AI 中文总结
本文通过 RSK 对应和 keys 刻画斜 Ferrers 形状的 Cauchy 核展开，给出 Demazure 原子与 key 多项式的恒等式，并推广到无限字母表。
AI 中文摘要
设 $\mu\subseteq\lambda\subseteq(m^n)$。我们刻画了在普通 Robinson--Schensted--Knuth 对应下，支撑在斜 Ferrers 图 $\lambda/\mu$ 上的矩阵的像。外边界决定了插入表右 key 的上界，而内边界决定了其左 key 的下界；两个界都依赖于记录表的 keys。这产生了斜 Ferrers Cauchy 核的表展开，使用 Lascoux 和 Schützenberger 的标准基多项式，由 Bruhat 序中的区间索引。证明首先处理普通 Ferrers 图。利用早期工作中右 key 的上确界刻画，我们通过单次 RSK 插入跟踪依赖于 $\lambda$ 的界。当 $\lambda$ 有重复部分时，这些弱列界不必构成半标准表。严格化确定了可容许弱组合的集合 $\operatorname{Comp}(\lambda)$，并且对于每个 $\alpha\in\operatorname{Comp}(\lambda)$，确定一个组合 $\alpha^\lambda$。普通 RSK 然后给出了展开式的保权双射实现 \\[ \prod_{(i,j)\in\lambda}\frac{1}{1-x_i y_j} = \sum_{\alpha\in\operatorname{Comp}(\lambda)} \hat K_\alpha(x)K_{\alpha^\lambda}(y), \\] 其中 $\hat K_\alpha$ 和 $K_\alpha$ 分别表示 Demazure 原子和 key 多项式。我们还给出了直接的可容许性判据和用于计算 $\alpha^\lambda$ 的停车过程。在转换约定后，这些与 Feigin、Khoroshkin 和 Makedonskyi 的可容许性条件和半冒泡排序构造一致。楼梯和截断楼梯恒等式作为特殊情况得出。最后，我们将弱界构造扩展到无限字母表，其中严格化不一定存在，并推导出 $m$-对称 Schur 函数的无限变量 Cauchy 恒等式。
英文摘要
Let $μ\subseteqλ\subseteq(m^n)$. We characterize the image under the ordinary Robinson--Schensted--Knuth correspondence of matrices supported on the skew Ferrers diagram $λ/μ$. The outer boundary determines an upper bound on the right key of the insertion tableau, while the inner boundary determines a lower bound on its left key; both bounds depend on the keys of the recording tableau. This yields tableau expansions of skew Ferrers Cauchy kernels using the standard basis polynomials of Lascoux and Schützenberger, indexed by intervals in Bruhat order. The proof first treats ordinary Ferrers diagrams. Using the supremum characterization of right keys from earlier work, we follow the $λ$-dependent bounds through single RSK insertions. When $λ$ has repeated parts, these weak column bounds need not form a semistandard tableau. Strictification determines a set $\operatorname{Comp}(λ)$ of admissible weak compositions and, for each $α\in\operatorname{Comp}(λ)$, a composition $α^λ$. Ordinary RSK then gives a weight-preserving bijective realization of the expansion \[ \prod_{(i,j)\inλ}\frac{1}{1-x_i y_j} = \sum_{α\in\operatorname{Comp}(λ)} \hat K_α(x)K_{α^λ}(y), \] where $\hat K_α$ and $K_α$ denote Demazure atoms and key polynomials, respectively. We also give a direct admissibility criterion and a parking procedure for computing $α^λ$. After translating conventions, these agree with the admissibility condition and half-bubble-sort construction of Feigin, Khoroshkin, and Makedonskyi. The staircase and truncated-staircase identities follow as special cases. Finally, we extend the weak-bound construction to an infinite alphabet, where strictification need not exist, and derive the infinite-variable Cauchy identity for the $m$-symmetric Schur functions.
Comments30 pages
