Last update: 17 September 2013
Fix $x\in \u2102\text{.}$ A Brauer diagram on $k$ dots is a graph on two rows of $k\text{-vertices,}$ one above the other, and $k$ edges such that each vertex is incident to precisely one edge. The product of two $k\text{-diagrams}$ ${d}_{1}$ and ${d}_{2}$ is obtained 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 $k$ paths and some number $c$ of closed loops. If $d$ is the $k\text{-diagram}$ with the edges that are the paths in this graph but with the closed loops removed, then the product ${d}_{1}{d}_{2}$ is given by ${d}_{1}{d}_{2}={\eta}^{c}d\text{.}$ For example, if $${d}_{1}=\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\end{array}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{d}_{2}=\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\end{array},$$ then $${d}_{1}{d}_{2}=\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\end{array}={x}^{2}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\end{array}\text{.}$$
The Brauer algebra ${B}_{k}\left(x\right)$ is the span of the $k\text{-diagrams}$ with multiplication given by the linear extension of the diagram multiplication. The dimension of the Brauer algebra is $$\text{dim}\left({B}_{k}\left(x\right)\right)=\left(2k\right)!!=(2k-1)(2k-3)\cdots 3\xb71,$$ since the number of $k\text{-diagrams}$ is $\left(2k\right)!!\text{.}$
The diagrams in ${B}_{k}\left(x\right)$ which have all their edges connecting top vertices to bottom vertices form a symmetric group ${S}_{k}\text{.}$ The elements $${s}_{i}=\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ni\ni+1\n\n\end{array}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{e}_{i}=\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ni\ni+1\n\n\end{array},$$ $1\le i\le k-1,$ generate the Brauer algebra ${B}_{k}\left(x\right)\text{.}$
Theorem B6.1. The Brauer algebra ${B}_{k}\left(x\right)$ has a presentation as an algebra by generators ${s}_{1},{s}_{2},\dots ,{s}_{k-1},$ ${e}_{1},{e}_{2},\dots ,{e}_{k-1}$ and relations $$\begin{array}{c}{s}_{i}^{2}=1,\phantom{\rule{1em}{0ex}}{e}_{i}^{2}=x{e}_{i},\phantom{\rule{1em}{0ex}}{e}_{i}{s}_{i}={s}_{i}{e}_{i}={e}_{i},\phantom{\rule{2em}{0ex}}1\le i\le k-1,\\ {s}_{i}{s}_{j}={s}_{j}{s}_{i},\phantom{\rule{1em}{0ex}}{s}_{i}{e}_{j}={e}_{j}{s}_{i},\phantom{\rule{1em}{0ex}}{e}_{i}{e}_{j}={e}_{j}{e}_{i},\phantom{\rule{2em}{0ex}}|i-j|>1,\\ {s}_{i}{s}_{i+1}{s}_{i}={s}_{i+1}{s}_{i}{s}_{i+1},\phantom{\rule{1em}{0ex}}{e}_{i}{e}_{i+1}{e}_{i}={e}_{i},\phantom{\rule{1em}{0ex}}{e}_{i+1}{e}_{i}{e}_{i+1}={e}_{i+1},\phantom{\rule{2em}{0ex}}1\le i\le k-2,\\ {s}_{i}{e}_{i+1}{e}_{i}={s}_{i+1}{e}_{i},\phantom{\rule{1em}{0ex}}{e}_{i+1}{e}_{i}{s}_{i+1}={e}_{i+1}{s}_{i},\phantom{\rule{2em}{0ex}}1\le i\le k-2\text{.}\end{array}$$
There are two different Brauer algebra analogues of the Schur Weyl duality theorem, Theorem A5.1. In the first one the orthogonal group $O(n,\u2102)$ plays the same role that $GL(n,\u2102)$ played in the ${S}_{k}\text{-case,}$ and in the second, the symplectic group $Sp(2n,\u2102)$ takes the $GL(n,\u2102)$ role.
Let $O(n,\u2102)=\{A\in {M}_{n}\left(\u2102\right)\hspace{0.17em}|\hspace{0.17em}A{A}^{t}=I\}$ be the orthogonal group and let $V$ be the usual $n\text{-dimensional}$ representation of the group $O(n,\u2102)\text{.}$ There is an action of the Brauer algebra ${B}_{k}\left(n\right)$ on ${V}^{\otimes k}$ which commutes with the action of $O(n,\u2102)$ on ${V}^{\otimes k}\text{.}$
Theorem B6.2.
(a) | The action of ${B}_{k}\left(n\right)$ on ${V}^{\otimes k}$ generates ${\text{End}}_{O\left(n\right)}\left({V}^{\otimes k}\right)\text{.}$ |
(b) | The action of $O(n,\u2102)$ on ${V}^{\otimes k}$ generates ${\text{End}}_{{B}_{k}\left(n\right)}\left({V}^{\otimes k}\right)\text{.}$ |
Let $Sp(2n,\u2102)$ be the symplectic group and let $V$ be the usual $2n\text{-dimensional}$ representation of the group $Sp(2n,\u2102)\text{.}$ There is an action of the Brauer algebra ${B}_{k}(-2n)$ on ${V}^{\otimes k}$ which commutes with the action of $Sp(2n,\u2102)$ on ${V}^{\otimes k}\text{.}$
Theorem B6.3.
(a) | The action of ${B}_{k}(-2n)$ on ${V}^{\otimes k}$ generates ${\text{End}}_{Sp(2n,\u2102)}\left({V}^{\otimes k}\right)\text{.}$ |
(b) | The action of $Sp(2n,\u2102)$ on ${V}^{\otimes k}$ generates ${\text{End}}_{{B}_{k}(-2n)}\left({V}^{\otimes k}\right)\text{.}$ |
Theorem B6.4. The Brauer algebra ${B}_{k}\left(x\right)$ is semisimple if $x\notin \{-2k+3,-2k+2,\dots ,k-2\}\text{.}$
Partial results for ${B}_{k}\left(x\right)$
The following results giving answers to the main questions (Ia-c) for the Brauer algebras hold when $x$ is such that ${B}_{k}\left(x\right)$ is semisimple.
(a) | How do we index/count them? |
There is a bijection $$\text{Partitions of}\hspace{0.17em}k-2h\text{,}\hspace{0.17em}h=0,1,\dots ,\lfloor k/2\rfloor \phantom{\rule{2em}{0ex}}\stackrel{1-1}{\u27f7}\phantom{\rule{2em}{0ex}}\text{Irreducible representations}\hspace{0.17em}{B}^{\lambda}\text{.}$$ | |
(b) | What are their dimensions? |
The dimension of the irreducible representation ${B}^{\lambda}$ is given by $$\begin{array}{ccc}\text{dim}\left({B}^{\lambda}\right)& =& \text{\# of up-down tableaux of shape}\hspace{0.17em}\lambda \hspace{0.17em}\text{and length}k\\ & =& \left(\genfrac{}{}{0ex}{}{k}{2h}\right)(2h-1)!!\frac{(k-2h)!}{{\prod}_{x\in \lambda}{h}_{x}},\end{array}$$ where ${h}_{x}$ is the hook length at the box $x$ in $\lambda \text{.}$ An up-down tableau of shape $\lambda $ and length $k$ is a sequence $(\varnothing ={\lambda}^{\left(0\right)},{\lambda}^{\left(1\right)},\cdots {\lambda}^{\left(k\right)}=\lambda )$ of partitions, such that each partition in the sequence differs from the previous one by either adding or removing a box. | |
(c) | What are their characters? |
A Murnaghan-Nakayama type rule for the characters of the Brauer algebras was given in [Ram1995]. |
References
(1) | The Brauer algebra was defined originally by R. Brauer [Bra1937] in 1937. H. Weyl treats it in his book [Wey1946]. |
(2) | The Schur-Weyl duality type theorems are due to Brauer [Bra1937], from his original paper. See also [Ram1995] for a detailed description of these Brauer algebra actions. |
(3) | The theorem giving values of x for which the Brauer algebra is semisimple is due to Wenzl, see [Wen1988-2]. |
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..