On the combinatorics of parabolic Schubert varieties

Abstract

In this thesis, we study different aspects of the combinatorics of parabolic Schubert varieties. Some of these aspects are well-known, others introduce some new results, and others are conjectural and consist of our recent research directions.

Firstly, we discuss the combinatorics of parabolic $R$-polynomials. Kazhdan--Lusztig polynomials $P_{x,y}(q)$ and $R$-polynomials are of fundamental importance in the representation theory of Lie theoretic objects and the topology and geometry of Schubert varieties. We will introduce chamber systems, Coxeter complexes, their connections to the Bruhat graph of an interval, and the corresponding formulas for the $R$-polynomials. We will explore some of the connections between Bruhat graphs and $R$-polynomials and how these notions extend to their parabolic analogs.

Secondly, we will explain our joint work with N. Libedinsky and D. Plaza \cite{BLP22}, studying the combinatorial invariance conjecture for $\tilde{A_2}$. The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer in the 1980s) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan--Lusztig polynomials are equal, that is, $P_{x,y}(q) = P_{x',y'}(q)$. We prove this conjecture for the affine Weyl group of type $\tilde{A_2}$. This is the first infinite group with non-trivial Kazhdan--Lusztig polynomials where the combinatorial invariance conjecture is proved.

Finally, we will explore the top-heaviness and unimodality of some parabolic Bruhat intervals in affine Weyl groups. Let $W$ be an affine Weyl group and let $W_f$ be its corresponding finite Weyl group. We will study the top-heaviness coefficients $\operatorname{th}(w)$ and we will study the unimodality of the sequence of Betti numbers associated with the parabolic Schubert varieties corresponding to a minimal length right $W_f$-coset representative $w$. We introduce a continuous analog of this sequence, where we can prove the unimodality theorem. We will explore the supremum of the top-heaviness coefficients $\operatorname{th}(w)$ in different types.