Last update: 10 September 2013
It was shown in [Ram1994] that the trace of the regular representation of the Brauer algebra was related to the character of the irreducible representation of the Brauer algebra labeled by the partition Here we will show that this result holds in a much more general context, for any centralizer algebra.
Let be a representation of a group and let Let be the representation of which is dual to and let There is a natural identification of with the algebra with the opposite multiplication. Let Clearly Let be the invariants. is a of Let be the trace of in this representation.
Let Let be the element which corresponds to The bitrace of the regular representation of is given by where the sum is over a basis of the algebra
This is a trivial consequence of the chain of isomorphisms Let be a basis of and let be a basis of Then and the corresponding element act on and respectively by Similarly, let act on and by Then the fact that and that implies that Then the element is invariant since Now let us consider the element acting on the invariant in corresponding to This is the invariant in given by which clearly corresponds to the element of given by
Remark. It follows that the bitrace of the regular representation of the Iwahori-Hecke algebra of type is the same as the character of the representation of the Kosuda algebra corresponding to the pair of partitions
This is a copy of the paper On the trace of the regular representation of a centralizer algebra by Arun Ram, Department of Mathematics, University of Wisconsin, Madison, WI 53706, February 14, 1994. This paper was supported in part by a National Science Foundation postdoctoral fellowship.