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Minimum quantum degrees with Maya diagrams

Ryan M. Shifler

TL;DR

This work develops a combinatorial framework based on Maya diagrams to analyze minimal quantum degrees in the small quantum cohomology of partial flag varieties $ ext{Fl}(I;n)$. By translating Bruhat order and chain computations into Maya-diagram language, the authors provide a canonical lower bound on minimal quantum degrees via row-wise degrees $ ext{deg}_y(v,w)$ and prove this bound is sharp through explicit chains constructed from generalized $(a,b)$-rim hooks. The approach yields a combinatorial proof of the uniqueness of minimal quantum degrees for partial flags and extends rim-hook techniques beyond Grassmannians, with a structured procedure for calculating these degrees. The results connect moment-graph chains, Hecke products, and Bruhat order through Maya diagrams, offering a practical, visual toolkit for calculating Gromov–Witten–type quantum degrees in type A flag varieties and suggesting analogous extensions to types B, C, and D.

Abstract

We use Maya diagrams to refine the criterion by Fulton and Woodward for the smallest powers of the quantum parameter $q$ that occur in a product of Schubert classes in the (small) quantum cohomology of partial flags. Our approach using Maya diagrams yields a combinatorial proof that the minimal quantum degrees are unique for partial flags. Furthermore, visual combinatorial rules are given to perform precise calculations.

Minimum quantum degrees with Maya diagrams

TL;DR

This work develops a combinatorial framework based on Maya diagrams to analyze minimal quantum degrees in the small quantum cohomology of partial flag varieties . By translating Bruhat order and chain computations into Maya-diagram language, the authors provide a canonical lower bound on minimal quantum degrees via row-wise degrees and prove this bound is sharp through explicit chains constructed from generalized -rim hooks. The approach yields a combinatorial proof of the uniqueness of minimal quantum degrees for partial flags and extends rim-hook techniques beyond Grassmannians, with a structured procedure for calculating these degrees. The results connect moment-graph chains, Hecke products, and Bruhat order through Maya diagrams, offering a practical, visual toolkit for calculating Gromov–Witten–type quantum degrees in type A flag varieties and suggesting analogous extensions to types B, C, and D.

Abstract

We use Maya diagrams to refine the criterion by Fulton and Woodward for the smallest powers of the quantum parameter that occur in a product of Schubert classes in the (small) quantum cohomology of partial flags. Our approach using Maya diagrams yields a combinatorial proof that the minimal quantum degrees are unique for partial flags. Furthermore, visual combinatorial rules are given to perform precise calculations.
Paper Structure (14 sections, 9 theorems, 50 equations)

This paper contains 14 sections, 9 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chains
  4. Quantum Cohomology
  5. Hecke Product
  6. Fulton and Woodward's formula for minimal quantum degrees
  7. Maya Diagrams
  8. Bruhat order with Maya diagrams
  9. Maya diagram combinatorics
  10. Generalized rim hook rule
  11. Chains in terms of Maya diagrams
  12. Maya diagrams and the Bruhat order
  13. Main Result
  14. Conflict of Interest

Key Result

Proposition 2.2

FW*Theorem 9.1 Let $v,w \in W^{P}$, and let $d$ be a degree. The following are equivalent:

Theorems & Definitions (40)

  • Remark 2.1
  • Proposition 2.2
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5
  • Example 3.6
  • Proposition 3.7
  • Definition 4.1
  • ...and 30 more