Boolean Schubert Structure Coefficients

TL;DR

Addresses the computation of Schubert structure constants in cohomology and equivariant cohomology for generalized flag varieties by focusing on boolean Weyl-group elements. It develops a boolean insertion framework that leverages the equivariant Chevalley formula and the Kostant–Kumar construction to obtain a subtraction-free, multiplicity-free path-sum formula for $d_{uv}^w$, with a cohomology analogue giving $c_{uv}^w$. In type $A$, the authors prove $c_{uv}^w\in\{0,1\}$ for boolean $u,v,w$ and provide an $O(n^2)$ algorithm to compute these constants for boolean permutations. The work clarifies the combinatorial structure of Schubert calculus on boolean loci and yields practical tools for computation and potential links to Pieri-type rules and boolean diagrams.

Abstract

The Schubert problem asks for combinatorial models to compute structure constants of the cohomology ring with respect to Schubert classes and has been an important open problem in algebraic geometry and combinatorics that guided fruitful research for decades. In this paper, we provide an explicit formula for the (equivariant) Schubert structure constants $c_{uv}^w$ across all Lie types when the elements $u,v,w$ are boolean. In particular, in type $A$, all Schubert structure constants on boolean elements are either $0$ or $1$.

Figures (11)

• Figure 1: The directed Dynkin diagram of type $C_3$.
• Figure 2: Left: the directed Dynkin diagram of type $E_7$. Right: the boolean diagram $B(w)$ for the boolean element $w=s_3s_2s_4s_5s_7$.
• Figure 3: The boolean diagram $B(u)$ for the boolean element $u=s_3s_5s_4s_7$.
• Figure 4: The boolean diagrams $B(v)$ for all the boolean elements $v$ satisfying $s_3s_5s_4s_7=u\overset{\alpha_4}{\rightsquigarrow}v$ with a non-equivariant insertion step.
• Figure 5: The boolean diagrams $B(v)$ for $s_3s_5s_4s_7=u\overset{\alpha_6}{\rightsquigarrow}v$.

Theorems & Definitions (37)

• Remark
• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Example 2.2
• Remark
• Proposition 2.3
• Theorem 2.4: Equivariant Chevalley formula
• Definition 2.5
• Lemma 2.6