Box splines, tensor product multiplicities and the volume function

Colin Mcswiggen

doi:10.5802/alco.164

Abstract

We study the relationship between the tensor product multiplicities of a compact semisimple Lie algebra 𝔤 and a special function 𝒥 associated to 𝔤, called the volume function. The volume function arises in connection with the randomized Horn’s problem in random matrix theory and has a related significance in symplectic geometry. Building on box spline deconvolution formulae of Dahmen–Micchelli and De Concini–Procesi–Vergne, we develop new techniques for computing the multiplicities from 𝒥, answering a question posed by Coquereaux and Zuber. In particular, we derive an explicit algebraic formula for a large class of Littlewood–Richardson coefficients in terms of 𝒥. We also give analogous results for weight multiplicities, and we show a number of further identities relating the tensor product multiplicities, the volume function and the box spline. To illustrate these ideas, we give new proofs of some known theorems.

Highlights

An important combinatorial problem in representation theory is the determination of tensor product multiplicities
The most widely studied case is g = su(n), where the multiplicities are usually called Littlewood–Richardson coefficients as they are described by the famous Littlewood–Richardson rule [27]
In this paper we study the relationship between the multiplicities Cλνμ and a special function J associated to g, called the volume function, which takes three arguments in a Cartan subalgebra t ⊂ g: J (α, β; γ), α, β, γ ∈ t

Summary

Introduction

An important combinatorial problem in representation theory is the determination of tensor product multiplicities. We give new proofs of three known results: the differentiability class of J (Corollary 4.1), the semiclassical asymptotics expressing J as a scaling limit of tensor product multiplicities (Corollary 4.5), and a theorem of Rassart [33] and Derksen–Weyman [14] stating that for fixed (λ, μ, ν), the Littlewood–Richardson coefficient CNNλν Nμ of su(n) is a polynomial in the variable N ∈ N (Theorem 4.6) These results are not essential to the rest of the paper, the proofs are helpful for illustrating our methods: the first two demonstrate the wide-ranging consequences of the De Concini–Procesi–Vergne convolution formula, while the third provides intuition for geometric arguments that we will use later in the proof of Theorem 5.22. We replace the volume function and the box spline with discrete approximations and study convolutions on the root lattice or the weight lattice rather than on the entire Cartan subalgebra This simplifies the deconvolution problem; for any given (λ, μ, ν), only finitely many values of J are needed to compute the tensor product multiplicity.

The volume function and the box spline

First applications

Discrete convolution and deconvolution

Weight multiplicities