arXiv subjects

Maki Nakasuji

Publications and source records attributed to Maki Nakasuji.

At least 19 recordsLinked to original sources

arXiv2026

Explicit formula for multi-indexed poly-Bernoulli numbers

The classical Bernoulli numbers $B_m$ can be expressed using Stirling numbers of the second kind, and M. Kaneko extended this framework by defining poly-Bernoulli numbers ${\mathbb B}_m^{(k)}$, for which explicit formulas using the Stirling numbers of the second kind and duality relations were obtained. Later, Kaneko and H. Tsumura introduced multi-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, \ldots, m_r}^{(k_1, \ldots, k_r)}$ using the multiple polylogarithm and reached their duality properties via an associated $\eta$-function. Explicit formulas for double-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, m_2}^{(k_1, k_2)}$ were obtained by Y. Baba, M. Nakasuji, and M. Sakata. In this article, we extend these results to general multi-indexed poly-Bernoulli numbers and use it to give an alternative proof of the duality of multi-indexed poly-Bernoulli numbers.

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arXiv2025

$SL_4$-Kloosterman sum via the Bruhat decomposition

We define the Kloosterman sum for $SL_4$ over the Kloosterman set via the Bruhat decomposition and stratify the Kloosterman set using the reduced word decomposition of the Weyl group element. The Kloosterman sum for an $SL_4$-long word is decomposed into finer parts (called the fine Kloosterman sum), and can be written as a finite sum of a product of two classical Kloosterman sums.

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arXiv2025

New determinant formulas of Giambelli-type for Schur multiple zeta-functions and their applications

In this article, we will prove the Giambelli formula for Schur multiple zeta-functions of extended shape which we call laced type, using the combinatorial method of proving the Giambelli formula for Schur function by Egecioglu and Remmel. Further we will obtain the Giambelli formula for Schur multiple zeta-functions of a certain skew type via the antipode on the set of quasi-symmetric functions. Combining these two Giambelli-type formulas, we will have new identities among Schur multiple zeta-functions.

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arXiv2025

Schur multiple zeta-functions of Hurwitz type

We study the Hurwitz-type analogue of Schur multiple zeta-functions involving shifting parameters. We extend various formulas, known for ordinary Schur multiple zeta-functions, to the case of Hurwitz type. We also mention unpublished results proved by Yamamoto and by Minoguchi. Further we present new formulas obtained by performing differentiation with respect to shifting parameters.

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arXiv2023

Expressions of content-parametrized Schur multiple zeta-functions via the Giambelli formula

In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.

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arXiv2022

Multi-indexed poly-Bernoulli numbers

As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index have been established. For the multi-indexed poly-Bernoulli numbers generalized by Kaneko-Tsumura, among such properties only the duality formula was obtained. In this paper, we restrict the double-indexed poly-Bernoulli numbers and show the explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index for them. Further, we define the variant of multiple-indexed poly-Bernoulli numbers using the star-version of multiple-indexed logarithms and obtain the relation between this kind of double and triple-indexed poly-Bernoulli numbers with multi-indexed poly-Bernoulli numbers ahead.

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arXiv2022

Symmetric Schur multiple zeta functions

We introduce the multiple zeta functions with structures similar to those of symmetric functions such as Schur $P$-, Schur $Q$-, symplectic and orthogonal functions in the representation theory. We first consider their basic properties such as a domain of absolute convergence. And then by restricting to the truncated multiple zeta functions, we obtain the pfaffian expression of the Schur $Q$-multiple zeta functions, the sum formula for Schur $P$- and Schur $Q$-multiple zeta functions, the determinant expressions of symplectic and orthogonal Schur multiple zeta functions under an assumption on variables. Finally, we generalize those to the quasi-symmetric functions.

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arXiv2022

An interpolation of the generalized duality formula for the Schur multiple zeta values to complex functions

One of the important research subjects in the study of multiple zeta functions is to clarify the linear relations and functional equations among them. The Schur multiple zeta functions are a generalization of the multiple zeta functions of Euler-Zagier type. Among many relations, the duality formula and its generalization are important families for both Euler-Zagier type and Schur type multiple zeta values. In this paper, following the method of previous works for multiple zeta values of Euler-Zagier type, we give an interpolation of the sums in the generalized duality formula, called Ohno relation, for Schur multiple zeta values. Moreover, we prove that the Ohno relation for Schur multiple zeta values is valid for complex numbers.

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arXiv2022

Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

We generalize the definition of Yang-Baxter basis of type $A$ Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75--90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of $p$-adic groups.

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arXiv2022

Shuffle product formula of the Schur multiple zeta values of hook type

We discuss the shuffle product of the Schur multiple zeta values, which are the special values of Schur multiple zeta functions. We first define $2$-labeled Schur posets to generalize Yamamoto's integral expression of the multiple zeta values and consider the product of hook-type Schur multiple zeta values by using these posets. Then, for the derived terms, we introduce a modified Hurwitz-type Schur multiple zeta function of hook type, named an elementary factorial Schur multiple zeta function. Furthermore, we generalize $2$-labeled Schur posets to consider the shuffle product of the elementary factorial Schur multiple zeta values and obtain an explicit formula for their shuffle product.

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arXiv2021

Duality formula and its generalization for Schur multiple zeta functions

In the study on multiple zeta values, the duality formula is one of the families of basic relations and plays an important role in the investigation of algebraic structure of the space spanned by all multiple zeta values along with the generalized duality formula (so called Ohno relation) obtained by the second author. In this article, we will discuss them for the Schur multiple zeta values which are the values at positive integers of the Schur multiple zeta function introduced by the first author, O. Phukswan and Y. Yamasaki.

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arXiv2021

The Pieri formulas for hook type Schur multiple zeta functions

We study the Pieri type formulas for the Schur multiple zeta functions along with those for the Schur polynomials. To formulate these formulas, we introduce a new insertion rule for adding boxes in the Young tableaux and obtain the results for the hook type Schur multiple zeta functions. For the proof, we show {certain} extended Jacobi-Trudi formulas for the Schur multiple zeta functions.

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arXiv2020

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta functions by an analogue of Weyl group multiple Dirichlet series in the sense of Bump et al. By combining with a result of Nakasuji, Phuksuwan and Yamasaki, our theorems yield a new method of finding functional relations among zeta-functions of root systems.

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arXiv2020

Parametrization of Kloosterman sets and $\mathrm{SL}_3$-Kloosterman sums

We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup $Γ_0(N)\subseteq \mathrm{SL}_3(\mathbb{Z})$. Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

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arXiv2018

Schur type poly-Bernoulli numbers

The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type $B$ and $C$, respectively. As a generalization of these poly-Bernoulli numbers, we introduce Schur type poly-Bernoulli numbers and investigate their properties. First, we define a generalization of Arakawa-Kaneko multiple zeta functions and obtain their expression in terms of Schur type Bernoulli numbers. Next, under the restriction to the hook type, we define a generalization of Kaneko-Tsumura multiple zeta functions and obtain similar expression in terms of Schur type Bernoulli numbers. Lastly, we study more properties such as a recurrence formula, a relation formula between Bernoulli numbers and a description in terms of the Stirling numbers.

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arXiv2018

On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.

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arXiv2017

Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials

A problem in representation theory of $p$-adic groups is the computation of the \textit{Casselman basis} of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

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arXiv2014

Factorial Schur functions and the Yang-Baxter equation

Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six vertex model). The proof is based on the Yang-Baxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=-1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we recover the description as a sum over tableaux. If $t=\infty$ we recover a description of Lascoux that was previously considered by McNamara. We also are able to prove using the Yang-Baxter equation the asymptotic symmetry of the factorial Schur functions in the shift parameters. Finally, we give a proof using our methods of the dual Cauchy identity for factorial Schur functions. Thus using our methods we are able to give thematic proofs of many of the properties of factorial Schur functions.

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