An inverse Grassmannian Littlewood-Richardson rule and extensions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929602526445568 |
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| author | Pechenik, Oliver Weigandt, Anna |
| author_facet | Pechenik, Oliver Weigandt, Anna |
| contents | Chow rings of flag varieties have bases of Schubert cycles $σ_u$, 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 $σ_u \cdot σ_v$ 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 $σ_u \cdot σ_v$ 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 $σ_u \cdot σ_v$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11185 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An inverse Grassmannian Littlewood-Richardson rule and extensions Pechenik, Oliver Weigandt, Anna Combinatorics 05E05, 05E14, 14M15, 14N15 Chow rings of flag varieties have bases of Schubert cycles $σ_u$, 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 $σ_u \cdot σ_v$ 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 $σ_u \cdot σ_v$ 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 $σ_u \cdot σ_v$ 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. |
| title | An inverse Grassmannian Littlewood-Richardson rule and extensions |
| topic | Combinatorics 05E05, 05E14, 14M15, 14N15 |
| url | https://arxiv.org/abs/2202.11185 |