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Presented By: Combinatorics Seminar - Department of Mathematics

Combinatorics Seminar -- Top layers of Grothendieck and Lascoux polynomials

Tianyi Yu (UCSD)

Tableaux for computing the degrees of Grothendieck and Lascoux polynomials Tableaux for computing the degrees of Grothendieck and Lascoux polynomials
Tableaux for computing the degrees of Grothendieck and Lascoux polynomials
The Schubert polynomials of the Symmetric group of n form a basis of the vector space they span. This space is well-studied with a dimension of n! and its Hilbert series being the q-analogue of n!. Key polynomials, which are characters of the Demazure modules, also form a basis for this space along with Schubert polynomials. Schubert and key polynomials are the ``bottom layers’’ of Grothendieck and Lascoux polynomials, respectively, which are two inhomogeneous polynomials. In this talk, we will focus on the space spanned by their ``top layers’’. We construct two bases using the top layer of Grothendieck and the top layer of Lascoux. We will also introduce a diagrammatic method to compute the degrees of these polynomials, involving drawing dark clouds and snowflakes. Finally, we will describe the Hilbert series of this space using a classical q-analogue of the Bell numbers.
Tableaux for computing the degrees of Grothendieck and Lascoux polynomials Tableaux for computing the degrees of Grothendieck and Lascoux polynomials
Tableaux for computing the degrees of Grothendieck and Lascoux polynomials

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