Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An equivariant Hochster's formula for $\mathfrak S_n$-invariant monomial
  ideals   Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and
let $I\subset R$ be a monomial ideal preserved by the natural action of the
symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to
determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$.
Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced
representations of tensor products of Specht modules associated to hook
partitions, and their multiplicities are determined by topological Betti
numbers of certain simplicial complexes. This result can be viewed as an
$\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of
monomial ideals. We apply our results to determine extremal Betti numbers of
$\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas
for their Castelnuovo--Mumford regularity and projective dimension. We also
give a concrete recipe for how the Betti numbers change as we increase the
number of variables, and in characteristic zero (or $>n$) we compute the
$\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of
$\mathrm{Tor}$ groups of the unsymmetrization of $I$.

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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th