Multiplicities, pictographs, and volumes

TL;DR

This paper explores pictographs used for calculating SU(n) multiplicities, discusses their semi-classical limits related to volume functions, and extends known polynomial decompositions to the case n=7.

Contribution

It introduces the case n=7 for Rn-polynomial decompositions, expanding the understanding of volume functions in representation theory.

Findings

Presentation of pictographs for SU(n) multiplicities

Analysis of semi-classical limits and volume functions

Extension of Rn-polynomial decomposition to n=7

Abstract

The present contribution is the written counterpart of a talk given in Yerevan at the SQS'2019 International Workshop "Supersymmetries and Quantum Symmetries" (SQS'2019, 26 August - August 31, 2019). After a short presentation of various pictographs (O-blades, metric honeycombs) that one can use in order to calculate SU(n) multiplicities (Littlewood-Richardson coefficients, Kostka numbers), we briefly discuss the semi-classical limit of these multiplicities in relation with the Horn and Schur volume functions and with the so-called Rn-polynomials that enter the expression of volume functions. For n < 7 the decomposition of the Rn-polynomials on Lie group characters is already known, the case n=7 is obtained here.