• Journal of the Korean Mathematical Society
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• v.56 no.3
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• pp.645-667
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• 2019

It has been little known when an Artinian point quotient has the strong Lefschetz property. In this paper, we find the Artinian point star configuration quotient having the strong Lefschetz property. We prove that if ${\mathbb{X}}$ is a star configuration in ${\mathbb{P}}^2$ of type s defined by forms (a-quadratic forms and (s - a)-linear forms) and ${\mathbb{Y}}$ is a star configuration in ${\mathbb{P}}^2$ of type t defined by forms (b-quadratic forms and (t - b)-linear forms) for $b=deg({\mathbb{X}})$ or $deg({\mathbb{X}})-1$, then the Artinian ring $R/(I{\mathbb_{X}}+I{\mathbb_{Y}})$ has the strong Lefschetz property. We also show that if ${\mathbb{X}}$ is a set of (n+ 1)-general points in ${\mathbb{P}}^n$, then the Artinian quotient A of a coordinate ring of ${\mathbb{X}}$ has the strong Lefschetz property.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.251-260
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• 2019

It has been little known when an Artinian (point) quotient has the strong Lefschetz property. In this paper, we find the Artinian complete intersection quotient having the SLP. More precisely, we prove that if ${\mathbb{X}}$ is a complete intersection in ${\mathbb{P}}^2$ of type (2, 2) and ${\mathbb{Y}}$ is a finite set of points in ${\mathbb{P}}^2$ such that ${\mathbb{X}}{\cup}{\mathbb{Y}}$ is a basic configuration of type (2, a) with $a{\geq}3$ or (3, a) with a = 3, 4, 5, 6, then $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP. We also show that if ${\mathbb{X}}$ is a complete intersection in ${\mathbb{P}}^2$ of type (3, 2) and ${\mathbb{Y}}$ is a finite set of points in ${\mathbb{P}}^2$ such that ${\mathbb{X}}{\cup}{\mathbb{Y}}$ is a basic configuration of type (3, 3) or (3, 4), then $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.401-415
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• 2020

It is well-known that if char𝕜 = 0, then an Artinian monomial complete intersection quotient 𝕜[x₁, …, x_n]/(x₁^a₁, …, x_n^a_n) has the strong Lefschetz property in the narrow sense, and it is decomposed by the direct sum of irreducible 𝖘𝖑₂-modules. For an Artinian ring A = 𝕜[x₁, x₂, x₃]/(x₁⁶, x₂⁶, x₃⁶), 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 S₃-module structure.

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.283-312
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• 2020

A sufficient condition for an Artinian complete intersection quotient S = 𝕜[x, y]/(x^m, yⁿ), 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 [3]. In contrast, we find a necessary and sufficient condition on m, n satisfying 3 ≤ m ≤ 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 ≤ n and m = 3, 4.

• The Pure and Applied Mathematics
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• v.25 no.4
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• pp.337-343
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• 2018

We investigate all kinds of the Hilbert function of the Artinian quotient of the coordinate ring of a linear star configuration in ${\mathbb{P}}^2$ of type 3 (or 3-general points in ${\mathbb{P}}^2$). As an application, we prove that such an Artinian quotient has the SLP.

• The Pure and Applied Mathematics
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• v.26 no.3
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• pp.209-214
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• 2019

We investigate all kinds of the Hilbert function of the Artinian quotient of the coordinate ring of a linear star configuration in ${\mathbb{P}}^n$ of type (n+1) (or (n+1)-general points in ${\mathbb{P}}^n$), which generalizes the result [7, Theorem 3.1].

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.763-783
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• 2018

In this paper, we study an Artinian point-configuration quotient having the SLP. We show that an Artinian quotient of points in $\mathbb{p}^n$ has the SLP when the union of two sets of points has a specific Hilbert function. As an application, we prove that an Artinian linear star configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ and $\mathbb{Y}$ are linear starconfigurations in $\mathbb{p}^2$ of type s and t for $s{\geq}(^t_2)-1$ and $t{\geq}3$. We also show that an Artinian $\mathbb{k}$-configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ is a $\mathbb{k}$-configuration of type (1, 2) or (1, 2, 3) in $\mathbb{p}^2$, and $\mathbb{X}{\cup}\mathbb{Y}$ is a basic configuration in $\mathbb{p}^2$.