Cauchy identities for skew Ferrers shapes via RSK and keys
Abstract
Let μ⊆λ⊆(mn). 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 Comp(λ) of admissible weak compositions and, for each α∈Comp(λ), a composition αλ. Ordinary RSK then gives a weight-preserving bijective realization of the expansion ∏(i,j)∈λ1/(1−xi yj) = ∑α∈Comp(λ) K̂α(x)Kαλ(y), where K̂α and Kα denote Demazure atoms and key polynomials, respectively. We also give a direct admissibility criterion and a parking procedure for computing αλ. 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.
Related
- Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
- Skew shapes, Ehrhart positivity, and beyond
- q-Zeta positivity for root and Ferrers-cell posets
- Puncturing-inclusion constructions of optimal Ferrers diagram rank-metric codes using semifield codes
- Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes
BibTeX
@misc{pena2026cauchy,
title = {{Cauchy identities for skew Ferrers shapes via RSK and keys}},
author = {Luis Pena},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15852},
eprint = {2609.15852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15852},
}
RIS
TY - UNPB
TI - Cauchy identities for skew Ferrers shapes via RSK and keys
AU - Luis Pena
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15852
UR - https://arxiv.org/abs/2609.15852
AB - Let μ⊆λ⊆(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 Comp(λ) of admissible weak compositions and, for each α∈Comp(λ), a composition α^λ. Ordinary RSK then gives a weight-preserving bijective realization of the expansion ∏_(i,j)∈λ1/(1−x_i y_j) = ∑_α∈Comp(λ) K̂_α(x)K_α^λ(y), where K̂_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.
ER -