Isotropic Subspaces of Schur Modules

TL;DR

This paper extends classical results about the vanishing of symmetric forms on subspaces to all symmetric types associated with Schur modules, providing a comprehensive understanding of isotropic subspaces in this broader context.

Contribution

It generalizes Tevelev's conditions for symmetric and skew-symmetric forms to all symmetric types linked to Schur modules, covering all partitions.

Findings

Extended classical vanishing conditions to all symmetric types.

Provided explicit criteria for isotropic subspaces in Schur modules.

Unified framework for symmetric multilinear forms and Schur modules.

Abstract

It is a well-known fact that over the complex numbers and for a fixed $k$ and $n$, a generic $s$ in $Sym^2V^*$ vanishes on some $k$-dimensional subspace of $V$ if and only if $n\geq 2k$. Tevelev found exact conditions for the extension of this statement for general symmetric and skew-symmetric multilinear forms, and we extend his work to all possible symmetric types, which corresponds to Schur modules for a general partition.