# How to Scale a Hypersimplex

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Amy Hanley, Ping Xu, Eyal Subag, Nigel Higson, Mathieu Stiénon.

**Speaker:** Nick Early, Penn State

**Abstract:** In a typical course in classical representation theory, it is shown early on that the finite-dimensional representations of a semisimple Lie algebra decompose into subspaces labeled by integer vectors in R^n called weights. The set of weights for a given representation spans a convex body which may be called a weight permutohedron. In particular, for sl(n) the weight permutohedra associated to the n-1 fundamental irreducible representations are called hypersimplices, the simplicial structure of which is still the subject of intensive research.
In this talk we ask the question, "what happens to the simplicial structure of a hypersimplex when it is scaled by an integer factor r?" The (partial) answer, which became the body of our thesis, enlists a new categorification of A. Postnikov's theory of generalized permutohedra. The geometric foundation for the categorification has been developed by A. Ocneanu in the past decade. Along the way, we prove an equivariant generalization of the classical Worpitzky identity in combinatorics and count solutions to a certain modular Diophantine equation in number theory.

## Room Reservation Information

**Room Number:** 104 Osmond Building

**Date:** 02/14/2017

**Time:** 2:35pm - 3:30pm