A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 13 March 2014

Irreducible Representations of the Iwahori-Hecke Algebras of Type A, the Birman-Wenzl Algebras and the Brauer Algebras

The Iwahori-Hecke Algebras of Type A, $H_m\left(q^2\right)$

The Iwahori-Hecke algebra of type A, denoted $H_m\left(q^2\right),$ is the algebra generated over $ℂ\left(q\right)$ by $1,g_1,\dots ,g_{m-1}$ subject to the relations

(B1)	$g_i{g}_{i+1}{g}_i=g_{i+1}{g}_i{g}_{i+1},$
(B2)	$g_i{g}_j=g_j{g}_i\text{if}|i-j|\ge 2,$
(IH)	$g_i^2=(q-q^{-1})g_i+1\text{.}$

The Iwahori-Hecke algebra $H_m\left(q\right)$ is often defined as the algebra generated over $ℂ\left(q\right)$ by $1,g_1^{\prime },\dots ,g_{m-1}^{\prime }$ subject to the relations

(B1)	$g_i^{\prime }{g}_{i+1}^{\prime }{g}_i^{\prime }=g_{i+1}^{\prime }{g}_i^{\prime }{g}_{i+1}^{\prime },$
(B2)	$g_i^{\prime }{g}_j^{\prime }=g_j^{\prime }{g}_i^{\prime }\text{if}|i-j|\ge 2,$
(IH)	${\left(g_i^{\prime }\right)}^2=(q-1)g_i^{\prime }+q\text{.}$

One can pass from one presentation to the other by setting $g_i=g_i^{\prime }/q\text{.}$

Let $𝔘$ be the Drinfel'd-Jimbo quantum group ${𝔘}_h\left(𝔰𝔩(r+1)\right)$ and let $V={\Lambda }_{{\omega }_1}$ be the irreducible $𝔘\text{-module}$ indexed by the fundamental weight ${\omega }_1\text{.}$ The centralizer ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ is a quotient of the Iwahori-Hecke algebra of type A, $H_m\left(q^2\right)\text{.}$

		Proof.
	This follows immediately from Proposition (4.4) and Corollary (4.15). $\square$

In fact, the classical Schur-Weyl duality gives that ${𝒵}_m$ is isomorphic to $H_m\left(q^2\right)$ if $𝔘={𝔘}_h\left(𝔰𝔩(r+1)\right),$ $r\ge m\text{.}$

The Young lattice $Y$ is the Bratteli diagram given in Fig. 1. The shapes of $Y$ which are on level $m$ are the partitions of $m\text{;}$ $${\hat{Y}}_m=\{\lambda ⊢m\}\text{.}$$ A partition $\lambda \in {\hat{Y}}_m$ is connected by an edge to a partition $\mu \in {\hat{Y}}_{m+1}$ if $\mu$ can be obtained by adding a box to $\lambda \text{.}$ Each tableau $T\in {𝒯}^{\lambda }$ in the Bratteli diagram $Y$ can be identified in a natural way with a standard tableau of shape $\lambda \text{.}$

Let $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)})$ be a standard tableau, i.e. a tableau in the Bratteli diagram $Y\text{.}$ Define $$\begin{array}{cc}{\diamond }_{m-1}(S,S)={\sigma }_k^{\left(m\right)}-{\sigma }_l^{(m-1)}-k+l,& \text{(6.2)}\end{array}$$ when ${\sigma }^{\left(m\right)}$ is obtained by adding a box to the $k\text{th}$ row of ${\sigma }^{(m-1)}$ and ${\sigma }^{(m-1)}$ is obtained by adding a box to the $l\text{th}$ row of ${\sigma }^{(m-2)}\text{.}$

The following result follows immediately from Theorem (4.8) and Corollary (6.1).

There is an identification of the Iwahori-Hecke algebras $H_m\left(q^2\right)$ with the path algebras corresponding to the Young lattice so that the generators $g_i$ are given by the formula $$g_i=\sum _{(S,T)\in {\Omega }_{i-1}^{i+1}}{\left(g_i\right)}_{ST}{E}_{ST},$$ where for each $S\in {𝒯}^m$ $${\left(g_i\right)}_{SS}=\frac{q^{{\diamond }_i(S,S)}}{\left[{\diamond }_i(S,S)\right]},$$ and for each pair $(S,T)\in {\Omega }_{i-1}^{i+1}$ such that $S\ne T$ we have $${\left(g_i\right)}_{ST}=\frac{\sqrt{[{\diamond }_i(S,S)-1][{\diamond }_i(S,S)+1]}}{\left[\left|{\diamond }_i(S,S)\right|\right]},$$ where ${\diamond }_i(S,S)$ is defined by (6.2).

The following corollaries are immediate consequences of the path algebra setup.

([Hoe1974], [Wen1988]) For each $\lambda \in {\hat{Y}}_m$ let $d_{\lambda }=\text{Card}\left({𝒯}^{\lambda }\right)$ be the number of standard tableaux of shape $\lambda \text{.}$ Define representations $$\begin{array}{cccc}{\pi }^{\lambda }:& H_m\left(q^2\right)& ⟶& M_{d_{\lambda }}\left(ℂ\left(q\right)\right)\\ & a& ⟼& {\left({\pi }^{\lambda }{\left(a\right)}_{ST}\right)}_{(S,T)\in {\Omega }^{\lambda }}\end{array}$$ of $H_m\left(q^2\right)$ by the following formulas: For each $S\in {𝒯}^{\lambda },$ $${\pi }^{\lambda }{\left(g_i\right)}_{SS}=\frac{q^{{\diamond }_i(S,S)}}{\left[{\diamond }_i(S,S)\right]}$$ and for each pair $(S,T)\in {\Omega }^{\lambda }$ such that $S\ne T,$ $${\pi }^{\lambda }{\left(g_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{[{\diamond }_i(S,S)-1][{\diamond }_i(S,S)+1]}}{\left[\left|{\diamond }_i(S,S)\right|\right]},& \text{if}{\sigma }^{\left(j\right)}={\tau }^{\left(j\right)}\hspace{0.17em}\text{for all}\hspace{0.17em}j\ne i,\\ 0,& \text{otherwise,}\end{array}$$ where $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}),$ $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m\right)})$ and ${\diamond }_i(S,S)$ is defined by (6.2). Then the representations ${\pi }^{\lambda },$ $\lambda \in {\hat{Y}}_m,$ are nonisomorphic irreducible representations of $H_m\left(q^2\right)\text{.}$

For each $\lambda \in {\hat{Y}}_m$ let ${𝒵}^{\lambda }$ be a vector space with basis $v_S,$ $S\in {𝒯}^{\lambda }\text{.}$ If $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}=\lambda )\in {𝒯}^{\lambda }$ then let $T$ be a tableau of the form $T=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)},\dots ,{\sigma }^{\left(m\right)})$ such that ${\tau }^{\left(i\right)}\ne {\sigma }^{\left(i\right)}\text{.}$ In view of (4.1), if $T$ exists then it is unique. Let ${\pi }^{\lambda }{\left(g_i\right)}_{SS}$ and ${\pi }^{\lambda }{\left(g_i\right)}_{ST}$ be as given in the previous corollary. Define an action of $H_m\left(q^2\right)$ on ${𝒵}^{\lambda }$ by defining $$g_i{v}_S=\{\begin{array}{cc}{\pi }^{\lambda }{\left(g_i\right)}_{SS}{v}_S+{\pi }^{\lambda }{\left(g_i\right)}_{ST}{v}_T,& \text{if}T\hspace{0.17em}\text{exists,}\\ {\pi }^{\lambda }{\left(g_i\right)}_{SS}{v}_S,& \text{if}T\hspace{0.17em}\text{does not exist,}\end{array}$$ for each $S\in {𝒯}^{\lambda }\text{.}$ Then the ${𝒵}^{\lambda },$ $\lambda \in {\hat{Y}}_m,$ are a complete set of nonisomorphic irreducible $H_m\left(q^2\right)\text{-modules.}$

The BirmanWenzl Algebras ${\text{BW}}_m(z,q)$

Let $z$ and $q$ be indeterminates. We define the Birman-Wenzl algebra ${\text{BW}}_m(z,q)$ (defined in [BWe1989] and [Mur1987]) as the algebra generated over $ℂ(z,q)$ by $1,g_1,g_2,\dots ,g_{m-1},$ which are assumed to be invertible, subject to the relations

(B1)	$g_i{g}_j=g_j{g}_i\text{if}|i-j|\ge 2,$
(B2)	$g_i{g}_{i+1}{g}_i=g_{i+1}{g}_i{g}_{i+1},$
(BW1)	$(g_i-z^{-1})(g_i+q^{-1})(g_i-q)=0,$
(BW2)	$e_i{g}_{i-1}^{\pm 1}{e}_i=z^{\pm 1}{e}_i$ and $e_i{g}_{i+1}^{\pm 1}{e}_i=z^{\pm 1}{e}_i,$

where $e_i$ is defined by the equation $$\begin{array}{cc}(q-q^{-1})(1-e_i)=g_i-g_i^{-1}\text{.}& \text{(6.6)}\end{array}$$ Letting $$\begin{array}{cc}x=\frac{z-z^{-1}}{q-q^{-1}}+1,& \text{(6.7)}\end{array}$$ one has the following relations $$\begin{array}{cccc}{e}_i^2& =& x{e}_i,& \text{(6.8)}\\ e_i{g}_i^{\pm 1}& =& g_i^{\pm 1}{e}_i=z^{\mp 1}{e}_i\text{.}& \text{(6.9)}\end{array}$$

Let $𝔘$ be the Drinfel'd-Jimbo quantum group ${𝔘}_h\left(𝔰𝔬(2r+1)\right)$ and let $V={\Lambda }_{{\omega }_1}$ be the irreducible $𝔘$ module indexed by the fundamental weight ${\omega }_1\text{.}$ Then centralizer ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ is isomorphic to a quotient of the Birman-Wenzl algebra, ${\text{BW}}_m(q^{2r},q)\text{.}$

		Proof.
	This follows immediately from Proposition (5.10), Corollary (5.22) and the definition of the Birman-Wenzl algebras. $\square$

Recall the Bratteli diagram $B$ given in Fig. 2. The shapes $\lambda \in {\hat{B}}_m$ of $B$ which are on level $m$ are the partitions in the set $${\hat{B}}_m=\{\lambda ⊢m-2k,0\le k\le \lfloor m/2\rfloor \}\text{.}$$ A partition $\lambda \in {\hat{B}}_m$ is connected by an edge to a partition $\mu \in {\hat{B}}_{m+1}$ if $\mu$ can be obtained from $\lambda$ by adding a box to $\lambda$ or removing a box from $\lambda \text{.}$ The tableaux $T\in {𝒯}^{\lambda }$ in the Bratteli diagram $B$ are called up-down tableaux since they are sequences of partitions in which each partition differs from the previous one by either adding or removing a box.

For the remainder of this section, unless otherwise specified, all paths and tableaux shall be from the Bratteli diagram $B\text{.}$

Let $y$ be a formal symbol and for each integer $d$ make the following notations: $$\begin{array}{cc}\begin{array}{cc}\begin{array}{ccc}\left[d\right]& =& \frac{q^d-q^{-d}}{q-q^{-1}},\\ [-y+d]& =& \frac{z^{-1}{q}^d-z{q}^{-d}}{q-q^{-1}},\\ \left[\frac{1}{y+d}\right]& =& \frac{z{q}^d}{[y+d]},\end{array}& \begin{array}{ccc}[y+d]& =& \frac{z{q}^d-z^{-1}{q}^{-d}}{q-q^{-1}},\\ \left[\frac{1}{d}\right]& =& \frac{q^d}{\left[d\right]},\\ \left[\frac{1}{-y+d}\right]& =& \frac{z^{-1}{q}^d}{[-y+d]}\text{.}\end{array}\end{array}& \text{(6.11)}\end{array}$$

Let $\lambda$ be a partition. Let ${\lambda }_i$ denote the length of the $i\text{th}$ column and let ${\lambda }_j^{\prime }$ denote the length of the $j\text{th}$ column. Define the hook length at a box $(i,j)\in \lambda$ to be $$h(i,j)={\lambda }_i-i+{\lambda }_j^{\prime }-j+1,$$ and, for each box $(i,j)\in \lambda ,$ define $$\begin{array}{cc}d(i,j)=\{\begin{array}{cc}{\lambda }_i+{\lambda }_j-i-j+1,& \text{if}i\le j,\\ -{\lambda }_i^{\prime }-{\lambda }_j^{\prime }+i+j-1,& \text{if}i>j\text{.}\end{array}& \text{(6.12)}\end{array}$$ Following [Wen1990] we define rational functions $Q_{\lambda }(z,q)$ as follows $$\begin{array}{cc}{Q}_{\lambda }(z,q)=\prod _{(j,j)\in \lambda }\frac{[y+{\lambda }_j-{\lambda }_j^{\prime }]+\left[h(j,j)\right]}{\left[h(j,j)\right]}\prod _{\underset{i\ne j}{(i,j)\in \lambda }}\frac{[y+d(i,j)]}{\left[h(i,j)\right]}\text{.}& \text{(6.13)}\end{array}$$ The important property of these functions ([Wen1990], Theorem 5.5) is that, if $𝔘={𝔘}_h𝔰𝔬(2r+1),$ then for all $\lambda \in \widehat{𝔘},$ $Q_{\lambda }(q^{2r},q)={\text{dim}}_q\left({\Lambda }_{\lambda }\right),$ where ${\Lambda }_{\lambda }$ is the irreducible $𝔘\text{-module}$ corresponding to the partition $\lambda \text{.}$ Thus, $Q_{\lambda }(z,q)$ is a two parameter version of the quantum dimension.

Now let us define two parameter versions of the constants ${\nabla }_m\left(S\right)$ and ${\diamond }_{m-1}(S,T)$ which are given in Proposition (5.13).

Let $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)})$ be a tableau in the Bratteli diagram $B\text{.}$ Define $$\begin{array}{cc}{\stackrel{\sim }{\nabla }}_m\left(S\right)=\{\begin{array}{cc}{q}^{2({\sigma }_k^{(m-1)}-k+1)},& \text{when}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}+{\epsilon }_k\text{;}\\ z^{-2}{q}^{2(-{\sigma }_k^{(m-1)}+k)},& \text{when}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}-{\epsilon }_k\text{.}\end{array}& \text{(6.14)}\end{array}$$ Let $S=({\sigma }^{(m-2)},{\sigma }^{(m-1)},{\sigma }^{\left(m\right)})$ and $T=({\sigma }^{(m-2)},{\tau }^{(m-1)},{\sigma }^{\left(m\right)})$ be such that $(S,T)\in {\Omega }_{m-2}^m\text{.}$ Then define $${\diamond }_{m-1}(S,T)=\{\begin{array}{cc}\pm ({\sigma }_k^{\left(m\right)}-k-{\tau }_l^{(m-1)}+l),& \text{if}{\tau }^{(m-1)}={\tau }^{(m-2)}\pm {\epsilon }_l\hspace{0.17em}\text{and}\hspace{0.17em}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}\pm {\epsilon }_k,\\ \pm (y+{\tau }_l^{(m-1)}-l+{\sigma }_k^{\left(m\right)}-k+1),& \text{if}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}\pm {\epsilon }_k\hspace{0.17em}\text{and}\hspace{0.17em}{\tau }^{(m-1)}={\tau }^{(m-2)}\mp {\epsilon }_l\text{.}\end{array}$$

There is an identification of the Birman-Wenzl algebras ${\text{BW}}_m(z,q)$ with the path algebras corresponding to the Bratteli diagram $B\text{.}$ With this identification:

(a)	The elements $D_m=g_{m-1}{g}_{m-2}\cdots g_1{g}_1\cdots g_{m-2}{g}_{m-1}$ are given by the formula, $$D_m=\sum _{S\in {𝒯}^m}{\left(D_m\right)}_{SS}{E}_{SS},\text{where}{\left(D_m\right)}_{SS}={\stackrel{\sim }{\nabla }}_m\left(S\right)\text{.}$$
(b)	The elements $e_i$ are given by the formula, $$e_i=\sum _{(S,T)\in {\Omega }_{i-1}^{i+1}}{\left(e_i\right)}_{ST}{E}_{ST}$$ where, if $S=({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)})$ and $T=({\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)}),$ then $${\left(e_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{Q_{{\sigma }^{\left(i\right)}}(z,q)Q_{{\tau }^{\left(u\right)}}(z,q)}}{Q_{{\sigma }^{(i-1)}}(z,q)}& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\\ 0& \text{otherwise.}\end{array}$$
(c)	The generators $g_i$ are given by the formula $$g_i=\sum _{(S,T)\in {\Omega }_{i-1}^{i+1}}{\left(g_i\right)}_{ST}{E}_{ST},$$ where, for each $S=({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)}),$ $${\left(g_i\right)}_{SS}=\{\begin{array}{cc}\left[\frac{1}{{\diamond }_i(S,S)}\right],& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ \left[\frac{1}{{\diamond }_i(S,S)}\right](1-\frac{Q_{{\sigma }^{\left(i\right)}}(z,q)}{Q_{{\sigma }^{(i-1)}}(z,q)}),& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)}\end{array}$$ and for each pair $(S,T)=(({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)}),({\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)}))\in {\Omega }_{i-1}^{i+1}$ such that $S\ne T,$ $${\left(g_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{[{\diamond }_i(S,S)-1][{\diamond }_i(S,S)+1]}}{\left[\left|{\diamond }_i(S,S)\right|\right]},& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ -\left[\frac{1}{{\diamond }_i(S,T)}\right]\frac{\sqrt{Q_{{\sigma }^{\left(i\right)}}(z,q)Q_{{\tau }^{\left(i\right)}}(z,q)}}{Q_{{\sigma }^{(i-1)}}(z,q)},& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\end{array}$$ where ${\diamond }_i(S,T)$ is given by (6.14).

		Proof.
	If $z$ is specialized to $q^{2r},$ $r>m,$ then the formulas given above coincide with the formulas given in (2.25), (5.9), and Theorem (5.14). In view of the results in Corollary (2.25), Theorem (3.12), and Theorem (5.14) it follows that this theorem holds whenever $z$ is specialized to $q^{2r},$ $r>m\text{.}$ Thus, for an infinite number of specializations of the parameter $z,$ the theorem holds. This is sufficient to guarantee that the theorem holds over $ℂ(z,q)\text{.}$ $\square$

For each $\lambda \in {\hat{B}}_m$ let $d_{\lambda }=\text{Card}\left({𝒯}^{\lambda }\right)$ be the number of up-down tableaux of shape $\lambda \text{.}$ Define representations $$\begin{array}{cccc}{\pi }^{\lambda }:& {\text{BW}}_m(z,q)& ⟶& M_{d_{\lambda }}\left(ℂ(z,q)\right)\\ & a& ⟼& {\left({\pi }^{\lambda }{\left(a\right)}_{ST}\right)}_{(S,T)\in {\Omega }^{\lambda }}\end{array}$$ of ${\text{BW}}_m(z,q)$ by defining For each $S\in {𝒯}^{\lambda },$ $${\pi }^{\lambda }{\left(g_i\right)}_{SS}=\{\begin{array}{cc}\left[\frac{1}{{\diamond }_i(S,S)}\right],& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ \left[\frac{1}{\frac{{\diamond }_i}{(S,S)}}\right](1-\frac{Q_{{\sigma }^{\left(i\right)}}(z,q)}{Q_{{\sigma }^{(i-1)}}(z,q)})& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\end{array}$$ and for each pair $(S,T)\in {\Omega }^{\lambda }$ such that $S\ne T,$ $${\pi }^{\lambda }{\left(g_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{[{\diamond }_i(S,S)-1][{\diamond }_i(S,S)+1]}}{\left[\left|{\diamond }_i(S,S)\right|\right]}& \text{if}{\sigma }^{\left(j\right)}={\tau }^{\left(j\right)}\hspace{0.17em}\text{for all}\hspace{0.17em}j\ne i\hspace{0.17em}\text{and}\hspace{0.17em}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ -\left[\frac{1}{{\diamond }_i(S,T)}\right]\frac{\sqrt{Q_{{\sigma }^{\left(i\right)}}(z,q)Q_{{\tau }^{\left(i\right)}}(z,q)}}{Q_{{\sigma }^{(i-1)}}(z,q)},& \text{if}\hspace{0.17em}{\sigma }^{\left(j\right)}={\tau }^{\left(j\right)}\hspace{0.17em}\text{for all}\hspace{0.17em}j\ne i\hspace{0.17em}\text{and}\hspace{0.17em}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\\ 0,& \text{otherwise,}\end{array}$$ where $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)}),$ $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m\right)})$ and ${\diamond }_i(S,S)$ is given by (6.14). Then the representations ${\pi }^{\lambda },$ $\lambda \in {\hat{B}}_m,$ are nonisomorphic irreducible representations of ${\text{BW}}_m(z,q)\text{.}$

Let $\lambda \in {\hat{B}}_m\text{.}$ If $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)})\in {𝒯}^{\lambda }$ such that ${\sigma }^{(i-1)}\ne {\sigma }^{(i+1)}$ then let $s_{i}S$ be the tableau $$s_{i}S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)},\dots ,{\sigma }^{\left(m\right)})$$ such that ${\tau }^{\left(i\right)}\ne {\sigma }^{\left(i\right)}\text{.}$ In view of Lemma (5.1), if $s_{i}S$ exists then it is unique. If $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)})\in {𝒯}^{\lambda }$ such that ${\sigma }^{(i-1)}={\sigma }^{(i+1)}$ then define $$e_{i}S=\{T=({\tau }^{\left(0\right)},\dots ,{\tau }^{\left(m\right)})\in {𝒯}^{\lambda }\hspace{0.17em}|\hspace{0.17em}S\ne T\hspace{0.17em}\text{and}\hspace{0.17em}{\tau }^{\left(j\right)}={\sigma }^{\left(j\right)}\hspace{0.17em}\text{for all}\hspace{0.17em}j\ne i\}\text{.}$$ With this notation we have the following.

For each $\lambda \in {\hat{B}}_m\left(r\right)$ let ${𝒵}^{\lambda }$ be a vector space with basis $v_S,$ $S\in {𝒯}^{\lambda }\text{.}$ Let constants ${\pi }^{\lambda }{\left(g_i\right)}_{ST},$ $(S,T)\in {\Omega }^{\lambda },$ be as given in Corollary (6.16). Define an action of ${\text{BW}}_m(z,q)$ on ${𝒵}^{\lambda }$ by defining $$g_i{v}_S=\{\begin{array}{cc}{\pi }^{\lambda }{\left(g_i\right)}_{SS}{v}_S+{\pi }^{\lambda }{\left(g_i\right)}_{S,s_{i}S}{v}_{s_{i}S},& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)}\hspace{0.17em}\text{and}\hspace{0.17em}{s}_{i}S\hspace{0.17em}\text{exists,}\\ {\pi }^{\lambda }{\left(g_i\right)}_{SS}{v}_S& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)}\hspace{0.17em}\text{and}\hspace{0.17em}{s}_{i}S\hspace{0.17em}\text{does not exist,}\\ {\pi }^{\lambda }{\left(g_i\right)}_{SS}{v}_S+\sum _{T\in e_{i}S}{\pi }^{\lambda }{\left(g_i\right)}_{ST}{v}_T,& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\end{array}$$ for each $S=({\sigma }^{\left(0\right)},\dots ,{\sigma }^{\left(m\right)})\in {𝒯}^{\lambda }\text{.}$ Then the ${𝒵}^{\lambda },$ $\lambda \in {\hat{B}}_m\left(r\right),$ are nonisomorphic irreducible ${\text{BW}}_m(z,q)\text{-modules.}$

The Brauer Algebra

An $m\text{-diagram}$ is a graph on two rows of $m\text{-vertices,}$ one above the other, and $m$ edges such that each vertex is incident to precisely one edge. The number of $m\text{-diagrams}$ is $\left(2m\right)!!=(2m-1)(2m-3)\cdots 3·1\text{.}$ We multiply two $m\text{-diagrams}$ $d_1$ and $d_2$ by placing $d_1$ above $d_2$ and identifying the vertices in the bottom row of $d_1$ with the corresponding vertices in the top row of $d_2\text{.}$ The resulting graph contains $m$ paths and some number $\gamma$ of closed cycles. Let $d$ be the $m\text{-diagram}$ whose edges are the paths in this graph (with the cycles removed). Then the product $d_1{d}_2$ is given by $d_1{d}_2=x^{\gamma }d\text{.}$ For example, if $$d_1=\begin{array}{c} \end{array}\text{and}{d}_2=\begin{array}{c} \end{array}$$ then $$d_1{d}_2=\begin{array}{c} \end{array}=x^2\begin{array}{c} \end{array}$$ Let $x$ be an indeterminate. The Brauer algebra $B_m\left(x\right)$ (defined originally by R. Brauer [Bra1937]) is the $ℂ\left(x\right)\text{-span}$ of the $m\text{-diagrams.}$ Diagram multiplication makes $B_m\left(x\right)$ an associative algebra whose identity ${\text{id}}_m$ is given by the diagram having each vertex in the top row connected to the vertex just below it in the bottom row. By convention $B_0\left(x\right)=B_1\left(x\right)=ℂ\left(x\right)\text{.}$

The group algebra $ℂ\left(x\right)\left[{𝒮}_m\right]$ of the symmetric group ${𝒮}_m$ is embedded in $B_m\left(x\right)$ as the span of the diagrams with only vertical edges. For $1\le i\le m-1,$, let $$s_i=\begin{array}{c} i i+1 ⋯ ⋯ \end{array}\text{and}{e}_i=\begin{array}{c} i i+1 ⋯ ⋯ \end{array}\text{.}$$ Then $e_i^2=x{e}_i,$ and the elements of the set $\{s_i,e_i\hspace{0.17em}|\hspace{0.17em}1\le i\le m-1\}$ generate $B_m\left(x\right)\text{.}$ Note that the $s_i$ correspond to the simple transpositions $(i,i+1)$ of ${𝒮}_m$ and that the $s_i,$ $1\le i\le m-1,$ generate $ℂ\left(x\right)\left[{𝒮}_m\right]\text{.}$

For each complex number $\xi \in ℂ$ one defines a Brauer algebra $B_m\left(\xi \right)$ over $ℂ$ as the linear span of $m\text{-diagrams}$ where the multiplication is given as above except with $x$ replaced by $\xi \text{.}$ R. Brauer [Bra1937] originally introduced the Brauer algebra $B_m\left(n\right)$ in his study of the centralizer of the tensor representation of the complex orthogonal group $O\left(n\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=I\}\text{.}$ Let $V={ℂ}^n$ be the standard or fundamental representation for $O\left(n\right)\text{.}$ The tensor space $V^{\otimes m}$ is a completely reducible $O\left(n\right)\text{-module}$ with irreducible summands labeled by partitions in the set $${\hat{B}}_m\left(n\right)=\{\lambda ⊢(m-2k)\hspace{0.17em}|\hspace{0.17em}0\le k\le \lfloor m/2\rfloor ,{\lambda }_1^{\prime }+{\lambda }_2^{\prime }\le n\}\text{.}$$ Note that when $n$ is sufficiently large ${\hat{B}}_m\left(n\right)={\hat{B}}_m$ where ${\hat{B}}_m$ is as defined in Section 5. Brauer gives an action of $B_m\left(n\right)$ on $V^{\otimes m}$ which commutes with the action of $O\left(n\right)\text{.}$ This action is such that $s_i$ is the permutation which transposes the $i\text{th}$ and the $(i+1)\text{st}$ tensor factors of $V^{\otimes m}$ and $e_1$ is $(2r+1)$ times the projection onto the invariants in the first two tensor factors of $V^{\otimes m}$ (see [Ram1995] for details). Brauer showed that the action of the Brauer algebra generates the full centralizer of the orthogonal group action on $V^{\otimes m}\text{.}$ Provided we assume that $r\ge m,$ all of these results hold if the group $O\left(n\right)$ is replaced by the group $SO(2r+1)\text{.}$

Let $r\ge 2$ and set $G=SO(2r+1)\text{.}$ Let $\stackrel{\sim }{V}={ℂ}^{2r+1}$ be the standard module for $G$ and let ${\stackrel{\sim }{𝒵}}_2={\text{End}}_G(\stackrel{\sim }{V}\otimes \stackrel{\sim }{V})\text{.}$ As $SO(2r+1)$ modules, $$\stackrel{\sim }{V}\otimes \stackrel{\sim }{V}\cong V^{\left(0\right)}\oplus V^{\left(1^2\right)}\oplus V^{\left(2\right)},$$ where $V^{\lambda }$ denotes the irreducible $G\text{-module}$ indexed by the partition $\lambda \text{.}$ Let ${\stackrel{\sim }{E}}_{\varnothing \varnothing },$ ${\stackrel{\sim }{E}}_{\left(1^2\right)\left(1^2\right)},$ and ${\stackrel{\sim }{E}}_{\left(2\right),\left(2\right)}$ be the $G\text{-invariant}$ projections onto the irreducible summands $V^{\varnothing },$ $V^{\left(1^2\right)},$ and $V^{\left(2\right)}$ respectively. We have chosen this notation so that it is suggestive of the identification of the centralizer algebra ${\stackrel{\sim }{𝒵}}_2$ with a path algebra corresponding to the Bratteli diagram $B\text{.}$ It can easily be shown that $B_2(2r+1)$ is isomorphic to ${\stackrel{\sim }{𝒵}}_2$ and that under this isomorphism $$\begin{array}{cc}\begin{array}{ccc}{e}_1& =& (2r+1){\stackrel{\sim }{E}}_{\varnothing \varnothing },\\ s_1& =& {\stackrel{\sim }{E}}_{\varnothing \varnothing }+{\stackrel{\sim }{E}}_{\left(2\right),\left(2\right)}-{\stackrel{\sim }{E}}_{\left(1^2\right)\left(1^2\right)}\text{.}\end{array}& \text{(6.19)}\end{array}$$ Let $𝔘={𝔘}_h\left(𝔰𝔬(2r+1)\right)$ and $V={\Lambda }_{{\omega }_1}$ be the irreducible $𝔘\text{-module}$ indexed by the fundamental weight ${\omega }_1\text{.}$ As $𝔘$ modules, $$V\otimes V\cong {\Lambda }_{\left(0\right)}\oplus {\Lambda }_{\left(1^2\right)}\oplus {\Lambda }_{\left(2\right)},$$ where ${\Lambda }_{\lambda }$ denotes the irreducible $𝔘\text{-module}$ indexed by the partition $\lambda \text{.}$ Let $E_{\varnothing \varnothing },$ $E_{\left(1^2\right)\left(1^2\right)},$ and $E_{\left(2\right),\left(2\right)}$ be the $𝔘\text{-invariant}$ projections onto the irreducible summands ${\Lambda }_{\varnothing },$ ${\Lambda }_{\left(1^2\right)},$ and ${\Lambda }_{\left(2\right)}$ respectively. It follows from (5.9) and Theorem (5.14), or by direct calculation, that the elements ${Ř}_1,$ ${Ě}_1\in {𝒵}_2={\text{End}}_{𝔘}(V\otimes V)$ are given by $$\begin{array}{cc}\begin{array}{ccc}{Ě}_1& =& (\left[2r\right]+1)E_{\varnothing \varnothing },\\ {Ř}_1& =& q^{-2r}{E}_{\varnothing \varnothing }+q{E}_{\left(2\right),\left(2\right)}-q^{-1}{E}_{\left(1^2\right)\left(1^2\right)}\text{.}\end{array}& \text{(6.20)}\end{array}$$ By comparing (6.19) and (6.20) we see that, at $q=1,$ the transformations ${Ř}_1$ and ${Ě}_1$ are the transformations $s_1$ and $e_1$ respectively. The transformations ${Ř}_i$ and ${Ě}_i$ in ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ are the same transformations as ${Ř}_1$ and ${Ě}_1$ respectively, except that they act on the $i\text{th}$ and the $(i+1)\text{st}$ tensor factors of $V^{\otimes m}$ instead of the first and second tensor factors. Similarly, the transformations ${š}_i$ and ${ě}_i$ in ${\stackrel{\sim }{𝒵}}_m={\text{End}}_G\left({\stackrel{\sim }{V}}^{\otimes m}\right)$ are the same transformations as $s_1$ and ${\epsilon }_1$ respectively, except that they act on the $i\text{th}$ and the $(i+1)\text{st}$ tensor factors of ${\stackrel{\sim }{V}}^{\otimes m}$ instead of the first and second tensor factors. Since, at $q=1,$ the transformations ${Ř}_1$ and ${Ě}_1$ are the same as $s_1$ and ${\epsilon }_1$ respectively, it follows that, at $q=1,$ ${Ř}_i$ and ${Ě}_i$ are the same as $s_i$ and $e_i$ respectively. Hence, at $q=1,$ the centralizer algebras ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ are the centralizer algebras ${\stackrel{\sim }{𝒵}}_m={\text{End}}_G\left({\stackrel{\sim }{V}}^{\otimes m}\right),$

Following [EKi1979], for each partition $\lambda ,$ define polynomials $$\begin{array}{cc}{P}_{\lambda }\left(x\right)=\prod _{(i,j)\in \lambda }\frac{x-1+d(i,j)}{h(i,j)},& \text{(6.21)}\end{array}$$ where the constants $d(i,j)$ and $h(i,j)$ are as given in (6.12). These polynomials have the important property that $P_{\lambda }(2r+1)=\text{dim}\left(V^{\lambda }\right),$ for each irreducible representation $V^{\lambda }$ of the orthogonal group $SO(2r+1)\text{.}$

Let $S-({\sigma }^{(m-2)},{\sigma }^{(m-1)},{\sigma }^{\left(m\right)})$ and $T=({\sigma }^{(m-2)},{\tau }^{(m-1)},{\sigma }^{\left(m\right)})$ be such that $(S,T)\in {\Omega }_{m-2}^m\text{.}$ Then define $${\diamond }_{m-1}(S,T)=\{\begin{array}{cc}\pm ({\sigma }_k^{\left(m\right)}-k-{\tau }_l^{(m-1)}+l),& \text{if}{\tau }^{(m-1)}={\tau }^{(m-2)}\pm {\epsilon }_l\hspace{0.17em}\text{and}\hspace{0.17em}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}\pm {\epsilon }_k,\\ \pm (x+{\tau }_l^{(m-1)}-l+{\sigma }_k^{\left(m\right)}-k)m& \text{if}{\sigma }^{\left(m\right)}={\sigma }^{(m-1)}\pm {\epsilon }_k\hspace{0.17em}\text{and}\hspace{0.17em}{\tau }^{(m-1)}={\tau }^{(m-2)}\mp {\epsilon }_l\text{.}\end{array}$$

There is an identification of the Brauer algebras $B_m\left(x\right)$ with the path algebras corresponding to the Bratteli diagram $B\text{.}$ With this identification:

(a)

The elements $e_i$ are given by the formula $$e_i=\sum _{(S,T)\in {\Omega }_{i-1}^{i+1}}{\left(e_i\right)}_{ST}{E}_{ST}$$ where, if $S=({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)})$ and $T=({\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)}),$ then $${\left(e_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{P_{{\sigma }^{\left(i\right)}}\left(x\right)P_{{\tau }^{\left(i\right)}}\left(x\right)}}{P_{{\sigma }^{(i-1)}}\left(x\right)}& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\\ 0& \text{otherwise.}\end{array}$$

(b)

The elements $s_i$ are given by the formula $$s_i=\sum _{(S,T)\in {\Omega }_{i-1}^{i+1}}{\left(s_i\right)}_{ST}{E}_{ST},$$ where, for each $S=({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)}),$ $${\left(s_i\right)}_{SS}=\{\begin{array}{cc}\frac{1}{{\diamond }_i(S,S)},& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ \frac{1}{{\diamond }_i(S,S)}(1-\frac{P_{{\sigma }^{\left(i\right)}}\left(x\right)}{P_{{\sigma }^{(i-1)}}\left(x\right)}),& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\end{array}$$ and for each pair $(S,T)=(({\sigma }^{(i-1)},{\sigma }^{\left(i\right)},{\sigma }^{(i+1)}),({\sigma }^{(i-1)},{\tau }^{\left(i\right)},{\sigma }^{(i+1)}))\in {\Omega }_{i-1}^{i+1}$ such that $S\ne T,$ $${\left(s_i\right)}_{ST}=\{\begin{array}{cc}\frac{\sqrt{({\diamond }_i(S,S)-1)({\diamond }_i(S,S)+1)}}{\left|{\diamond }_i(S,S)\right|},& \text{if}{\sigma }^{(i-1)}\ne {\sigma }^{(i+1)},\\ -\frac{1}{{\diamond }_i(S,T)}\frac{\sqrt{P_{{\sigma }^{\left(i\right)}}\left(x\right)P_{{\tau }^{\left(i\right)}}\left(x\right)}}{P_{{s_i}^{(i-1)}}\left(x\right)},& \text{if}{\sigma }^{(i-1)}={\sigma }^{(i+1)},\end{array}$$ where ${\diamond }_i(S,T)$ is as given just before Theorem (6.22).

		Proof.
	It follows from the discussion above that, at $q=1,$ the centralizer algebras ${𝒵}_m$ are the same as the centralizer algebras ${\stackrel{\sim }{𝒵}}_m\text{.}$ Note that the formulas in Theorem (5.14) all specialize to well defined rational numbers at $q=1\text{.}$ Thus, there is an identification of the centralizer algebras ${\stackrel{\sim }{𝒵}}_m$ with the path algebras corresponding to the centralizer algebras so that the elements $e_i$ and $s_i$ are given by the formulas in Theorem (5.14) evaluated at $q=1\text{.}$ These specializations are well defined and are equal to the formulas in the statement of Theorem (6.15) except with $x$ replaced by $2r+1\text{.}$ The centralizer algebras ${\stackrel{\sim }{𝒵}}_m$ are quotients of the Brauer algebras $B_m(2r+1)\text{.}$ If $r>m$ these algebras are isomorphic [Bra1937]. Thus, it follows from the previous paragraph that there is an identification of the Brauer algebras $B_m(2r+1),$ $r>m,$ with the path algebras corresponding to the centralizer algebras so that the elements $e_i$ and $s_i$ are given as in the above statement except with $x$ replaced by $2r+1\text{.}$ So Theorem (6.15) is true for an infinite number of specializations of the parameter $x\text{.}$ The result follows. $\square$

Notes and References

This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{†}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
${}^{†}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.

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