[Paper Review] On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions

Maki Nakasuji, Ouamporn Phuksuwan+1 more|arXiv (Cornell University)|Apr 27, 2017

Advanced Mathematical Identities22 references3 citations

TL;DR

This paper introduces Schur multiple zeta functions (SMZFs) as a combinatorial generalization of multiple zeta functions, using skew Young diagrams and quasi-symmetric functions. It establishes Jacobi-Trudi, Giambelli, and dual Cauchy formulas under a key analytic assumption, and provides iterated integral representations of Ribenhoist type, enabling a duality principle for SMZFs.

ABSTRACT

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.

Motivation & Objective

• To generalize multiple zeta functions using combinatorial structures such as skew Young diagrams and Schur-type quasi-symmetric functions.
• To establish algebraic and analytic formulas—such as Jacobi-Trudi and Giambelli—for Schur multiple zeta functions under a convergence condition on the diagonal variables.
• To extend the theory of multiple zeta functions to a framework involving the Hopf algebra of quasi-symmetric functions (QSym).
• To derive iterated integral representations of Ribenhoist type for Schur multiple zeta values, enabling a duality principle.
• To clarify the role of the diagonal assumption in avoiding error terms in structural formulas.

Proposed method

• Define Schur-type quasi-symmetric functions Sν(α) associated with skew Young diagrams ν within the Hopf algebra QSym.
• Use the antipode S of QSym to relate Sν(α) to its dual under anti-diagonal transpose ν#, establishing duality.
• Apply the Jacobi-Trudi and Giambelli formulas in the context of SMZFs, with H-type and E-type formulations corresponding to essential and monomial quasi-symmetric functions.
• Derive meromorphic continuation of SMZFs to C^{s+r-1} using known continuation properties of MZFs and MZSFs.
• Construct iterated integral representations of Ribenhoist type by adapting methods from [Kaneko-Yamamoto, Yamamoto], enabling variable substitution for duality.
• Utilize the specialization of Macdonald's n-th variation of Schur functions (via Nakagawa et al.) to realize SMZFs under the diagonal assumption.

Experimental results

Research questions

• RQ1 How can multiple zeta functions be generalized using combinatorial structures such as skew Young diagrams?
• RQ2 What algebraic formulas—like Jacobi-Trudi or Giambelli—hold for Schur multiple zeta functions under analytic constraints?
• RQ3 Can iterated integral representations of Ribenhoist type be constructed for Schur multiple zeta values?
• RQ4 What is the role of the diagonal variable assumption in ensuring clean structural formulas without error terms?
• RQ5 How does the antipode in the Hopf algebra QSym relate Schur-type quasi-symmetric functions to their duals under anti-diagonal transpose?

Key findings

• The paper establishes H-type and E-type Jacobi-Trudi formulas for Schur multiple zeta functions under the assumption that variables on diagonal lines satisfy ℜ(a), ℜ(b), ℜ(d) > 1 and ℜ(c) ≥ 1.
• The Giambelli formula and dual Cauchy formula are proven for SMZFs under the same convergence condition, extending classical identities to the Schur-type setting.
• SMZFs admit meromorphic continuation to C^{s+r-1} due to the known continuation of multiple zeta functions and their starred variants.
• The Schur-type quasi-symmetric functions Sν(α) are realized as specializations of Macdonald's n-th variation of Schur functions, under the diagonal assumption.
• Iterated integral representations of Ribenhoist type are constructed, allowing a duality principle via variable substitution when the dual value remains of Ribenhoist type.
• Without the diagonal assumption, extraneous error terms arise in structural formulas, which are set for clarification in future work.