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Stability of products of double Grothendieck polynomials

Andrew Hardt, David Wallach

TL;DR

This work analyzes products of double Grothendieck polynomials as representatives of equivariant $K$-theory classes for Schubert varieties, proving that forward- and back-stability numbers align with those for Schubert polynomials across several stability notions. The authors establish explicit equalities $\mathop{\mathrm{FS}}(u,v)=\Xi(u,v)$ and $\mathop{\mathrm{BS}}(u,v)=\Omega(u,v)$, with analogous results for back-stable and $K$-theoretic variants, where $\Xi$ and $\Omega$ are combinatorial functions defined via Lehmer codes and Rothe/dual-Rothe diagrams. The proofs combine Grothendieck expansion formulas (Lenart, Fomin–Kirillov, Lam–Lee–Shimozono), Hardt–Wallach-type stability arguments, and positivity results (Brion, Anderson), yielding a new proof of Lam–Lee–Shimozono’s finite-expansion conjecture for back-stable double Grothendieck polynomials. Additionally, the paper provides a detailed description of the integer support of double Grothendieck products, giving precise conditions on which simple reflections can occur and highlighting cases for dominant and Grassmannian permutations. Overall, the results deepen the understanding of stability phenomena in equivariant $K$-theory of flag varieties and offer concrete, computable criteria for the appearance of terms in Grothendieck product expansions.

Abstract

We prove that products of double Grothendieck polynomials have the same back- and forward-stability numbers as products of Schubert polynomials, characterize which simple reflections appear in such products, and also give a new proof of a finiteness conjecture of Lam-Lee-Shimozono on products of back-stable Grothendieck polynomials which was first proved by Anderson. To do this, we use the main theorems from our recent work, as well as expansion formulas of Lenart, Fomin-Kirillov, and Lam-Lee-Shimozono.

Stability of products of double Grothendieck polynomials

TL;DR

This work analyzes products of double Grothendieck polynomials as representatives of equivariant -theory classes for Schubert varieties, proving that forward- and back-stability numbers align with those for Schubert polynomials across several stability notions. The authors establish explicit equalities and , with analogous results for back-stable and -theoretic variants, where and are combinatorial functions defined via Lehmer codes and Rothe/dual-Rothe diagrams. The proofs combine Grothendieck expansion formulas (Lenart, Fomin–Kirillov, Lam–Lee–Shimozono), Hardt–Wallach-type stability arguments, and positivity results (Brion, Anderson), yielding a new proof of Lam–Lee–Shimozono’s finite-expansion conjecture for back-stable double Grothendieck polynomials. Additionally, the paper provides a detailed description of the integer support of double Grothendieck products, giving precise conditions on which simple reflections can occur and highlighting cases for dominant and Grassmannian permutations. Overall, the results deepen the understanding of stability phenomena in equivariant -theory of flag varieties and offer concrete, computable criteria for the appearance of terms in Grothendieck product expansions.

Abstract

We prove that products of double Grothendieck polynomials have the same back- and forward-stability numbers as products of Schubert polynomials, characterize which simple reflections appear in such products, and also give a new proof of a finiteness conjecture of Lam-Lee-Shimozono on products of back-stable Grothendieck polynomials which was first proved by Anderson. To do this, we use the main theorems from our recent work, as well as expansion formulas of Lenart, Fomin-Kirillov, and Lam-Lee-Shimozono.
Paper Structure (11 sections, 7 theorems, 69 equations, 3 figures)

This paper contains 11 sections, 7 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Permutations
  4. Proof of the main theorem
  5. Grothendieck stability
  6. Double Grothendieck stability
  7. The integer support of the double Grothendieck product
  8. Examples
  9. Maximal and minimal support of $P$
  10. Dominant permutations
  11. Grassmannian permutations

Key Result

Theorem 1

For all $u,v\in S_{\mathbb{Z}_+}$, where $\Xi(u,v), \Omega(u,v)$ are given by eq:Xi-Omega-def.

Figures (3)

  • Figure 1: Top: Rothe diagrams for the permutations $u = 2147653$ and $v = 1547236$, along with the computation $\Omega(u,v) = 2$. Bottom: dual Rothe diagrams for $u$ and $v$, along with the computation $\Xi(u,v) = 10$.
  • Figure 2: Rothe and dual Rothe diagrams for the dominant permutation $w = 4563712\in S_7$. The corresponding partition is $\lambda=(3,3,3,2,2)$, and the conjugate partition in $\lambda' = (5,5,3)$. The parts of $\lambda$ appear as the number of boxes in each row of the Rothe diagram, while the parts of $\lambda'$ appear in the rows of the dual Rothe diagram given by \ref{['eq:dominant-example']}.
  • Figure 3: Rothe and dual Rothe diagrams for the $4$-Grassmannian permutation $w = 23671458\in S_8$. The corresponding partition is $\lambda=(3,3,1,1)$, and the conjugate partition is $\lambda'=(4,2,2)$. The parts of $\lambda$ appear as the number of boxes in the first $k$ rows of the Rothe diagram, while the parts of $\lambda'$ appear as the number of boxes in the last $n-k$ rows of the dual Rothe diagram.

Theorems & Definitions (20)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Corollary 1: LamLeeShimiozono-grothendieck, Anderson-strong-positivity
  • proof
  • Definition 2
  • Example 1
  • Lemma 1
  • proof : Proof of Theorem \ref{['thm:double-grothendieck-stability']}(c), forward stability part
  • Theorem 2: Lenart-noncommutative
  • ...and 10 more