Contingency tables and the generalized Littlewood–Richardson coefficients

Abstract:

The Littlewood–Richardson coefficients $c^\lambda _{\mu \nu }$ give the multiplicity of an irreducible polynomial $\operatorname {GL}_n$-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.