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Deligne--Lusztig varieties, toric orbifolds, and the $q$-Klyachko algebra

Ruizhen Liu

Abstract

We investigate the geometry behind the $q$-Klyachko algebra, introduced by Nadeau--Tewari. When $q$ is a prime power, we show that the $q$-Klyachko algebra is the image of the pullback map on Chow rings $\mathrm{CH}(\mathrm{Fl}_{n+1})\to\mathrm{CH}(\mathrm{DL}_n)$, where $\mathrm{DL}_n\subseteq \mathrm{Fl}_n$ is a compactified Deligne--Lusztig variety inside the complete flag variety $\mathrm{Fl}_{n+1}$. When $q$ is a positive rational number, we establish a Kähler package for the $q$-Klyachko algebra through inputs from toric geometry.

Deligne--Lusztig varieties, toric orbifolds, and the $q$-Klyachko algebra

Abstract

We investigate the geometry behind the -Klyachko algebra, introduced by Nadeau--Tewari. When is a prime power, we show that the -Klyachko algebra is the image of the pullback map on Chow rings , where is a compactified Deligne--Lusztig variety inside the complete flag variety . When is a positive rational number, we establish a Kähler package for the -Klyachko algebra through inputs from toric geometry.
Paper Structure (9 sections, 13 theorems, 27 equations, 2 figures, 1 table)

This paper contains 9 sections, 13 theorems, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation from probability theory
  3. Statements of the main results
  4. The Deligne--Lusztig variety of a Coxeter element
  5. The toric orbifold $X_{\Sigma_{n,q}}$
  6. A $q$-deformation of the type $A$ Cartan matrix
  7. The simplicial fan $\Sigma_{n,q}$: construction and properties
  8. Cohomology of the toric orbifold $X_{\Sigma_{n,q}}$
  9. A Kähler package for the $q$-Klyachko algebra

Key Result

Theorem 1.2

For $1\le i\le n$, we set $L_i$ to be the pullback of line bundle on $\mathrm{Fl}_{n+1}$ associated to the $i$th fundamental weight for the type $A_n$ root system. Then, we have the following isomorphism of graded $\mathbf Q$-algebras,

Figures (2)

  • Figure 1: An application of the random displacement rule at $i$.
  • Figure 2: Illustration of the fan $\Sigma_{2,2}$, maximal cones labelled by two-part partitions of $[2]$.

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Proposition 2.3: Langer
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • ...and 16 more