We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.
Inspired by a recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi-Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.
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.
We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.
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…
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
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.
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 fun…
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.
We use earlier defined notion of $n$- determinant to investigate sub-determinants of an extended Vandermonde matrix. Firstly, we demonstrate our method on a number of particular cases. Then we prove that all these results may be stated in terms of Schur's polynomials. In our main result, we prove that Schur polynomials are equal to minors of a fixed matrix, which entries are formed of elementary symmetric polynomials. Such a formula is known as the second Jaccobi-Trudi identity.
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
