Matrix Units in the Symmetric Group Algebra, and Unitary Integration

TL;DR

This paper constructs matrix units in the symmetric group algebra, establishing an isomorphism with the Young graph's path algebra, and derives new formulas for integrating polynomials over the unitary group.

Contribution

It introduces explicit matrix units in the symmetric group algebra and provides alternative formulas for unitary integrals, expanding computational tools in representation theory.

Findings

Explicit isomorphism between symmetric group algebra and Young graph path algebra

New formulas for polynomial integration over the unitary group

Closed-form expressions for moments of the first k rows of the unitary group

Abstract

In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this construction, we obtain new formulas, alternative to Weingarten calculus, for the integral of a polynomial over the unitary group with respect to the Haar measure. In particular, we obtain a closed formula for the law of moments of the first k rows of the unitary group.