An inverse Grassmannian Littlewood-Richardson rule and extensions

TL;DR

This paper develops a new combinatorial rule for calculating structure constants in the Chow rings of flag varieties, extending known Littlewood-Richardson rules to broader classes of permutations using novel backstable clans.

Contribution

It introduces backstable clans and establishes linear relations to extend Littlewood-Richardson rules to inverse Grassmannian permutations in flag varieties.

Findings

Derived a positive combinatorial rule for products of inverse Grassmannian permutations.

Extended the rule to permutations covered in weak Bruhat order by inverse Grassmannian permutations.

Established new linear relations among structure constants.

Abstract

Chow rings of flag varieties have bases of Schubert cycles $\sigma_u$, indexed by permutations. A major problem of algebraic combinatorics is to give a positive combinatorial formula for the structure constants of this basis. The celebrated Littlewood-Richardson rules solve this problem for special products $\sigma_u \cdot \sigma_v$ where $u$ and $v$ are $p$-Grassmannian permutations. Building on work of Wyser, we introduce backstable clans to prove such a rule for the problem of computing the product $\sigma_u \cdot \sigma_v$ when $u$ is $p$-inverse Grassmannian and $v$ is $q$-inverse Grassmannian. By establishing several new families of linear relations among structure constants, we further extend this result to obtain a positive combinatorial rule for $\sigma_u \cdot \sigma_v$ in the case that $u$ is covered in weak Bruhat order by a $p$-inverse Grassmannian permutation and $v$ is a $q$-inverse Grassmannian permutation.