Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes

Leonardo C. Mihalcea, Hiroshi Naruse, Changjian Su

Abstract

We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold $G/P$ by the class of any line bundle $\mathcal{L}_λ$. Our formula is given in terms of the $λ$-chains of Lenart and Postnikov. Its proof relies on a change of basis formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action on torus-equivariant K-theory of the complete flag manifold $G/B$ via left Demazure--Lusztig operators. We revisit some wall-crossing formulae for the stable envelopes in $T^*(G/B)$. We use our Chevalley formula, and the equivalence between motivic Chern classes of Schubert cells and K-theoretic stable envelopes in $T^*(G/B)$, to give formulae for the change of polarization, and for the change of slope for stable envelopes. We prove several additional applications, including Serre, star, and Dynkin, dualities of the Chevalley coefficients, new formulae for the Whittaker functions, and for the Hall--Littlewood polynomials. We also discuss positivity properties of Chevalley coefficients, and properties of the coefficients arising from multiplication by minuscule weights.

Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes

Abstract

We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold by the class of any line bundle . Our formula is given in terms of the -chains of Lenart and Postnikov. Its proof relies on a change of basis formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action on torus-equivariant K-theory of the complete flag manifold via left Demazure--Lusztig operators. We revisit some wall-crossing formulae for the stable envelopes in . We use our Chevalley formula, and the equivalence between motivic Chern classes of Schubert cells and K-theoretic stable envelopes in , to give formulae for the change of polarization, and for the change of slope for stable envelopes. We prove several additional applications, including Serre, star, and Dynkin, dualities of the Chevalley coefficients, new formulae for the Whittaker functions, and for the Hall--Littlewood polynomials. We also discuss positivity properties of Chevalley coefficients, and properties of the coefficients arising from multiplication by minuscule weights.
Paper Structure (34 sections, 45 theorems, 191 equations, 2 figures)

This paper contains 34 sections, 45 theorems, 191 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Affine Hecke algebra via alcove walk algebra
  3. Affine Hecke algebra
  4. Alcove walk algebra
  5. Alcoves
  6. Alcove walk algebra
  7. A $\lambda$-chain formula for the transition coefficients $c_{u,\mu}^{w,\lambda}$
  8. Alcove paths and $\lambda$-chains
  9. $\lambda$-chain formulae
  10. Motivic Chern classes of Schubert cells
  11. Definition of the Motivic Chern classes
  12. Complete flag variety case
  13. Partial flag variety case
  14. Chevalley formulae for the motivic Chern classes
  15. Chevalley coefficients
  16. ...and 19 more sections

Key Result

Theorem 1.1

The following Chevalley formula holds in $K_T(G/B)[y]$: where the Chevalley coefficients are given by and the sum is over subsets $J = \{ j_1 < \ldots < j_t \} \subset \{ 1, 2, \ldots, l \}$ such that $u < u s_{\beta_{j_1}} < u s_{\beta_{j_1}} s_{\beta_{j_2}} < \ldots < u s_{\beta_{j_1}} s_{\beta_{j_2}} \cdot \ldots \cdot s_{\beta_{j_t}} =w$; the Weyl group element $\tilde{r}_{J_>}$ is defined i

Figures (2)

  • Figure 1: Alcove walk $\color{red} p_{\lambda}$ from $\mathop{\mathrm{A_\circ}}\nolimits$ to $\mathop{\mathrm{A_\circ}}\nolimits+\lambda$$\color{blue}p_w=c^{-}_2 c^{-}_1$ , $\color{red} p_\lambda=c^{+}_{0} c^{+}_{2} c^{+}_{1} c^{+}_{0} c^{+}_{2} c^{+}_{0}$
  • Figure 2: Alcove walk $\color{red} p_{-\lambda}$ from $\mathop{\mathrm{A_\circ}}\nolimits$ to $\mathop{\mathrm{A_\circ}}\nolimits-\lambda$$\color{blue}p_w=c^{-}_2 c^{-}_1$, $\color{red} p_{-\lambda}=c^{-}_{2} c^{-}_{1} c^{-}_{2} c^{-}_{0} c^{-}_{1} c^{-}_{2}$

Theorems & Definitions (107)

  • Theorem 1.1
  • Example 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Example 2.5
  • Remark 2.6
  • Lemma 2.7
  • Remark 2.8
  • proof
  • ...and 97 more