Last update: 17 September 2013
A is viewed as two rows of vertices, one above the other, and 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: We multiply and using the concatenation product given by identifying the vertices in the top row of with the corresponding vertices in the bottom row of to obtain the product tangle Then we allow the following “moves.”
Reidemeister moves II and III:
Fix The Birman-Murakami-Wenzl algebra is the span of the tangles produced by tracing the Brauer diagrams with multiplication determined by the tangle multiplication and the Reidemeister moves and the following tangle identities. The Reidemeister moves and the tangle identities can be applied in any appropriate local portion of the tangle.
Theorem B7.1. Fix The Birman-Murakami-Wenzl algebra is the algebra generated over by which are assumed to be invertible, subject to the relations where is defined by the equation
The BMW-algebra is a of the Brauer algebra in the same sense that the Iwahori-Hecke algebra of type is a of the group algebra of the symmetric group. If we allow ourselves to be imprecise (about the limit) we can write 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 is not a root of unity and for any
Partial results for
The following results hold when and are such that is semisimple.
|(a)||How do we index/count them?|
|There is a bijection|
|(b)||What are their dimensions?|
|The dimension of the irreducible representation is given by where is the hook length at the box in 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].|
|(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].|
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..