Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)

TL;DR

This paper derives explicit quasipolynomial formulas for Kronecker coefficients indexed by two two-row shapes, revealing their structure and enabling characterization of zeros, thus disproving a related conjecture about their stretching functions.

Contribution

It provides the first explicit quasipolynomial formulas for these specific Kronecker coefficients and characterizes their vanishing conditions, advancing understanding of their combinatorial structure.

Findings

Explicit quasipolynomial formulas for the coefficients

Characterization of zero Kronecker coefficients

Disproof of Mulmuley's conjecture on stretching functions

Abstract

We show that the Kronecker coefficients (the Clebsch-Gordan coefficients of the symmetric group) indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.