A dual Littlewood-Richardson rule and extensions

• Published in: arXiv.org
• Main Authors: Pechenik, Oliver, Weigandt, Anna
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 02.08.2022
• Subjects: Coefficients; Combinatorial analysis; Homology; Rings (mathematics)
• ISSN: 2331-8422

Littlewood-Richardson coefficients are nonnegative integers that play a pivotal role in combinatorial algebra, bridging the representation theory of symmetric and general linear groups, the Chow rings of Grassmannians, and eigenvalues of Hermitian matrices. Littlewood-Richardson rules are positive combinatorial formulas for Littlewood-Richardson coefficients that manifest their nonnegativity. More generally, Chow rings of flag varieties have bases of Schubert cycles \(\sigma_u\), indexed by permutations. Here, the Littlewood-Richardson coefficients describe special products \(\sigma_u \cdot \sigma_v\) where \(u\) and \(v\) are \(p\)-Grassmannian permutations. Building on work on Wyser, we introduce backstable clans to prove a "dual" Littlewood-Richardson rule for the product \(\sigma_u \cdot \sigma_v\) when \(u\) and \(v\) are inverse to \(p\)-Grassmannian permutations. Moreover, our positive combinatorial rule extends to the case where \(u\) is \(p\)-inverse Grassmannian and \(v\) is \(q\)-inverse Grassmannian. We further give a positive combinatorial rule for \(\sigma_u \cdot \sigma_v\) in the case that \(u\) is \(p\)-inverse Grassmannian and \(v\) is covered in weak Bruhat order by a \(q\)-inverse Grassmannian permutation.