Centre of the Schur Algebra

TL;DR

This paper provides a combinatorial basis for the center of the Schur algebra derived from symmetric group conjugacy classes, linking it to Schur-Weyl duality, character tables, and primitive idempotents.

Contribution

It introduces a new basis for the Schur algebra's center based on symmetric group conjugacy classes and character theory, with explicit combinatorial descriptions.

Findings

Explicit basis for the center of the Schur algebra

Connection between basis elements and symmetric group characters

Non-singularity result for submatrices of the symmetric group character table

Abstract

We describe a basis of the centre of the Schur algebra which comes from conjugacy classes in the symmetric group via Schur-Weyl duality. We give a combinatorial description of expansions of these basis elements in terms of the basis originally used by Schur. The primitive central idempotents of the Schur algebra can be written down using this basis and the character table of the symmetric group. Along the way we prove a result on the non-singularity of the submatrix of the character table matrix of a symmetric group obtained by taking rows and columns indexed by partitions with at most n parts for any n.