Characters of Brauer’s centralizer algebras

Abstract

Brauer's centralizer algebras are finite dimensional algebras with a distinguished basis. Each Brauer centralizer algebra contains the group algebra of a symmetric group as a subalgebra and the distinguished basis of the Brauer algebra contains the permutations as a subset. In view of this containment it is desirable to generalize as many known facts concerning the group algebra of the symmetric group to the Brauer algebras as possible. This paper studies the irreducible characters of the Brauer algebras in view of the distinguished basis. In particular we define an analogue of conjugacy classes, and derive Frobenius formulas for the characters of the Brauer algebras. Using the Frobenius formulas we derive formulas for the irreducible character of the Brauer algebras in terms of the irreducible characters of the symmetric groups and give a combinatorial rule for computing these irreducible characters.Introduction. Classically, Frobenius [Fr] determined the characters of the symmetric group by exploiting the connection between the power symmetric functions and the Schur functions. Schur [Scl, Sc2] later showed that this connection arises from the fact that the general linear group and the symmetric group each generate the full centralizer of each other on tensor space, now referred to as the Schur-Weyl duality. In his landmark book [Wy], Weyl used this duality as the principal algebraic tool for studying the representations of the classical groups. In 1937 R. Brauer [Br] gave a nice basis for the centralizer algebra of the action of the orthogonal and symplectic groups on tensor space.In [Rl] we gave a formula for the characters of the Hecke algebra of type A in the same spirit as the original formula of Frobenius for the characters of the symmetric group. This formula was then used to derive a combinatorial rule for computing the characters of the

Linear functionals of eigenvalues of random matrices

“…Ram that was cited in [DS94] for the relevant representation theory of these groups has now appeared as [Ram95] -see also [Ram97]). We outline the relevant arguments in §8.…”

“…The relevant moment formulae were established in [DS94] using the thesis work of Arun Ram that has now been published in [Ram95,Ram97]. We refer to these papers for background and details.…”

“…From pp. 186-187 of [Ram95] we have Integrating over the orthogonal group and using orthogonality of characters shows that…”

A ‘Second Orthogonality Relation’ for Characters of Brauer Algebras

“…9) . In some sense , takes a diagram , splits it down the horizontal centerline , puts a hinge on the right-hand side , and opens it up to be twice as long .…”

“…This approach also gives a new proof of the Frobenius formula for the Brauer algebras . This proof is quite elementary and the 'second orthogonality relations' are used in place of the Weyl character formula which was used in the original proof [9] .…”