# Combinatorics and complexity of Kronecker coefficients

November 3 to November 7, 2014

at the

American Institute of Mathematics, San Jose, California

organized by

Igor Pak, Greta Panova, and Ernesto Vallejo

## Original Announcement

This workshop will be devoted to the study of Kronecker coefficients which describe the decomposition of tensor products of irreducible representations of a symmetric group into irreducible representations. We concentrate on their combinatorial interpretation, computational aspects and applications to other fields.

The workshop will focus on:

• Finding combinatorial interpretation for the Kronecker coefficients. In terms of complexity theory this amounts to working on resolving whether the problem $KRON$ is in $\#P$. The aim will be to use complexity theory to find evidence for or against that.
• Determining the complexity of deciding the problem $KP$ of positivity of the Kronecker coefficients. Mulmuley's conjecture states that $KP$ is in $P$. The goal will be to either prove this conjecture or else show that, for example, $KP$ is $NP$--hard.
• Resolving combinatorial special cases. Among them are proving the Saxl conjecture that for every large enough symmetric group has an irreducible representation whose tensor square contains every irreducible representation as a constituent. Other interesting combinatorial aspects include the application of Kronecker coefficients to solving combinatorial problems of different origins, specifically proving unimodality results.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Proof of Stembridge's conjecture on stability of Kronecker coefficients
by  Steven V Sam and Andrew Snowden,  J. Algebraic Combin. 43 (2016), no. 1, 1-10  MR3439297
Membership in moment polytopes is in NP and coNP
by  Peter Bürgisser, Matthias Christandl, Ketan D. Mulmuley and Michael Walter,  SIAM J. Comput. 46 (2017), no. 3, 972–991  MR3662037