An Artinian Ring Having the Strong Lefschetz Property and Representation Theory

Commun. Korean Math. Soc. Published online January 20, 2020

Yong-Su Shin
Sungshin Women University

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 ${\rm sl}_2$-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.