arXiv subjects

Shota Maehara

Publications and source records attributed to Shota Maehara.

4 recordsLinked to original sources

arXiv2026

Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices

In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of $2$-dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of $2$-dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of $B_2$-exponents, and we provide an alternative proof for some known results on the exponents.

math.CO↗

arXiv2025

On Universal derivations for multiarrangements

The study of universal derivations for arbitrary multiarrangements and multiplicity functions was initiated by Abe, Röhrle, Stump, and Yoshinaga in 2024 which focused on arrangements arising from (well-generated) reflection groups. In this paper we provide a criterion for determining whether a derivation is universal along with a characterization of universal derivations for arbitrary 2-multiarrangements. As an application we give descriptions of universal derivations for several multiarrangements, including the so-called deleted $A_3$ arrangement. This is the first known example of a non-reflection arrangement that admits a universal derivation distinct from the Euler derivation.

math.CO↗

arXiv2025

Extendability of the $B_2$-arrangement

Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank.

math.CO↗

arXiv2023

Explicit description of a basis for derivations of a Coxeter multiarrangement of type $B_2$

In this article, we consider the multiarrangements whose underlying arrangements are the Coxeter arrangement of type $B_2$. For some special multiplicities, we give an explicit description of bases for the derivation modules. As an application, we also describe the lower derivations of bases for the derivation modules of some Coxeter multiarrangements of type $A_2$, which are different from ones given by Wakamiko.

math.CO↗