Last update: 22 December 2013
A Markov trace on the affine braid group is a trace functional which respects the inclusions where More precisely, a Markov trace on the affine braid group with parameters is a sequence of functions such that
If is a finite dimensional module and the quantum trace of on (see [LRa1977, §3] and [CPr1994, Def. 4.2.9]) is the trace of the action of on is the quantum dimension of The first step of the standard argument for proving Weyl’s dimension formula [BtD1985, VI Lemma 1.19], shows that the quantum dimension of the finite dimensional irreducible is where and for a positive integer
Let be dominant integral weights. Let and and let be the representation of defined in Proposition 3.1. Then the functions form a Markov trace on the affine braid group with parameters and where the positive integers and the sum in the expression for are as in the tensor product decomposition
The fact that as defined in the statement of the Theorem satisfies (1)–(4) in the definition of a Markov trace follows exactly as in [LRa1977, Theorem 3.10c]. The formula for the parameter is derived in [LRa1977, (3.9) and Thm. 3.10(2)].
It remains to check (5). The proof is a combination of the argument used in [Ore1999, Theorem 5.3] and the argument in the proof of [LRa1977, Theorem 3.10c]. Let be given by where If is simple then is the unique projection onto the invariants in Pictorially, The argument of [LRa1977, Theorem 3.10b] shows that Since is a homomorphism from to and, since is simple, Schur’s lemma implies that Let Then and This last calculation is more palatable in a pictorial format, It remains to calculate the constant By (2.14), where and is the projection onto the component in the decomposition of Thus
There is another formula [TWe1217386, Lemma (3.51)], for the constant in Theorem 5.1, namely, where is the Weyl group of
Let be as in Theorem 5.1 and let Then is the restriction of the linear functional to Since is a finite dimensional semisimple module is a finite dimensional semisimple algebra. The weights of the Markov trace mt are the constants defined by where are the irreducible characters of
Let and be finite dimensional irreducible The weights of the Markov trace on the affine braid group defined in Theorem 5.1 are
Since is finite dimensional and semisimple the algebra is a finite dimensional semisimple algebra. Schur’s lemma can be used to show that, as a bimodule, where the are the irreducible modules. In the notation of (4.1), and is the character of Taking the quantum trace on both sides of (5.9) gives The result follows by dividing both sides by
This is a typed version of the paper Affine Braids, Markov Traces and the Category by Rosa Orellana and Arun Ram*.
*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.
This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.