Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups

TL;DR

This paper compares Gelfand-Tsetlin bases for alternating and symmetric groups, providing a recursive algorithm to express basis vectors of alternating groups in terms of symmetric group bases, implemented in Sage.

Contribution

It introduces a recursive algorithm for expanding Gelfand-Tsetlin basis vectors of alternating groups in symmetric group bases, with implementation in Sage.

Findings

Algorithm successfully computes basis expansions

Implementation verified on various group representations

Facilitates analysis of alternating group representations

Abstract

Young's orthogonal basis is a classical basis for an irreducible representation of a symmetric group. This basis happens to be a Gelfand-Tsetlin basis for the chain of symmetric groups. It is well-known that the chain of alternating groups, just like the chain of symmetric groups, has multiplicity-free restrictions for irreducible representations. Therefore each irreducible representation of an alternating group also admits Gelfand-Tsetlin bases. Moreover, each such representation is either the restriction of, or a subrepresentation of, the restriction of an irreducible representation of a symmetric group. In this article, we describe a recursive algorithm to write down the expansion of each Gelfand-Tsetlin basis vector for an irreducible representation of an alternating group in terms of Young's orthogonal basis of the ambient representation of the symmetric group. This algorithm is implemented with the Sage Mathematical Software.