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Affine Schubert calculus and double coinvariants

Erik Carlsson, Alexei Oblomkov

TL;DR

This work constructs a geometric realization of the diagonal double coinvariant algebra $DR_n$ as an action on the equivariant Borel–Moore homology of the affine flag variety in type $A$, with $x_i$ acting by Chern classes and $y_i$ defined from left/right affine-Weyl actions. The action preserves the affine Springer fiber homology, and for the case $m=n+1$ yields an isomorphism with DR_n via the Schubert class $ar{ riangle}_n$, linking to a Cherednik-algebra action to first order. A geometric filtration by the Garsia–Stanton descent order recovers a monomial basis for $DR_n$ and provides an independent, geometric proof of the Shuffle Theorem. The paper also develops a Hessenberg-paving framework for the affine Springer fiber, connecting combinatorial parking-function statistics with fixed-point data and Hessenberg varieties, and discusses extensions to broader affine settings and root systems.

Abstract

We define an action of the double coinvariant algebra $DR_n$ on the equivariant Borel-Moore homology of the affine flag variety $\widetilde{Fl}_n$ in type $A$, which has an explicit form in terms of the left and right action of the (extended) affine Weyl group and multiplication by Chern classes. Up to first order in the augmentation ideal, we show that it coincides with the action of the Cherednik algebra on the equivariant homology of the homogeneous affine Springer fiber $\widetilde{S}_{n,m} \subset \widetilde{Fl}_n$ due to Yun and the second author, and therefore preserves the non-equivariant Borel-Moore homology groups $H_*(\widetilde{S}_{n,m})\hookrightarrow H_*(\widetilde{Fl}_n)$. We then define a geometric filtration $F_{a} H_*(\widetilde{S}_{n,n+1})=H_*(\widetilde{S}(a))$ by closed subspaces $\widetilde{S}(a)\subset \widetilde{S}_{n,n+1}$, which we prove recovers the Garsia-Stanton descent order on $DR_n$. We use this to deduce an explicit monomial basis of $DR_n$, as well as an independent proof of the (non-compositional) Shuffle Theorem.

Affine Schubert calculus and double coinvariants

TL;DR

This work constructs a geometric realization of the diagonal double coinvariant algebra as an action on the equivariant Borel–Moore homology of the affine flag variety in type , with acting by Chern classes and defined from left/right affine-Weyl actions. The action preserves the affine Springer fiber homology, and for the case yields an isomorphism with DR_n via the Schubert class , linking to a Cherednik-algebra action to first order. A geometric filtration by the Garsia–Stanton descent order recovers a monomial basis for and provides an independent, geometric proof of the Shuffle Theorem. The paper also develops a Hessenberg-paving framework for the affine Springer fiber, connecting combinatorial parking-function statistics with fixed-point data and Hessenberg varieties, and discusses extensions to broader affine settings and root systems.

Abstract

We define an action of the double coinvariant algebra on the equivariant Borel-Moore homology of the affine flag variety in type , which has an explicit form in terms of the left and right action of the (extended) affine Weyl group and multiplication by Chern classes. Up to first order in the augmentation ideal, we show that it coincides with the action of the Cherednik algebra on the equivariant homology of the homogeneous affine Springer fiber due to Yun and the second author, and therefore preserves the non-equivariant Borel-Moore homology groups . We then define a geometric filtration by closed subspaces , which we prove recovers the Garsia-Stanton descent order on . We use this to deduce an explicit monomial basis of , as well as an independent proof of the (non-compositional) Shuffle Theorem.

Paper Structure

This paper contains 32 sections, 31 theorems, 245 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Combinatorial notation and preliminaries
  3. Combinatorial notations
  4. Coinvariant algebras
  5. Rational parking functions
  6. Schedules
  7. Restricted permutations
  8. Hessenberg paving combinatorics
  9. Lattice description of parking functions
  10. Geometric preliminaries
  11. Root systems
  12. The affine flag variety
  13. The affine Springer fiber
  14. Action of the Cherednik algebra
  15. Affine Schubert calculus
  16. ...and 17 more sections

Key Result

Proposition 2.1

(Allen allen1994descent) For any composition $\mathbf{a}$, there exists a partition $\mu$ and a composition $\mathbf{c}$ such that where $m_\mu(\mathbf{y})$ is the monomial symmetric function. Furthermore, $\mu$ is the empty partition if and only if $\mathbf{a}$ is a descent composition, that is $\mathbf{y}^\mathbf{a}=\mathbf{y}^{\mathbf{maj}(\sigma)}$ for some $\sigma\in S_n$.

Figures (3)

  • Figure 1: A rational parking function $P=(\pi,\sigma)\in \mathop{\mathrm{PF}}\nolimits(4,7)$. Then we have $\mathbf{area}(\pi)=(0,1,1,1)$, $\mathbf{coarea}(\pi)=(0,0,2,4)$.
  • Figure 2: A parking function $P=(\pi,\sigma)\in \mathop{\mathrm{PF}}\nolimits(7)$ with $\sigma=(6,7,2,3,5,1,4)$, $\mathop{\mathrm{dinv}}\nolimits(P)=4$, $\mathbf{area}(P)=(0, 1, 1, 2, 2, 1, 0)$, $\mathbf{coarea}(P)=(0, 0, 1, 1, 2, 4, 6)$, $\mathop{\mathrm{word}}\nolimits(P)=(5, 3, 1, 2, 7, 4, 6)$, and $\mathbf{maj}(P)=(1, 1, 2, 0, 2, 0, 1)$, the descent composition whose $i$th entry is the area in the row containing $\sigma_i$.
  • Figure 3: The elements of $\mathop{\mathrm{cars}}\nolimits(\tau)$ for $\tau=(3,1,2,5,4)$, as in Figure 4 of haglund2008catalan.

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1: Shuffle Theorem haglund2005combinatorialcarlsson2015proof
  • Definition 2.5
  • Proposition 2.2
  • proof
  • Example 1
  • ...and 69 more