An equivariant Hochster's formula for Sn-invariant monomial ideals

Satoshi Murai*, Claudiu Raicu

*この研究の対応する著者

研究成果: Article査読

抄録

Let (Formula presented.) be a polynomial ring over a field (Formula presented.) and let (Formula presented.) be a monomial ideal preserved by the natural action of the symmetric group (Formula presented.) on (Formula presented.). We give a combinatorial method to determine the (Formula presented.) -module structure of (Formula presented.). Our formula shows that (Formula presented.) 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 (Formula presented.) -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of (Formula presented.) -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 (Formula presented.)) we compute the (Formula presented.) -invariant part of (Formula presented.) in terms of (Formula presented.) groups of the unsymmetrization of (Formula presented.).

本文言語English
ページ(範囲)1974-2010
ページ数37
ジャーナルJournal of the London Mathematical Society
105
3
DOI
出版ステータスPublished - 2022 4月

ASJC Scopus subject areas

  • 数学 (全般)

フィンガープリント

「An equivariant Hochster's formula for Sn-invariant monomial ideals」の研究トピックを掘り下げます。これらがまとまってユニークなフィンガープリントを構成します。

引用スタイル