An inverse Grassmannian Littlewood-Richardson rule and extensions

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.