Contingency tables and the generalized Littlewood-Richardson coefficients

TL;DR

This paper introduces a new method for computing generalized Littlewood-Richardson coefficients using contingency tables, extending the classical combinatorial approach to multiple tensor products and other classical groups.

Contribution

It generalizes Littlewood-Richardson coefficients to r-fold tensor products and develops a contingency table-based computation method, extending to orthogonal and symplectic groups.

Findings

Method reduces to counting contingency tables in special cases

Provides a new combinatorial approach for generalized coefficients

Extends results to orthogonal and symplectic groups

Abstract

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.