Positivity conditions for plethystic coefficients

by Álvaro Gutiérrez Cáceres (University of Bristol)

Thursday 7 Dec 2023, 15:00 → 16:00 Europe/Rome

Aula Riunioni (Department of Mathematics)

The composition of two Schur functors is known as the plethysm of the respective representations. Understanding the coefficients of plethysm (that is, the decomposition into irreducible representations of the plethysm of two irreducible representations) is a notoriously difficult problem, featured in Stanley's list of important problems in algebraic combinatorics. In particular, Stanley's asks for a combinatorial model from which non-negativity of these coefficients can be deduced. Such a model may not exist, as argued by Pak and Ikenmeyer, since deciding positivity of plethystic coefficients is NP hard.

In this talk, we present the definition of plethysm and the coefficients of plethysm in the language of symmetric functions, and a condition for the positivity of the coefficients. This condition is of polynomial complexity and conceptual relevance.

Joint work with Mercedes Rosas.