Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2020; 35(2): 401-415

Online first article January 20, 2020      Printed April 30, 2020

https://doi.org/10.4134/CKMS.c190093

Copyright © The Korean Mathematical Society.

An Artinian ring having the strong Lefschetz property and representation theory

Yong-Su Shin

KIAS

Abstract

It is well-known that if ${\rm char}\k=0$, then an Artinian monomial complete intersection quotient $\k[x_1,\dots,x_n]/(x_1^{a_1},\dots,x_n^{a_n})$ has the strong Lefschetz property in the narrow sense, and it is decomposed by the direct sum of irreducible $\sl_2$-modules. For an Artinian ring $A=\k[x_1,x_2,x_3]/(x_1^6,x_2^6,x_3^6)$, by the Schur-Weyl duality theorem, there exist $56$ trivial representations, $70$ standard representations, and $20$ sign representations inside $A$. In this paper we find an explicit basis for $A$, which is compatible with the $S_3$-module structure.

Keywords: The strong Lefschetz property, representation theory, Artinian monomial complete intersection quotients, Hilbert functions

MSC numbers: Primary 13A02; Secondary 20C99

Supported by: This paper was supported by a grant from Sungshin Women's University.

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