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Signed puzzles for Schubert coefficients

Igor Pak, Colleen Robichaux

TL;DR

We present a fully combinatorial, signed puzzle rule for Schubert coefficients $c^w_{u,v}$ in type $A$, derived from Knutson's recurrence and implemented via a region $\Gamma(u,v,w)$ tiled by pieces from a finite set $\mathcal{T}_n$. The signed-count of puzzles equals $c^w_{u,v}$, with the sign determined by the number of red (negative) pieces, and the inductive argument encodes Knutson's four cases along the top row. A major application shows that the sums $\gamma_k(n)=\sum_{u,v,w:\operatorname{inv}(w)=k} c^w_{u,v}$ are polynomials in $n$ for fixed $k$, proven via a reduction to counting lattice points in unimodular polyhedra using Ehrhart theory. The results provide a novel signed puzzle framework for Schubert structure constants, bridging combinatorial tilings, recurrence relations, and polyhedral counting, and opening avenues for extensions to other cohomology theories and signed-interpretation questions.

Abstract

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded number of inversions.

Signed puzzles for Schubert coefficients

TL;DR

We present a fully combinatorial, signed puzzle rule for Schubert coefficients in type , derived from Knutson's recurrence and implemented via a region tiled by pieces from a finite set . The signed-count of puzzles equals , with the sign determined by the number of red (negative) pieces, and the inductive argument encodes Knutson's four cases along the top row. A major application shows that the sums are polynomials in for fixed , proven via a reduction to counting lattice points in unimodular polyhedra using Ehrhart theory. The results provide a novel signed puzzle framework for Schubert structure constants, bridging combinatorial tilings, recurrence relations, and polyhedral counting, and opening avenues for extensions to other cohomology theories and signed-interpretation questions.

Abstract

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded number of inversions.

Paper Structure

This paper contains 19 sections, 3 theorems, 14 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Standard definitions and notation
  3. Knutson's recurrence
  4. The construction
  5. The region
  6. Puzzle pieces
  7. Labels and indicators
  8. Constraints
  9. Summary
  10. Proof of Theorem \ref{['t:main']}
  11. Example
  12. Proof of Theorem \ref{['t:app']}
  13. The setup
  14. Relative placements and labelings
  15. Actual labelings
  16. ...and 4 more sections

Key Result

Theorem 1.1

For every integer $n$, let $\mathcal{T}_n$ be a set of $O(n^9)$ puzzle pieces defined in Section s:puzzle. Let $u,v,w\in S_n$ be permutations with $\operatorname{{\rm inv}}(u)+\operatorname{{\rm inv}}(v)=\operatorname{{\rm inv}}(w)$, and denote $\ell=\binom{n}{2}-\operatorname{{\rm inv}}(u)$. Let $\

Figures (16)

  • Figure 4.1: Region $\Gamma$.
  • Figure 4.2: White, shaded and dark triangles. Three colors of dark triangles: dark yellow, dark blue and dark red.
  • Figure 4.3: Possible docket numbers of dark triangles.
  • Figure 4.4: Indicators $\circ$ and $\ast$ on the left and right edges of three types of triangles.
  • Figure 4.5: Permutation labels on three types of triangles.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1: Knutson's recurrence Knu03
  • Remark 5.1