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Structure constants of Peterson Schubert calculus

Tao Gui, Yuqi Jia, Xinkai Yu, Zhexi Zhang, Yuchen Zhu

Abstract

We give an explicit, positive, and type-uniform formula for all equivariant structure constants of the Peterson Schubert calculus in arbitrary Lie types, using only the Cartan matrix of the corresponding root system $Φ$. As an application, we derive a type-uniform formula for the mixed $Φ$-Eulerian numbers.

Structure constants of Peterson Schubert calculus

Abstract

We give an explicit, positive, and type-uniform formula for all equivariant structure constants of the Peterson Schubert calculus in arbitrary Lie types, using only the Cartan matrix of the corresponding root system . As an application, we derive a type-uniform formula for the mixed -Eulerian numbers.

Paper Structure

This paper contains 7 sections, 20 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Flag varieties and Schubert varieties
  4. Peterson varieties and some geometric constructions
  5. Cohomology ring of Peterson varieties and Peterson Schubert classes
  6. Structure constants of Peterson Schubert calculus
  7. Applications to mixed Eulerian numbers

Key Result

Theorem 1.2

Under the isomorphism eq-HHMpre, the Peterson Schubert class $p_I$, $I \subset \Delta$, is represented by the monomial where $\operatorname{det}\left(C_I\right)$ is the determinant of the Cartan sub-matrix $C_I$ determined by $I\subset\Delta$ and $\left|W_I\right|$ is the order of the parabolic subgroup $W_I$ of the Weyl group $W$ determined by $I\subset\Delta$.

Theorems & Definitions (40)

  • Theorem 1.2: Peterson Schubert monomials
  • Theorem 1.3: Structure constants of equivariant Peterson Schubert calculus
  • Theorem 1.4
  • Theorem 1.5
  • Example 2.1: Type $A_{n-1}$ Peterson variety
  • Proposition 2.2: See Goldin2024positivity and lam2016total
  • Proposition 2.3
  • Theorem 2.4: harada2015equivariant, Theorem 4.1
  • Theorem 2.5: Goldin2024positivity
  • Theorem 2.6: Intersection multiplicity formula, goldin2021equivariant
  • ...and 30 more