论文标题
逆向格拉斯曼尼亚利特伍德 - 里查森（Richardson）的规则和扩展
An inverse Grassmannian Littlewood-Richardson rule and extensions
论文作者
论文摘要
国旗品种的食物环具有舒伯特周期$σ_U$的基础，由排列索引。代数组合主义者的一个主要问题是为此基础结构常数提供一个积极的组合公式。著名的Littlewood-Richardson规则解决了特殊产品的问题$σ_U\ CDOTσ_V$其中$ u $和$ v $是$ P $ -Grassmannian排列。 在Wyser的工作基础上，我们介绍了Backstable Clans，以证明计算产品$σ_U\ CDOTσ_V$的问题时，当$ u $是$ u $是$ p $ -p $ -inverse Grassmannian和$ v $是$ q $ q $ $ q $ - inverse-inverse grassmannian。通过在结构常数之间建立几个新的线性关系家族，我们进一步扩展了此结果，以获得$σ_U\ cdotσ_v$的积极组合规则，在这种情况下，$ u $ $ u $以薄弱的bruhat订单覆盖了$ p $ - inverse-inverse grassmannian permuty，$ v $ $ v $是$ q $ q $ - $ q $ inverse-inverse-inverse-inverse-inversemanderneversnian pentuart。
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.
