Updated on 2024/07/14

写真a

 
MURAI, Satoshi
 
Affiliation
Faculty of Education and Integrated Arts and Sciences, School of Education
Job title
Professor

Research Experience

  • 2019.04
    -
    Now

    Waseda University   Faculty of Education and Integrated Arts and Sciences

  • 2018.04
    -
    2019.03

    Waseda University   Faculty of Education and Integrated Arts and Sciences

  • 2014.04
    -
    2018.03

    Osaka University   Graduate School of Information Science and Technology

  • 2013.10
    -
    2014.03

    Yamaguchi University   of Science and Engineering, Graduate School

  • 2009.10
    -
    2013.09

    Yamaguchi University   of Science and Engineering, Graduate School

Committee Memberships

  • 2018
    -
    Now

    Algebraic Combinatorics  Editors-in-Chief

Professional Memberships

  •  
     
     

    THE MATHEMATICAL SOCIETY OF JAPAN

Research Areas

  • Applied mathematics and statistics / Basic mathematics / Algebra

Research Interests

  • Combinatorial Topology

  • Commutative Algebra

  • Algebraic Combinatorics

Awards

  • 日本数学会 建部賢弘賞奨励賞

    2008.08  

    Winner: 村井 聡

 

Papers

  • An equivariant Hochster's formula for S<inf>n</inf>-invariant monomial ideals

    Satoshi Murai, Claudiu Raicu

    Journal of the London Mathematical Society   105 ( 3 ) 1974 - 2010  2022.04

     View Summary

    Let (Formula presented.) be a polynomial ring over a field (Formula presented.) and let (Formula presented.) be a monomial ideal preserved by the natural action of the symmetric group (Formula presented.) on (Formula presented.). We give a combinatorial method to determine the (Formula presented.) -module structure of (Formula presented.). Our formula shows that (Formula presented.) is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an (Formula presented.) -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of (Formula presented.) -invariant monomial ideals, and in particular recover formulas for their Castelnuovo–Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or (Formula presented.)) we compute the (Formula presented.) -invariant part of (Formula presented.) in terms of (Formula presented.) groups of the unsymmetrization of (Formula presented.).

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    4
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  • A note on the reducedness and Gröbner bases of Specht ideals

    Satoshi Murai, Hidefumi Ohsugi, Kohji Yanagawa

    Communications in Algebra   50 ( 12 ) 5430 - 5434  2022

     View Summary

    The Specht ideal of shape λ, where λ is a partition, is the ideal generated by all Specht polynomials of shape λ. Haiman and Woo proved that these ideals are reduced and found their universal Gröbner bases. In this short note, we give a short proof for these results.

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  • A filtration on the cohomology rings of regular nilpotent Hessenberg varieties

    Megumi Harada, Tatsuya Horiguchi, Satoshi Murai, Martha Precup, Julianna Tymoczko

    Mathematische Zeitschrift   298 ( 3-4 ) 1345 - 1382  2021.08

     View Summary

    Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in GL(n, C) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in GL(n- 1 , C) / B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce a new set of polynomials which are closely related to the relations in the above presentation and obtain a sequence of equivalence relations they satisfy; this allows us to derive our filtration. In addition, we obtain the following three corollaries. First, we give an inductive formula for the Poincaré polynomial of these varieties. Second, we give an explicit monomial basis for the cohomology rings of regular nilpotent Hessenberg varieties with respect to the presentation mentioned above. Third, we derive a basis of the set of linear relations satisfied by the images of the Schubert classes in the cohomology rings of regular nilpotent Hessenberg varieties. Finally, our methods and results suggest many directions for future work; in particular, we propose a definition of “Hessenberg Schubert polynomials” in the context of regular nilpotent Hessenberg varieties, which generalize the classical Schubert polynomials. We also outline several open questions pertaining to them.

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    1
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  • Strictness of the log-concavity of generating polynomials of matroids

    Satoshi Murai, Takahiro Nagaoka, Akiko Yazawa

    Journal of Combinatorial Theory. Series A   181  2021.07

     View Summary

    Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

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  • Betti tables of monomial ideals fixed by permutations of the variables

    Satoshi Murai

    Transactions of the American Mathematical Society   373 ( 10 ) 7087 - 7107  2020.10

     View Summary

    Let Sn be a polynomial ring with n variables over a field and {In}n≥1 a chain of ideals such that each In is a monomial ideal of Sn fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of In for all large intergers n from the Zmgraded Betti tables of Im for some small integers m. Our main result shows that the projective dimension and the regularity of In eventually become linear functions on n, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.

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    6
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  • Betti numbers of symmetric shifted ideals

    Jennifer Biermann, Hernán de Alba, Federico Galetto, Satoshi Murai, Uwe Nagel, Augustine O'Keefe, Tim Römer, Alexandra Seceleanu

    Journal of Algebra   560   312 - 342  2020.10  [Refereed]

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    14
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  • Hessenberg varieties and hyperplane arrangements

    Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, Satoshi Murai, Takashi Sato

    Journal für die reine und angewandte Mathematik (Crelles Journal)   2020 ( 764 ) 241 - 286  2020.07  [Refereed]

     View Summary

    <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.

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  • Exceptional Balanced Triangulations on Surfaces

    Steven Klee, Satoshi Murai, Yusuke Suzuki

    Graphs and Combinatorics   35 ( 6 ) 1361 - 1373  2019.11

     View Summary

    Izmestiev, Klee and Novik proved that any two balanced triangulations of a closed surface F2 can be transformed into each other by a sequence of six operations called basic cross flips. Recently Murai and Suzuki proved that among these six operations only two operations are almost sufficient in the sense that, with for finitely many exceptions, any two balanced triangulations of a closed surface F2 can be transformed into each other by these two operations. We investigate such finitely many exceptions, called exceptional balanced triangulations, and obtain the list of exceptional balanced triangulations of closed surfaces with low genera. Furthermore, we discuss the subsets O of the six operations satisfying the property that any two balanced triangulations of the same closed surface can be connected through a sequence of operations from O.

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  • Solomon–Terao algebra of hyperplane arrangements

    Takuro ABE, Toshiaki MAENO, Satoshi MURAI, Yasuhide NUMATA

    Journal of the Mathematical Society of Japan   71 ( 4 ) 1027 - 1047  2019.10  [Refereed]

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    3
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  • The Numbers of Edges of 5-Polytopes with a Given Number of Vertices

    Kusunoki Takuya, Murai Satoshi

    ANNALS OF COMBINATORICS   23 ( 1 ) 89 - 101  2019.03  [Refereed]

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  • Local h-Vectors of Quasi-Geometric and Barycentric Subdivisions

    Martina Juhnke-Kubitzke, Satoshi Murai, Richard Sieg

    Discrete and Computational Geometry   61 ( 2 ) 1 - 16  2018.04  [Refereed]

     View Summary

    In this paper, we answer two questions on local h-vectors, which were asked by Athanasiadis. First, we characterize all possible local h-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local (Formula presented.)-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. Along the way, we derive a new recurrence formula for the derangement polynomials.

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  • Balanced generalized lower bound inequality for simplicial polytopes

    Martina Juhnke-Kubitzke, Satoshi Murai

    Selecta Mathematica, New Series   24 ( 2 ) 1677 - 1689  2018.04  [Refereed]

     View Summary

    A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the h-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial d-polytopes, that is simplicial d-polytopes whose underlying graphs are d-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their h-numbers.

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  • Balanced subdivisions and flips on surfaces

    Satoshi Murai, Yusuke Suzuki

    Proceedings of the American Mathematical Society   146 ( 3 ) 939 - 951  2018  [Refereed]

     View Summary

    In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere.

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  • LEFSCHETZ PROPERTIES OF BALANCED 3-POLYTOPES

    Cook David II, Juhnke-Kubitzke Martina, Murai Satoshi, Nevo Eran

    ROCKY MOUNTAIN JOURNAL OF MATHEMATICS   48 ( 3 ) 769 - 790  2018  [Refereed]

    DOI

  • A generalized lower bound theorem for balanced manifolds

    Martina Juhnke-Kubitzke, Satoshi Murai, Isabella Novik, Connor Sawaske

    Mathematische Zeitschrift   289 ( 3-4 ) 1 - 22  2017.11  [Refereed]

     View Summary

    A simplicial complex of dimension (Formula presented.) is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if (Formula presented.) is an arbitrary balanced triangulation of any closed homology manifold of dimension (Formula presented.), then (Formula presented.), thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag (Formula presented.)-vectors.

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  • On stacked triangulated manifolds

    Basudeb Datta, Satoshi Murai

    ELECTRONIC JOURNAL OF COMBINATORICS   24 ( 4 )  2017.10  [Refereed]

     View Summary

    We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension d &gt;= 4, if Delta is a tight connected closed homology d-manifold whose ith homology vanishes for 1 &lt; i &lt; d - 1, then Delta is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.

  • Face numbers and the fundamental group

    Satoshi Murai, Isabella Novik

    ISRAEL JOURNAL OF MATHEMATICS   222 ( 1 ) 297 - 315  2017.10  [Refereed]

     View Summary

    We resolve a conjecture of Kalai asserting that the g(2)-number of any (finite) simplicial complex Delta that represents a normal pseudomanifold of dimension d &gt;= 3 is at least as large as ((d+2)(2))m(Delta), where m(Delta) denotes the minimum number of generators of the fundamental group of Delta. Furthermore, we prove that a weaker bound, h(2)(Delta) &gt;= ((d+1)(2))m(Delta), applies to any d-dimensional pure simplicial poset Delta all of whose faces of co-dimension &gt;= 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Psi all of whose vertex links satisfy Serre's condition (S-r), we establish lower bounds on h(1)(Psi) ,..., h(r)(Psi) in terms of the mu-numbers introduced by Bagchi and Datta.

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  • Face Numbers of Manifolds with Boundary

    Satoshi Murai, Isabella Novik

    INTERNATIONAL MATHEMATICS RESEARCH NOTICES   ( 12 ) 3603 - 3646  2017.06  [Refereed]

     View Summary

    We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's sand mu-numbers for the case of relative simplicial complexes and prove stronger versions of the above statements with the Betti numbers replaced by the mu-numbers. Our results provide natural generalizations of known theorems and conjectures for closed manifolds and appear to be new even for the case of a ball.

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  • A duality in Buchsbaum rings and triangulated manifolds

    Satoshi Murai, Isabella Novik, Ken-ichi Yoshida

    ALGEBRA & NUMBER THEORY   11 ( 3 ) 635 - 656  2017  [Refereed]

     View Summary

    Let 1 be a triangulated homology ball whose boundary complex is partial derivative Delta. A result of Hochster asserts that the canonical module of the Stanley-Reisner ring F[Delta] of Delta is isomorphic to the Stanley-Reisner module F[Delta, partial derivative Delta] of the pair (Delta, partial derivative Delta]. This result implies that an Artinian reduction of F[Delta, partial derivative Delta] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of F[Delta]. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the h ''-numbers of Buchsbaum complexes and use it to prove the monotonicity of h ''-numbers for pairs of Buchsbaum complexes as well as the unimodality of h ''-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.

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  • Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

    Naoyuki Matsuoka, Satoshi Murai

    JOURNAL OF ALGEBRA   455   14 - 31  2016.06  [Refereed]

     View Summary

    In this paper, we study simplicial complexes whose Stanley-Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen-Macaulay simplicial complexes. A d-dimensional simplicial complex Delta is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet F of Delta, the simplicial complex Delta \ {F} is Cohen-Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension &lt;= 2. (C) 2016 Elsevier Inc. All rights reserved.

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  • ON HILBERT FUNCTIONS OF GENERAL INTERSECTIONS OF IDEALS

    Giulio Caviglia, Satoshi Murai

    NAGOYA MATHEMATICAL JOURNAL   222 ( 1 ) 61 - 73  2016.06  [Refereed]

     View Summary

    Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of Green's hyperplane section theorem.

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  • Tight combinatorial manifolds and graded Betti numbers

    Satoshi Murai

    COLLECTANEA MATHEMATICA   66 ( 3 ) 367 - 386  2015.09  [Refereed]

     View Summary

    In this paper, we study the conjecture of Kuhnel and Lutz, who state that a combinatorial triangulation of the product of two spheres with is tight if and only if it has exactly vertices. To approach this conjecture, we use graded Betti numbers of Stanley-Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

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    19
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  • REGULARITY BOUNDS FOR KOSZUL CYCLES

    Aldo Conca, Satoshi Murai

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   143 ( 2 ) 493 - 503  2015.02  [Refereed]

     View Summary

    We study the Castelnuovo-Mumford regularity of the module of Koszul cycles Z(t)(I, M) of a homogeneous ideal I in a polynomial ring S with respect to a graded module M in the homological position t is an element of N. Under mild assumptions on the base field we prove that reg Z(t)(I, S) is a subadditive function of t when dim S/I = 0. For Borel-fixed ideals I, J we prove that reg Z(t)(I, S/J) &lt;= t(1 + reg I) + reg S/J, a result already announced by Bruns, Conca and Romer.

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  • Invariance of Pontrjagin classes for Bott manifolds

    Suyoung Choi, Mikiya Masuda, Satoshi Murai

    ALGEBRAIC AND GEOMETRIC TOPOLOGY   15 ( 2 ) 965 - 986  2015  [Refereed]

     View Summary

    A Bott manifold is the total space of some iterated CP1 -bundles over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are Z/2 -trivial, where a Bott manifold is called Z/2 -trivial if its cohomology ring with Z/2 -coefficients is isomorphic to that of a product of copies of CP1.

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  • Squarefree P-modules and the cd-index

    Satoshi Murai, Kohji Yanagawa

    ADVANCES IN MATHEMATICS   265   241 - 279  2014.11  [Refereed]

     View Summary

    In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen-Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley-Reisner ring of the barycentric subdivision of an odd dimensional Cohen Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function. (C) 2014 Elsevier Inc. All rights reserved.

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  • THE FLAG f-VECTORS OF GORENSTEIN* ORDER COMPLEXES OF DIMENSION 3

    Satoshi Murai, Eran Nevo

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   142 ( 5 ) 1527 - 1538  2014.05  [Refereed]

     View Summary

    We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f-vectors of order complexes triangulating rational homology 3-spheres, and show they are also the characterization of the flag f-vectors of the subfamily of regular CW-complexes homeomorphic to the 3-sphere. As a corollary, we characterize the f-vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization gives rise to a speculated intimate connection between the f-vectors of flag homology spheres and the f-vectors of Gorenstein* order complexes.

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  • On r-stacked triangulated manifolds

    Satoshi Murai, Eran Nevo

    JOURNAL OF ALGEBRAIC COMBINATORICS   39 ( 2 ) 373 - 388  2014.03  [Refereed]

     View Summary

    The notion of r-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the r-stackedness for triangulated homology manifolds and study its basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal.

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  • On ideals with the Rees property

    Juan Migliore, Rosa M. Miro-Roig, Satoshi Murai, Uwe Nagel, Junzo Watanabe

    ARCHIV DER MATHEMATIK   101 ( 5 ) 445 - 454  2013.11  [Refereed]

     View Summary

    A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal is said to be -full if for some , where is the graded maximal ideal of . It was proved by one of the authors that -full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.

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  • FACE VECTORS OF SIMPLICIAL CELL DECOMPOSITIONS OF MANIFOLDS

    Satoshi Murai

    ISRAEL JOURNAL OF MATHEMATICS   195 ( 1 ) 187 - 213  2013.06  [Refereed]

     View Summary

    In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd-dimensional manifolds without boundary.

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  • On the generalized lower bound conjecture for polytopes and spheres

    Satoshi Murai, Eran Nevo

    ACTA MATHEMATICA   210 ( 1 ) 185 - 202  2013.03  [Refereed]

     View Summary

    In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h (0), h (1), aEuro broken vertical bar, h (d) ) satisfies . Moreover, if h (r-1) = h (r) for some then P can be triangulated without introducing simplices of dimension a parts per thousand currency signd - r.
    The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.

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  • Regularity bounds for binomial edge ideals

    Kazunori Matsuda, Satoshi Murai

    Journal of Commutative Algebra   5 ( 1 ) 141 - 149  2013  [Refereed]

     View Summary

    We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices. © 2013 Rocky Mountain Mathematics Consortium.

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  • Sharp upper bounds for the Betti numbers of a given Hilbert polynomial

    Giulio Caviglia, Satoshi Murai

    ALGEBRA & NUMBER THEORY   7 ( 5 ) 1019 - 1064  2013  [Refereed]

     View Summary

    We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

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  • H-vectors of simplicial cell balls

    Satoshi Murai

    Transactions of the American Mathematical Society   365 ( 3 ) 1533 - 1550  2013  [Refereed]

     View Summary

    A simplicial cell ball is a simplicial poset whose geometric realization is homeomorphic to a ball. Recently, Samuel Kolins gave a series of necessary conditions and sufficient conditions on h-vectors of simplicial cell balls, and characterized them up to dimension 6. In this paper, we extend Kolins' results. We characterize all possible h-vectors of simplicial cell balls in arbitrary dimension. © 2012 American Mathematical Society.

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  • Hilbert schemes and Betti numbers over Clements-Lindstrom rings

    Satoshi Murai, Irena Peeva

    COMPOSITIO MATHEMATICA   148 ( 5 ) 1337 - 1364  2012.09  [Refereed]

     View Summary

    We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over a Clements-Lindstrom ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. This is an analogue of Hartshorne's theorem that Grothendieck's Hilbert scheme is connected. We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.

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  • On the cd-index and gamma-vector of S*-shellable CW-spheres

    Satoshi Murai, Eran Nevo

    MATHEMATISCHE ZEITSCHRIFT   271 ( 3-4 ) 1309 - 1319  2012.08  [Refereed]

     View Summary

    We show that the gamma-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable CW-spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing c = 1 and considering the resulted polynomial in d, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.

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  • LEFSCHETZ PROPERTIES AND THE VERONESE CONSTRUCTION

    Martina Kubitzke, Satoshi Murai

    MATHEMATICAL RESEARCH LETTERS   19 ( 5 ) 1043 - 1053  2012  [Refereed]

     View Summary

    In this paper, we investigate Lefschetz properties of Veronese subalgebras. We show that, for a sufficiently large r, the rth Veronese subalgebra of a Cohen-Macaulay standard graded K-algebra has properties similar to the weak and strong Lefschetz properties, which we call the 'quasi-weak' and 'almost strong' Lefschetz properties. By using this result, we obtain new results on h- and g-polynomials of Veronese subalgebras.

  • Spheres arising from multicomplexes

    Satoshi Murai

    JOURNAL OF COMBINATORIAL THEORY SERIES A   118 ( 8 ) 2167 - 2184  2011.11  [Refereed]

     View Summary

    In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex Delta on the vertex set V with Delta not equal 2(v) the deleted join of Delta with its Alexander dual Delta(v) is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable. which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory. (C) 2011 Elsevier Inc. All rights reserved.

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  • The Lex-Plus-Powers Conjecture holds for pure powers

    Jeff Mermin, Satoshi Murai

    ADVANCES IN MATHEMATICS   226 ( 4 ) 3511 - 3539  2011.03  [Refereed]

     View Summary

    We prove Evans&apos; Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence. (C) 2010 Elsevier Inc. All rights reserved.

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    18
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  • FREE RESOLUTIONS OF LEX IDEALS OVER A KOSZUL TORIC RING

    Satoshi Murai

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   363 ( 2 ) 857 - 885  2011.02  [Refereed]

     View Summary

    In this paper we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular we study in which toric ring R all lex-ideals are compare twis linear. We give a certain necessity aid sufficiently condition for this property and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from the Segre product P(1) x ... x P(1), every Hilbert function of graded ideal is attained by a lex-ideals and that lex-ideals have the greatest graded Betti numbers among all ideals having the same Hilbert function.

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  • Hilbert schemes and maximal Betti numbers over veronese rings

    Vesselin Gasharov, Satoshi Murai, Irena Peeva

    MATHEMATISCHE ZEITSCHRIFT   267 ( 1-2 ) 155 - 172  2011.02  [Refereed]

     View Summary

    Macaulay&apos;s Theorem (Macaulay in Proc. Lond Math Soc 26:531-555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P (r-1)). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne&apos;s Theorem (Hartshorne in Math. IHES 29:5-48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.

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  • REGULARITY OF CANONICAL AND DEFICIENCY MODULES FOR MONOMIAL IDEALS

    Manoj Kummini, Satoshi Murai

    PACIFIC JOURNAL OF MATHEMATICS   249 ( 2 ) 377 - 383  2011.02  [Refereed]

     View Summary

    We show that the Castelnuovo-Mumford regularity of the canonical or a deficiency module of the quotient of a polynomial ring by a monomial ideal is bounded by its dimension.

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  • Applications of mapping cones over Clements-Lindstrom rings

    Vesselin Gasharov, Satoshi Murai, Irena Peeva

    JOURNAL OF ALGEBRA   325 ( 1 ) 34 - 55  2011.01  [Refereed]

     View Summary

    We prove that Gotzmann's Persistence Theorem holds over every Clements-Lindstrom ring. We also construct the infinite minimal free resolution of a square-free Borel ideal over such a ring. (C) 2010 Elsevier Inc. All rights reserved.

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    3
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  • Betti numbers of lex ideals over some Macaulay-Lex rings

    Jeff Mermin, Satoshi Murai

    JOURNAL OF ALGEBRAIC COMBINATORICS   31 ( 2 ) 299 - 318  2010.03  [Refereed]

     View Summary

    Let A=K[x (1),aEuro broken vertical bar,x (n) ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R=A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Furedi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.

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  • Betti numbers of chordal graphs and f -vectors of simplicial complexes

    Takayuki Hibi, Kyouko Kimura, Satoshi Murai

    JOURNAL OF ALGEBRA   323 ( 6 ) 1678 - 1689  2010.03  [Refereed]

     View Summary

    Let G he a chordal graph and I(G) its edge ideal. Let beta(I(G)) = (beta(0), beta(1).....beta(p)) denote the Betti sequence of I(G), where beta(1) stands for the ith total Betti number of I(G) and where p is the projective dimension of I(G). It will be shown that there exists a simplicial complex Delta of dimension p whose f-vector f(Delta) = (f(0). f(1).....f(p)) coincides with beta(I(G)). (C) 2010 Elsevier Inc, All rights reserved.

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  • Algebraic shifting of strongly edge decomposable spheres

    Satoshi Murai

    JOURNAL OF COMBINATORIAL THEORY SERIES A   117 ( 1 ) 1 - 16  2010.01  [Refereed]

     View Summary

    Recently, Nevo introduced the notion of strongly edge decomposable spheres. in this paper, we characterize algebraic shifted complexes of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley-Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0. (C) 2009 Elsevier Inc. All rights reserved.

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    10
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  • ON FACE VECTORS OF BARYCENTRIC SUBDIVISIONS OF MANIFOLDS

    Satoshi Murai

    SIAM JOURNAL ON DISCRETE MATHEMATICS   24 ( 3 ) 1019 - 1037  2010  [Refereed]

     View Summary

    We study face vectors of barycentric subdivisions of simplicial homology manifolds. Recently, Kubitzke and Nevo proved that the g-vector of the barycentric subdivision of a Cohen-Macaulay simplicial complex is an M-vector, which in particular proves the g-conjecture for barycentric subdivisions of simplicial homology spheres. In this paper, we prove an analogue of this result for Buchsbaum simplicial posets and simplicial homology manifolds.

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  • H-VECTORS OF SIMPLICIAL COMPLEXES WITH SERRE&apos;S CONDITIONS

    Satoshi Murai, Naoki Terai

    MATHEMATICAL RESEARCH LETTERS   16 ( 5-6 ) 1015 - 1028  2009.09  [Refereed]

     View Summary

    We study h-vectors of simplicial complexes which satisfy Serre&apos;s condition (S(r)). Let r be a positive integer. We say that a simplicial complex Delta satisfies Serre&apos;s condition (S(r)) if (H) over tilde (i)(lk(Delta)(F); K) = 0 for all F is an element of Delta and for all i &lt; min {r-1, dim lk(Delta)(F)}, where lk(Delta) (F) is the link of Delta with respect to F and where &lt;(H)over tilde&gt;(i) (Delta; K) is the reduced homology groups of Delta over a field K. The main result of this paper is that if Delta satisfies Serre&apos;s condition (S(r)) then (i) h(k) (Delta) is non-negative for k = 0, 1, ... , r and (ii) Sigma(k &gt;= r) h(k) (Delta) is non-negative.

  • Face vectors of two-dimensional Buchsbaum complexes

    Satoshi Murai

    ELECTRONIC JOURNAL OF COMBINATORICS   16 ( 1 )  2009.05  [Refereed]

     View Summary

    In this paper, we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes.

  • ALGEBRAIC SHIFTING AND GRADED BETTI NUMBERS

    Satoshi Murai, Takayuki Hibi

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   361 ( 4 ) 1853 - 1865  2009  [Refereed]

     View Summary

    Let S = K[x(1),...,x(n),] denote the polynomial ring in n variables over a field K with each deg x(i) = 1. Let Delta be a simplicial complex on [n] = {1,...,n} and I(Delta) subset of S its Stanley-Reisner ideal. We write Delta(e) for the exterior algebraic shifted complex of Delta and Delta(c) for a combinatorial shifted complex of Delta. Let beta(ii+j) (I(Delta)) = dim(K) Tor(i) (K, I(Delta))(i+j) denote the graded Betti numbers of I(Delta). In the present paper it will be proved that (i) beta(ii+j) (I(Delta e)) &lt;= beta(ii+j) (I(Delta c)) for all i and j, where the base field is infinite, and (ii) beta(ii+j) (I(Delta)) &lt;= beta(ii+j) (I(Delta c)) for all i and j, where the base field is arbitrary. Thus in particular one has beta(ii+j) (I(Delta)) &lt;= beta(ii+j) (I(Delta lex)) for all i and j, where Delta(lex) is the unique lexsegment simplicial complex with the same f-vector as Delta and where the base field is arbitrary.

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  • Gotzmann ideals of the polynomial ring

    Satoshi Murai, Takayuki Hibi

    MATHEMATISCHE ZEITSCHRIFT   260 ( 3 ) 629 - 646  2008.11  [Refereed]

     View Summary

    Let A = K[x(1),..., x (n)] denote the polynomial ring in n variables over a field K. We will classify all the Gotzmann ideals of A with at most n generators. In addition, we will study Hilbert functions H for which all homogeneous ideals of A with the Hilbert function H have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notion of segmentwise critical Hilbert functions and show that segmentwise critical Hilbert functions are inflexible.

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  • Borel-plus-powers monomial ideals

    Satoshi Murai

    JOURNAL OF PURE AND APPLIED ALGEBRA   212 ( 6 ) 1321 - 1336  2008.06  [Refereed]

     View Summary

    Let S = K[x(1), ..., x(n)] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition p in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing x(1)(p), ..., x(n)(p), where p is a prime number. (C) 2007 Elsevier B.V. All rights reserved.

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  • Rigidity of linear strands and generic initial ideals

    Satoshi Murai, Pooja Singla

    NAGOYA MATHEMATICAL JOURNAL   190   35 - 61  2008.06  [Refereed]

     View Summary

    Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. We study rigidity properties of the graded Betti numbers of graded ideals in S and E when passing to their generic initial ideals. First, we prove that if the graded Betti numbers beta(S)(ii+k)(S/&gt;I) = beta(S)(ii+k)(S/Gin(I)) for all q &gt;= i, where I subset of S is a graded ideal. Second, we show that if beta(E)(ii+k)(E/I) = beta(E)(ii+k)(E/Gin(I)) for all q &gt;= 1, where I subset of E is a graded ideal. In addition, it will be shown that the graded Betti numbers beta(R)(ii+k)(R/I) = beta(R)(ii+k)(R/Gin(I)) for all i &gt;= 1 if and only if I-&lt; k &gt; and I &lt; k+1 &gt; have a linear resolution. Here (&lt; d &gt;)is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra.

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  • Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals

    Satoshi Murai

    JOURNAL OF ALGEBRAIC COMBINATORICS   27 ( 3 ) 383 - 398  2008.05  [Refereed]

     View Summary

    In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.

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  • Componentwise linear ideals with minimal or maximal Betti numbers

    Juergen Herzog, Takayuki Hibi, Satoshi Murai, Yukihide Takayama

    ARKIV FOR MATEMATIK   46 ( 1 ) 69 - 75  2008.04  [Refereed]

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    We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.

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  • A combinatorial proof of Gotzmann's persistence theorem for monomial ideals

    Satoshi Murai

    EUROPEAN JOURNAL OF COMBINATORICS   29 ( 1 ) 322 - 333  2008.01  [Refereed]

     View Summary

    Gotzmann proved the persistence for minimal growth of Hilbert functions of homogeneous ideals. His theorem is called Gotzmann's persistence theorem. In this paper, based on the combinatorics of binomial coefficients, a simple combinatorial proof of Gotzmann's persistence theorem in the special case of monomial ideals is given. (c) 2006 Elsevier Ltd. All rights reserved.

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  • The depth of an ideal with a given Hilbert function

    Satoshi Murai, Takayuki Hibi

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   136 ( 5 ) 1533 - 1538  2008  [Refereed]

     View Summary

    Let A = K[x(1),...,x(n)] denote the polynomial ring in n variables over a field K with each deg x(i) = 1. Let I be a homogeneous ideal of A with I not equal A and H-A/I the Hilbert function of the quotient algebra A/I. Given a numerical function H:N -&gt; N satisfying H = H-A/I for some homogeneous ideal I of A, we write A(H) for the set of those integers 0 &lt;= r &lt;= n such that there exists a homogeneous ideal I of A with H-A/I=H and with depth A/I = r. It will be proved that one has either AH = {0, 1,...,b} for some 0 &lt;= b &lt;= n or |A(H)| = 1.

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  • Kruskal-Katona type theorems for clique complexes arising from chordal and strongly chordal graphs

    Juergen Herzog, Takayuki Hibi, Satoshi Murai, Ngo Viet Trung, Xinxian Zheng

    COMBINATORICA   28 ( 3 ) 315 - 323  2008  [Refereed]

     View Summary

    A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal-Katona type theorems for forests, quasi-forests, pure forests and pure quasi-forests will be presented.

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  • Hilbert functions of d-regular ideals

    Satoshi Murai

    JOURNAL OF ALGEBRA   317 ( 2 ) 658 - 690  2007.11  [Refereed]

     View Summary

    In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p &gt;= 0 and d &gt; 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dirn(I) is smaller than or equal to p and whose regularity reg(I) is smaller than or equal to d. then there exists a monomial ideal L having the maximal graded Belli numbers among graded ideals J which have the same Hilbert function as 1 and which satisfy proj dim(J) &lt;= p and reg(J) &lt;=, d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals. (c) 2007 Elsevier Inc. All rights reserved.

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  • Generic initial ideals and squeezed spheres

    Satoshi Murai

    ADVANCES IN MATHEMATICS   214 ( 2 ) 701 - 729  2007.10  [Refereed]

     View Summary

    In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley-Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed (d-1)-sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley-Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Gamma having the weak Lefschetz property a squeezed sphere Sq(Gamma), and show that this operation increases graded Betti numbers. (c) 2007 Elsevier Inc. All rights reserved.

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  • Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes

    Satoshi Murai

    ARKIV FOR MATEMATIK   45 ( 2 ) 327 - 336  2007.10  [Refereed]

     View Summary

    In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let sigma and tau be simplicial complexes and sigma * tau be their join. Let J(sigma) be the exterior face ideal of sigma and Delta(sigma) the exterior algebraic shifted complex of sigma. Assume that sigma * tau is a simplicial complex on [n] = {1, 2,..., n}. For any d-subset S subset of [n], let m &lt;=(rev) s (sigma) denote the number of d-subsets R is an element of sigma which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that m &lt;=(rev)s (Delta (sigma*tau))&gt; m &lt;= S-rev (Delta (Delta(sigma)* Delta (tau))) for all S subset of[n]. To prove this fact, we also prove that m &lt;=(rev)s(Delta(sigma))&gt;-m &lt;= S-rev(Delta(Delta(phi)(sigma))) for all S subset of[n] and for all nonsingular matrices phi, where Delta(phi)(sigma) is the simplicial complex defined by J(Delta phi)(sigma)=in(phi(J sigma)).

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  • Algebraic shifting of cyclic polytopes and stacked polytopes

    Satoshi Murai

    DISCRETE MATHEMATICS   307 ( 14 ) 1707 - 1721  2007.06  [Refereed]

     View Summary

    Gil Kalai introduced the shifting-theoretic upper bound relation as a method to generalize the g-theorem for simplicial spheres by using algebraic shifting. We will study the connection between the shifting-theoretic upper bound relation and combinatorial shifting. Also, we will compute the exterior algebraic shifted complex of the boundary complex of the cyclic d-polytope as well as of a stacked d-polytope. It will turn out that, in both cases, the exterior algebraic shifted complex coincides with the symmetric algebraic shifted complex. (C) 2006 Elsevier B.V. All rights reserved.

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  • Maximal Betti numbers of Cohen-Macaulay complexes with a given f-vector

    Satoshi Murai, Takayuki Hibi

    ARCHIV DER MATHEMATIK   88 ( 6 ) 507 - 512  2007.06  [Refereed]

     View Summary

    Given the f-vector f = (f(0), f(1),...) of a Cohen-Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Delta(f) with f (Delta(f)) = f such that, for any Cohen-Macaulay simplicial complex A with f(Delta) = f, one has beta(ij)(I Delta) &lt;= beta(ij)(I-Delta f) for all i and j, where f(Delta) is the f-vector of A and where beta(ij)(IA) are graded Betti numbers of the Stanley-Reisner ideal I-Delta of Delta.

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  • Algebraic shifting of finite graphs

    Satoshi Murai

    COMMUNICATIONS IN ALGEBRA   35 ( 10 ) 3071 - 3094  2007  [Refereed]

     View Summary

    In the present article, for bipartite graphs and chordal graphs, their exterior algebraic shifted graph and their symmetric algebraic shifted graph are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that for a &gt;= 3 and b &gt;= 3, the exterior algebraic shifted graph of the complete bipartite graph K-a,K-b of size a, b is different from the symmetric algebraic shifted graph of K-a,K-b. Second, we will show that the exterior algebraic shifted graph of any chordal graph G coincides with the symmetric algebraic shifted graph of G. In addition, it will be shown that the exterior algebraic shifted graph of any chordal graph G is equal to some combinatorial shifted graph of G.

    DOI

    Scopus

    1
    Citation
    (Scopus)
  • Gotzmann monomial ideals

    Murai Satoshi

    ILLINOIS JOURNAL OF MATHEMATICS   51 ( 3 ) 843 - 852  2007  [Refereed]

  • Gin and Lex of certain monomial ideals

    Satoshi Murai, Takayuki Hibi

    MATHEMATICA SCANDINAVICA   99 ( 1 ) 76 - 86  2006  [Refereed]

     View Summary

    Let A = K[x(1),..., x(n)] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg x(i) = 1. Given arbitrary integers i and j with 2 &lt;= i &lt;= n and 3 &lt;= j &lt;= n, we will construct a monomial ideal I subset of A such that (i) beta k(I) &lt; beta(k)(Gin(I)) for all k &lt; i, (ii) beta(i) (I) = beta(i) (Gin (I)), (iii) beta(l) (Gin (I)) &lt; beta(l) (Lex (I)) for all l &lt; j and (iv) beta(j) (Gin (I)) = beta(j) (Lex (I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x(1) &gt;... &gt; x(n) and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.

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Presentations

  • 多様体の単体分割の持つ組合せ論的・代数的対称性

    村井 聡

    2017年度日本数学会秋季総合分科会 特別講演 

    Presentation date: 2017.09

  • 凸多面体の面の数え上げ論の近況

    村井 聡

    2017年度日本数学会年会 応用数学分科会特別セッション「凸多面体の数え上げ論の近況」 

    Presentation date: 2017.03

  • 多様体の単体分割の組合せ論と代数

    村井 聡

    第63回トポロジーシンポジウム 

    Presentation date: 2016.07

  • 多様体の三角形分割の組合せ論と可換代数,

    村井 聡

    第60回代数学シンポジウム 

    Presentation date: 2015.08

  • 単体的セル複体の面の数え上げの話

    村井 聡

    第56回代数学シンポジウム 

    Presentation date: 2011.08

Research Projects

  • 可換環論・数え上げ組合せ論・組合せトポロジーの間の相互関係の研究

    日本学術振興会  科学研究費助成事業

    Project Year :

    2021.04
    -
    2025.03
     

    村井 聡

     View Summary

    本研究の目的は可換環論と組合せ論の間の相互関係を発展させることである。特に、現在知られている「(a) 可換環論における単項式イデアルの研究」、「(b) 凸多面体の数え上げ組合せ論」、「(c) 単体的複体の組合せトポロジー」、の間の相互関係を更に発展させることを目指している。
    本年度は、Specht イデアルと呼ばれるイデアルの代数的性質に関する研究を中心行った。Specht イデアルは Specht 多項式と呼ばれる多項式から生成されるイデアルであるが、ヤング図形の組合せ論、部分空間配置の組合せ論、表現論などとも関連する代数・組合せ論の両方の面から興味がもたれているイデアルである。
    具体的な研究成果として、Haiman と Woo によって発見された Specht イデアルがラディカルイデアルであること、および、そのグレブナー基底が組合せ論的な手法で簡単に計算できることの簡単な別証明を与えることに成功した。本研究成果は、既知の結果の別証明ではあるが、Specht イデアルに関する最も基本的な性質に初等的な証明を与えるものであり、今後の Specht イデアルの更なる研究における基礎理論の一つを与えるものになると考えている。
    また、国際研究集会「MFO-RIMS Tandem Workshop Symmetries on polynomial ideals and varieties」(2021年9月開催)や研究集会「オンライン可換環論セミナー2021」(2021年7月開催)をオンライン形式で開催し、可換環論および、可換環論と関わる組合せ論の研究に関する情報交換の場を設けた。

  • Triangulations of polytopes and manifolds with nice coloring structure

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2016.04
    -
    2020.03
     

    Murai Satoshi

     View Summary

    In this research, we study combinatorial structures of triangulated manifolds having a nice coloring structure, called balanced. Here are main achievement.
    <BR>
    [1] We give an affirmative answer to the conjecture of Novik and Klee on lower bounds of the face numbers of balanced triangulated manifolds. [2] We show that any (1,1,1)-balanced 3-polytope has the Lefschetz property with respect to a colored linear system of parameters, while (2,1)-balanced 3-polytopes do not always have this property. [3] We solve a problem posed by Izmestiev, Klee and Novik, on pentagon moves for balanced triangulated surfaces.

  • Face enumeration of convex polytopes and cell complexes

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2013.04
    -
    2016.03
     

    Murai Satoshi

     View Summary

    In this research project, we study the face numbers of polytopes and cell complexes. The followings are main results of this project.
    (1) We find new method to study the cd-index of a convex polytope by using commutative algebra, and by applying this new method, we find new upper bounds of the cd-index.
    (2) We find new applications of polyhedral Morse inequality and graded Betti numbers to the study of face numbers of triangulated manifolds.

  • A ring theoretic approach to face numbers of triangulated manifolds

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2010.04
    -
    2014.03
     

    MURAI Satoshi

     View Summary

    The study of face vectors of simplicial complexes and simplicial cell complexes is a current trend in combinatorics. In particular, face vectors of triangulated spheres and manifolds has been of great interest in this research area. In this research project, we get the following results on this topic. (1) characterizations of face vectors of simplicial cell decompositions of products of spheres and those of balls (2) solution of the generalized lower bound conjecture for simplicial polytopes.
    The second result sovles a conjecture of McMullen and Walkup in 1971.

  • Algebraic shiftingとg-予想に関する研究

    日本学術振興会  科学研究費助成事業

    Project Year :

    2009
     
     
     

    村井 聡

     View Summary

    平成21年度は主に低次元の単体的複体のalgebraic shifting及び面の個数の数え上げについての研究を行った。
    単体的複体の面の個数の研究は、組合せ論の分野における主要な研究課題の一つである。本研究では、algebraic shiftingと呼ばれる可換代数の手法を主な研究手段として、代数的なアプローチから単体的複体の面の個数の分類問題についての研究を行った。
    本年度は、先ずBuchsbaum単体的複体についての研究を行った。Buchsbaum単体的複体は多様体の三角形分割の概念を代数的に一般化したもので、特に、Buchsbaum単体的複体の面の個数のとりうる値を完全に分類することが懸案の課題とされている。この問題に関し、1996年、寺井直樹教授は2次元Buchsbaum単体的複体の面の個数の分類に関するある予想を提唱した。この予想に対し、計算機を使い、様々なBuchsbaum単体的複体のalgebraic shiftingを計算し、その計算結果からBuchsbaum単体的複体の面の個数の性質を捕らえる、というアプローチからBuchsbaum単体的複体の面の個数についての研究を行った。その結果、上記の2次元Buchsbaum単体的複体の面の個数に関する寺井直樹教授の予想を肯定的に解決し、特に、2次元Buchsbaum単体的複体の面の個数を完全に分類することに成功した。
    また、その後、寺井直樹教授との共同研究で、上記の手法をさらに一般化することにより、セール条件と呼ばれる代数的な性質を持つ単体的複体のh-列が部分的に非負になるという結果を得た。この結果はCohen-Macaulayと呼ばれる性質を持つ単体的複体のh-列が非負になるというスタンレーの古典的な重要な結果の精密化となっており、今後のh-列の研究に大きな影響を与えることが期待できる結果である。

  • 有限自由分解とgeneric initial idealの研究

    日本学術振興会  科学研究費助成事業

    Project Year :

    2006
    -
    2008
     

    村井 聡

     View Summary

    平成20年度は主にalgebraic shiftingと単項式イデアルの次数付きベッチ数についての研究を推進した.
    Algebraic shiftingと単項式イデアルの次数付きベッチ数に関する研究において最も重要な問題の一つは,Herzogの予想と呼ばれるベッチ数の不等式に関する4つの予想である.当該研究課題ではこのHerzogの予想についての研究を進めて来たが,今年度,combinatorial shiftingと呼ばれるalgebraic shiftingに近い作用を取ることで次数付ベッチ数は減少しない,ということを証明し,特に,これにより上記の一連の不等式のうち未解決であったものの一つを肯定的に解決した.
    2006年BabsonとNovikは従来のalgebraic shiftingの一般化として,colored algebraic shiftingと呼ばれる作用を導入した.このcolored algebraic shiftingがalgebraic shiftingが持つような様々な良い性質を持つか?ということが注目されているが,今年度の研究成果の一つとして,colored algebraic shiftingを取る事で次数付ベッチ数は減少しないという結果を証明し,colored algebraic shiftingの基礎理論の構築に貢献した.
    組合せ論における重要な研究問題の一つに,与えられた単体的複体のクラスに対しそのクラスに入る単体的複体の面の個数の特徴付けを与えよ,という問題がある.今回,algebraic shiftingについての最先端の理論を応用することで,コーダルグラフに付随するflag complexの面の個数の特徴付けを与えるという結果を得た.

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Other

  • Interactions Between Topological Combinatorics and Combinatorial Commutative Algebra (BIRS)

    2023.04
     
     

     View Summary

    組織委員

  • INdAM Meeting The Strong and Weak Lefschetz Properties (イタリア, Cortona)

    2022.09
     
     

     View Summary

    組織委員

  • MFO-RIMS Tandem Workshop Symmetries on polynomial ideals and varieties (MFO & RIMS)

    2021.09
     
     

     View Summary

    組織委員

  • The Japanese Conference on Combinatorics and its Applications in Sendai

    2018
     
     
  • Lefschetz Properties in Algebra, Geometry and Combinatorics

    2017
     
     
  • The Japanese Conference on Combinatorics and its Applications

    2016
     
     
  • The 26th International Conference on Formal Power Series and Algebraic Combinatorics

    2014
     
     
  • Japan Conference on Graph Theory and Combinatorics

    2014
     
     
  • The 24th International Conference on Formal Power Series and Algebraic Combinatorics

    2012
     
     

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Syllabus

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Sub-affiliation

  • Faculty of Education and Integrated Arts and Sciences   Graduate School of Education

Internal Special Research Projects

  • 対称群の作用で固定される単項式イデアルの研究

    2020   Claudiu

     View Summary

    イデアルの自由分解に関する研究は、数学の可換環論の分野における主要な研究課題の一つである。本研究では、単項式イデアルであって、対称群の作用で固定されるものについて、その自由分解に関する研究を行った。 本研究の研究成果として、Claudiu Raicu(University of Notre Dame)との共同研究として、対称群の作用で固定される単項式イデアルのベッチ数を組合せ論的に記述することに成功した。ベッチ数は一般には計算が難しい量であるが、本研究結果は、この難しい不変量を組合せ論的な手法により簡単に計算する方法を与えるものである。

  • Hessenberg多様体のコホモロジ一環の環構造に関する研究

    2019   村井 聡

     View Summary

    Hessenberg多様体は、表現論、代数幾何、組合せ論などの様々な数学の分野と関連し、近年注目されている研究対象である。本研究では、Hessenberg多様体のコホモロジー環の環構造に関する研究を行い、Megumi Harada, Tatsuya Horiguchi, Martha Precup, Julianna Tymoczkoらとの共同研究によって、regular nilpotent Hessenberg多様体のコホモロジー環が綺麗なfiltration構造を持つことを発見し、flag多様体のコホモロジー環の持つ"monomial基底"に相当する概念をHessenberg多様体に一般化した。

  • Hessenberg多様体のコホモロジ一環の基底に関する研究

    2018   堀口達也

     View Summary

    Hessenberg多様体は、表現論、代数幾何、組合せ論などの様々な数学の分野と関連し、近年注目されている研究対象であり、特に、現在Hessenberg多様体のコホモロジー環に関する研究が盛んに行われている。 本研究では、regular nilpotent Hessenberg多様体のコホモロジー環の基底に関する研究を行い、その研究成果として、 HaradaとTymoczko らによって予想されたコホモロジー環の基底の候補となるシューベルト多項式の族が、型が(n-1,...,n-1,n,...,n)の形をしているHessenberg多様体の場合に実際に基底となることを証明することに成功した。