Canonical idempotents of multiplicity-free families of algebras

TL;DR

This paper introduces methods to compute canonical primitive idempotents in multiplicity-free algebra families, illustrating their application to symmetric group algebras and Brauer algebras, enhancing understanding of their algebraic structure.

Contribution

It provides new computational techniques for canonical idempotents in multiplicity-free families, including explicit calculations for Brauer algebras.

Findings

Canonical idempotents correspond to Young's seminormal idempotents in symmetric groups.

Methods for computing idempotents are applicable to various algebra families.

Explicit examples for Brauer algebras demonstrate the methods' effectiveness.

Abstract

Any multiplicity-free family of finite dimensional algebras has a canonical complete set of of pairwise orthogonal primitive idempotents in each level. We give various methods to compute these idempotents. In the case of symmetric group algebras over a field of characteristic zero, the set of canonical idempotents is precisely the set of seminormal idempotents constructed by Young. As an example, we calculate the canonical idempotents for semisimple Brauer algebras.