Predicting Linear Factors for the Kostant Vector Partition of the Root System A_(n-1)
By: Luis Alberto Perez
Department: Mathematics
Faculty Advisor: Dr. Federico Ardila
The Kostant partition function of the root system $A_{n-1}$ counts the number of ways of expressing an integral vector in $\mathbf{R}^n$ as a positive integral combination of the positive roots of $A_{n-1}$. Combinatorially, it counts the number of integer flows in a network with given inputs and outputs. Many important quantities in the representation theory of Lie algebras, such as weight multiplicities and tensor product multiplicities, are expressed in terms of the Kostant partition function of $A_{n-1}$. The Kostant partition function of $A_{n-1}$ is a piece-wise polynomial function. The regions of polynomiality are called chambers, and form the chamber complex. Finding factorization patterns of polynomials representing the Kostant partition function is an active area of study. De Loera, Sturmfels, and several others have observed that many of these polynomials have a surprising amount of linear factors. We find a way to predict linear factors by enlarging each chamber in the chamber complex of the Kostant partition function for A_(n-1). The enlarged chambers are a result of a localization theorem by de Concini-Procesi-Vergne.