Equivariant Hilbert Series of Subspace Arrangements

Abstract: To an hyperplane arrangement we can associate a combinatorial object, a matroid. Similarly, to a subspace arrangement we associate a polymatroid. Each subspace in the arrangement can be viewed algebraically as a linear ideal. We can study the product of these linear ideals by using the combinatorial data of the polymatroid. In particular, by tensoring the ambient vector space with an $n$-dimensional vector space, we introduce an action of $GL_n$ on the subspace arrangement and consequently on the product of linear ideals. The equivariant Hilbert series of the product of the linear ideals can be recursively constructed from the polymatroid of the subspace arrangement. Moreover, considering the transpose functor on the category of polynomial representations, we produce an ideal in the exterior algebra associated to the subspace arrangement. The corresponding effect of this functor on the equivariant Hilbert series is the involution that maps the Schur symmetric function $s_{\lambda}$ to $s_{\lambda’}$, the function indexed by the transpose of the original partition.

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