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A sufficient condition for an Artinian complete intersection quotient $S=\k[x,y]/(x^m,y^n)$, where $\k$ is an algebraically closed field of a prime characteristic, to have the strong Lefschetz property (SLP) was proved by S. B. Glasby, C. E. Praezer, and B. Xia in \cite{GPX}. In contrast, we find a necessary and sufficient condition on $m$, $n$ satisfying $3 \le m \le n$ and $p > 2m-3$ for $S$ to fail to have the SLP. Moreover we find the Jordan types for $S$ failing to have SLP for $m \le n$ and $m = 3, 4$.
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참고문헌 (19)
참고문헌 신청
A. C. Aitken / 1935 / The Normal Form of Compound and Induced Matrices / Proc. London Math. Soc. 38(2):354~376
D. Cook, II / 2012 / The Lefschetz properties of monomial complete intersections in positive characteristic / J. Algebra 369:42~58
S. P. Glasby / 2016 / Decomposing modular tensor products, and periodicity of `Jordan partitions' / J. Algebra 450:570~587
J. A. Green / 1962 / The modular representation algebra of a nite group / Illinois J. Math. 6:607~619
T. Harima / 2013 / The Lefschetz properties / Lecture Notes in Mathematics 2080
D. Cook, II / 2012 / The Lefschetz properties of monomial complete intersections in positive characteristic / J. Algebra 369:42~58
S. P. Glasby / 2016 / Decomposing modular tensor products, and periodicity of `Jordan partitions' / J. Algebra 450:570~587
J. A. Green / 1962 / The modular representation algebra of a nite group / Illinois J. Math. 6:607~619
T. Harima / 2013 / The Lefschetz properties / Lecture Notes in Mathematics 2080
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