Combinatorics@Sendai

Abstract : 

Euler showed that the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. MacMahon showed that the number of partitions of n for which no part occurs exactly once is equal to the number of partitions of n into parts divisible by 2 or 3. Both these results are instances of a general phenomenon based on the fact that certain polynomials are the product of cyclotomic polynomials. After discussing this assertion, we explain how it can be extended to such topics as counting certain polynomials over finite fields and obtaining Dirichlet series generating functions for certain classes of integers. We also discuss a connection with numerical semigroups.

Abstract : Schur(-type) multiple zeta values has been introduced by Nakasuji, Phuksuwan and Yamasaki, as an extension of multiple zeta values and an analogue of Schur polynomials. In this talk, I will introduce the duality formula for Schur multiple zeta values. It is a joint work with Maki Nakasuji (Sophia University).

Abstract : 

It is well known that the monodromy preserving deformation equation for a rank two rational connection on the Riemann sphere gives rise to the Painlevé VI  equation, which is a non-autonomous Hamilton system with affine Weyl group symmetry of type D_4^{(1)}. Recent progress in mathematical physics motivates its discretization as well as quantization. 

There are two ways to do this; 1) quantize the affine Weyl group symmetry, and 2) use a certain quantum group representations and employ the idea of solvable lattice statistical models. It turns out that both approaches are quite successful and gives the identical system, but the symmetry structure etc are not well-understood clearly yet.  This talk will be based on my previous papers [0703036, 1210.0915] and recent collaborations [2211.16772, 2309.15364].  

Abstract : 

The counting of the plane partitions appears in many branches, ranging from combinatorics, representation theory, geometry and physics, for example. In this talk I will discuss combinatorics of certain generalizations of plane partitions, wherein the hexagonal dimer model arising from the projection of a plane partition is replaced by more general periodic dimer models.

ABSTRACT: We consider the strong Lefschetz property for standard graded Artinian Gorenstein algebras. Such an algebra has a presentation of the quotient algebra of the ring of the differential polynomials modulo the annihilator of some homogeneous polynomial. There is a characterization of the strong Lefschetz property for such an algebra by the non-degeneracy of the higher Hessian matrix of the homogeneous polynomial.

    Maeno and Numata conjectured that if such an algebra is defined by the basis generating polynomial of any matroid, then it has the strong Lefschetz property.

    For this conjecture, we give counterexamples that are associated with graphic matroids. We prove the degeneracy of the higher Hessian matrix by constructing a non-zero element in the kernel of that matrix.

Abstract : In 1927 Emanuel Sperner proved that if $S_1,\dots,S_m$ are distinct subsets of an $n$-element set such that we never have $S_i\subsetS_j$, then $m\leq {n\choose\lfloor n/2\rfloor}$. Moreover, equality is achieved by taking all subsets of $S$ with $\lfloor n/2\rfloor$

elements. This result spawned a host of generalizations, most conveniently stated in the language of partially ordered sets (posets). We will survey some of the highlights of this subject, including the use of linear algebra and of the hard Lefschetz theorem applied to the cohomology of complex projective varieties with a cellular decomposition. The most notable class of such varieties are the generalized flag varieties. We will conclude by discussing two recent proofs of a 1984 conjecture of Anders Bj\"orner on the weak Bruhat order of the symmetric group $S_n$.