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PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

Part of: Theory of modules and ideals Representation theory of groups General commutative ring theory

Published online by Cambridge University Press:  09 December 2021

CHRIS MCDANIEL Open the ORCID record for CHRIS MCDANIEL [Opens in a new window]  and
JUNZO WATANABE Open the ORCID record for JUNZO WATANABE [Opens in a new window]
CHRIS MCDANIEL
Affiliation:
Department of Mathematics, Endicott College, 376 Hale Street, Beverly, MA 01915, USA cmcdanie@endicott.edu
JUNZO WATANABE
Affiliation:
Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan watanabe.junzo@tokai-u.jp
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Abstract

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

MSC classification

Primary: 13A50: Actions of groups on commutative rings; invariant theory
Secondary: 13C14: Cohen-Macaulay modules 20C30: Representations of finite symmetric groups

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Type
Article
Information
Nagoya Mathematical Journal , Volume 247 , September 2022 , pp. 690 - 730
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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References

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