Cauchy identities for skew Ferrers shapes via RSK and keys

Abstract

Let $\mu \subseteq \lambda \subseteq (m^n)$. We characterize the image under the ordinary Robinson--Schensted--Knuth correspondence of matrices supported on the skew Ferrers diagram $\lambda /\mu$. 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 $\lambda$-dependent bounds through single RSK insertions. When $\lambda$ has repeated parts, these weak column bounds need not form a semistandard tableau. Strictification determines a set $\mathrm{Comp}(\lambda )$ of admissible weak compositions and, for each $\alpha \in \mathrm{Comp}(\lambda )$, a composition ${\alpha }^{\lambda }$. 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 ${\hat{K}}_{\alpha }$ and $K_{\alpha }$ denote Demazure atoms and key polynomials, respectively. We also give a direct admissibility criterion and a parking procedure for computing ${\alpha }^{\lambda }$. 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.

30 pages