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

Abstract

For every irreducible complex representation $$\pi _\lambda $$ of the symmetric group $${{\mathfrak {S}}}_n$$ , we construct, in a canonical way, a so-called intrinsic hyperplane arrangement $${{\mathcal {A}}}_{\lambda }$$ 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 $${{\mathfrak {S}}}_n$$ ), has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties.