# Kronecker coefficients -- combinatorics, complexity and beyond

# Kronecker coefficients -- combinatorics, complexity and beyond

The Kronecker coefficients $g(\lambda,\mu,\nu)$ are defined as the multiplicity of an irreducible representation $S_\lambda$ of the symmetric group $S_n$ in the tensor product of two other irreducibles, $S_\mu \otimes S_\nu$. Finding a positive combinatorial formula for these nonnegative integers or even criteria for their positivity has been a 75+ year old problem in representation theory and algebraic combinatorics. Recently, the Kronecker coefficients appeared as central objects in the field of Geometric Complexity Theory and more questions about their computational complexity emerged. In this talk we will discuss a few problems of different characters involving these coefficients -- the Saxl conjecture on the tensor square $S_\delta \otimes S_\delta$ where $\delta$ is the staircase partition, the combinatorial side with the new proof of Sylvester's theorem on the unimodality of the q-binomial coefficients as polynomials in q giving effective bounds on both, and some complexity results.