Seminar of Algebra

Vector partition functions appear in a wide variety of contexts such as Statistics, Number Theory, Algebraic Geometry, and Representation Theory. In this talk we illustrate how vector partition functions arise from Representation Theory problems and then show how this approach can be exploited in order to compute and better understand the arising Algebraic Combinatorial coefficients. We give vector partition function interpretations for the Kronecker coefficients (due to Mishna, Rosas, and Sundaram) and Littlewood-Richardson coefficients (due to Rassart). In the case of Kronecker coefficients, we give computational results, some vanishing conditions, and recent progress towards a conjectured bound. For Littlewood-Richardson coefficients, we give a determinantal formula and explain some known stability results. Additionally, we explain how one can understand the linear symmetries from the geometry.