Combinatorial Representation Theory

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

Last update: 17 September 2013

Appendix B

B7. The Birman-Murakami-Wenzl algebras $BM{W}_k(r,q)$

A $k\text{-tangle}$ is viewed as two rows of $k$ vertices, one above the other, and $k$ strands that connect vertices in such a way that each vertex is incident to precisely one strand. Strands cross over and under each other in three-space as they pass from one vertex to the next. For example, the following are 7-tangles: $$t_1=\begin{array}{} \end{array}{t}_2=\begin{array}{} \end{array}\text{.}$$ We multiply $k\text{-tangles}$ $t_1$ and $t_2$ using the concatenation product given by identifying the vertices in the top row of $t_2$ with the corresponding vertices in the bottom row of $t_1$ to obtain the product tangle $t_1{t}_2\text{.}$ Then we allow the following “moves.”

Reidemeister moves II and III:

(R2)	$\begin{array}{c} \end{array}⟷\begin{array}{c} \end{array}⟷\begin{array}{c} \end{array}$
(R3)	$\begin{array}{c} \end{array}⟷\begin{array}{c} \end{array}$

Given a Brauer diagram $d$ we will make a tangle $T_d$ tracing the edges in order from left to right across the top row and then from left to right across the bottom row. Any time an edge that we are tracing crosses and edge that has been already traced we raise the pen briefly so that the edge being traced goes under the edge which is already there. Applying this process to all of the Brauer diagrams on $k$ dots produces a set of $\left(2k\right)!!$ tangles. $$d=\begin{array}{} \end{array}$ T_d$=\begin{array}{} \end{array}$$

Fix $r,q\in ℂ\text{.}$ The Birman-Murakami-Wenzl algebra $BM{W}_k(r,q)$ is the span of the $\left(2k\right)!!$ tangles produced by tracing the Brauer diagrams with multiplication determined by the tangle multiplication and the Reidemeister moves and the following tangle identities. $$\begin{array}{c}\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})\text{.}\\ \begin{array}{c} \end{array}=r^{-1}\begin{array}{c} \end{array},\begin{array}{c} \end{array}=r\begin{array}{c} \end{array}\text{.}\\ \begin{array}{c} \end{array}=x,\text{where}x=\frac{r-r^{-1}}{q-q1}+1\text{.}\end{array}$$ The Reidemeister moves and the tangle identities can be applied in any appropriate local portion of the tangle.

Theorem B7.1. Fix $r,q\in ℂ\text{.}$ The Birman-Murakami-Wenzl algebra $BM{W}_k(r,q)$ is the algebra generated over $ℂ$ by $1,g_1,g_2,\dots ,g_{k-1},$ which are assumed to be invertible, subject to the relations $$\begin{array}{c}{g}_i{g}_{i+1}{g}_i=g_{i+1}{g}_i{g}_{i+1},\\ g_i{g}_j=g_j{g}_i\text{if}\hspace{0.17em}|i-j|\ge 2,\\ (g_i-r^{-1})(g_i+q^{-1})(g_i-q)=0,\\ E_i{g}_{i-1}^{\pm 1}{E}_i=r^{\pm 1}{E}_i\text{and}{E}_i{r}_{i+1}^{\pm 1}{E}_i=r^{\pm 1}{E}_i,\end{array}$$ where $E_i$ is defined by the equation $$(q-q^{-1})(1-E_i)=g_i-g_i^{-1}\text{.}$$

The BMW-algebra is a $q\text{-analogue}$ of the Brauer algebra in the same sense that the Iwahori-Hecke algebra of type $A$ is a $q\text{-analogue}$ of the group algebra of the symmetric group. If we allow ourselves to be imprecise (about the limit) we can write $$\underset{q\to 1}{\text{lim}}{BMW}_k(q^{n+1},q)=B_k\left(n\right)\text{.}$$ It would be interesting to sharpen the following theorem to make it an if and only if statement.

Theorem B7.2 (Wz3). The Birman-Murakami-Wenzl algebra is semisimple if $q$ is not a root of unity and $r\ne q^{n+1}$ for any $n\in \mathbb{Z}\text{.}$

Partial results for $BM{W}_k(r,q)$

The following results hold when $r$ and $q$ are such that $BM{W}_k(r,q)$ is semisimple.

I. What are the irreducible $BM{W}_k(r,q)\text{-modules?}$

(a)	How do we index/count them?
	There is a bijection $$\text{Partitions of}\hspace{0.17em}k-2h\text{,}\hspace{0.17em}h-0,1,\dots ,\lfloor k/2\rfloor \stackrel{1-1}{⟷}\text{Irreducible representations}\hspace{0.17em}{W}^{\lambda }\text{.}$$
(b)	What are their dimensions?
	The dimension of the irreducible representation $W^{\lambda }$ is given by $$\begin{array}{ccc}\text{dim}\left(W^{\lambda }\right)& =& \text{\# of up-down tableaux of shape}\hspace{0.17em}\lambda \hspace{0.17em}\text{and length}\hspace{0.17em}k\\ & =& \left(\genfrac{}{}{0ex}{}{k}{2h}\right)(2h-1)!!\frac{(k-2h)!}{{\prod }_{x\in \lambda }{h}_x},\end{array}$$ where $h_x$ is the hook length at the box $x$ in $\lambda ,$ and up-down tableaux is as in the case of the Brauer algebra, see Section B6 (Ib).
(c)	What are their characters?
	A Murnaghan-Nakayama rule for the irreducible characters of the BMW-algebras was given in [HRa1995].

References

(1)	The Birman-Murakami-Wenzl algebra was defined independently by Birman and Wenzl in [BWe1989] and by Murakami in [Mur1987]. See [CPr1994] for references to the analogue of Schur-Weyl duality for the BMW-algebras. The articles [HRa1995], [LRa1977], [Mur1990], [Rem1992], and [Wen1990] contain further important information about the BMW-algebras.
(2)	Although the tangle description of the BMW algebra was always in everybody’s minds it was Kaufmann that really made it precise see [Kau1990].

Notes and references

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..

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