Papers by Masato Kobayashi
This thesis discusses intersections of the Schubert varieties in the flag variety associated to a... more This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders
From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhda... more From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15
arXiv: Combinatorics, 2018
We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-side... more We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si... more As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).
The aim of this article is to show one clear difference between single (left or right) cosets and... more The aim of this article is to show one clear difference between single (left or right) cosets and double cosets in Coxeter groups. Let (W, S) be a Coxeter system and I ⊆ S. As is wellknown, (say) left cosets wWI and xWI are always isomorphic as sets (and moreover as graded posets under right weak order) as long as we concern the same index I. On the contrary, we show that when it comes to double cosets, WIwWJ and WIxWJ are not necessarily isomorphic even as sets. As an application, we show that each lower interval has a decomposition by certain double cosets.
The Electronic Journal of Combinatorics, 2010
Bigrassmannian permutations are known as permutations which have precisely one left descent and o... more Bigrassmannian permutations are known as permutations which have precisely one left descent and one right descent. They play an important role in the study of Bruhat order. Fulton introduced the essential set of a permutation and studied its combinatorics. As a consequence of his work, it turns out that the essential set of bigrassmannian permutations consists of precisely one element. In this article, we generalize this observation for essential sets of arbitrary permutations. Our main theorem says that there exists a bijection between bigrassmanian permutations maximal below a permutation and its essential set. For the proof, we make use of two equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.
Order, 2019
We generalize the author's formula (2011) on weighted counting of inversions on permutations to o... more We generalize the author's formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).
Linear Algebra and its Applications, 2017
We introduce a new directed graph structure into the set of alternating sign matrices. This inclu... more We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, q-analogs (which we name qTNN and qSFL properties). As a by-product, we obtain a new kind of permutation statistic, the signed bigrassmannian statistics, using Dodgson's condensation on determinants.
International Mathematical Forum, 2013
We develop combinatorics of Fulton's essential set particularly with a connection to Baxter permu... more We develop combinatorics of Fulton's essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret Eriksson-Linusson's characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of Bruhat-Ehresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.
Order, 2010
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precis... more In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schützenberger and Reading.
Fuzzy Sets and Systems, 2014
Following Aguiló-Suñer-Torrens (2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-T... more Following Aguiló-Suñer-Torrens (2008), Kolesárová-Mesiar-Mordelová-Sempi (2006) and Mayor-Suñer-Torrens (2005), we continue to develop a theory of matrix representation for discrete copulas. To be more precise, we give characterizations of meet-irreducible discrete copulas from an order-theoretical aspect: we show that the set of all irreducible discrete copulas is a lattice in analogy with Nelsen andÚbeda-Flores (2005). Moreover, we clarify its lattice structure related to Kendall's τ and Spearman's ρ borrowing ideas from Coxeter groups.
As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, si... more As an application of linear algebra for enumerative combinatorics, we introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian determinant. First, a signed bigrassmannian polynomial is a variant of the statistic given by the number of bigrassmannian permutations below a permutation in Bruhat order as Reading suggested (2002) and afterward the author developed (2011). Second, bigrassmannian determinant is a q-analog of the determinant with respect to our statistic. It plays a key role for a determinantal expression of those polynomials. We further show that bigrassmannian determinant satisfies weighted condensation as a generalization of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986)
Journal of Combinatorial Theory, Series A, 2013
From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhda... more From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15
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Papers by Masato Kobayashi