Wednesday, February 1st, 2012
3:00pm - 4:00pm, in Science 2-064

Todor Milev

UMass Boston

Computing vector partition functions and branching laws of generalized Verma modules

Abstract: Given a finite set L of vectors with non-negative integral coordinates in n dimensions, the vector partition function with respect to L is the number of ways to express the vector (x_1,..., x_n) as an integral linear combination with non-negative coefficients of the elements of L. As a function of (x_1,..., x_n), the vector partition function with respect to L is a piecewise quasi-polynomial (i.e. it is quasi-polynomial over a finite set of closed polyhedra with walls with normals with rational coordinates). Given a semisimple complex Lie algebra g and a reductive in g subalgerba g', the Weyl character formula implies that, for compatible parabolic subalgebras of g and g', the branching laws of generalized Verma g-modules over g' are computed in closed form as piecewise quasi-polynomials via vector partition functions. In the first part of the talk we will present an algorithm for computing vector partition functions and demonstrate the software written for the purpose; in the second part of the talk we will discuss branching laws of generalized Verma modules (both the multiplicity questions and the explicit computations of singular vectors realizing the branching). We will conclude with a short discussion of the Lie representation theory software used in the talk.