Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.5802/alco.340

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Scuola Normale Superiore - Edizioni della Normale  

DOI： 10.2422/2036-2145.202105_038

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s00209-023-03426-z

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1016/j.jalgebra.2022.06.028

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1093/imrn/rnaa145

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.jpaa.2021.106791

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1016/j.topol.2021.107990

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：ACM  

A distribution on the permutations over a fixed finite set is called a ranking distribution. Modelling ranking distributions is one of the major topics in preference learning as such distributions appear as the ranking data produced by many judges. In this paper, we propose a geometric model for ranking distributions. Our idea is to use hyper-surface arrangements in a metric space as the representation space, where each component cut out by hyper-surfaces corresponds to a total ordering, and its volume is proportional to the probability. In this setting, the union of components corresponds to a partial ordering and its probability is also estimated by the volume. Similarly, the probability of a partial ordering conditioned by another partial ordering is estimated by the ratio of volumes. We provide a simple iterative algorithm to fit our model to a given dataset. We show our model can represent the distribution of a real-world dataset faithfully and can be used for prediction and visualisation purposes.

DOI： 10.1145/3447548.3467253

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s10468-020-09961-1

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1093/imrn/rnz099

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Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s10801-020-00986-9

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Other Link： http://link.springer.com/article/10.1007/s10801-020-00986-9/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s10801-020-00979-8

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Other Link： http://link.springer.com/article/10.1007/s10801-020-00979-8/fulltext.html

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Walter de Gruyter GmbH  

<title>Abstract</title>Given a semisimple complex linear algebraic group <inline-formula id="j_crelle-2018-0039_ineq_9999_w2aab3b7e1517b1b6b1aab1c14b1b1Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_1056.png" /><tex-math>{ { G } }</tex-math></alternatives></inline-formula> and a lower ideal <italic>I</italic> in positive roots of <italic>G</italic>, three objects arise:
the ideal arrangement <inline-formula id="j_crelle-2018-0039_ineq_9998_w2aab3b7e1517b1b6b1aab1c14b1b7Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0652.png" /><tex-math>{\mathcal{A}_{I } }</tex-math></alternatives></inline-formula>, the regular nilpotent Hessenberg variety <inline-formula id="j_crelle-2018-0039_ineq_9997_w2aab3b7e1517b1b6b1aab1c14b1b9Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula>, and the regular semisimple Hessenberg variety <inline-formula id="j_crelle-2018-0039_ineq_9996_w2aab3b7e1517b1b6b1aab1c14b1c11Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0713.png" /><tex-math>{\operatorname{Hess}(S,I)}</tex-math></alternatives></inline-formula>.
We show that
a certain graded ring derived from the logarithmic derivation module of <inline-formula id="j_crelle-2018-0039_ineq_9995_w2aab3b7e1517b1b6b1aab1c14b1c13Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0652.png" /><tex-math>{\mathcal{A}_{I } }</tex-math></alternatives></inline-formula> is isomorphic to
<inline-formula id="j_crelle-2018-0039_ineq_9994_w2aab3b7e1517b1b6b1aab1c14b1c15Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0431.png" /><tex-math>{H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> and <inline-formula id="j_crelle-2018-0039_ineq_9993_w2aab3b7e1517b1b6b1aab1c14b1c17Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0432.png" /><tex-math>{H^{*}(\operatorname{Hess}(S,I))^{W } }</tex-math></alternatives></inline-formula>,
the invariants in <inline-formula id="j_crelle-2018-0039_ineq_9992_w2aab3b7e1517b1b6b1aab1c14b1c19Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0433.png" /><tex-math>{H^{*}(\operatorname{Hess}(S,I))}</tex-math></alternatives></inline-formula> under an action of the Weyl group <italic>W</italic> of <italic>G</italic>.
This isomorphism is shown
for general Lie type,
and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of <italic>W</italic> is isomorphic to the cohomology ring of the flag variety <inline-formula id="j_crelle-2018-0039_ineq_9991_w2aab3b7e1517b1b6b1aab1c14b1c27Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0410.png" /><tex-math>{G/B}</tex-math></alternatives></inline-formula>.

This surprising connection between Hessenberg varieties and hyperplane
arrangements enables us to produce a number of interesting
consequences. For instance, the surjectivity of the restriction map
<inline-formula id="j_crelle-2018-0039_ineq_9990_w2aab3b7e1517b1b6b1aab1c14b2b1Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0428.png" /><tex-math>{H^{*}(G/B)\to H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> announced by Dale
Peterson
and an affirmative answer to
a conjecture of Sommers and Tymoczko are immediate consequences. We also
give an explicit ring presentation of <inline-formula id="j_crelle-2018-0039_ineq_9989_w2aab3b7e1517b1b6b1aab1c14b2b3Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0431.png" /><tex-math>{H^{*}(\operatorname{Hess}(N,I))}</tex-math></alternatives></inline-formula> in
types <italic>B</italic>, <italic>C</italic>, and <italic>G</italic>. Such a presentation was already known in type
<italic>A</italic> and when <inline-formula id="j_crelle-2018-0039_ineq_9988_w2aab3b7e1517b1b6b1aab1c14b2c13Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula> is the Peterson variety. Moreover, we find
the volume polynomial of <inline-formula id="j_crelle-2018-0039_ineq_9987_w2aab3b7e1517b1b6b1aab1c14b2c15Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula> and see that the hard
Lefschetz property and the Hodge–Riemann relations hold for
<inline-formula id="j_crelle-2018-0039_ineq_9986_w2aab3b7e1517b1b6b1aab1c14b2c17Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2018-0039_eq_0711.png" /><tex-math>{\operatorname{Hess}(N,I)}</tex-math></alternatives></inline-formula>, despite the fact that it is a singular variety in general.

DOI： 10.1515/crelle-2018-0039

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Other Link： https://www.degruyter.com/downloadpdf/journals/crll/2020/764/article-p241.xml

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1016/j.jalgebra.2020.02.007

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1090/proc/14592

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

We introduce a new algebra associated with a hyperplane arrangement A, called the Solomon-Terao algebra ST(A, eta), where eta is a homogeneous polynomial. It is shown by Solomon and Terao that ST(A, eta) is Artinian when eta is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that ST(A, eta) is a complete intersection if and only if A is free. We also give a factorization formula of the Hilbert polynomials of ST(A, eta) when A is free, and pose several related questions, problems and conjectures.

DOI： 10.2969/jmsj/79957995

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s00208-018-1713-9

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s10801-017-0806-y

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1142/S0129167X18500556

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In [10], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan or Shi arrangements in [12]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A(2). (C) 2017 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2017.09.024

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

The Ish arrangement was introduced by Armstrong to give a new interpretation of the q, t-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free. (C) 2016 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jcta.2016.09.008

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Language：English   Publishing type：Research paper (international conference proceedings)   Publisher：Springer International Publishing  

This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in Abe (Invent. Math. 204(1), 317–346, 2016). The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division theorem can be regarded as a modified converse of the Orlik’s conjecture with a combinatorial condition, i.e., an arrangement is free if the restriction is free and the characteristic polynomial of the restriction divides that of an arrangement. In this article we recall, summarize, pose and re-formulate some of results and problems related to the division theorem based on Abe (Invent. Math. 204(1), 317–346, 2016), and study the modified Orlik’s conjecture with partial answers.

DOI： 10.1007/978-3-319-58971-8_14

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, Terao's open problem asking whether freeness depends only on combinatorics is true. A long standing problem whether all free arrangements are recursively free or not was settled by the second author and Hoge very recently, by giving a free but non-recursively free plane arrangement consisting of 27 planes. In this paper, we construct free but non-recursively free plane arrangements consisting of 13 and 15 planes, and show that the example with 13 planes is the smallest in the sense of the cardinality of planes. In other words, all free plane arrangements consisting of at most 12 planes are recursively free. To show this, we completely classify all free plane arrangements in terms of inductive freeness and three exceptions when the number of planes is at most 12. Several properties of the 15 plane arrangement are proved by computer programs. Also, these two examples solve negatively a problem posed by Yoshinaga on the moduli spaces, (inductive) freeness and, rigidity of free arrangements. (C) 2016 Published by Elsevier B.V.

DOI： 10.1016/j.disc.2015.12.017

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

Let be the triple of hyperplane arrangements. We show that the freeness of and the division of by imply the freeness of . This "division theorem" improves the famous addition-deletion theorem, and it has several applications, which include a definition of "divisionally free arrangements". It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free.

DOI： 10.1007/s00222-015-0615-7

J-GLOBAL

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

In this article, we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual partition formula. Then, it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each subarrangement of the filtration satisfies the dual partition formula. This generalizes the main result in Abe et al. (J. Eur. Math. Soc., to appear) which affirmatively settled a conjecture by Sommers and Tymoczko (Trans. Am. Math. Soc. 358:3493-3509, 2006).

DOI： 10.1007/s10801-015-0624-z

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：EUROPEAN MATHEMATICAL SOC  

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.

DOI： 10.4171/JEMS/615

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：FRENCH MATHEMATICAL SOC  

We consider deformations of the Weyl arrangement of type A(2), which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.

DOI： 10.24033/bsmf.2726

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Language：English   Publishing type：Research paper (scientific journal)  

In this paper, we specify a class of mathematical problems, which we refer to as "Function Density Problems" (FDPs, in short), and point out novel connections of FDPs to the following two cryptographic topics; theoretical security evaluations of keyless hash functions (such as SHA-1), and constructions of provably secure pseudorandom generators (PRGs) with some enhanced security property introduced by Dubrov and Ishai (STOC 2006). Our argument aims at proposing new theoretical frameworks for these topics (especially for the former) based on FDPs, rather than providing some con-crete and practical results on the topics. We also give some examples of mathematical discussions on FDPs, which would be of independent interest from mathematical viewpoints. Finally, we discuss possible directions of future research on other crypto-graphic applications of FDPs and on mathematical studies on FDPs themselves.

DOI： 10.1142/S0129054115500100

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, we know very little about the bases themselves except their existence. In this article, we prove the unique existence of two distinguished bases which we call the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-). They are characterized by nice divisibility properties relative to the simple roots. (C) 2014 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2014.09.011

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.5427/jsing.2014.8h

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Language：English   Publishing type：Research paper (scientific journal)  

Ziegler showed that the multirestriction of a free arrangement is also free. After Ziegler's work, several results concerning the "reverse direction", i.e., characterizing freeness of an arrangement via that of its multirestriction, have appeared. In this paper, we prove a new characterization of freeness in which the second Betti number of the arrangement plays a crucial role. © 2013 Springer-Verlag Berlin Heidelberg.

DOI： 10.1007/s00209-013-1165-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We give a lower bound on the number of chambers divided by a family of lines in the real plane. Also, a sufficient condition for 3-arrangements to be free is given in terms of chambers. To prove them, we give the upper bound on differences of exponents for balanced 2-multiarrangements in terms of the cardinality of hyperplanes.

DOI： 10.1007/s10801-012-0393-x

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MICHIGAN MATHEMATICAL JOURNAL  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Let A be an irreducible Coxeter arrangement and W be its Coxeter group. Then W naturally acts on A. A multiplicity in : A -&gt; Z is said to be equivariant when in is constant on each W-orbit of A. in this article, we prove that the multi-derivation module D(A, m) is a free module whenever in is equivariant by explicitly constructing a basis, which generalizes the main theorem of Terao (2002) [12]. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the W-invariant part D(A, m)(W) for any multiplicity m is a free module over the W-invariant subring. (C) 2012 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2012.04.015

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.

DOI： 10.1007/s10801-011-0291-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON, INC  

We prove a characterization of freeness, conjectured by Athanasiadis, for the family of hyperplane arrangements, which lie between the Coxeter and the Catalan arrangement of the type A(l). One direction was already proved in [2]. Here we prove the other direction

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD  

Let W be a finite Weyl group and A be the corresponding Weyl arrangement. A deformation of A is an affine arrangement which is obtained by adding to each hyperplane H is an element of A several parallel translations of H by the positive root (and its integer multiples) perpendicular to H. We say that a deformation is W-equivariant if the number of parallel hyperplanes of each hyperplane H is an element of A depends only on the W-orbit of H. We prove that the conings of the W-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga&apos;s theorem conjectured by Edelman-Reiner. (C) 2011 Elsevier Ltd. All rights reserved.

DOI： 10.1016/j.ejc.2011.06.005

J-GLOBAL

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We define primitive derivations for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the primitive filtrations of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration was studied in Abe and Terao (2010) [2], which generalizes the Hodge filtration introduced by K. Saito (e.g., Saito, 2003 [6]). (C) 2010 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2010.12.010

CiNii Article

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

Let W be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.

DOI： 10.1007/s00209-009-0489-8

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable. (C) 2009 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.jalgebra.2009.07.025

J-GLOBAL

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OXFORD UNIV PRESS  

We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signed-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs.

DOI： 10.1112/jlms/jdp019

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TAYLOR & FRANCIS INC  

We introduce the family of B2-type arrangements as a generalization of the classical Coxeter arrangement of type B2 and consider the stability and the freeness of it. We show the freeness and (semi)stability are determined by the combinatorics. Moreover, we give a partial answer to the 4-shift problem, which is a conjecture on the combinatorics and geometry induced from the B2-type arrangements.

DOI： 10.1080/00927870802465969

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

A central arrangement A of hyperplanes in an l-dimensional vector space V is said to be totally free if a multiarrangement (A, m) is free for any multiplicity m : A -&gt; Z(&gt; 0). It has been known that A is totally free whenever l &lt;= 2. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.

DOI： 10.1090/S0002-9939-08-09755-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

The addition-deletion theorems for hyperplane arrangements, which were originally shown by Terao [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 293-320.], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements, including supersolvable arrangements and the Coxeter arrangement of type A(3), to construct free and non-free multiarrangements.

DOI： 10.1112/jlms/jdm110

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We give a criterion for a reflexive sheaf to split into a direct sum of line bundles.

DOI： 10.1090/S0002-9939-08-09367-2

J-GLOBAL

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free. (c) 2007 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2007.04.019

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：JAPAN ACAD  

We give the first complete classification of free and non-free multiplicities on an arrangement, called the deleted A(3) arrangement, which admits both of them.

DOI： 10.3792/pjaa.83.99

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

DOI： 10.1090/S0002-9947-07-04161-X

J-GLOBAL

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Language：English  

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

We give a necessary and sufficient condition for the stability and the freeness of the family of A2-type arrangements. Moreover, we determine explicitly when the normalization of the sheafification of its module of reduced logarithmic vector fields is isomorphic to T-p2(-2).

DOI： 10.1215/kjm/1250281752

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Event date： 15 1 2024 - 15 1 2024

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：Ruhr University Bochum  

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Event date： 11 9 2023 - 15 9 2023

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 3 7 2023 - 3 7 2023

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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Event date： 14 2 2023 - 17 2 2023

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：RIMS  

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Event date： 16 1 2023 - 20 1 2023

Language：English   Presentation type：Oral presentation (general)  

Venue：Kyushu University  

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Event date： 24 10 2022 - 24 10 2022

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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Event date： 18 10 2022 - 21 10 2022

Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Event date： 16 2 2022 - 16 2 2022

Language：Japanese   Presentation type：Oral presentation (general)  

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Event date： 6 12 2021 - 10 12 2021

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 17 11 2021 - 17 11 2021

Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

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Event date： 11 10 2021 - 15 10 2021

Language：English   Presentation type：Oral presentation (invited, special)  

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Event date： 12 7 2021 - 12 7 2021

Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：online  

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Event date： 13 8 2020 - 14 8 2020

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：online  

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Event date： 26 6 2020 - 26 6 2020

Language：English   Presentation type：Oral presentation (general)  

Venue：online  

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Language：English   Presentation type：Oral presentation (general)  

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Event date： 11 1 2020 - 12 1 2020

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：山口県宇部市  

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Event date： 2 12 2019 - 6 12 2019

Language：English   Presentation type：Oral presentation (general)  

Venue：University of Tokyo  

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Language：English   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：Sevilla  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Leibniz Universität Hannover, Germany  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：University of Wisconsin-Madison, Madison  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

Venue：London, Canada  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Hanoi  

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Language：English   Presentation type：Oral presentation (general)  

Venue：北海道大学  

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Language：English   Presentation type：Oral presentation (general)  

Venue：University Paris Diderot  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Osaka City University  

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Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：Shinshu University  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：BIRS, Banff, Canada  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：Okayama University  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Ruhr University Bochum, Bochum, Germany.  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Beijing, China  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Bucharest, Romania  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Northeastern University, Boston, USA  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Sapporo, Japan  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Oberwolfach, Germany  

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Language：English   Presentation type：Oral presentation (general)  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

Venue：University of Pau and Pays de l'Adour, Pau, France  

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Language：English   Presentation type：Oral presentation (general)  

Venue：University of Pau and Pays de l'Adour, Pau, France  

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Language：English   Presentation type：Oral presentation (general)  

Venue：University of Bremen, Bremen, Germany  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Centre Mont-Royal, Montreal, Canada  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Kyoto  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：Saga University  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Sapporo, Japan  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Hannover, Germany  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Kaiserslautern, Germany  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Tokyo Metropolitan University  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Pisa, Italy  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Bremen, Germany  

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Language：English   Presentation type：Oral presentation (general)  

Venue：Dijon, France  

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Language：Japanese   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Hokkaido University  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：University of Hokkaido  

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Language：English   Presentation type：Oral presentation (invited, special)  

Venue：The University of the Basque Country  

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Language：English   Presentation type：Other  

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Language：English   Presentation type：Oral presentation (general)  

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Language：English   Presentation type：Oral presentation (general)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Other  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Other  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：Japanese   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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Language：English   Presentation type：Oral presentation (invited, special)  

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