Box splines, tensor product multiplicities and the volume function

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.

On the Minor Problem and Branching Coefficients

“…The aim of this note is to review these questions and to make explicit the link, by use of orbital integrals. It is thus in the same vein as recent works on the Horn [8][9][10][11][12] or Schur-Horn [13] problems.…”

“…with dim V α " ∆npα`ρnq ∆npρq . Plugging in (12) the expression (31) and the analogous one for supn ´1q leads to s Kpα `ρn ; γ `ρn´1 q "…”

Multiplicities, pictographs, and volumes

“…It is interesting to investigate the differentiability class of the volume functions, as well as the nature of singularities, but we shall say nothing about this problem here, we only notice that this is one of the subjects studied in [3], [5], [6], and we refer the interested reader to these articles. The other aspects of I and J briefly mentioned in the previous paragraphs, in particular their relations with the Horn and the Schur problems, will not be discussed in the present paper either, we refer the reader to the same articles, as well as to [2,4,17], and to the forthcoming thesis [18].…”

“…[2]. Inverting the above formulae (and their analog for the Schur volume function) is a very interesting question that has been addressed in [17].…”

On Schur problem and Kostka numbers

“…Remark. Inverting the I-multiplicity formula (29) is an interesting question that has been addressed in [17], sect. 6.…”