Skew Plücker Relations
Kazuya Aokage, Eriko Shinkawa, Hiro-Fumi Yamada
TL;DR
The paper addresses generalizing relative Plücker relations, traditionally satisfied by Schur functions, to the realm of skew Schur functions. It introduces skew Plücker relations, expressed as a double sum over an index i and hook partitions, with the skew Schur terms $S_{k_0\\dots \\ell_i/ H^{a-1}_{b}}(u)$ and $S_{\\ell_0\\ell_1\\dots \\widehat{\\ell_i} \\dots \\ell_{n+m}/ H^{c-1}_{d}}(u)$, and proves the identity for even $m$ using determinant-derived quantities $\\xi(H^{a-1}_{b},H^{c-1}_{d})$. It derives differential versions of these relations and situates the results within the framework of Hirota bilinear equations and the KP / modified KP hierarchies, highlighting connections to $\\tau$-functions and potential extensions to BKP via $Q$-functions. The work provides a bridge between representation-theoretic Plücker relations and integrable systems, with implications for $\ au$-function formalisms and algebraic geometry of flag varieties.
Abstract
Schur functions satisfy the relative Plücker relations which describe the projective embedding of the flag varieties and the Hirota bilinear equations for the modified KP hierarchies. These relative Plücker relations are generalized to the skew Schur functions.