Generalised Temperley-Lieb algebras of type $G(r,1,n)$

The Temperley-Lieb algebra was originally introduced in [29] in connection with transition matrices in statistical mechanics. Since then, it has been shown to have links with many diverse areas of mathematics, such as operator algebras, quantum groups, categorification and representation theory. It can be defined in several different ways,including as a quotient of the Hecke algebra and as an associative diagram algebra. Steinberg pointed out in 1982 during a lecture at UCLA that the Temperley-Lieb algebra $TL_{n}(q)$ is a quotient of the Hecke algebra $H_{n}(q)$ of type $A_{n-1}$ by the ideal generated by some idempotents corresponding to two non-commuting generators. It plays a pivotal role in the polynomial invariant of knots and links discovered by Jones in [18]. In 1990, Kauffman found a diagrammatic presentation for the Temperley-Lieb algebra in [19]. It is defined as an associative diagram algebra modulo planar isotopy for classical unoriented knots and links. In [6], Bernstein, Frenkel and Khovanov ”categorify” the Temperley-Lieb algebra, which is the commutant of the quantum enveloping algebra $U_{q}(\mathfrak{sl}_{2})$ in its action on the $n^{th}$ tensor power $V_{1}^{\bigotimes n}$ of the natural two dimensional representation $V_{1}$. More precisely, they realise the Temperley-Lieb algebra via the projective and Zuckerman functors in the usual category $\mathcal{O}$ for $\mathfrak{gl}_{n}$. Moreover, the Temperley-Lieb algebra also plays an important role in statistical physics models, and readers can find more references in [5],[2] and [1].

The “blob algebra” $B_{n}(q,Q)$ was introduced in [22] by Martin and Saleur in connection with statistical mechanics. It has a similar diagrammatic representation to the original Temperley-Lieb algebra. Algebraically, it can also be identified with a quotient of $HB_{n}(q,Q)$ the two-parameter Hecke algebra of type $B_{n}$. For these reasons, the blob algebra is considered as a type-$B_{n}$ analogue of $TL_{n}(q)$ and is also denoted by $TLB_{n}(q,Q)$. In analogy to the original Temperley-Lieb algebra, $TLB_{n}(q,Q)$ also fits into a category. In [17], Iohara, Lehrer and Zhang construct a family of equivalences between the Temperley-Lieb category of type B and a certain category of infinite dimensional representations of $U_{q}(\mathfrak{sl}_{2})$ as a generalization of the Temperley-Lieb category $TL(q)$.

Cellularity is a property of associative algebras which permits their deformation and enables the computation of their decomposition numbers. The above algebras are shown to be cellular in [13]. Using this result, the standard modules and decomposition numbers are well-known, see [23],[12] and [14].

In this paper, we define a generalised Temperley-Lieb algebra $TL_{r,1,n}$ in Definition 3.1 corresponding to the imprimitive unitary reflection group $G(r,1,n)$, which can be regarded as a generalisation of the Coxeter groups of type $A_{n-1}$ and $B_{n}$. This generalisation strengthens the connection between Temperley-Lieb algebras and cyclotomic Hecke algebras and brings potential innovations in knot theory, quantum enveloping algebras and statistical physics models. We also construct a cellular structure of our generalised Temperley-Lieb algebra $TL_{r,1,n}$ in Theorem 4.9, which provides clues for a potential diagrammatic realisation. Using this cellular basis, we give a condition equivalent to the semisimplicity of $TL_{r,1,n}$ in Theorem 5.2. Moreover, we determine the associative decomposition matrix in Theorem 6.15.

Our generalisation is inspired by the construction of generalised blob algebras by Martin and Woodcock. They show in [24] that $TLB_{n}(q,Q)$ can be obtained as the quotient of $HB_{n}(q,Q)$ by the idempotents corresponding to the irreducible modules associated with the bipartitions $((2),\emptyset)$ and $(\emptyset,(2))$ and generalised this construction to all cyclotomic Hecke algebras $H(r,1,n)$.

Following this method, we define in Definition 3.1 the generalised Temperley-Lieb algebra $TL_{r,1,n}$ as a quotient of the corresponding cyclotomic Hecke algebra $H(r,1,n)$ by the two-sided ideal generated by certain idempotents. The Temperley-Lieb algebras of both type $A_{n-1}$ and $B_{n}$ are special cases of $TL_{r,1,n}$. Therefore, we refer to this quotient of the cyclotomic Hecke algebra $H(r,1,n)$ as the generalised Temperley-Lieb algebra of type $G(r,1,n)$.

As a direct consequence of the definition, our generalised Temperley-Lieb algebra $TL_{r,1,n}$ is a quotient of the generalised blob algebra $B_{r,n}$. We use this property to construct a graded cellular structure on it.

Lobos and Ryom-Hansen [25] give a graded cellular basis associated with one-column multipartitions of $n$ for the generalised blob algebras $B_{r,n}$. In [7], Bowman gives the decomposition numbers for the standard modules of the generalised blob algebras $B_{r,n}$. In Lemma 4.8, we show that the ideal of $B_{r,n}$ corresponding to its quotient $TL_{r,1,n}$ is exactly spanned by the elements corresponding to a downward closed subset of the poset in the cellular basis given by Lobos and Ryom-Hansen. Therefore, a graded cellular structure of $TL_{r,1,n}$ can be obtained by a truncation of that of $B_{r,n}$, which is described in Theorem 4.9. The graded cellular basis of $TL_{r,1,n}$ is associated with one-column $r$-partitions of $n$ which consist of at most two non-empty components.

Using this graded cellular basis, we investigate the representations of the generalised Temperley-Lieb algebra $TL_{r,1,n}$. Following the calculation of the dimensions of the cell modules, we concentrate on the bilinear form $\phi_{\lambda}(,)$ on each cell module. For a one-column multipartition $\lambda$, let $W(\lambda)$ be the corresponding cell module of $TL_{r,1,n}$. Define a bilinear form $\phi_{\lambda}(,)$ on $W(\lambda)$ in the usual way.According to Theorem 3.4 in [13], $W(\lambda)$ is simple if and only if $rad(\phi_{\lambda})=0$ . Using this method, we obtain a criterion for the semisimplicity of $TL_{r,1,n}$ in Theorem 5.2 and interpret it as a restriction on the parameters in Corollary 5.3.

As the main result in this paper, we determine the decomposition numbers for the cell modules using this bilinear form. We first give a method to find out the radical of the bilinear form $\phi_{\lambda}(,)$ in section 6.2. By showing that this method is not sensitive to the parameter $r$, we demonstrate that the dimension of a simple $TL_{r,1,n}$-module equals that of a corresponding $TLB_{n}(q,Q)$-module with the parameters chosen properly. By showing that the cell and simple modules of $TL_{r,1,n}$ can be realised as those of a corresponding Temperley-Lieb algebra of type $B_{n}$, we transform the decomposition matrix of $TL_{r,1,n}$ into a combination of those of $TLB_{n}(q,Q)$ in Lemma 6.12.

The non-graded decomposition numbers of the Temperley-Lieb algebras of type $B_{n}$ are given by Martin and Woodcock in [24] and Graham and Lehrer in [14] and the graded ones are given by Plaza in [27]. Following their results, we give the decomposition numbers of $TL_{r,1,n}$ in Theorem 6.15.

A partition of $n$ is a sequence $\sigma=(\sigma_{1}\geq\sigma_{2}\geq\dots)$ of non-negative integers $\sigma_{i}$ such that $|\sigma|=\sum_{i\geq 1}\sigma_{i}=n$. Denote $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{k})$ if $\sigma_{i}=0$ for $i>k$ and $\sigma_{k}>0$. A multipartition (more specifically, an $r$-partition) of $n$ is an ordered $r$-tuple $\lambda=(\lambda^{(1)},\dots,\lambda^{(r)})$ of partitions with $|\lambda^{(1)}|+\dots+|\lambda^{(r)}|=n$; if the $i^{th}$ component of $\lambda$ is empty, $|\lambda^{(i)}|=0$. Denote by $\mathfrak{P}_{n}^{(r)}$ the set of $r$-partitions of $n$.

The Young diagram of a multipartition $\lambda\in\mathfrak{P}_{n}^{(r)}$ is the set of 3-tuples:

For fixed $l$, the subset $[\lambda^{(l)}]:=\{(a,b,l)|a>0,1\leq b\leq\lambda_{a}^{(l)}\}$ is the $l^{th}$ component of this Young diagram of the multipartition $\lambda$. It can be regarded as a Young diagram of the partition $\lambda^{(l)}$. If $|\lambda^{(l)}|=0$, we call it an empty component of the multipartition $\lambda$. As there is a unique Young diagram $[\lambda]$ for each multipartition $\lambda\in\mathfrak{P}_{n}^{(r)}$ and vice versa, we do not distinguish $[\lambda]$ and $\lambda$ in the following sections of this thesis. For each 3-tuple $(a,b,l)\in[\lambda]$, we call it the node in the $a^{th}$ row and $b^{th}$ column of the $l^{th}$ component. A Young tableau is obtained by filling in the nodes with numbers $1,2,\dots,n$. More precisely, a Young tableau of shape $\lambda$ is a bijective map:

For example, let $n=9$, $r=3$ and $\lambda=((4,3),0,(1,1))$. Figure-1 shows a Young tableau of shape $\lambda$.

A standard Young tableau is a Young tableau in which the numbers increase along each row and column in every component. Denote by $Std(\lambda)$ the set of standard tableaux of shape $\lambda$. The tableau in Figure-1 is a standard Young tableau of shape $((4,3),\emptyset,(1,1))$.

Next we introduce the definition of one-column multipartitions which are the main tool we use to describe the representations of our generalised Temperley-Lieb algebras. A multipartition $\lambda\in\mathfrak{P}_{n}^{(r)}$ is called a one-column multipartition if each non-empty component consists of only one column. In other words, $\lambda$ is called a one-column multipartition if $\lambda_{a}^{(l)}\leq 1$ for all $a>0,1\leq l\leq r$. Figure 2 gives an example of the one-column multipartition and one of its standard tableaux. Denote by $\mathfrak{B}_{n}^{(r)}$ the set of one-column $r$-partitions of $n$. If $\gamma=(i,1,k)$ and $\gamma^{\prime}=(i^{\prime},1,k^{\prime})$ are two nodes of a one-column multipartition $\lambda$, we say $\gamma<\gamma^{\prime}$ if either $i<i^{\prime}$ or $i=i^{\prime}$ and $k<k^{\prime}$. We write $\gamma\leq\gamma^{\prime}$ if $\gamma<\gamma^{\prime}$ or $\gamma=\gamma^{\prime}$. For example, let $t$ be the tableau in Figure 2, then we have $t(1)<t(2)<t(4)<t(3)<t(5)<t(6)<t(7)<t(8)<t(9)$. In fact, $\leq$ is a total order on the set of nodes $N=\{(a,1,l)|a>0,1\leq l\leq r\}$.

For $\lambda,\mu\in\mathfrak{B}_{n}^{(r)}$, we say $\lambda\unlhd\mu$ if for each $\gamma_{0}\in\mathbb{N}\times\{1\}\times\{1,2\dots,r\}$ we have

Thus, roughly speaking, a multipartition is smaller if its diagram has more ‘small’ nodes with respect to the total order $\leq$. The relation $\unlhd$ defines a partial order on $\mathfrak{B}_{n}^{(r)}$. The cellular structure of our generalised Temperley-Lieb algebra $TL_{r,1,n}$ will involve this poset. As an example, the diagram in Figure-3 shows the partial order on $\mathfrak{B}_{3}^{(3)}$.

For $1\leq k\leq r$, denote by $\mathfrak{B}_{n}^{(r)}(k)$ the subset of $\mathfrak{B}_{n}^{(r)}$ consisting of the multipartitions containing exactly $k$ non-empty components. We next discuss some consequences of the definition of the partial order.

The converse of this lemma is not always true. Counterexamples can be found in Figure-3.

To study the cellular structure of our generalized Temperley-Lieb algebras, we are particularly interested in those multipartitions with at most two non-empty components. The following lemmas give descriptions of the partial order on the elements of $\mathfrak{B}_{n}^{(r)}(1)$ and $\mathfrak{B}_{n}^{(r)}(2)$.

Denote by $\lambda_{k_{1},k_{2}}^{[a]}\in\mathfrak{B}_{n}^{(r)}(2)$ ($k_{1}<k_{2}$, $|a|<n$ and $a\equiv n$ $mod$ 2 ) the multipartition $\lambda$ of which $\lambda^{(k_{1})}$ and $\lambda^{(k_{2})}$ are the two non-empty components and $|\lambda^{(k_{1})}|-|\lambda^{(k_{2})}|=a$. The following lemma gives a description of the partial order on $\mathfrak{B}_{n}^{(r)}(2)$.

We next introduce a special standard tableau $t^{\lambda}$ for each multipartition $\lambda$ which will be used in Section 4 to define the cellular basis of our generalised Temperley-Lieb algebra $TL_{r,1,n}$. Let $t^{\lambda}$ be the tableau of shape $\lambda$ such that $t^{\lambda}(i)<t^{\lambda}(j)$ if $1\leq i<j\leq n$. As $\leq$ is a total order on the nodes, $t^{\lambda}$ is unique for $\lambda\in\mathfrak{B}_{n}^{(r)}$. Moreover, for each $t\in Std(\lambda)$, denote by $d(t)\in\mathfrak{S}_{n}$ the unique permutation such that $t=t^{\lambda}\circ d(t)$ where $\circ$ is the natural $\mathfrak{S}_{n}$-action on the tableaux.

We next recall the definition of cyclotomic Hecke algebras and description of their representations following [4] by Ariki and Koike. The irreducible representations are described in the language of multipartions. Our generalised Temperley-Lieb algebras will be defined as quotients of cyclotomic Hecke algebras by certain ideals generated by a set of idempotents corresponding to a set of irreducible representations.

We first recall the presentation of the imprimitive reflection group $G(r,1,n)$.

The generators $s_{0},s_{1},s_{2},\dots,s_{n-1}$ are called simple reflections and the relations above can be represented in the Dynkin diagram in Figure-4.

The reflection group $G(r,1,n)$ can 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. Moreover, if $\zeta\in\mathbb{C}^{*}$ is a primitive $r^{th}$ root of unity, the complex reflection group $G(r,1,n)$ is generated by the following elements:

where $E_{i,j}$ is the elementary $n\times n$ matrix with the $(i,j)$-entry nonzero. Viewed thus, the generators are realised as (pseudo)-reflections in the $n$ dimensional space over $\mathbb{C}$, exhibiting $G(r,1,n)$ as a complex reflection group.

Let $R=\mathbb{Z}[q,q^{-1},v_{1},v_{2},\dots,v_{r}]$ where $q,v_{1},v_{2},\dots,v_{r}$ are indeterminates over $\mathbb{C}$. The cyclotomic Hecke algebra corresponding to $G(r,1,n)$ over $R$ is defined as follows:

The irreducible representations of $H(r,1,n)$ are described in the language of $r$-partitions of $n$. Let $\lambda$ be an $r$-partition of $n$ and $K$ be the quotient field of $R$. Denote by $f_{\lambda}$ the number of standard tableaux of shape $\lambda$ and by $\mathfrak{Y}^{\lambda}=\{\mathbb{Y}_{1},\mathbb{Y}_{2},\dots,\mathbb{Y}_{f_{\lambda}}\}$ the set of all the standard tableaux of shape $\lambda$. Let $\mathbb{V}_{\lambda}$ be the vector space over $K$ with the basis $\mathbb{Y}_{1},\mathbb{Y}_{2},\dots,\mathbb{Y}_{f_{\lambda}}$; namely,

where the sum runs over all the standard tableaux $\mathbb{Y}_{i}$ of shape $\lambda$.

Let $\gamma=(a,b,i)$ be a node in $\lambda^{(i)}$ for some $i$. The content $c(\lambda:\gamma)$ of $\gamma$ in $\lambda$ is defined to be the content of $\gamma$ in the Young diagram $\lambda^{(i)}$, namely, $c(\lambda:\gamma)=c(\lambda^{(i)}:\gamma)=b-a$.

For any indeterminate $x$ and an integer $k$, let $\Delta(k,x)$ be the polynomial $\Delta(k,x)=1-q^{k}x$ in $q$ and $x$ with coefficients in $K$ and define $M(k,x)$ as follows:

With the notations above, Ariki and Koike describe a complete set of the irreducible representations of $H(r,1,n)$:

Further, if $R$ is specialised to a field $F$, the following theorem states exactly when $H(r,1,n)$ is semisimple:

In the case when the cyclotomic Hecke algebra $H(r,1,n)$ is semisimple, each one-dimensional module corresponds to a central idempotent in $H(r,1,n)$. We will use these idempotents to define our generalised Temperley-Lieb algebra $TL_{r,1,n}$ in the next section.

To show the graded cellularity of our generalised Temperley-Lieb algebras, we shall make use of the cyclotomic KLR algebras, which are isomorphic to the cyclotomic Hecke algebras $H(r,1,n)$. In the next section, we will use this isomorphism to give a second interpretation of our generalised Temperley-Lieb algebra $TL_{r,1,n}$.

The KLR algebra was first introduced independently by Khovanov and Lauda in [20] and by Rouquier in [28] to categorify the quantum deformation of the universal enveloping algebra of the lower-triangular subalgebra of a Kac-Moody algebra $\mathfrak{g}$. Brundan and Kleshchev have shown in [10] that the KLR algebras of type $A_{n-1}$ are isomorphic to the cyclotomic Hecke algebras corresponding to the complex reflection groups $G(r,1,n)$ or their rational degenerations. In this section, we recall the special cyclotomic KLR algebra which is isomorphic to $H(r,1,n)$. We refer readers to [10] for a broader discussion.

Let $F$ be a fixed ground field and $q\in F^{*}$. Let $e$ be the smallest positive integer such that $1+q+\dots+q^{e-1}=0$, setting $e:=0$ if no such integer exists. Let $\Gamma$ be the quiver with vertex set $I:=\mathbb{Z}/e\mathbb{Z}$, and a directed edge from $i$ to $j$ if $j=i+1$. Thus $\Gamma$ is the quiver of type $A_{\infty}$ if $e=0$ or $A_{e-1}^{(1)}$, with the following orientation:

Let $(c_{ij})$ be the Cartan matrix associated with $\Gamma$, so that

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

Let $P_{+}=\oplus_{i\in I}\mathbb{N}\Lambda_{i}$ be the dominant weights in the weight lattice and $Q_{+}=\oplus_{i\in I}\mathbb{N}\alpha_{i}$ be the positive roots. For $\Lambda\in P_{+}$, define the length of $\Lambda$, $l(\Lambda):=\sum_{i\in I}(\Lambda,\alpha_{i})$ and for $\alpha\in Q_{+}$, let the height of $\alpha$ be $ht(\alpha):=\sum_{i\in I}(\alpha,\Lambda_{i})$. Fix an $\alpha\in Q_{+}$, define $I^{\alpha}$ as follows:

Fixing a dominant weight $\Lambda$, we next recall the definition of the cyclotomic KLR algebra of type $A_{n-1}$ associated with the weight $\Lambda$. The following definition is originally from the work of Khovanov and Lauda in [20] and Rouquier in [28].