An Artinian ring having the strong Lefschetz property and representation theory
Commun. Korean Math. Soc. 2020 Vol. 35, No. 2, 401-415
Published online April 6, 2020
Yong-Su Shin
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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