##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Greta Panova

#### Department of Mathematics, UCLA

## Combinatorics and positivity of Kronecker coefficients

##### Abstract:

Kronecker coefficients count the multiplicities of irreducible representations in the tensor product of two irreducible representations of the symmetric group. While their study was initiated almost 75 years, very little is still known about them, and one of the major problems of algebraic combinatorics is to find a positive combinatorial interpretation for the Kronecker coefficients. Recently this problem found a new meaning in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the ``P vs NP'' Millennium problem. In this talk we will describe several problems and some results on different aspects of the Kronecker coefficients. A conjecture of Jan Saxl states that the tensor square $S_\rho$ $ \otimes$ $S_\rho$, where $\rho$ is the staircase partition, contains every irreducible representation of $S_n$. We will present results towards this conjecture, as well as a tool for determining the positivity of certain Kronecker coefficients. We will also explore the combinatorial aspect of the problem and show how to prove certain unimodality results using Kronecker coefficients, including Sylvester's theorem on the unimodality of $q$-binomial coefficients (as polynomials in $q$) and some new extensions thereof, like strict unimodality. The results are based on joint work with Igor Pak and Ernesto Vallejo.

Host: Jeff Remmel

### October 1, 2013

### 4:00 PM

### AP&M 7321

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