The Brauer algebra and the symplectic Schur algebra

TL;DR

This paper explores the properties and block structures of symplectic and orthogonal Schur algebras over fields of positive characteristic, providing new proofs and formulas related to the Brauer algebra.

Contribution

It introduces the symplectic Schur functor, establishes basic properties, and derives a modified Jantzen sum formula and block results under specific conditions, connecting to the Brauer algebra.

Findings

Derived a modified Jantzen sum formula for symplectic Schur algebra.

Established block results for symplectic and orthogonal Schur algebras in small characteristic.

Provided a new proof of the geometric description of Brauer algebra blocks in characteristic 0.

Abstract

Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\ge1, r\ge0 and m\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. Finally, we deduce from the previous results a new proof of the geometric description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin.