Discrete Math Seminar: Travis Scrimshaw

The cohomology ring of the Grassmannian, the set of k-dimensional subspaces in n-dimensional space, can be described by Schur functions, a symmetric function that are characters of the special linear Lie group. To study the K-theory ring, the corresponding objects we use are (symmetric) Grothendieck polynomials. Demazure characters can be considered as partial Schur functions and are characters of representations of the subgroup of upper triangular matrices. The K-theoretic analog of Demazure characters are known as Lascoux polynomials, but they currently have no representation theoretic or geometric interpretation. In joint work with Valentin Buciumas and Katherine Weber, we give the first known combinatorial interpretation for Lascoux polynomials by describing a colored version of the 5-vertex model of Motegi and Sakai. In this talk, we will discuss Lascoux polynomials, the colored 5-vertex model, and the corresponding combinatorial interpretation from our result. No knowledge of the material will be assumed.