The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group

TL;DR

This paper introduces a canonical construction of an intrinsic hyperplane arrangement for every irreducible complex representation of the symmetric group, generalizing the classical braid arrangement with notable invariant properties.

Contribution

It provides a new, canonical hyperplane arrangement associated with each irreducible symmetric group representation, extending classical arrangements and linking to invariant subspaces of Young subgroups.

Findings

Constructed a canonical hyperplane arrangement for each irreducible representation.

Generalized the classical braid arrangement to arbitrary irreducible representations.

Described the arrangement in terms of invariant subspaces of Young subgroups.

Abstract

For every irreducible complex representation~$\pi_\lambda$ of the symmetric group~$\S_n$, we construct, in a canonical way, a so-called intrinsic hyperplane arrangement~$\A_{\lambda}$ in the space of~$\pi_\lambda$. This arrangement is a direct generalization of the classical braid arrangement (which is the special case of our construction corresponding to the natural representation of~$\S_n$), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.