Matrix Units in the Symmetric Group Algebra, and Unitary Integration

2013/07/17 by Timothy Cioppa, Cioppa, Timothy, Benoit Collins +2 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.1307.4766

arxiv created 2013/07/17 · openalex publication_date 2013/07/17 · arxiv updated 2013/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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.

Cited by

• Weingarten calculus with virtual isometries
• Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

Related

• Linear algebra and group theory
• The characteristic polynomial on compact groups with Haar measure : some equalities in law
• Vertex operators and the class algebras of symmetric groups
• Matrix integrals over unitary groups: An application of Schur-Weyl duality
• Weingarten calculus with virtual isometries