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

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

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 [email protected]
JUNZO WATANABE
Affiliation:
Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan [email protected]
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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
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

REFERENCES

Etingof, P., Gorsky, E., and Losev, I., Representations of Cherednik algebras with minimal support and torus knots , Adv. Math. 277 (2015), 124–180.Google Scholar
Fröberg, R. and Shapiro, B., On Vandermonde varieties , Math. Scand. 119 (2016), no. 1, 76–91.CrossRefGoogle Scholar
Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry.Google Scholar
Griffeth, S., Sam, S. V., and Zamaere, C. B., Jack polynomials as fractional quantum hall states and the Betti numbers of the (k + 1)-equals ideal , Comm. Math. Phys. 330 (2014), 415–434.Google Scholar
Harima, T., Maeno, T., Morita, H., Numata, Y., Wachi, A., and Watanabe, J., The Lefschetz Properties, Lecture Notes in Math. 2080, Springer, Berlin-Heidelberg, 2013.CrossRefGoogle Scholar
Hochster, M. and Eagon, J., Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci , Amer. J. Math. 93 (1971), no. 4, 1020–1058.Google Scholar
James, G. D. and Peel, M. H., Specht series for skew representations of symmetric groups , J. Algebra 56 (1979), no. 2, 343–364.CrossRefGoogle Scholar
Kustin, A. and Vraciu, A., The weak Lefschetz property for monomial complete intersections in positive characteristic , Trans. Amer. Math. Soc. 366 (2014), 4571–4601.CrossRefGoogle Scholar
Li, S. and Li, W., Independence numbers of graphs and generators of ideals , Combinatorica 1 (1981), no. 1, 55–61.CrossRefGoogle Scholar
Matsumura, H., Commutative Ring Theory, Cambridge Stud. Adv. Math. 8, Cambridge University Press, New York, 2006.Google Scholar
Migliore, J., Miró-Roig, R., and Nagel, U., Monomial ideals, almost complete intersections and the weak Lefschetz property , Trans. Amer. Math. Soc. 363 (2011), 229–257.Google Scholar
Nagata, M., Local Rings, Interscience Publishers, New York, 1962.Google Scholar
Reisner, G. A., Cohen–Macaulay quotients of polynomial rings , Adv. Math. 21 (1976), 30–49.CrossRefGoogle Scholar
Sagan, B., The Symmetric Group Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd ed., Grad. Texts in Math. 203, Springer, New York–Heidelberg, 2001.Google Scholar
Watanabe, J., “ The Dilworth number of Artinian rings and finite posets with rank function ” in Commutative Algebra and Combinatorics, Kyoto, 1985, (eds. Nagata, M. and Matsumura, H.), Adv. Stud. Pure Math. 11, North-Holland, Amsterdam; Kinokuniya, 1987, 303–312.CrossRefGoogle Scholar
Watanabe, J. and Yanagawa, K., Vandermonde determinantal ideals , Math. Scand. 125 (2019), no. 2, 179–184.CrossRefGoogle Scholar
Yanagawa, K., When is a Specht ideal Cohen–Macaulay? J. Commut. Algebra, to appear.Google Scholar