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Yasuhide Numata

Publications and source records attributed to Yasuhide Numata.

31 records · Page 2Linked to original sources
arXiv2011

On computation of the characteristic polynomials of the discriminantal arrangements and the arrangements generated by generic points

In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the discriminantal arrangement with the parameter (n,k) is isomorphic to the intersection lattice of the hyperplane arrangement generated by n generic points in the d-dimensional vector space where d=n-k-1. The combinatorics of the discriminantal arrangements is very hard, except for the special cases of the Boolean arrangements (k=0) and the braid arrangements (k=1). We review some results on the intersection lattices of the arrangements generated by generic points and use them to obtain some computational results on the characteristic polynomials of the discriminantal arrangements.

math.CO↗
arXiv2010

On intersection lattices of hyperplane arrangements generated by generic points

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.

math.CO↗
arXiv2010

Graph presentations for moments of noncentral Wishart distributions and their applications

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and $2\times 2$ Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.

math.ST↗
arXiv2008

Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups

For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

math.RT↗
arXiv2008

A bijective proof of a factorization formula for Macdonald polynomials at roots of unity

We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis.

math.CO↗
arXiv2007

An Algorithm to Construct A Basis for the Module of Logarithmic Vector Fields

We consider logarithmic vector fields parametrized by finite collections of weighted hyperplanes. For a finite collection of weighted hyperplanes in a two-dimensional vector space, it is known that the set of such vector fields is a free module of rank two whose basis elements are homogeneous. We give an algorithm to construct a homogeneous basis for the module.

math.CO↗
arXiv2007

An extended Schur's lemma and its application

The Springer modules have a combinatorial property called ``coincidence of dimensions,'' i.e., the Springer modules are naturally decomposed into submodules with common dimensions. Morita and Nakajima proved the property by giving modules with common dimensions whose induced modules are isomorphic to the submodules of Springer modules. They proved that the induced modules are isomorphic to the submodules, by showing the coincidence of their characters. Our aim is to construct isomorphisms between the induced modules and their corresponding submodules in a combinatorial manner. For this purpose, we show lemmas, which are equivalent to the classical Schur's lemma in special cases. We also give a procedure to construct isomorphisms, and explicitly construct isomorphisms in the case of the Springer modules corresponding to Young diagrams of two rows.

math.CO↗
arXiv2006

Pieri's Formula for Generalized Schur Polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's formula to generalized Schur polynomials.

math.CO↗
arXiv2006

An example of generalized Schur operators involving planar binary trees

Young's lattice is a prototypical example of differential posets. Differential posets have the Robinson correspondence, the correspondence between permutations and pairs of standard tableaux with the same shape, as in the case of Young's lattice. Fomin introduced generalized Schur operators to generalize the method of Robinson correspondence in differential posets to the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard tableaux with the same shape. In this paper, we introduce operators on the vector space whose basis is the set of planar binary trees. To prove that the operators are generalized Schur operators, we construct a correspondence, which is an extension of Fomin's r-correspondence for them.

math.CO↗
arXiv2006

Tabloids and Weighted Sums of Characters of Certain Modules of the Symmetric Groups

We consider certain modules of the symmetric groups whose basis elements are called tabloids. Some of these modules are isomorphic to subspaces of the cohomology rings of subvarieties of flag varieties as modules of the symmetric groups. We give a combinatorial description for some weighted sums of their characters, i.e., we introduce combinatorial objects called $(ρ,\Ll)$-tabloids and rewrite weighted sums of characters as the numbers of these combinatorial objects. We also consider the meaning of these combinatorial objects, i.e., we construct a correspondence between $(ρ,\Ll)$-tabloids and tabloids whose images are eigenvectors of the action of an element of cycle type $ρ$ in quotient modules.

math.CO↗