$ where $I=\mathbb{Z}/p\mathbb{Z}$ and $J=\mathbb{Z}/e\mathbb{Z}$, and there
is a directed edge from $(i_{1},j_{1})$ to $(i_{1},j_{1}+1)$. As there is no
edge between $(i,j)$ and $(i^{\prime},j^{\prime})$ if $i\neq i^{\prime}$ in
the quiver $\Gamma_{e,p}$, it decomposes as $p$ layers which are isomorphic to
each other. For example, $\Gamma_{0,3}$ consists of three layers, each of them
is a quiver of type $A_{\infty}$:

${\Gamma_{0,3}:}$${\dots}$${(0,-1)}$${(0,0)}$${(0,1)}$${(0,2)}$${\dots}$${\dots}$${(1,-1)}$${(1,0)}$${(1,1)}$${(1,2)}$${\dots}$${\dots}$${(2,-1)}$${(2,0)}$${(2,1)}$${(2,2)}$${\dots}$

Let $(c_{(i,j)(i^{\prime},j^{\prime})})$ be the Cartan matrix associated with
$\Gamma_{e,p}$, so that

(2.11) $\displaystyle c_{(i,j)(i^{\prime},j^{\prime})}=\begin{cases}2&\textit{
if }(i,j)=(i^{\prime},j^{\prime});\\\ -1&\textit{ if }e\neq
2,i=i^{\prime}\textit{ and }j=j^{\prime}\pm 1;\\\ -2&\textit{ if
}e=2,i=i^{\prime}\textit{ and }j=j^{\prime}\pm 1;\\\ 0&\textit{
otherwise.}\end{cases}$

Let $\\{\alpha_{(i,j)}|(i,j)\in K\\}$ be the associated set of simple roots
and $\\{\Lambda_{(i,j)}|(i,j)\in K\\}$ be the set of fundamental weights. Let
$(,)$ be the bilinear form determined by

$\displaystyle(\alpha_{(i,j)},\alpha_{(i^{\prime},j^{\prime})})=c_{(i,j)(i^{\prime},j^{\prime})}\textit{
and
}(\Lambda_{(i,j)},\alpha_{(i^{\prime},j^{\prime})})=\delta_{(i,j)(i^{\prime},j^{\prime})}.$

Let $P_{+}=\oplus_{(i,j)\in K}\mathbb{N}\Lambda_{(i,j)}$ be the dominant
weight lattice and $Q_{+}=\oplus_{(i,j)\in K}\mathbb{N}\alpha_{(i,j)}$ be the
positive root lattice. For $\Lambda\in P_{+}$, define the length of $\Lambda$,
$l(\Lambda):=\sum_{(i,j)\in K}(\Lambda,\alpha_{(i,j)})$ and for $\alpha\in
Q_{+}$, the height of $\alpha$, $ht(\alpha):=\sum_{(i,j)\in
K}(\alpha,\Lambda_{(i,j)})$.

Fix a dominant weight $\Lambda$ such that $l(\Lambda)=r$ and
$(\Lambda,\alpha_{(i,j)})=(\Lambda,\alpha_{(i^{\prime},j)})$ for all
$i,i^{\prime}\in I\text{ and }j\in J$. Let $\zeta$ be a $p^{th}$ primitive
root of unity and choose the parameters $v_{1},v_{2},\dots,v_{r}$ so that the
equation $(\ref{ordering relation})$ becomes:

(2.12) $\prod_{i\in I}\prod_{j\in
J}(T_{0}-\zeta^{i}q^{j})^{(\Lambda,\alpha_{(i,j)})}=0,$

which can be written as

(2.13) $\prod_{j\in J}(T_{0}^{p}-q^{pj})^{(\Lambda,\alpha_{(i,j)})}=0.$

Denote by $H_{n}^{\Lambda}(q,\zeta)$ the algebra $H(r,1,n)$ specialised as
above. Then $H_{n}^{\Lambda}(q,\zeta)$ can be also regarded as a
specialisation of $H_{n}(q,u_{k})$ which is given by $(\ref{eq8.3})$ with
$w_{k}=q^{j_{k}}$ for some $j_{k}\in J$. Then we have an automorphism
$\sigma:H_{n}^{\Lambda}(q,\zeta)\mapsto H_{n}^{\Lambda}(q,\zeta)$ such that:

(2.14) $\sigma(T_{0})=\zeta T_{0},\sigma(T_{i})=T_{i}\text{ for all }1\leq
i\leq n-1.$

Similarly, choose the parameters $u_{1},\dots,u_{d}$ such that the first
equation in Definition 2.2 becomes

(2.15) $\prod_{j\in J}(S-q^{pj})^{(\Lambda,\alpha_{(i,j)})}=0$

and denote by $H_{p,n}^{\Lambda}(q)$ the new algebra with this specialization.
As a direct consequence of Proposition 2.5, the algebra $H_{p,n}^{\Lambda}(q)$
is isomorphic to $H_{n}^{\Lambda}(q,\zeta)^{\sigma}$ via the automorphism
$\phi$ given in Proposition 2.4.

We next recall the interpretation of the automorphism $\sigma$ as that of the
cyclotomic KLR algebra by Rostam in [11]. Let
$\sigma_{0}:\Gamma_{e,p}\mapsto\Gamma_{e,p}$ be the quiver automorphism given
by

$\sigma_{0}((i,j))=(i-1,j)$

and $\sigma_{1}$ be the map on $K^{n}$ induced by $\sigma_{0}$, that is the
map such that

(2.16) $\sigma_{1}(\mathbf{k})_{l}=\sigma_{0}(\mathbf{k}_{l})$

for all $\mathbf{k}\in K^{n}$ and $1\leq l\leq n$. Let $\sigma^{\prime}$ be
the algebra automorphism of the cyclotomic KLR algebra
$R_{n}^{\Lambda}(\Gamma_{e,p})$ given by

$\displaystyle\sigma^{\prime}(e(\mathbf{k}))$
$\displaystyle=e(\sigma_{1}(\mathbf{k}));$
$\displaystyle\sigma^{\prime}(y_{i})=y_{i}$ $\displaystyle\text{ for }1\leq
i\leq n$ $\displaystyle\sigma^{\prime}(\psi_{i})=\psi_{i}$
$\displaystyle\text{ for }1\leq i\leq n-1.$

Then we have

######  Theorem 2.6.

([11],Theorem 4.14) Let $f:R_{n}^{\Lambda}(\Gamma_{e,p})\to
H_{n}^{\Lambda}(q,\zeta)$ be the isomorphism between the cyclotomic KLR
algebra and Hecke algebra given in Theorem 1.1 in [3] and $\sigma$ be the
isomorphism of $H_{n}^{\Lambda}(q,\zeta)$ defined in (2.14). Let
$\sigma^{\prime}$ be the isomorphism of $R_{n}^{\Lambda}(\Gamma_{e,p})$
defined above. Then we have $f\circ\sigma^{\prime}=\sigma\circ f$.

The following corollary is a direct consequence of this theorem:

######  Corollary 2.7.

The Hecke algebra $H_{p,n}^{\Lambda}(q)$ is isomorphic to
$R_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}$, the fixed point subalgebra of
$R_{n}^{\Lambda}(\Gamma_{e,p})$ under $\sigma$.

We shall not distinguish below between the two automorphisms $\sigma$ and
$\sigma^{\prime}$.

##  3\. The generalised Temperley-Lieb algebra $TL_{r,1,n}$

In this section, we recall the definition of the generalised Temperley-Lieb
algebra $TL_{r,1,n}$ constructed in [8]. Our new generalised Temperley-Lieb
algebra $TL_{r,p,n}$ will later be defined as the subalgebra of $TL_{r,1,n}$
fixed by automorphism induced by $\sigma$.

Let $H(r,1,n)$ be the cyclotomic Hecke algebra defined in Definition 2.3 and
$H_{i,i+1}$ be the subalgebra of $H(r,1,n)$ which is generated by two non-
commuting generators, $T_{i}$ and $T_{i+1}$ where $0\leq i\leq n-2$. We call
$H_{i,i+1}$ a parabolic subalgebra of $H(r,1,n)$. The parabolic subalgebra
$H_{0,1}$ has $2r$ 1-dimensional representations corresponding to the
multipartitions $(0,0,\dots,(2),\dots,0)$ and $(0,0,\dots,(1,1),\dots,0)$
where $(2)$ and $(1,1)$ are in the $j^{th}$ component of the $r$-partition
with $1\leq j\leq r$. Denote by $E_{0}^{(j)}$ and $F_{0}^{(j)}$ the
corresponding primitive central idempotents in $H_{0,1}$. These exist because
of our choice of parameters, which implies semisimplicity.

For $i$ such that $1\leq i\leq n-2$, the parabolic subalgebra $H_{i,i+1}$ is a
Hecke algebra of type $A_{2}$. It has 2 1-dimensional representations and the
corresponding primitive central idempotents are

$\displaystyle E_{i}$
$\displaystyle=a(T_{i}T_{i+1}T_{i}+T_{i}T_{i+1}+T_{i+1}T_{i}+T_{i}+T_{i+1}+1);$
$\displaystyle F_{i}$
$\displaystyle=b(T_{i}T_{i+1}T_{i}-qT_{i}T_{i+1}-qT_{i+1}T_{i}+q^{2}T_{i}+q^{2}T_{i+1}-q^{3}),$

where $a=(q^{3}+2q^{2}+2q+1)^{-1}$ and $b=-a$. The generalised Temperley-Lieb
algebra $TL_{r,1,n}$ is defined as the quotient of $H(r,1,n)$ by the two-sided
ideal generated by half of the central idempotents listed above; specifically,
the definition is as follows.

######  Definition 3.1.

([8], Definition 3.3) Let $H(r,1,n)$ be the cyclotomic Hecke algebra defined
in 2.3 and $E_{i}$( $E_{i}^{(j)}$, if $i=0$) be the primitive central
idempotents of $H_{i,i+1}$ listed above. The generalised Temperley-Lieb
algebra $TL_{r,1,n}$ is

$TL_{r,1,n}:=H(r,1,n)/\langle
E_{0}^{(1)},\dots,E_{0}^{(r)},E_{1},\dots,E_{n-2}\rangle.$

According to Theorem 1.1 in [3], the Hecke algebra $H(r,1,n)$ is isomorphic to
a cyclotomic KLR algebra $\mathcal{R}_{n}^{\Lambda}$ with $\Lambda$ a dominant
weight decided by the parameters $q,v_{1},v_{2},\dots,v_{r}$. Therefore, we
have the following alternative definition of $TL_{r,1,n}$ as a quotient of the
cyclotomic KLR algebra $\mathcal{R}_{n}^{\Lambda}$:

######  Theorem 3.2.

[8, Theorem 3.24] Let $TL_{r,1,n}$ be the generalised Temperley-Lieb algebra
in Definition 3.1 and $\mathcal{R}_{n}^{\Lambda}$ be the cyclotomic KLR
algebra isomorphic to the corresponding Hecke algebra $H(r,1,n)$. Then

(3.1) $TL_{r,1,n}\cong\mathcal{R}_{n}^{\Lambda}/\mathcal{J}_{n},$

where the ideal $\mathcal{J}_{n}$ is defined by

(3.2) $\displaystyle\mathcal{J}_{n}(\Lambda)=$
$\displaystyle\sum_{\mathbf{i}\in I^{n-2},(\alpha_{i},\Lambda)>0}\langle
e(i,i+1,\mathbf{i})\rangle_{\mathcal{R}_{n}^{\Lambda}}$
$\displaystyle+\sum_{\mathbf{i}\in I^{n-3},(\alpha_{i_{j}},\Lambda)>0\text{
for }j=1,2,3,i_{j}\neq i_{k}\textit{ for }j\neq k}\langle
e(i_{1},i_{2},i_{3},\mathbf{i})\rangle_{\mathcal{R}_{n}^{\Lambda}},$

with $e(\mathbf{i})$ the KLR generators of $\mathcal{R}_{n}^{\Lambda}$ (cf.
Definition 2.27 in [8]).

##  4\. The definition of $TL_{r,p,n}$

In this section, we define the generalised Temperley-Lieb algebra $TL_{r,p,n}$
corresponding to $G(r,p,n)$ using the automorphism $\sigma$. In analogy with
the case $p=1$, we will show that $TL_{r,p,n}$ is a quotient of the
corresponding Hecke algebra $H(r,p,n)$.

Denote by $TL_{r,1,n}(q,\zeta)$ the specialisation of the generalised
Temperley-Lieb algebra $TL_{r,1,n}$ corresponding to
$H_{n}^{\Lambda}(q,\zeta)$.

By Theorem 3.24 in [8],
$TL_{r,1,n}(q,\zeta)=\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})/\mathcal{J}$
where $\mathcal{J}$ is the two-sided ideal generated by all the
$e(\mathbf{k}),\mathbf{k}\in K^{n}$ such that:

(4.1) $i_{1}=i_{2},j_{1}=j_{2}-1\text{ if }\mathbf{k}_{1}=(i_{1},j_{1})\text{
and }\mathbf{k}_{2}=(i_{2},j_{2})$

or

(4.2) $(\alpha_{\mathbf{k}_{l}},\Lambda)>0\text{ for }l=1,2,3.$

As $\sigma_{0}((i,j))=(i-1,j)$, $e(\mathbf{k})$ satisfies $(\ref{con1})$ if
and only if $\sigma(e(\mathbf{k}))$ does. The restriction on $\Lambda$ implied
by $(\Lambda,\alpha_{(i,j)})=(\Lambda,\alpha_{(i^{\prime},j)})$ guarantees
that $e(\mathbf{k})$ satisfies $(\ref{con2})$ if and only if
$\sigma(e(\mathbf{k}))$ does. Therefore, we have

$\sigma(\mathcal{J})=\mathcal{J}$

which implies that $\sigma$ induces an automorphism of $TL_{r,1,n}(q,\zeta)$,
which we denote by $\sigma_{TL}$. In other words, $\sigma=\sigma_{TL}$ is the
automorphism of $TL_{r,1,n}(q,\zeta)$ such that

(4.3)
$\displaystyle\sigma_{TL}(e((i_{1},j_{1}),(i_{2},j_{2}),\dots,(i_{n},j_{n})))$
$\displaystyle=e((i_{1}-1,j_{1}),(i_{2}-1,j_{2}),\dots,(i_{n}-1,j_{n}));$
$\displaystyle\sigma_{TL}(y_{i})=y_{i}$ $\displaystyle\text{ for }1\leq i\leq
n$ $\displaystyle\sigma_{TL}(\psi_{i})=\psi_{i}$ $\displaystyle\text{ for
}1\leq i\leq n-1.$

Now we can state the main theorem in this section.

######  Theorem 4.1.

Let $\zeta\in R$ be a primitive root of unity of order $p$ and
$TL_{r,1,n}(q,\zeta)$ be the specialization of the generalised Temperley-Lieb
algebra $TL_{r,1,n}^{\Lambda}$ such that the relation $(\ref{ordering
relation})$ is transformed into (2.12). Let $\sigma_{TL}$ be the automorphism
of $TL_{r,1,n}(q,\zeta)$ defined above, $TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$ be
the fixed point subalgebra under $\sigma_{TL}$ and let
$\mathcal{J}^{\sigma}=\mathcal{J}\cap\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}$.
Then we have

(4.4)
$TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}\cong\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}/\mathcal{J}^{\sigma}.$

Thus the fixed subalgebra $TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$ is a quotient of
$\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}$, which is isomorphic to the
cyclotomic Hecke algebra of type $G(r,p,n)$.

###### Proof.

Let
$f:\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}/\mathcal{J}^{\sigma}\to
TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$ be the map such that
$f(a+\mathcal{J}^{\sigma})=a+\mathcal{J}$ for all
$a\in\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}$. We only need to prove
the surjectivity.

For any $a+\mathcal{J}\in TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$, we have
$a+\mathcal{J}=\sigma(a+\mathcal{J})=\sigma(a)+\mathcal{J}$. Thus,
$a+\mathcal{J}=\sigma(a)+\mathcal{J}=\dots=\sigma^{p-1}(a)+\mathcal{J}$. As
the characteristic of the field is 0 ,
$a^{\prime}=\frac{a+\sigma(a)+\dots+\sigma^{p-1}(a)}{p}$ is an element in
$\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}$ and
$a^{\prime}+\mathcal{J}=a+\mathcal{J}$. Further,
$a^{\prime}+\mathcal{J}^{\sigma}\in\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}/\mathcal{J}^{\sigma}$
and $f(a^{\prime}+\mathcal{J}^{\sigma})=a+\mathcal{J}$. So $f$ is a
surjection.

So
$f:\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}/\mathcal{J}^{\sigma}\mapsto
TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$ is an isomorphism. Therefore, we have
$TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}\cong\mathcal{R}_{n}^{\Lambda}(\Gamma_{e,p})^{\sigma}/\mathcal{J}^{\sigma}$.
∎

We now define our generalised Temperley-Lieb algebra of type $G(r,p,n)$ as
follows:

######  Definition 4.2.

Let $R$ be a field of characteristic $0$ and let $TL_{r,1,n}(q,\zeta)$ be the
generalised Temperley-Lieb algebra defined over $R$ as in subsection 4. Let
$\sigma_{TL}$ be the automorphism of $TL_{(}r,1,n)(q,\zeta)$ as in Theorem
4.1. We define the fixed subalgebra $TL_{r,1,n}(q,\zeta)^{\sigma_{TL}}$ as
$TL_{r,p,n}$ and refer to it as the Temperley-Lieb algebra corresponding to
the complex reflection group $G(r,p,n)$.

Since the map $\sigma_{TL}$ is trivial when $p=1$, $TL_{r,1,n}(q,\zeta)$ can
be regarded as a special case of the generalised Temperley-Lieb algebra
$TL_{r,p,n}$. In particular, the Temperley-Lieb algebras of types $A_{n-1}$
and $B_{n}$ are both special cases of our $TL_{r,p,n}$, where $r=p=1$ and
$r=2,\;p=1$, respectively. As another special case of $TL_{r,p,n}$ with
$r=p=2$, the algebra $TL_{2,2,n}$ is a quotient of the Temperley-Lieb algebra
of type-$D_{n}$ in the sense of [12] and [9]. This quotient is called a forked
Temperley-Lieb algebra. We refer readers to [6] for details.

##  5\. 3-dimensional multipartitions and the cellular structure of
$TL_{r,1,n}$

As a specialization of the Temperley-Lieb algebra $TL_{r,1,n}$,
$TL_{r,1,n}(q,\zeta)$ is a cellular algebra according to Theorem 4.19 in [8].
In this section we focus on this specialisation. As we see from the
interpretation above, this specialisation transforms the quiver with a single
layer in section 2.4 in [8] into one with $p$ layers. Accordingly, we need to
rearrange the partitions in a multipartition. More precisely, we introduce
3-dimensional multipartitions and tableaux. They are the primary tool we will
later use to prove the cellularity of the generalised Temperley-Lieb algebra
$TL_{r,p,n}$.

Let $\mathcal{K}=I^{\prime}\times L$ be an index set where
$I^{\prime}=\\{0,1,\dots,p-1\\}$ and $L=\\{0,1,\dots,d-1\\}$. Note that this
index set $\mathcal{K}$ is different from the vertex set $K$ of the quiver
$\Gamma_{e,p}$. We first construct a table with $d$ rows and $p$ columns and
label the boxes in the table with the elements of $\mathcal{K}$. We call this
table the floor of a 3D multipartition. Figure-1 shows the floor with $p=5$
and $d=3$:

(0,0)(1,0)(2,0)(3,0)(4,0)(0,1)(1,1)(2,1)(3,1)(4,1)(0,2)(1,2)(2,2)(3,2)(4,2)
Figure 1. The floor of $5\times 3$

A 3D multipartition of $n$ consists of a floor labelled with $\mathcal{K}$ and
$n(i,l)$ nodes in a column in the box $(i,l)$ on the floor such that

$\sum_{i=0}^{p-1}\sum_{l=0}^{d-1}n(i,l)=n.$

Denote by $\mathfrak{P}_{n}$ the set of 3D multipartitions of $n$ in which at
most two boxes on the floor are non-empty. Figures 2 and 3 give two examples
of such 3D multipartitions. We next introduce some notation relating to the 3D
multipartitions in $\mathfrak{P}_{n}$.

For $\lambda\in\mathfrak{P}_{n}$, if $\lambda$ has only one non-empty
component, which is in the box $(i,l)$, call $\lambda$ a single
multipartition, and denote it by $\lambda_{(i,l)}$. Otherwise, let
$(i_{1},l_{1})$ and $(i_{2},l_{2})$ be the two non-empty boxes with
$l_{1}<l_{2}$ or $l_{1}=l_{2}$ and $i_{1}<i_{2}$. If there are $a$ nodes in
the partition in box $(i_{1},l_{1})$ and $b$ nodes in $(i_{2},l_{2})$, denote
$\lambda$ by $\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a-b]}$. As $\lambda$ is
a multipartition of $n$, we have $a+b=n$. Therefore, the multipartition
$\lambda$ is determined by the indices $(i_{1},l_{1}),(i_{2},l_{2})$ and the
number ${a-b}$. We are now in a position to define the reducibility of a 3D
multipartition.

######  Definition 5.1.

Let $\lambda\in\mathfrak{P}_{n}$ be a 3D multipartition. If $\lambda$ is of
the form $\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a-b]}$ with $i_{1}\neq
i_{2}$, we call it an irreducible multipartition. Otherwise, we call it
reducible.

In other words, if the two non-empty boxes are in the same column of the
floor, we call the multipartition a reducible one; otherwise it is
irreducible. In the last section of this chapter, we will show that the cell
modules of $TL_{r,p,n}$ corresponding to irreducible multipartitions are
simple. Figure-2 gives an irreducible multipartition of $5$ and Figure-3 gives
a reducible one.

01234012 Figure 2.  The irreducible multipartition
$\lambda_{(2,0),(4,2)}^{[-1]}$ 01234012 Figure 3.  The reducible
multipartition $\lambda_{(2,0),(2,2)}^{[-3]}$

We next define the residue of each node in a multipartition. This residue
provides a connection between the multipartitions and the quivers we
introduced above. Let $j_{l}\in J=\mathbb{Z}/e\mathbb{Z}$ be such that the
dominant weight $\Lambda$ associated with the parameters in (2.12) is of the
following form

(5.1) $\Lambda=\sum_{i=0}^{p-1}\sum_{l=0}^{d-1}\Lambda_{(i,j_{l})}.$

For $\lambda\in\mathfrak{P}_{n}$, let $\gamma=(a,1,(i,l))$ be the node in the
$a^{th}$ row of the $(i,l)^{th}$ component of $\lambda$. We denote

$Res^{\Lambda}(\gamma)=(i,1-a+j_{l})\in K$

where $K=\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/e\mathbb{Z}$ is the vertex set
of the quiver $\Gamma_{e,p}$.

A tableau corresponding to a 3D multipartition of $n$ is defined as a filling
of the multipartition with numbers $1,2,\dots,n$; we call it standard if these
numbers increase from the floor table along each column. For a 3D
multipartition $\lambda$, denote by $Tab(\lambda)$ the set of tableaux of
shape $\lambda$ and by $Std(\lambda)$ the standard ones. For $t\in
Tab(\lambda)$ and $1\leq m\leq n$, set
$Res_{t}^{\Lambda}(m)=Res^{\Lambda}(\gamma)$, where $\gamma$ is the unique
node such that $t(m)=\gamma$. Define the residue sequence of $t$ as follows:

(5.2)
$res^{\Lambda}(t):=(Res_{t}^{\Lambda}(1),Res_{t}^{\Lambda}(2),\dots,Res_{t}^{\Lambda}(n))\in
K^{n}.$

For two nodes $\gamma=(a,1,(i,l))$ and
$\gamma^{\prime}=(a^{\prime},1,(i^{\prime},{l^{\prime}}))$ in a 3D
multipartition $\lambda$ we say $\gamma<\gamma^{\prime}$ if one of the
following conditions holds:

(i) $a<a^{\prime}$;

(ii) $a=a^{\prime}$ and $l<l^{\prime}$;

(iii)$a=a^{\prime}$, $l=l^{\prime}$ and $i<i^{\prime}$.

We write $\gamma\leq\gamma^{\prime}$ if $\gamma<\gamma^{\prime}$ or
$\gamma=\gamma^{\prime}$. The order $\leq$ is a total order on the set of
nodes. Denote by $t^{\lambda}$ the unique tableau of shape $\lambda$ such that
$t^{\lambda}(i)<t^{\lambda}(j)$ if $1\leq i<j\leq n$. Let
$e_{\lambda}=e(res^{\Lambda}(t^{\lambda}))$ be the idempotent in
$R_{n}^{\Lambda}(\Gamma_{e,p})$.

Now the total order $\leq$ on nodes leads to a partial order $\unlhd$ on
$\mathfrak{P}_{n}$, the set of 3D-multipartitions in the following way: for
$\lambda$ and $\mu\in\mathfrak{P}_{n}$, we say $\lambda\unlhd\mu$ if for each
$\gamma_{0}\in\mathbb{N}\times\\{1\\}\times\mathcal{K}$ we have

(5.3)
$|\\{\gamma\in[\lambda]:\gamma\leq\gamma_{0}\\}|\geq|\\{\gamma\in[\mu]:\gamma\leq\gamma_{0}\\}|,$

where $\leq$ is the total order on nodes.

Let $s,t\in Std(\lambda)$ and fix reduced expressions
$d(s)=s_{i_{1}}s_{i_{2}}\dots s_{i_{k}}$ and $d(t)=s_{j_{1}}s_{j_{2}}\dots
s_{j_{m}}$ such that $s=t^{\lambda}\circ d(s)$ and $t=t^{\lambda}\circ d(t)$
where $\circ$ is the natural $\mathfrak{S}_{n}$-action on the tableaux. Let
$*$ be the unique $R$-linear anti-automorphism of the KLR algebra
$\mathcal{R}_{n}^{\Lambda}$ introduced by Brundan and Kleshchev in section 4.5
of [4] which fixes all the generators of KLR type.

For any 3D-multipartition $\lambda\in\mathfrak{P}_{n}$ and $s,t\in
Std(\lambda)$, let

(5.4) $C_{s,t}^{\lambda}=\psi_{d(s)}^{*}e_{\lambda}\psi_{d(t)}.$

Then we have

######  Theorem 5.2.

([8], Theorem 4.19) The set
$\\{C_{s,t}^{\lambda}|\lambda\in\mathfrak{P}_{n},s,t\in Std(\lambda)\\}$ forms
a (graded) cellular basis of the generalised Temperley-Lieb algebra
$TL_{r,1,n}(q,\zeta)$ with respect to the partial order $\unlhd$.

##  6\. Skew cellular algebras

In this section, we review some general results on graded skew cellular
algebras, which were introduced by Hu, Mathas and Rostam in [7] as a slight
generalisation of the cellular algebras by Graham and Lehrer in [5]. We will
demonstrate the cellularity of our generalised Temperley-Lieb algebra
$TL_{r,p,n}$ using the theory of skew cellularity.

In a similar way to cellular algebras, skew cellular algebras are defined in
terms of a skew cell datum, which consists of the terms of a cell datum as
well as a poset involution.

######  Definition 6.1.

Let $\mathfrak{P}$ be a finite poset with $\trianglelefteq$ as the partial
order. A poset automorphism of $(\mathfrak{P},\trianglelefteq)$ is a
permutation $\sigma$ of $\mathfrak{P}$ such that

$\lambda\trianglelefteq\mu\text{ if and only if
}\sigma(\lambda)\trianglelefteq\sigma(\mu)$

for all $\lambda,\mu\in\mathfrak{P}$. If $\sigma=\iota$ is an involution, we
say that $\iota$ is a poset involution of $\mathfrak{P}$.

Graded skew cellular algebras are defined as follows.

######  Definition 6.2.

([7], Definition 2.2) Let $R$ be an integral domain and $A$ be a
$\mathbb{Z}$-graded algebra over $R$ which is a free module of finite rank
over $R$.

$A$ is a graded skew cellular algebra if it has a graded skew cellular datum
$(\mathfrak{P},\iota,T,C,deg)$ where $(\mathfrak{P},\trianglelefteq)$ is a
poset, $\iota$ is a poset involution and for each $\lambda\in\mathfrak{P}$,
there is a finite set $T(\lambda)$ together with a bijection
$\iota_{\lambda}:T(\lambda)\mapsto T(\iota(\lambda))$ such that

$\iota_{\iota_{\lambda}}\circ\iota_{\lambda}=id_{T(\lambda)}.$

The injection

$\displaystyle C:\cup_{\lambda\in\mathfrak{P}}T(\lambda)\times T(\lambda)$
$\displaystyle\mapsto A$ $\displaystyle(S,T)$ $\displaystyle\mapsto
C_{S,T}^{\lambda}$

and the degree map

$deg:\cup_{\lambda\in\mathfrak{P}}T(\lambda)\mapsto\mathbb{Z}$

satisfy the following conditions:

($C_{1}$) $C_{S,T}^{\lambda}$ is homogeneous of degree $deg(S)+deg(T)$;

($C_{2}$)The image of $C$ forms an $R$-basis of $A$;

($C_{3}$) Let $A_{\triangleleft\lambda}$ be the $R$-span of all the elements
of form $C_{X,Y}^{\mu}$ with $\mu\triangleleft\lambda$ in the poset. Then for
all $a\in A$,

(6.1) $aC_{S,T}^{\lambda}=\sum_{S^{\prime}\in
T(\lambda)}r_{a}(S^{\prime},S)C_{S^{\prime},T}^{\lambda}\text{ }mod\text{
}A_{\triangleleft\lambda}$

with the coefficients $r_{a}(S^{\prime},S)$ independent of $T$.

($C_{4}$) There is a unique anti-isomorphism $*:A\mapsto A$ such that

(6.2)
$(C_{S,T}^{\lambda})^{*}=C_{\iota_{\lambda}(T),\iota_{\lambda}(S)}^{\iota(\lambda)}.$

The basis $\\{C_{S,T}^{\lambda}|\lambda\in\mathfrak{P},S,T\in T(\lambda)\\}$
is called a $\mathbb{Z}$-graded skew cellular basis of $A$.

The following is a baby example of the concept of a skew cellular structure.

######  Example 6.3.

([7, Example 2.5]) Let $R$ be a ring and $x,y$ be two indeterminates. For any
integer $m\geq 1$, let $A=R[x]/(x^{m})\oplus R[y]/(y^{m})$. Let
$\mathfrak{P}=\mathbb{Z}_{2}\times\\{0,1,\dots,m-1\\}$ and $\trianglelefteq$
be the partial order on $\mathfrak{P}$ such that
$(i_{1},k_{1})\trianglelefteq(i_{2},k_{2})$ only if $i_{1}=i_{2}$ and
$k_{1}\leq k_{2}$. Let $\iota$ be the poset involution on $\mathfrak{P}$ such
that $\iota(i,k)=(i+1,k)$. Define $\mathcal{T}(i,k)=\\{k\\}$ and $deg(k)=k$,
that is, there exists a unique ‘tableau’ of each element in $\mathfrak{P}$.
Define

$C_{kk}^{(i,k)}=\begin{cases}x^{k}&\textit{ if }i=0;\\\ y^{k}&\textit{ if
}i=1.\end{cases}$

Then $(\mathfrak{P},\iota,\mathcal{T},C,deg)$ is a graded cellular datum of
$A$.

By applying the anti-isomorphism $*$ to (6.1) and relabelling, we obtain

(6.3) $C_{S,T}^{\lambda}a=\sum_{T^{\prime}\in
T(\lambda)}r_{a^{*}}(\iota_{\lambda}(T^{\prime}),\iota_{\lambda}(T))C_{S,T^{\prime}}^{\lambda}\text{
}mod\text{ }A_{\triangleleft\lambda}$

where the coefficients
$r_{a^{*}}(\iota_{\lambda}(T^{\prime}),\iota_{\lambda}(T))$ are the same ones
as in (6.1) and do not depend on $S$.

It should be remarked that if the poset involution $\iota=id_{\mathfrak{P}}$
and $\iota_{\lambda}=id_{T(\lambda)}$, the graded skew cellular algebra is
graded cellular. We will show in this way that the apparent skew cellularity
of our $TL_{r,p,n}$ is actually cellularity.

Similarly to cellular structures, skew cellular ones are also a useful tool to
study the representations of algebras which are not semisimple. Since our
generalised Temperley-Lieb algebra $TL_{r,p,n}$ will later turn out to be a
cellular algebra, we omit details of skew cell modules and decomposition
matrices. We refer the reader to [7] for further information.

Next recall the definition of shift automorphisms which provide a general
method for the construction of skew cellular algebras from cellular ones.

######  Definition 6.4.

([7, Definition 2.22]) Let $A$ be a $\mathbb{Z}$-graded cellular algebra with
graded cell datum $(\mathfrak{P},T,C,deg)$. A shift automorphism of $A$ is a
triple of automorphisms
$\sigma=(\sigma_{A},\sigma_{\mathfrak{P}},\sigma_{\mathcal{T}})$ where
$\sigma_{A}$ is an algebra automorphism of $A$, $\sigma_{\mathfrak{P}}$ is a
poset automorphism of $\mathfrak{P}$ and $\sigma_{\mathcal{T}}$ is a bijection
on $\mathcal{T}=\bigsqcup_{\lambda\in\mathfrak{P}}T(\lambda)$ such that:

($a$) For $S\in T(\lambda)$, $\sigma_{\mathcal{T}}(S)\in
T(\sigma_{\mathfrak{P}}(\lambda))$ and $deg(\sigma_{\mathcal{T}}(S))=deg(S)$;

($b$) For $S,T\in T(\lambda)$,
$\sigma_{A}(C_{S,T}^{\lambda})=C_{\sigma_{\mathcal{T}}(T),\sigma_{\mathcal{T}}(S)}^{\sigma_{\mathfrak{P}}(\lambda)}$;

($c$) For $S,T\in T(\lambda)$, $\sigma_{\mathcal{T}}^{k}(T)=T$ if and only if
$\sigma_{\mathcal{T}}^{k}(S)=S$.

Before stating the main theorem of this section, we introduce some notation.

Denote by $A^{\sigma}$ the subalgebra of $A$ consisting of $\sigma_{A}$-fixed
points. Let $p$ be the order of $\sigma_{A}$ and $p^{\prime}$ be the order of
$\sigma_{\mathfrak{P}}$. Let $\mathfrak{P}_{\sigma}$ be a set of
representatives for the $\langle\sigma_{\mathfrak{P}}\rangle$-orbits in
$\mathfrak{P}$. Define a partial order $\trianglelefteq_{\sigma}$ on
$\mathfrak{P}_{n}$ by

$\lambda\triangleleft_{\sigma}\mu\text{ if and only if
}\sigma_{\mathfrak{P}}^{k}(\lambda)\triangleleft\mu\text{ for some
}k\in\mathbb{Z}.$

Denote by $o_{\lambda}$ the size of the
$\langle\sigma_{\mathfrak{P}}\rangle$-orbit through $\lambda$. Let
$\sigma_{\lambda}=\sigma_{\mathcal{T}}^{o_{\lambda}}$. Similarly, let
$T_{\sigma}(\lambda)$ be a set of representatives for the
$\langle\sigma_{\lambda}\rangle$-orbit on $T(\lambda)$. The condition ($c$)
above implies that for a $\lambda\in\mathfrak{P}$, all the orbits in
$T(\lambda)$ are of the same size. Denote this common value by
$o_{T}(\lambda)$. Then $o_{\lambda}=o_{\mu}$ and $o_{T}(\lambda)=o_{T}(\mu)$
for $\lambda$ and $\mu$ in the same
$\langle\sigma_{\mathfrak{P}}\rangle$-orbit on $\mathfrak{P}$.

Let
$\mathfrak{P}_{\sigma,p}:=\\{(\lambda,k)|\lambda\in\mathfrak{P}_{\sigma},k\in\mathbb{Z}/o_{T}(\lambda)\mathbb{Z}\\}$
be the poset with partial order $\triangleleft_{\sigma}$ given by

$(\lambda,k)\trianglelefteq_{\sigma}(\mu,l)\text{ if and only if
}(\lambda,k)=(\mu,l)\text{ or }\lambda\triangleleft_{\sigma}\mu,$

for all $(\lambda,k),(\mu,l)\in\mathfrak{P}_{\sigma,p}$.

Define $T_{\sigma}(\lambda,k)=T_{\sigma}(\lambda)$ for
$(\lambda,k)\in\mathfrak{P}_{\sigma,p}$. Assuming there exists a primitive
$p^{th}$ root of unity $\zeta$, set

$C_{\sigma}(S,T)=\sum_{j=0}^{o_{T}(\lambda)-1}\sum_{l=0}^{p-1}\zeta_{\lambda}^{kj}\sigma_{A}^{l}(C_{S,\sigma_{\lambda}^{j}(T)}^{\lambda})$

where $\zeta_{\lambda}=\zeta^{p/o_{T}(\lambda)}$ and $S,T\in
T_{\sigma}(\lambda,k)$. Let $deg_{\sigma}(S)=deg(S)$. Finally, let
$\iota_{\sigma}$ be the poset involution such that
$\iota_{\sigma}(\lambda,k)=(\lambda,-k)$ and
$(\iota_{\sigma})_{\lambda,k}:T_{\sigma}(\lambda,k)\mapsto
T_{\sigma}(\lambda,-k)$ be the map given by the identity map on
$T_{\sigma}(\lambda)$.

Then we have a shift automorphism that leads to a skew cellular algebra. More
precisely,

######  Theorem 6.5.

([7, Theorem 2.28]) Suppose $A$ is a $\mathbb{Z}$-graded cellular algebra with
graded cell datum $(\mathfrak{P},T,C,deg)$ over the integral domain $R$. Let
$\sigma=(\sigma_{A},\sigma_{\mathfrak{P}},\sigma_{\mathcal{T}})$ be a shift
automorphism of $A$. Denote by $p$ the order of $\sigma_{A}$. If $R$ contains
a primitive $p^{th}$ root of unity and $p\in R^{*}$, then $A^{\sigma}$ is a
graded skew cellular algebra with skew cellular datum
$(\mathfrak{P}_{\sigma,p},\iota_{\sigma},T_{\sigma},C_{\sigma},deg_{\sigma})$.

##  7\. The skew cellularity of $TL_{r,p,n}$

In [7], Hu, Mathas and Rostam introduced skew cellular algebras as a
generalisation of the original cellular algebras. They define the notion of a
“shift automorphism” of a cellular algebra and show that the fixed subalgebra
has a skew cellular structure (cf. Theorem 6.5 above). In this section, we
prove that $TL_{r,p,n}$ is skew cellular by constructing a shift automorphism
on $TL_{r,1,n}(q,\zeta)$ and identifying $TL_{r,p,n}$ as the subalgebra
consisting of the fixed points of the shift automorphism.

We first define the shift automorphism on the specialised Temperley-Lieb
algebra $TL_{r,1,n}(q,\zeta)$. As shown in Theorem 4.1, $TL_{r,p,n}$ is the
fixed-point subalgebra of $TL_{r,1,n}(q,\zeta)$ under $\sigma_{TL}$, which is
induced by the automorphism $\sigma^{\prime}$ of the cyclotomic KLR algebra
$R_{n}^{\Lambda}(\Gamma_{e,p})$ defined above Theorem 2.6. So we choose the
algebra automorphism $\sigma_{TL}$ as the first term in the shift automorphism
$\sigma$.

Condition (b) in Definition 6.4 indicates that the poset automorphism
$\sigma_{\mathfrak{P}}$ and the bijection $\sigma_{\mathcal{T}}$ are
determined by the cellular structure of $A$ and the algebra automorphism
$\sigma_{A}$. However, the poset map $\sigma_{\mathfrak{P}}$ induced by the
algebra automorphism $\sigma_{TL}$ with respect to the cellular structure in
Theorem 5.2 is not an automorphism of the poset $(\mathfrak{P}_{n},\unlhd)$
with the partial order $\unlhd$ given by (5.3). As a counterexample, let
$\lambda_{(0,0)}$ and $\lambda_{(1,0)}\in\mathfrak{P}_{n}$ be the 3D
multipartitions of $n$ with the $(0,0)^{th}$ and $(1,0)^{th}$ components are
the only non-empty ones respectively. We have
$\lambda_{(0,0)}\unlhd\lambda_{(1,0)}$ but

$\sigma_{\mathfrak{P}}(\lambda_{(1,0)})=\lambda_{(0,0)}\unlhd\lambda_{(p-1,0)}=\sigma_{\mathfrak{P}}(\lambda_{(0,0)}).$

We therefore introduce a new cellular structure on the specialised algebra
$TL_{r,1,n}(q,\zeta)$. It should be pointed out that the following
construction does not work for the original $TL_{r,1,n}$ without the
specialisation to the case we treat. Let $\mathfrak{P}_{n}$ be the set of 3D
multipartitions with at most two non-empty components. We first introduce a
new partial order on the indices on the floor table.

######  Definition 7.1.

Let $\mathcal{K}=\\{0,1,\dots,p-1\\}\times\\{0,1,\dots,d-1\\}$ be the index
set introduced in Section 5. Define $\leq$ as the total order on
$\mathcal{K}$, such that for $(i_{1},l_{1}),(i_{2},l_{2})\in\mathcal{K}$:

(7.1) $(i_{1},l_{1})\leq(i_{2},l_{2})\text{ if and only if }l_{1}<l_{2}\text{
or }l_{1}=l_{2}\text{ and }i_{1}\leq i_{2}.$

And define $\leq^{\prime}$ as the partial order on $\mathcal{K}$, such that
for $(i_{1},l_{1}),(i_{2},l_{2})\in\mathcal{K}$:

(7.2) $(i_{1},l_{1})\leq^{\prime}(i_{2},l_{2})\text{ if and only if
}l_{1}<l_{2}\text{ or }l_{1}=l_{2}\text{ and }i_{1}=i_{2}.$

It is obvious that $(i_{1},l_{1})\leq^{\prime}(i_{2},l_{2})$ implies
$(i_{1},l_{1})\leq(i_{2},l_{2})$. Similarly to the notation in section 5,
denote a multipartition $\lambda\in\mathfrak{P}_{n}$ by:

(1) $\lambda_{(i,l)}$, if the $(i,l)^{th}$ component is the only non-empty one
in $\lambda$. Equivalently, $\lambda$ consists of a column of $n$ nodes in the
$(i,l)^{th}$ position.

(2) $\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$, if the $(i_{1},l_{1})^{th}$
and $(i_{2},l_{2})^{th}$ components in $\lambda$ are non-empty with
$(i_{1},l_{1})<(i_{2},l_{2})$ in $(\mathcal{K},\leq)$ and
$a=a_{i_{1},l_{1}}-a_{i_{2},l_{2}}$ where $a_{i,l}$ is the number of nodes in
the $(i,l)^{th}$ component. It is obvious that $2-n\leq a\leq n-2$ and
$a\equiv n$ ($mod$ 2).

We next define two partial orders $\trianglelefteq$ and
$\trianglelefteq^{\prime}$ on $\mathfrak{P}_{n}$, which respectively
correspond to $\leq$ and $\leq^{\prime}$ on $\mathcal{K}$, as follows:

######  Definition 7.2.

Let $\mathfrak{P}_{n}$ be the set of 3D multipartitions of $n$, each
containing at most two non-empty components. Define the partial order
$\trianglelefteq$ (resp. $\trianglelefteq^{\prime}$) to be the unique partial
order satisfying:

(1) If $\lambda$ and $\mu$ are of the form $\lambda_{(i_{1},l_{1})}$ and
$\mu_{(i_{2},l_{2})}$, then

$\lambda_{(i_{1},l_{1})}\trianglelefteq(resp.\trianglelefteq^{\prime})\mu_{(i_{2},l_{2})}\text{
if and only if }(i_{1},l_{1})\leq(resp.\leq^{\prime})(i_{2},l_{2})\text{ in
}\mathcal{K}.$

(2) If $\lambda$ and $\mu$ are of the form
$\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$ and $\mu_{(i_{3},l_{3})}$, then

$\lambda_{(i_{1},l_{1})(i_{2},l_{2})}^{[a]}\trianglelefteq(resp.\trianglelefteq^{\prime})\mu_{(i_{3},l_{3})}\text{
if and only if }(i_{1},l_{1})\leq(resp.\leq^{\prime})(i_{3},l_{3})\text{ in
}\mathcal{K}.$

(3) If $\lambda$ and $\mu$ are of the form
$\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$ and
$\mu_{(i_{3},l_{3}),(i_{4},l_{4})}^{[b]}$, then
$\lambda_{(i_{1},l_{1})(i_{2},l_{2})}^{[a]}{\trianglelefteq(resp.\trianglelefteq^{\prime})}\mu_{(i_{3},l_{3}),(i_{4},l_{4})}^{[b]}$
if and only if $(i_{1},l_{1})\leq(resp.\leq^{\prime})(i_{3},l_{3})$ ,
$(i_{2},l_{2})\leq(\leq^{\prime})(i_{4},l_{4})$ in $\mathcal{K}$ and one of
the following cases holds:

($i$) $|a|<|b|$;

($ii$) $|a|=|b|$ and $a\geq b$;

($iii$) $|a|=|b|$, $a<b$ and
$(i_{2},l_{2})\leq(resp.\leq^{\prime})(i_{3},l_{3})$.

Theorem 5.2 implies $\\{C_{s,t}^{\lambda}\\}$ is a cellular basis with respect
to the partial order $\trianglelefteq$. We next use the other partial order
$\trianglelefteq^{\prime}$ to define a new cellular structure on
$TL_{r,1,n}(q,\zeta)$. As a direct consequence of the fact that
$\leq^{\prime}$ is covered by $\leq$, $\lambda\trianglelefteq^{\prime}\mu$
implies $\lambda\trianglelefteq\mu$ for any $\lambda,\mu\in\mathfrak{P}_{n}$.
The converse is not always true. Nevertheless, we have the following two
lemmas:

######  Lemma 7.3.

Let $\lambda\triangleleft\mu\in\mathfrak{P}_{n}$ be two 3D multipartitions. If
there exists a standard tableau $t$ of shape $\lambda$ such that

$e(t)=e_{\mu},$

where $e_{\mu}=e(res^{\Lambda}(t^{\mu}))$, then we have
$\lambda\triangleleft^{\prime}\mu\in\mathfrak{P}_{n}$ where
$\triangleleft^{\prime}$ is the finer partial order defined above.

###### Proof.

As $\mu\in\mathfrak{P}_{n}$, it has one of the following three shapes:

(a) $\mu$ has only one non-empty component, that is $\mu=\mu_{(i,l)}$. In this
case, we have

(7.3) $res^{\Lambda}(t^{\mu})=((i,j_{l}),(i,j_{l}-1),\dots,(i,j_{l}-n)).$

So
$res^{\Lambda}(t)=res^{\Lambda}(t^{\mu})=((i,j_{l}),(i,j_{l}-1),\dots,(i,j_{l}-n))$,
which implies that the shape of $t$ is either of the form
$\lambda_{(i,l_{1})}$ or $\lambda_{(i,l_{2}),(i,l_{3})}^{[a]}$. By Definition
7.2, $\lambda\triangleleft\mu$ implies $(i,l_{1})\leq(i,l)$ in the first case
and $(i,l_{2})\leq(i,l)$ in the second. Comparing the two orders in Definition
7.1, we have $(i,l_{1})\leq^{\prime}(i,l)$ in the first case and
$(i,l_{2})\leq^{\prime}(i,l)$ in the second. As a immediate consequence,
$\lambda\triangleleft^{\prime}\mu$ in both cases.

(b) $\mu$ has two non-empty components and is of the form
$\mu=\mu_{(i,l_{1}),(i,l_{2})}^{[a]}$. With the same argument as in the first
case, we have $\lambda=\lambda_{(i,l_{3}),(i,l_{4})}^{[b]}$. As the first
terms in all the indices are the same, the two partial orders on the indices,
$\leq$ and $\leq^{\prime}$, are equivalent. Therefore,
$\lambda_{(i,l_{3}),(i,l_{4})}^{[b]}\triangleleft\mu_{(i,l_{1}),(i,l_{2})}^{[a]}$
implies
$\lambda_{(i,l_{3}),(i,l_{4})}^{[b]}\triangleleft^{\prime}\mu_{(i,l_{1}),(i,l_{2})}^{[a]}$.

(c) $\mu$ is of the form $\mu=\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$. There
are $\dfrac{1}{2}(n+a)$ terms of the form $(i_{1},x)$ and $\dfrac{1}{2}(n-a)$
terms of the form $(i_{2},y)$ in the residue of $t^{\mu}$. As
$res^{\Lambda}(t)=res^{\Lambda}(t^{\mu})$,the number of nodes in $\lambda$
with residue of the form $(i_{1},x)$ is $\dfrac{1}{2}(n+a)$ and that of the
form $(i_{2},y)$ is $\dfrac{1}{2}(n-a)$. The restriction on the parameters
imposed in (2.1) guarantees that the nodes in the first two rows of a 3D
multipartition have different residues. In other words, the nodes in the first
two rows are uniquely determined by their residues. As $t$ is a standard
tableau of shape $\lambda$, the number $1$ in $t$ is in the first row.
Further, $res^{\Lambda}(t)=res^{\Lambda}(t^{\mu})$ implies that
$Res_{t}(1)=Res_{t^{\mu}}(1)$. Therefore,
$t(1)=t^{\mu}(1)=(1,1,(i_{1},l_{1}))$ where $t(1)$ is the node labelled with
$1$ in $t$. Similarly, $t(2)=t^{\mu}(2)=(1,1,(i_{2},l_{2}))$. As $\lambda$
consists of at most two non-empty components, we have
$\lambda=\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}=\mu$ which contradicts
$\lambda\triangleleft\mu$. So there is no standard tableau $t$ of shape
$\lambda$ such that $e(t)=e_{\mu}$ in this case.

Therefore, if $\lambda\triangleleft\mu\in\mathfrak{P}_{n}$ and there exists a
standard tableau $t$ of shape $\lambda$ such that

$e(t)=e_{\mu},$

the 3D multipartition $\mu$ is of the form either $\mu_{(i,l)}$ or
$\mu_{(i,l_{1}),(i,l_{2})}^{[a]}$. In both of these cases,
$\lambda\triangleleft^{\prime}\mu\in\mathfrak{P}_{n}$ where
$\triangleleft^{\prime}$ is the finer partial order in Definition 7.2. ∎

######  Lemma 7.4.

Let $\lambda\triangleleft\mu\in\mathfrak{P}_{n}$ be two 3D multipartitions. We
have $\lambda\triangleleft^{\prime}\mu$ if there exists a standard tableau $t$
of shape $\lambda$ and a Garnir tableau $g$ of shape $\mu$ such that

$e(t)=e(g).$

###### Proof.

The proof of cases (a) and (b) is the same as in the last lemma. We claim that
the third case does not occur. If $\mu$ is of form
$\mu=\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$. There are $\dfrac{1}{2}(n+a)$
terms of the form $(i_{1},x)$ and $\dfrac{1}{2}(n-a)$ terms of the form
$(i_{2},y)$ in the residue of $t^{g}$. Let $i$ be such that $Res_{g}(1)=(i,j)$
and $c$ be the smallest number such that $Res_{g}(c)=(i^{\prime},j^{\prime})$
where $i^{\prime}\neq i$. As $g$ is a Garnir tableau, $s_{k}\circ g$ is
standard for some $1\leq k\leq n-1$. If $k=1$ (or $c$), $1$( or $c$ ) is in
the second row and $2$ (or $c+1$) is in the first row of the same component,
otherwise it is in the first row of $g$. As
$res^{\Lambda}(t)=res^{\Lambda}(g)$, $Res_{t}(1)=Res_{g}(1)$ and
$Res_{t}(c)=Res_{g}(c)$. Both $1$ and $c$ should be in the first row of $t$ as
$t$ is standard. As the nodes in the first two rows are uniquely determined by
their residue, the nodes containing $1$ and $c$ are in the same position in
$t$ and $g$. So we have
$\lambda=\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}=\mu$, which contradicts
$\lambda\triangleleft\mu$. So there is no standard tableau $t$ of shape
$\lambda\triangleleft\mu$ such that $e(t)=e_{g}$ for some Garnir tableau $g$
of shape $\mu$. ∎

By Theorem 5.2, for $\lambda\in\mathfrak{P}_{n}$ and $s,t\in Std(\lambda)$,
let $C_{s,t}^{\lambda}\in TL_{r,1,n}(q,\zeta)$ be the element defined in
$(\ref{eq104})$. Then $\\{C_{s,t}^{\lambda}|\lambda\in\mathfrak{P}_{n},s,t\in
Std(\lambda)\\}$ is a graded cellular basis of $TL_{r,1,n}(q,\zeta)$ with
respect to the partial order $\trianglelefteq$. We next show that this
cellularity still holds with respect to the finer partial order
$\trianglelefteq^{\prime}$, that is,

######  Theorem 7.5.

Let $TL_{r,1,n}(q,\zeta)$ be the specialization of $TL_{r,1,n}$ in section 2.
Let $(\mathfrak{P}_{n},\trianglelefteq^{\prime})$ be the poset defined in
Definition 7.2 and $C_{s,t}^{\lambda}$ be the element defined in
$(\ref{eq104})$. Then $\\{C_{s,t}^{\lambda}|\lambda\in\mathfrak{P}_{n},s,t\in
Std(\lambda)\\}$ is a graded cellular basis of $TL_{r,1,n}(q,\zeta)$ over $R$
with respect to with respect to the cell datum $(\mathfrak{P}_{n},Std,C,deg)$.

###### Proof.

Comparing with the cellular structure in Theorem 5.2, it is enough to show
that for any $a\in TL_{r,1,n}(q,\zeta)$, $\lambda\in\mathfrak{P}_{n}$ and
$s,t\in Std(\lambda)$, we have

(7.4) $aC_{s,t}^{\lambda}=\sum_{s^{\prime}\in
Std(\lambda)}r_{a}(s^{\prime},s)C_{s^{\prime},t}^{\lambda}+\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{a}(s,t,u,v)C_{u,v}^{\mu}$

where $r_{a}(s^{\prime},s)\in R$ does not depend on $t$ and $c_{a}(s,t,u,v)\in
R$. We only need to check the cases where $a$ is in the generating set of
$TL_{r,1,n}(q,\zeta)$ as a KLR algebra.

If $a=e(i)$, $aC_{s,t}^{\lambda}$ is either 0 or $C_{s,t}^{\lambda}$ because
all $e(i)$’s are orthogonal idempotents.

If $a=y_{k}$, the following equation can be obtained by direct calculation
using the generating relations of KLR algebras.

(7.5)
$y_{k}C_{s,t}^{\lambda}=y_{i}\psi_{d(s)}^{*}e_{\lambda}\psi_{d(t)}=\psi_{d(t)}^{*}y_{j}e_{\lambda}\psi_{d(t)}+\sum_{l(V)<l({d(t)})}c(V)\psi_{V}^{*}e_{\lambda}\psi_{d(t)}$

where $c(V)$ is 1 or -1. The first term is in the two-sided ideal generated by
$y_{j}e_{\lambda}$. The other terms can be transformed by Lemma 6.7 in [8] as
follows:

(7.6) $\psi_{V}^{*}e_{\lambda}\psi_{d(t)}=\sum_{s^{\prime}\in
Std(\lambda)}r_{a}(s^{\prime},s)C_{s^{\prime},t}^{\lambda}+\sum_{i=1}^{n}Y_{i}+\sum_{g\in
Gar(\lambda)}D_{g}$

where $Y_{i}$ is in the two-sided ideal generated by $y_{i}e_{\lambda}$ and
$D_{g}$ is in the one generated by $e(g)$ with $g$ running over the Garnir
tableaux of shape $\lambda$.

If $a=\psi_{k}$, Lemma 6.7 in [8] indicates that

(7.7) $aC_{s,t}^{\lambda}=\sum_{s^{\prime}\in
Std(\lambda)}r_{a}(s^{\prime},s)C_{s^{\prime},t}^{\lambda}+\sum_{i=1}^{n}Y_{i}+\sum_{g\in
Gar(\lambda)}D_{g}$

where $Y_{i}$ is in the two-sided ideal generated by $y_{i}e_{\lambda}$ and
$D_{g}$ is in the one generated by $e(g)$ with $g$ running over the Garnir
tableaux of shape $\lambda$.

Therefore, it is enough to show that

(7.8) $y_{i}e_{\lambda}=\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{i}(u,v)C_{u,v}^{\mu}$

for $1\leq i\leq n$, and

(7.9) $e(g)=\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{g}(u,v)C_{u,v}^{\mu}$

for $g\in Gar(\lambda)$. We treat $y_{i}e_{\lambda}$ first. By Lemma 17 and
Lemma 37 in [10], we have

(7.10) $y_{i}e_{\lambda}=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{i}(u,v)C_{u,v}^{\mu}.$

As $y_{i}e_{\lambda}=e_{\lambda}y_{i}$ and $e_{\lambda}$ is an idempotent, we
have

$\displaystyle y_{i}e_{\lambda}$ $\displaystyle=e_{\lambda}y_{i}e_{\lambda}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{i}(u,v)e_{\lambda}C_{u,v}^{\mu}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{i}(u,v)e_{\lambda}\psi_{d(u)}^{*}e_{\mu}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{i}(u,v)e_{\lambda}e(u)\psi_{d(u)}^{*}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu),e_{\lambda}=e(u)}c_{i}(u,v)e(u)\psi_{d(u)}^{*}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu),e_{\lambda}=e(u)}c_{i}(u,v)C_{u,v}^{\mu}$
$\displaystyle=\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{i}(u,v)C_{u,v}^{\mu}$

where the last step is by Lemma 7.3 and the coefficients $c_{i}(u,v)$ are zero
if $e_{\lambda}\neq e(u)$ in the last summand.

For $e(g)$, Lemma 35 in [10] implies that

$e(g)=\sum_{\mu\triangleleft\lambda,u,v\in Std(\mu)}c_{g}(u,v)C_{u,v}^{\mu}.$

As $e(g)$ is an idempotent, we have

$\displaystyle e(g)$ $\displaystyle=e(g)^{2}$
$\displaystyle=e(g)\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{g}(u,v)C_{u,v}^{\mu}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{g}(u,v)e(g)\psi_{d(u)}^{*}e_{\mu}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu)}c_{g}(u,v)e(g)e(u)\psi_{d(u)}^{*}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu),e(u)=e(g)}c_{g}(u,v)e(u)\psi_{d(u)}^{*}\psi_{d(v)}$
$\displaystyle=\sum_{\mu\triangleleft\lambda,u,v\in
Std(\mu),e(u)=e(g)}c_{g}(u,v)C_{u,v}^{\mu}$
$\displaystyle=\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{g}(u,v)C_{u,v}^{\mu}$

where the last step is by Lemma 7.4 and the coefficients $c_{g}(u,v)$ are zero
if $e_{g}\neq e(u)$ in the last sum.

So for any $a\in TL_{r,1,n}(q,\zeta)$, $\lambda\in\mathfrak{P}_{n}$ and
$s,t\in Std(\lambda)$, we have

(7.11) $aC_{s,t}^{\lambda}=\sum_{s^{\prime}\in
Std(\lambda)}r_{a}(s^{\prime},s)C_{s^{\prime},t}^{\lambda}+\sum_{\mu\triangleleft^{\prime}\lambda,u,v\in
Std(\mu)}c_{a}(s,t,u,v)C_{u,v}^{\mu}.$

Therefore, $\\{C_{s,t}^{\lambda}|\lambda\in\mathfrak{P}_{n},s,t\in
Std(\lambda)\\}$ is a graded cellular basis of $TL_{r,1,n}(q,\zeta)$ over $R$
with respect to $\trianglelefteq^{\prime}$. ∎

Now consider the corresponding poset automorphism $\sigma_{\mathfrak{P}_{n}}$
on the poset $(\mathfrak{P}_{n},\trianglelefteq^{\prime})$. Let
$\sigma_{\mathfrak{P}_{n}}:\mathfrak{P}_{n}\mapsto\mathfrak{P}_{n}$ be the
bijective map given by:

$\displaystyle\sigma_{\mathfrak{P}_{n}}(\lambda_{(i,l)})$
$\displaystyle=\lambda_{(i-1,l)},$
$\displaystyle\sigma_{\mathfrak{P}_{n}}(\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]})$
$\displaystyle=\lambda_{(i_{1}-1,l_{1}),(i_{2}-1,l_{2})}^{[a]},$

where $i-1=p-1$ when $i=0$. In other words, $\sigma_{\mathfrak{P}_{n}}$ shifts
the 3D-multipartitions forward by one column. We may now prove:

######  Lemma 7.6.

$\sigma_{\mathfrak{P}_{n}}$ is a poset automorphism of
$(\mathfrak{P}_{n},\trianglelefteq^{\prime})$.

###### Proof.

The bijectivity is obvious. We prove that $\sigma_{\mathfrak{P}_{n}}$
preserves the partial order $\trianglelefteq^{\prime}$. For any
$(i_{1},l_{1})<^{\prime}(i_{2},l_{2})\in\mathcal{K}$, we have $l_{1}<l_{2}$
which implies that $(i_{1}-1,l_{1})<^{\prime}(i_{2}-1,l_{2})$. Therefore,
$(i_{1},l_{1})\leq^{\prime}(i_{2},l_{2})$ implies
$(i_{1}-1,l_{1})\leq^{\prime}(i_{2}-1,l_{2})$. By Definition 7.2,
$\lambda\triangleleft^{\prime}\mu\in\mathfrak{P}_{n}$ implies
$\sigma_{\mathfrak{P}_{n}}(\lambda)\triangleleft^{\prime}\sigma_{\mathfrak{P}_{n}}(\mu)$.
Therefore, $\sigma_{\mathfrak{P}_{n}}$ is a poset automorphism of
$(\mathfrak{P}_{n},\trianglelefteq^{\prime})$. ∎

For $t\in Std(\lambda)$ where $\lambda\in\mathfrak{P}_{n}$, define
$\sigma_{\mathcal{T}}(t)$ as the standard tableau with shape
$\sigma_{\mathfrak{P}_{n}}(\lambda)$ and all the numbers in the same relative
positions as those in $t$.

######  Theorem 7.7.

Let $\sigma_{TL}$ be the automorphism of $TL_{r,1,n}(q,\zeta)$ defined by
$(\ref{eq817})$, $\sigma_{\mathfrak{P}_{n}}$ and $\sigma_{\mathcal{T}}$ be the
corresponding automorphisms described above. Then
$\sigma=(\sigma_{TL},\sigma_{\mathfrak{P}_{n}},\sigma_{\mathcal{T}})$ is a
shift automorphism ($cf.$ Definition 6.4) with respect to the cell datum
$(\mathfrak{P}_{n},Std,C,deg)$ of $TL_{r,1,n}(q,\zeta)$ where
$\mathfrak{P}_{n}$ is the poset with $\trianglelefteq^{\prime}$ (cf. Theorem
7.5).

###### Proof.

By the argument above, $\sigma_{TL}$, $\sigma_{\mathfrak{P}_{n}}$ and
$\sigma_{\mathcal{T}}$ are well defined. The conditions ($a$) and ($b$) in
Definition 6.4 can be checked directly. For any $t\in Std(\lambda)$,
$\sigma_{\mathcal{T}}^{k}(t)=t$ if and only if $p|k$. So (c) holds for all
$\lambda\in\mathfrak{P}_{n}$. ∎

We next identify the skew cell datum of the point-wise fixed subalgebra
$TL_{r,p,n}$ under the shift automorphism $\sigma$. Let $\equiv$ be the
equivalence relation on $\mathfrak{P}_{n}$ such that

(7.12) $\displaystyle\lambda_{(i,l)}$ $\displaystyle\equiv\lambda_{(i+k,l)}$
$\displaystyle\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$
$\displaystyle\equiv\mu_{(i_{1}+k,l_{1}),(i_{2}+k,l_{2})}^{[a]},$

for all $k\in I$ and
$\lambda_{(i,l)},\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}\in\mathfrak{P}_{n}$.
Notice that $\lambda\equiv\sigma_{\mathfrak{P}_{n}}(\lambda)$ for all
$\lambda\in\mathfrak{P}_{n}$. Denote by $[\lambda]$ the equivalence class in
$\mathfrak{P}_{n}$ containing $\lambda$. Then $[\lambda]$ is the orbit of
$\lambda$ under $\sigma_{\mathfrak{P}_{n}}$. Define the set of orbits

(7.13) $\mathfrak{P}_{n,p}:=\\{[\lambda]|\lambda\in\mathfrak{P}_{n}\\}.$

Define $[\lambda]\trianglelefteq_{p}[\mu]\in\mathfrak{P}_{n,p}$ if and only if
there exist $\lambda_{1}\equiv\lambda$ and $\mu_{1}\equiv\mu$ such that
$\lambda_{1}\trianglelefteq^{\prime}\mu_{1}\in\mathfrak{P}_{n}$. We should
remind readers that $[\lambda]\trianglelefteq_{p}[\mu]$ does not imply
$\lambda\trianglelefteq^{\prime}\mu$, they may not be comparable with respect
to $\trianglelefteq^{\prime}$. We next give a counterexample.

######  Example 7.8.

Let $r=9$, $p=3$ and $n=5$. Then $\lambda_{(1,1),(2,3)}^{[1]}$ and
$\mu_{(0,1)}$ are two 3D multipartitions in $\mathfrak{P}_{5}$. We have

$[\lambda_{(1,1),(2,3)}^{[1]}]\trianglelefteq_{p}[\mu_{(0,1)}]$

because
$\lambda_{(1,1),(2,3)}^{[1]}\equiv\lambda_{(0,1),(1,3)}^{[1]}\trianglelefteq^{\prime}\mu_{(0,1)}$
according to $(2)$ in Definition 7.2. But $\lambda_{(1,1),(2,3)}^{[1]}$ and
$\mu_{(0,1)}$ are incomparable with respect to $\trianglelefteq^{\prime}$.

By definition, for any $\lambda\in\mathfrak{P}_{n}$ and $t\in Std(\lambda)$,
the size of the $\langle\sigma_{\mathfrak{P}_{n}}\rangle$-orbit containing
$\lambda$ and the size of the $\langle\sigma_{\mathcal{T}}\rangle$-orbit
containing $t$ are both $p$. Therefore, the $\sigma_{\lambda}$ in the general
theory before Theorem 6.5 is identity which implies that the poset involution
$\iota_{p}=id_{\mathfrak{P}_{n}}$. This implies the skew cellular structure we
construct is actually cellular.

In analogy to the equivalence relation among the multipartitions, we define an
equivalence relation $\equiv$ on
$\mathcal{T}=\bigsqcup_{\lambda\in\mathfrak{P}}T(\lambda)$ as follows:

For $s,t\in\mathcal{T}$, we say $s\equiv t$ if and only if $shape(s)\equiv
shape(t)$ and the numbers in $s$ are in the same relative positions as those
in $t$. Denote by $[t]$ the equivalence class containing $t$ in $\mathcal{T}$.
For $s\equiv t\in\mathcal{T}$, as the numbers in $s$ are in the same relative
positions as those in $t$, we have $d(s)=d(t)$ and $deg(s)=deg(t)$, where
$d(s)$ is the element in $\mathfrak{S}_{n}$ transforming $s$ to the unique
standard tableau $t^{\lambda}$. Therefore, the element $d([t])$ can be defined
as $d(s)$ and $deg_{p}([t])$ can be defined as $deg(s)$ for any $s\equiv t$.
For $[\lambda]\in\mathfrak{P}_{n,p}$, define:

(7.14) $T_{p}([\lambda]):=\\{[t]|t\in Std(\lambda)\\}.$

Finally, for $[s],[t]\in T_{p}([\lambda])$, define

(7.15)
$C_{p}^{[\lambda]}([s],[t])=\psi_{d([s])}^{*}(\sum_{\mu\equiv\lambda}e_{\mu})\psi_{d([t])}.$

The following Theorem is a direct consequence of Theorem 6.5:

######  Theorem 7.9.

Let $TL_{r,p,n}$ be the Temperley-Lieb algebra of type $G(r,p,n)$ defined in
Definition 4.2, $\mathfrak{P}_{n,p}$ be the poset of equivalence classes of
multipartitions defined in (7.13 and $T_{p},C_{p},deg_{p}$ be as described
above. Then $TL_{r,p,n}$ is a skew cellular algebra with skew cell datum
$(\mathfrak{P}_{n,p},id,T_{p},C_{p},deg_{p})$. Moreover, since the poset
involution is trivial, $TL_{r,p,n}$ is a cellular algebra with cell datum
$(\mathfrak{P}_{n,p},T_{p},C_{p},deg_{p})$.

##  8\. The representations of $TL_{r,p,n}$

In this section, we study the representations of $TL_{r,p,n}$ from the
cellular point of view. We first calculate the dimensions of the cell modules
and define reducible and irreducible elements in the poset
$\mathfrak{P}_{n,p}$. These names come from the fact that the cell modules
corresponding to the irreducible elements are simple. Then we show that the
cell modules of $TL_{r,p,n}$ corresponding to reducible elements in
$\mathfrak{P}_{n,p}$ can be regarded as cell modules of some $TL_{d,1,n}$
where $d=\frac{r}{p}$ with special parameters. Finally we calculate the
decomposition numbers of the reducible cell modules.

###  8.1. The cell modules of $TL_{r,p,n}$ and irreducible cells

In this subsection, we give a description of the cell modules of $TL_{r,p,n}$.
Let $[\lambda]\in\mathfrak{P}_{n,p}$ be an equivalence class of
multipartitions in $\mathfrak{P}_{n}$ with respect to the equivalence relation
$\equiv$ defined in (7.12). The (left) cell module of $TL_{r,p,n}$
corresponding to $[\lambda]$, denoted $W([\lambda])$, is the free module over
$R$ with basis $\\{C_{[s]}|[s]\in T([\lambda])\\}$. The $TL_{r,p,n}$-action is
defined by

(8.1) $aC_{[s]}=\sum_{[s^{\prime}]\in
T([\lambda])}r_{a}([s^{\prime}],[s])C_{[s^{\prime}]}$

for all $a\in TL_{r,p,n}$ and $[s]\in T([\lambda])$, where
$r_{a}([s^{\prime}],[s])$ is the coefficient uniquely defined by

(8.2) $aC_{[s],[t]}^{[\lambda]}=\sum_{[s^{\prime}]\in
T([\lambda])}r_{a}([s^{\prime}],[s])C_{[s^{\prime}],[t]}^{[\lambda]}+\sum_{[\mu]\triangleleft_{p}[\lambda],[u],[v]\in
T([\mu])}dC_{[u],[v]}^{[\mu]}.$

The right cell modules of $TL_{r,p,n}$ are defined similarly. We next
calculate the dimensions of the cell modules. For
$[\lambda]\in\mathfrak{P}_{n,p}$, the dimension of the cell module
$W([\lambda])$ equals to the number of standard tableaux of shape $\lambda$.
Therefore, for
$\lambda_{(i,l)},\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}\in\mathfrak{P}_{n}$,
we have

$\displaystyle dim(W([\lambda_{(i,l)}]))=$ $\displaystyle 1$ $\displaystyle
dim(W([\mu_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}]))=$
$\displaystyle\binom{n}{\frac{n-a}{2}}.$

Further, denote by $TL_{\triangleleft_{p}[\lambda]}$ and
$TL_{\trianglelefteq_{p}[\lambda]}$ the subspaces spanned by
$\\{C_{[s],[t]}^{[\mu]}|[\mu]\triangleleft_{p}[\lambda]\\}$ and
$\\{C_{[s],[t]}^{[\mu]}|[\mu]\trianglelefteq_{p}[\lambda]\\}$ respectively.
These are $TL_{r,p,n}$-bimodules by definition of a cell datum. Define the
$TL_{r,p,n}$-bimodule corresponding to $[\lambda]$ as

(8.3)
$TL([\lambda]):=TL_{\trianglelefteq_{p}[\lambda]}/TL_{\triangleleft_{p}[\lambda]}.$

Then we have

(8.4) $dim(TL([\lambda]))=\begin{cases}1&\textit{ if
}\lambda=\lambda_{(i,l)};\\\ \binom{n}{\frac{n-a}{2}}^{2}&\textit{ if
}\lambda=\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}.\end{cases}$

This permits calculation of the dimension of $TL_{r,p,n}$:

$\displaystyle dim(TL_{r,p,n})$
$\displaystyle=\sum_{[\lambda]\in\mathfrak{P}_{n,p}}dim(TL([\lambda]))$
$\displaystyle=\sum_{[\lambda]\in\mathfrak{P}_{n,p}(1)}dim(TL([\lambda]))+\sum_{[\lambda]\in\mathfrak{P}_{n,p}(2)}dim(TL([\lambda]))$
$\displaystyle=d+\frac{1}{p}\binom{r}{2}\left(\sum_{a_{1}=1}^{n-1}\binom{n}{a_{1}}^{2}\right)$
$\displaystyle=d+\frac{1}{p}\binom{r}{2}\left(\binom{2n}{n}-2\right)$
$\displaystyle=\frac{1}{p}\left(\binom{r}{2}\binom{2n}{n}-r^{2}+2r\right)$

where $\mathfrak{P}_{n,p}(i)$ consists of the equivalence classes of 3D
multipartitions with exactly $i$ non-empty components. By Definition 5.1, the
reducibility of $\lambda$ is equivalent to that of any
$\lambda^{\prime}\in[\lambda]$. So we can say $[\lambda]\in\mathfrak{P}_{n,k}$
is reducible if $\lambda\in\mathfrak{P}_{n}$ is.

The poset $\mathfrak{P}_{n,p}$ into two parts, viz. those subsets
$\mathfrak{P}_{n,p}^{0}$ consisting of the reducible orbits and the
$\mathfrak{P}_{n,p}^{1}$ consisting of the irreducible orbits. We will show
that the cell modules corresponding to these two parts have different
properties. We begin by showing that the bimodules corresponding to the
irreducible orbits are irreducible two-sided ideals of $TL_{r,p,n}$.

######  Lemma 8.1.

Let $[\lambda]\in\mathfrak{P}_{n,p}$ be irreducible. For any
$[\mu]\neq[\lambda]$, $[s],[t]\in T([\mu])$ and $[u],[v]\in T([\lambda])$, we
have

$C_{p}^{[\mu]}([s],[t])C_{p}^{[\lambda]}([u],[v])=0.$

###### Proof.

By (7.15), we have

$\displaystyle C_{p}^{[\mu]}([s],[t])C_{p}^{[\lambda]}([u],[v])$
$\displaystyle=$
$\displaystyle\psi_{d([s])}^{*}(\sum_{\mu^{\prime}\equiv\mu}e_{\mu^{\prime}})\psi_{d([t])}\psi_{d([u])}^{*}(\sum_{\lambda^{\prime}\equiv\lambda}e_{\lambda^{\prime}})\psi_{d([v])}$
$\displaystyle=$
$\displaystyle\psi_{d([s])}^{*}\psi_{d([t])}(\sum_{t^{\prime}\equiv
t}e(t^{\prime}))(\sum_{u^{\prime}\equiv
u}e(u^{\prime}))\psi_{d([u])}^{*}\psi_{d([v])}.$

It suffices to show

$(\sum_{t^{\prime}\equiv t}e(t^{\prime}))(\sum_{u^{\prime}\equiv
u}e(u^{\prime}))=0.$

As the distinct $e(\mathbf{i})$ are mutually orthogonal, we need only show
that $e(t)\neq e(u)$ for all $u\in Std(\lambda)$ and $t\in Std(\mu)$ with
$\mu\not\equiv\lambda$. Let $\lambda$ be of the form
$\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$ with $i_{1}\neq i_{2}$ and
$e(u)=e(k_{1},k_{2},\dots,k_{n})$. As $u$ is a standard tableau of shape
$\lambda$, $\frac{n+a}{2}$ of the $k$’s are of the form $(i_{1},x)$ and
$\frac{n-a}{2}$ $k$’s are of the form $(i_{2},x)$. If $e(t)=e(u)$, there
should be $\frac{n+a}{2}$ nodes in $\mu$ with residues of the form $(i_{1},x)$
and $\frac{n-a}{2}$ nodes in $\mu$ with residues of the form $(i_{2},x)$. As
$\mu$ has at most 2 non-empty components,
$\mu=\mu_{(i_{1},l_{3}),(i_{2},l_{4})}^{[a]}$.

We shall show that $l_{3}=l_{1}$ and $l_{4}=l_{2}$. Denote by $k_{a}$ the
first $k$ of the form $(i_{1},x)$ and $k_{b}$ the first of the form
$(i_{2},x)$. As $u$ is a standard tableau, the numbers $a$ and $b$ are in the
first row. For the same reason, the numbers $a$ and $b$ are in the first row
of $t$. According to the restriction on the parameters in (2.1), the nodes in
the first row are uniquely determined by their residues. Therefore, $a$ and
$b$ are in the same position in $u$ and $t$, which implies that $l_{3}=l_{1}$
and $l_{4}=l_{2}$. Thus $\mu=\lambda$, contradicting our assumption that
$\mu\not\equiv\lambda$. So $e(t)\neq e(u)$, so that $e(t)e(u)=0$.

So for any standard tableau $t^{\prime}\equiv t$, we have
$Shape(t^{\prime})\equiv\mu\not\equiv\lambda$. Thus,

$\left(\sum_{t^{\prime}\equiv t}e(t^{\prime})\right)e(u^{\prime})=0.$

But for any standard tableau $u^{\prime}\equiv u$,
$Shape(u^{\prime})\equiv\lambda\not\equiv\mu^{\prime}$. We have

$\left(\sum_{t^{\prime}\equiv
t}e(t^{\prime})\right)\left(\sum_{u^{\prime}\equiv u}e(u^{\prime})\right)=0.$

Therefore, $C_{p}^{[\mu]}([s],[t])C_{p}^{[\lambda]}([u],[v])=0$. ∎

By applying the anti-involution of $TL_{r,p,n}$ which fixes the KLR
generators, we further obtain:

######  Lemma 8.2.

Let $[\lambda]\in\mathfrak{P}_{n,p}$ be irreducible. For any
$[\mu]\neq[\lambda]$, $[s],[t]\in T([\mu])$ and $[u],[v]\in T([\lambda])$, we
have

$C_{p}^{[\lambda]}([u],[v])C_{p}^{[\mu]}([s],[t])=0.$

The following proposition is a direct consequence of the two lemmas above:

######  Proposition 8.3.

Let $[\lambda]\in\mathfrak{P}_{n,p}$ be irreducible and $W([\lambda])$ be the
cell module of $TL_{r,p,n}$ corresponding to $\lambda$. For any
$[\mu]\neq[\lambda]$ and $[s],[t]\in T([\mu])$, we have

$C_{p}^{[\mu]}([s],[t])W([\lambda])=0.$

The following proposition is where the name “irreducible multipartition” comes
from. Let $L([\lambda])$ be the simple module corresponding to
$[\lambda]\in\mathfrak{P}_{n,p}$. We have:

######  Proposition 8.4.

Let $[\lambda]\in\mathfrak{P}_{n,p}$ be an irreducible multipartition, then
$W([\lambda])=L([\lambda])$. In other words, the cell module $W([\lambda])$ is
simple.

###### Proof.

Let $[\mu]\in\mathfrak{P}_{n,p}$ be a equivalence class different from
$[\lambda]$. By Proposition 8.3, $C_{p}^{[\mu]}([s],[t])W([\lambda])=0$ for
all $[s],[t]\in T([\mu])$. On the other hand, since
$L([\mu])=W([\mu])/rad(\phi_{[\mu]})$, for any $x\in L([\mu])$, there exist
$[s],[t]\in T([\mu])$ such that $C_{p}^{[\mu]}([s],[t])x\neq 0$. Therefore,we
have

$[W([\lambda]):L([\mu])]=0$

for all $[\mu]\neq[\lambda]$. So $W([\lambda])$ is simple and
$W([\lambda])=L([\lambda])$. ∎

Note that the converse of the proposition is not true. As a counterexample,
the multipartitions with exactly one non-empty component are reducible by
definition, but the corresponding cell modules are always simple since they
are of rank 1.

###  8.2. The reducible cells and their decomposition numbers

As shown in the last subsection, the cell modules corresponding to irreducible
orbits are simple. To determine the decomposition numbers for all the cell
modules, we concentrate on those corresponding to reducible orbits in this
subsection.

Let $\Lambda$ be the dominant weight we choose for the specialisation
$H_{n}^{\Lambda}(q,\zeta)$ ($cf.$ 2.12) and $TL_{r,1,n}^{\Lambda}(q,\zeta)$ be
the corresponding Temperley-Lieb algebra. For $d=\frac{r}{p}$, let
$\Lambda^{0}$ be the dominant weight of length $d$ such that
$(\Lambda^{0},\alpha_{(0,j)})=(\Lambda,\alpha_{(0,j)})$ for all $j\in J$. Let
$\mathcal{R}_{n}^{\Lambda^{0}}$ be the cyclotomic KLR algebra corresponding to
$\Lambda^{0}$ and $TL_{d,1,n}^{\Lambda^{0}}$ be the Temperley-Lieb quotient we
define in Theorem 3.2. The following lemma indicates that
$TL_{d,1,n}^{\Lambda^{0}}$ is a quotient of $TL_{r,1,n}^{\Lambda}(q,\zeta)$:

######  Lemma 8.5.

Let $TL_{r,1,n}^{\Lambda}(q,\zeta)$ be the Temperley-Lieb quotient of
$H_{n}^{\Lambda}(q,\zeta)$ (cf. Theorem 3.2) and $TL_{d,1,n}^{\Lambda^{0}}$ be
the Temperley-Lieb algebra as defined above. Then $TL_{d,1,n}^{\Lambda^{0}}$
is a quotient of $TL_{r,1,n}^{\Lambda}(q,\zeta)$ by the two-sided ideal
generated by all
$e(\mathbf{i})=e(\mathbf{i}_{1},\mathbf{i}_{2},\dots,\mathbf{i}_{n})$ where
$\mathbf{i}_{j}$ are in the vertex set
$K=\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/e\mathbb{Z}$ and
$(\Lambda^{0},\alpha_{\mathbf{i}_{1}})=0$.

###### Proof.

Denote by $\mathcal{R}_{n}^{\Lambda^{0}}$ the KLR algebra with respect to the
dominant weight $\Lambda^{0}$. Let $\mathcal{R}_{n}^{\Lambda}$ be the KLR
algebra isomorphic to $H_{n}^{\Lambda}(q,\zeta)$ (cf.Theorem 1.1 in [3]). By
comparing the generators and relations, we notice that
$\mathcal{R}_{n}^{\Lambda^{0}}$ is a quotient of $\mathcal{R}_{n}^{\Lambda}$
by the two-sided ideal generated by all $e(\mathbf{i})$ such that
$(\Lambda^{0},\alpha_{\mathbf{i}_{1}})=0$ where $\mathbf{i}\in K^{n}$ and
$\alpha_{\mathbf{i}_{1}}$ is the simple root in section 2.

By Theorem 3.2, $TL_{r,1,n}^{\Lambda}(q,\zeta)$ is a quotient of
$\mathcal{R}_{n}^{\Lambda}$ by the two-sided ideal $\mathcal{J}_{n}(\Lambda)$
in (3.2), and $TL_{d,1,n}^{\Lambda^{0}}$ is a quotient of
$\mathcal{R}_{n}^{\Lambda^{0}}$ by the two-sided ideal
$\mathcal{J}_{n}(\Lambda^{0})$ in (3.2). We only need to show that generators
of $\mathcal{J}_{n}(\Lambda)$ in (3.2) are in the two-sided ideal of
$\mathcal{R}_{n}^{\Lambda}$ generated by all $e(\mathbf{i})$ such that
$(\Lambda^{0},\alpha_{\mathbf{i}_{1}})=0$ and $\mathcal{J}_{n}(\Lambda^{0})$.
This is obvious except for the idempotent $e(i_{1},i_{2},i_{3},\mathbf{i})$
where $(\Lambda^{0},\alpha_{i_{1}})>0$ but $(\Lambda^{0},\alpha_{i_{2}})=0$ or
$(\Lambda^{0},\alpha_{i_{3}})=0$. Without losing generality, assume
$(\Lambda^{0},\alpha_{i_{2}})=0$. In this case, we use the fact,
$e(i_{1},i_{2},i_{3},\mathbf{i})=\psi_{1}e(i_{2},i_{1},i_{3},\mathbf{i})\psi_{1}$,
to show that $e(i_{1},i_{2},i_{3},\mathbf{i})$ is in the two-sided ideal
generated by all $e(\mathbf{i})$ such that
$(\Lambda^{0},\alpha_{\mathbf{i}_{1}})=0$.

Therefore, the ideal of $\mathcal{R}_{n}^{\Lambda}$ corresponding to
$TL_{r,1,n}^{\Lambda}(q,\zeta)$ is in that corresponding to
$TL_{d,1,n}^{\Lambda^{0}}$. So $TL_{d,1,n}^{\Lambda^{0}}$ is a quotient of
$TL_{r,1,n}^{\Lambda}(q,\zeta)$. Moreover, we have:

(8.5) $\displaystyle TL_{d,1,n}^{\Lambda^{0}}$
$\displaystyle=\mathcal{R}_{n}^{\Lambda^{0}}/\mathcal{J}_{n}(\Lambda^{0})$
$\displaystyle=\mathcal{R}_{n}^{\Lambda}/\langle
e(\mathbf{i}),\mathcal{J}_{n}(\Lambda^{0})\rangle$
$\displaystyle=\mathcal{R}_{n}^{\Lambda}/\langle
e(\mathbf{i}),\mathcal{J}_{n}(\Lambda)\rangle$
$\displaystyle=TL_{r,1,n}^{\Lambda}(q,\zeta)/\langle e(\mathbf{i})\rangle,$

where $\mathbf{i}$ runs over all $\mathbf{i}\in K^{n}$ such that
$(\Lambda^{0},\alpha_{\mathbf{i}_{1}})=0$. ∎

Let $f:TL_{r,1,n}^{\Lambda}(q,\zeta)\to TL_{d,1,n}^{\Lambda^{0}}$ be the
natural quotient map. We have

(8.6) $\displaystyle f(y_{k})$ $\displaystyle=y_{k},\textit{ for }1\leq k\leq
n;$ $\displaystyle f(\psi_{l})$ $\displaystyle=\psi_{l},\textit{ for }1\leq
l\leq n-1;$ $\displaystyle f(e_{\lambda})$
$\displaystyle=\begin{cases}e_{\lambda},\textit{ if
}\lambda\in\mathfrak{P}_{n}^{1};\\\ 0,\textit{ if
}\lambda\in\mathfrak{P}_{n}-\mathfrak{P}_{n}^{1},\end{cases}$

where $\mathfrak{P}_{n}$ is the set of 3D multipartitions of $n$ with at most
two non-empty components, and $\mathfrak{P}_{n}^{1}$ is the subset consisting
of those 3D multipartitions with all the non-empty components in the first
column of the floor table. Since $TL_{r,p,n}^{\Lambda}$ is a subalgebra of
$TL_{r,1,n}^{\Lambda}(q,\zeta)$, $f(TL_{r,p,n}^{\Lambda})$ is a subalgebra of
$TL_{d,1,n}^{\Lambda^{0}}$. We show the map $f$ is surjective. By Theorem 5.2,
$TL_{d,1,n}^{\Lambda^{0}}$ has a cellular basis $\\{C_{s,t}^{\lambda}\\}$
corresponding to the cell datum
$(\mathfrak{B}_{n}^{(d)}-\mathfrak{D}_{n}^{(d)},Std,C,deg)$. For any
$\lambda_{0}\in\mathfrak{B}_{n}^{(d)}-\mathfrak{D}_{n}^{(d)}$ and
$s_{0},t_{0}\in Std(\lambda_{0})$, let
$j_{1},j_{2},\dots,j_{n}\in\mathbb{Z}/e\mathbb{Z}$ be such that
$e_{\lambda_{0}}=e((0,j_{1}),(0,j_{2}),\dots,(0,j_{n}))$. We have
$C_{p}^{[\lambda_{0}]}([s_{0}],[t_{0}])\in TL_{r,p,n}^{\Lambda}$ and

(8.7) $\displaystyle f(C_{p}^{[\lambda_{0}]}([s_{0}],[t_{0}]))$
$\displaystyle=f(\psi_{d([s_{0}])}^{*}(\sum_{\lambda^{\prime}\equiv\lambda_{0}}e_{\lambda^{\prime}})\psi_{d([t_{0}])})$
$\displaystyle=\psi_{d([s_{0}])}^{*}f(\sum_{\lambda^{\prime}\equiv\lambda_{0}}e_{\lambda^{\prime}})\psi_{d([t_{0}])}$
$\displaystyle=\psi_{d([s_{0}])}^{*}f(\sum_{i=0}^{p-1}e((i,j_{1}),(i,j_{2}),\dots,(i,j_{n})))\psi_{d([t_{0}])}$
$\displaystyle=\psi_{d([s_{0}])}^{*}e((0,j_{1}),(0,j_{2}),\dots,(0,j_{n}))\psi_{d([t_{0}])}$
$\displaystyle=C_{s_{0},t_{0}}^{\lambda_{0}}.$

Therefore, the restriction of the natural map $f$ on $TL_{r,p,n}^{\Lambda}$ is
surjective, that is $f(TL_{r,p,n}^{\Lambda})=TL_{d,1,n}^{\Lambda^{0}}$.

The next lemma shows that the kernel of this map is the subspace spanned by
the elements corresponding to irreducible orbits in the cellular basis in
$(\ref{cbrpn})$.

######  Lemma 8.6.

Let $TL_{r,p,n}^{1}$ be the subspace spanned by all elements
$C_{p}^{[\lambda]}([s],[t])$ in the cellular basis of $TL_{r,p,n}^{\Lambda}$
in (7.15), where $[\lambda]$ is irreducible. Then $TL_{r,p,n}^{1}$ is the
kernel of $f|_{TL_{r,p,n}^{\Lambda}}$ where $f$ is the natural quotient map
defined above. Moreover, $TL_{r,p,n}^{1}$ is an ideal of
$TL_{r,p,n}^{\Lambda}$ and

(8.8) $TL_{d,1,n}^{\Lambda^{0}}\cong TL_{r,p,n}^{\Lambda}/TL_{r,p,n}^{1}$

###### Proof.

We first show that $f(C_{p}^{[\lambda]}([u],[v]))=0$ if $[\lambda]$ is
irreducible. Using the notation introduced before Definition 7.2, let
$\lambda=\lambda_{(i_{1},l_{1}),(i_{2},l_{2})}^{[a]}$. As $\lambda$ is
irreducible, we have $i_{1}\neq i_{2}$ and
$e_{\lambda}=e((i_{1},j_{l_{1}}),(i_{2},j_{l_{2}}),\mathbf{k})$ with
$\mathbf{k}\in K^{n-2}$. We have

(8.9) $\displaystyle e_{[\lambda]}$
$\displaystyle=\sum_{\lambda^{\prime}\equiv\lambda}e_{\lambda}$
$\displaystyle=\sum_{i=0}^{p-1}e((i_{1}-i,j_{l_{1}}),(i_{2}-i,j_{l_{2}}),\sigma_{1}^{i}(\mathbf{k})),$

where $\sigma_{1}$ is the map defined in (2.16). For $i\neq i_{1}$,
$e((i_{1}-i,j_{l_{1}}),(i_{2}-i,j_{l_{2}}),\sigma_{1}^{i}(\mathbf{k}))$ is
obviously in the ideal described in Lemma 8.5. If $i=i_{1}$, we have
$i_{2}-i\neq 0$ and
$e((i_{1}-i,j_{l_{1}}),(i_{2}-i,j_{l_{2}}),\sigma_{1}^{i}(\mathbf{k})=\psi_{1}e((i_{2}-i,j_{l_{2}}),(i_{1}-i,j_{l_{1}}),\sigma_{1}^{i}(\mathbf{k})\psi_{1}$.
So $e((i_{1}-i,j_{l_{1}}),(i_{2}-i,j_{l_{2}}),\sigma_{1}^{i}(\mathbf{k})$ is
in that ideal. Then we have $e_{[\lambda]}\in ker(f)$, thus
$C_{p}^{[\lambda]}([u],[v])\in ker(f)$ for any $u,v\in Std(\lambda)$.
Therefore, we have $TL_{r,p,n}^{1}\subseteq ker(f)$.

On the other hand, for any reducible orbit $[\mu]\in\mathfrak{P}_{n,p}$ and
$[s],[t]\in T_{p}([\mu])$(cf.7.14), equation (8.7) shows that
$C_{p}^{[\mu]}([s],[t])$ is not in the kernel of $f$. So
$TL_{r,p,n}^{1}=ker(f|_{TL_{r,p,n}^{\Lambda}})$. Since
$f|_{TL_{r,p,n}^{\Lambda}}$ is a surjective algebra homomorphism,
$TL_{r,p,n}^{1}$ is an ideal of $TL_{r,p,n}^{\Lambda}$ and

$TL_{d,1,n}^{\Lambda^{0}}\cong TL_{r,p,n}^{\Lambda}/TL_{r,p,n}^{1}.$

∎

The next lemma is an immediate consequence of the definition of cellular
algebras and shows that $TL_{d,1,n}^{\Lambda^{0}}$ inherits a cellular
structure from $TL_{r,p,n}^{\Lambda}$ as a quotient.

######  Lemma 8.7.

Let the algebras $TL_{r,p,n}^{\Lambda}$ and $TL_{r,p,n}^{1}$ be as defined
above. Then $TL_{r,p,n}^{\Lambda}/TL_{r,p,n}^{1}$ is a graded cellular algebra
with cellular datum $(\mathfrak{P}_{n,p}^{0},T_{p},\mathfrak{C}_{p},deg_{p})$,
where $\mathfrak{P}_{n,p}^{0}$ is the subset of $\mathfrak{P}_{n,p}$
consisting of all the reducible orbits, $T_{p},deg_{p}$ are the same as the
ones in Theorem 7.9 and
$\mathfrak{C}_{p}^{[\mu]}([s],[t])=C_{p}^{[\mu]}([s],[t])+TL_{r,p,n}^{1}$ for
all $[\mu]\in\mathfrak{P}_{n,p}^{0}$ and $[s],[t]\in T_{p}([\mu])$.

For $[\mu]\in\mathfrak{P}_{n,p}^{0}$ and $[\lambda]\in\mathfrak{P}_{n,p}^{1}$,
Lemma 8.2 implies

(8.10) $C_{p}^{[\lambda]}([u],[v])W([\mu])=0.$

Thus the kernel of the map $f$ above acts trivially on $W([\mu])$, whence the
action of $TL_{r,p,n}$ on the cell module $W([\mu])$ is exactly the same as
that of $TL_{d,1,n}^{\Lambda^{0}}$. Hence we need only to determine the
decomposition numbers for the cell modules $W([\mu])$, regarded as
$TL_{d,1,n}^{\Lambda^{0}}$-modules.

By Theorem 5.2, the generalised Temperley-Lieb algebra
$TL_{d,1,n}^{\Lambda^{0}}$ has another cell datum
$(\mathfrak{B}_{n}^{(d)}-\mathfrak{D}_{n}^{(d)},Std,C,deg)$ where
$\mathfrak{B}_{n}^{(d)}-\mathfrak{D}_{n}^{(d)}$ consists of the 3D
multipartitions with at most two non-empty components and both of them are in
the first column of the floor table. The decomposition numbers corresponding
to this cell datum are given in the last chapter. To show that these two cell
data are the same, we first choose a special representative in each reducible
orbit.

######  Definition 8.8.

For any reducible orbit $[\mu]\in\mathfrak{P}_{n,p}^{0}$, define the original
representative $\mu_{0}$ in $[\mu]$ as the multipartition such that all the
non-empty components are in the first column of the floor table.

Since the non-empty components in a reducible multipartition are in the same
column, and the map $\sigma_{\mathfrak{P}_{n}}$ moves multipartitions one
column ahead, there exists a unique original representative $\mu_{0}$ in each
reducible orbit $[\mu]$. For $[t]\in\mathcal{T}_{p}([\mu])$, denote by
$t_{0}\in[t]$ the standard tableau of shape $\mu_{0}$. Let
$f:TL_{r,p,n}^{\Lambda}\to TL_{d,1,n}^{\Lambda^{0}}$ be the natural quotient
map.Then we have

$\displaystyle f(C_{p}^{[\mu]}([s],[t]))$
$\displaystyle=f(\psi_{d([s])}^{*}(\sum_{\mu^{\prime}\equiv\mu_{0}}e_{\mu^{\prime}})\psi_{d([t])}$
$\displaystyle=\psi_{d([s])}^{*}f(\sum_{i=0}^{p-1}e_{\mu_{i}})\psi_{d([t])}$
$\displaystyle=\psi_{d(s_{0})}^{*}f(e_{\mu_{0}})\psi_{d(t_{0})}$
$\displaystyle=C^{\mu_{0}}(s_{0},t_{0}).$

Therefore, we have

(8.11) $\mathfrak{C}_{p}^{[\mu]}([s],[t])=C^{\mu_{0}}(s_{0},t_{0}),$

for all $[\mu]\in\mathfrak{P}_{n,p}^{0}$ and $[s],[t]\in T_{p}([\mu])$. As a
direct consequence, we have

######  Lemma 8.9.

For any reducible orbit $[\mu]$, let $W([\mu])$(resp. $L([\mu])$) be the cell
( resp. simple) module of $TL_{d,1,n}^{\Lambda^{0}}$ with respect to the cell
datum $(\mathfrak{P}_{n,p}^{0},T_{p},\mathfrak{C}_{p},deg_{p})$. Denote by
$\mu_{0}$ the original representative in $[\mu]$. Let $W(\mu_{0})$
($L(\mu_{0})$) be the cell (resp. simple) module with respect to the cell
datum $(\mathfrak{B}_{n}^{(d)}-\mathfrak{D}_{n}^{(d)},Std,C,deg)$. Then we
have

$W([\mu])\cong W(\mu_{0});L([\mu])\cong L(\mu_{0}).$

Moreover, let $[\lambda],[\mu]$ be two reducible orbits. As
$TL_{d,1,n}^{\Lambda^{0}}$-modules, we have

(8.12) $[W([\lambda]):L([\mu])]=[W(\lambda_{0}):L(\mu_{0})].$

Since $TL_{r,p,n}^{1}$ acts trivially on $W([\lambda])$ where $[\lambda]$ is
reducible, the left hand side equals to the decomposition numbers for the
corresponding cell modules of $TL_{r,p,n}$. The right-hand side can be
obtained from Theorem 6.23 in [8]. We are now in a position to describe the
decomposition numbers of $TL(r,p,n)$.

######  Theorem 8.10.

Let $TL_{r,p,n}$ be the Temperley-Lieb algebra for the imprimitive complex
reflection group $G(r,p,n)$ as defined in Definition 4.2. According to Theorem
7.9, $TL_{r,p,n}$ is a cellular algebra with respect to the poset
$\mathfrak{P}_{n,p}$ in $(\ref{poset831})$. For an orbit of multipartitions
$[\lambda]\in\mathfrak{P}_{n,p}$, let $W([\lambda])$ and $L([\lambda])$ be the
cell and simple modules corresponding to $[\lambda]$, respectively.

If $\lambda$ is irreducible (cf. Definition 5.1), then

(8.13) $W([\lambda])=L([\lambda]).$

If $\lambda$ is reducible, $W([\lambda])$ has no decomposition factors of the
form $L([\mu])$ where $\mu$ is irreducible.

For any reducible orbit $[\mu]\in\mathfrak{P}_{n,p}$, let $\lambda_{0}$ and
$\mu_{0}$ be the original representatives as defined in Definition 8.8. Then

(8.14) $[W([\lambda]),L([\mu])]=\begin{cases}1&\text{ if
}\lambda_{0}\unlhd\mu_{0}\text{ and there exists }t_{0}\in
Std(\lambda_{0})\text{ such that }e(t_{0})=e_{\mu_{0}};\\\ 0&\text{
otherwise,}\end{cases}$

where $e(t_{0})$ and $e_{\mu_{0}}$ are the KLR generators.

###### Proof.

If $\lambda$ is irreducible, $W([\lambda])$ is simple by Proposition 8.4, so
we have

$W([\lambda])=L([\lambda]).$

If $\lambda$ is reducible but $\mu$ is irreducible, Lemma 8.2 implies that

$C_{p}^{[\lambda]}([s],[t])W([\mu])=0,$

for all $[s],[t]\in T([\lambda])$. On the other hand, for any non-zero element
$x\in L([\lambda])$, there exist $[s],[t]\in T([\lambda])$ such that
$C_{p}^{[\lambda]}([s],[t])x\neq 0$. Therefore, we have

$[W([\mu]):L([\lambda])]=0.$

If both $\lambda$ and $\mu$ are reducible, then $(\ref{finaltool})$ applies.
Hence by Theorem 6.23 in [8],

$\displaystyle[W([\lambda]),L([\mu])]$
$\displaystyle=[W(\lambda_{0}):L(\mu_{0})]$
$\displaystyle=\begin{cases}1&\text{ if }\lambda_{0}\unlhd\mu_{0}\text{ and
there exists }t_{0}\in Std(\lambda_{0})\text{ such that
}e(t_{0})=e_{\mu_{0}};\\\ 0&\text{ otherwise.}\end{cases}$

∎

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