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A Gröbner basis for Kazhdan-Lusztig ideals of the flag variety of affine type A

Balázs Elek, Daoji Huang

Abstract

A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via Bott-Samelson maps, and give explicit equations that generate the Kazhdan-Lusztig ideals in these coordinates. Furthermore, our equations form a Gröbner basis for the Kazhdan-Lusztig ideals. Our result generalizes a result of Woo-Yong that gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A flag variety.

A Gröbner basis for Kazhdan-Lusztig ideals of the flag variety of affine type A

Abstract

A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via Bott-Samelson maps, and give explicit equations that generate the Kazhdan-Lusztig ideals in these coordinates. Furthermore, our equations form a Gröbner basis for the Kazhdan-Lusztig ideals. Our result generalizes a result of Woo-Yong that gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A flag variety.

Paper Structure

This paper contains 7 sections, 16 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Matrix realization of the affine flag variety in type A
  3. Equations for opposite Schubert conditions
  4. Equations for Kazhdan-Lusztig vareties in $Fl_0(V)$
  5. A preferred Bott-Samelson parametrization for $\mathcal{X}_\circ^w$
  6. Equations for Kazhdan-Lusztig varieties in $Fl_0(V)$
  7. Gröbner basis for Kazhdan-Lusztig ideals

Key Result

Lemma \oldthetheorem

For all $(i,j)\in \mathbb{Z}\times \mathbb{Z}$, $\dim(E^i\cap L_j)\ge |v\mathbb{Z}_{\le j}\cap \mathbb{Z}_{>i}|$ if and only if there exists $l\ge l_v(i,j)$ such that all minors $\det (M_{I,J})$ where $I\subset \mathbb{Z}_{\le i}$, $J\subseteq [j-l+1, j]$, and $|I|=|J| =n_v(i,j,l)+1$ vanish.

Figures (5)

  • Figure 1: Example for $l_v(i,j)$ and $n_v(i,j,l)$
  • Figure 2: Computing $\mathcal{X}^w_\circ \cap \mathcal{X}_v$ for $w=[-5,5,0,10]$ and $v=[-1,1,6,4]$
  • Figure 3: Example computation for Case (a)
  • Figure 4: Example computation for Case (b)
  • Figure 5: Example computation for Case (c)

Theorems & Definitions (41)

  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 31 more