A combinatorial formula for fusion coefficient

TL;DR

This paper presents a new combinatorial formula for fusion coefficients using cylindric tableaux, extending previous work on Kostka matrices and providing positive formulas under specific conditions.

Contribution

It introduces a novel combinatorial expression for fusion coefficients via cylindric tableaux, with conditions for positivity and a modified involution proof.

Findings

Fusion coefficients expressed as an alternating sum over cylindric tableaux.

Positive combinatorial formulas obtained when skew shape has a cutting point or weight has two parts.

Modified sign-reversing involution used in the proof.

Abstract

Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general.