The Crystal Structure of the Plethysm of Schur Functions (18frg224)


(York University)

(Université du Québec à Montréal)


The Banff International Research Station will host the "The crystal structure of the plethysm of Schur functions" workshop in Banff from April 1 to April 8, 2018.

At the beginning of the $20^{th}$ century, mathematicians like Schur, Frobenius and Young laid the foundations for one of the largest branches of algebra known as representation theory. One of the main problems in representation theory deals with the decomposition of representations into the smallest possible components. The objective of this BIRS Focused Research Group is to study the decomposition of the representation obtained when we compose two representations of the general linear group. Special cases of this problem have been solved in the decades immediately following the 1950's, but even with close to 70 years of attempts, the problem remains elusive.

Our goal is to apply the theory of crystals and define a crystal graph for the composition of two general linear group representations. We will build on and adapt a construction by Loehr and Warrington that has already yielded conjectured special cases. The solution of this problem will have important applications in physics, invariant theory and quantum computing.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).