Characteristic maps for the Brauer algebra

TL;DR

This paper explores two analogues of the classical characteristic map for the Brauer algebra, focusing on the second map's images of central idempotents and deriving classical group representation formulas.

Contribution

It computes the images of Brauer algebra idempotents under the second characteristic map and derives hook dimension formulas for classical group representations.

Findings

Calculated images of central idempotents in terms of Schur polynomials.

Derived hook dimension formulas from properties of primitive idempotents.

Extended the characteristic map framework to the Brauer algebra context.

Abstract

The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We consider the second characteristic map both in the orthogonal and symplectic case, and calculate the images of central idempotents of the Brauer algebra in terms of the Schur polynomials. The calculation is based on the Okounkov--Olshanski binomial formula for the classical Lie groups. We also reproduce the hook dimension formulas for representations of the classical groups by deriving them from the properties of the primitive idempotents of the symmetric group and the Brauer algebra.