Schur Multiple Zeta Functions

• Schur multiple zeta functions are multivariable Dirichlet series indexed by Young diagrams that generalize Euler–Zagier zeta functions by encoding tableau combinatorics and interpolation between strict and weak inequality regimes.
• Their structure is underpinned by determinant identities such as Jacobi–Trudi, Giambelli, and dual Cauchy formulas, which transform content-parametrized settings into hook-type building blocks with analytic continuation conditions.
• Recent developments extend these functions to include Hurwitz-type variants and symplectic, orthogonal, Schur P/Q forms, linking classical zeta results with root-system methods and quasi-symmetric function frameworks.

Schur multiple zeta functions are multivariable Dirichlet series indexed by Young diagrams, or more generally skew Young diagrams, in which the summation runs over semi-standard Young tableaux and the summand records a boxwise system of exponents. Introduced by Nakasuji, Phuksuwan, and Yamasaki, they form a combinatorial generalization of Euler–Zagier multiple zeta functions and their star variants, while retaining structural features familiar from Schur-function theory, such as Jacobi–Trudi, Giambelli, and dual Cauchy identities under suitable diagonal assumptions (Nakasuji et al., 2017). Subsequent work developed their duality and Ohno-type relations, checkerboard and ribbon evaluations, Pieri and Littlewood–Richardson rules, Hurwitz-shifted versions, and analogues attached to Schur $P$-, $Q$-, symplectic, and orthogonal functions (Nakasuji et al., 2021, Bachmann et al., 2017, Nakaoka, 2023, Matsumoto et al., 19 Mar 2025, Nakasuji et al., 2022).

1. Definition and analytic framework

Let $\lambda/\mu$ be a skew Young diagram with box set $D(\lambda/\mu)$, and let $\mathrm{SSYT}(\lambda/\mu)$ denote the semi-standard Young tableaux of that shape, with entries weakly increasing along rows and strictly increasing down columns. For a complex exponent array $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$, the Schur multiple zeta function is

$$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$

Absolute convergence holds when $\Re s_{ij}\ge 1$ on non-corner boxes and $\Re s_{ij}>1$ on corner boxes; for positive-integer exponents this is often encoded by the admissibility condition that corner entries are at least $2$ (Nakasuji et al., 2021).

For straight shapes, the theory interpolates the two classical Euler–Zagier regimes. Under the convention used in the foundational paper, $Q$0 recovers the classical multiple zeta function and $Q$1 recovers the multiple zeta-star function (Nakasuji et al., 2017). This interpolation is not merely formal: the tableau combinatorics directly encode the pattern of strict and weak inequalities that distinguishes $Q$2 from $Q$3.

A basic specialization already exhibits the symmetric-function origin of the theory. If every box carries the same complex parameter $Q$4 with $Q$5, then

$Q$6

so the Schur multiple zeta function becomes an evaluation of the Schur function. The power-sum expansion then gives

$Q$7

and in particular $Q$8 (Nakasuji et al., 2017). Thus even the constant-parameter case already ties SMZFs to both symmetric-group character theory and classical special values of the Riemann zeta function.

2. Determinantal structure and Schur-function analogues

The central structural feature of SMZFs is that, under a diagonal or content condition, the standard determinant identities for Schur functions lift to zeta-theoretic identities. The basic assumption is that the exponent attached to a box depends only on its content $Q$9, equivalently $\lambda/\mu$0 or $\lambda/\mu$1. Under this hypothesis, Nakasuji–Phuksuwan–Yamasaki established Jacobi–Trudi determinant formulas in terms of truncated Euler–Zagier multiple zeta and multiple zeta-star functions, together with Giambelli and dual Cauchy formulas (Nakasuji et al., 2017).

In Frobenius notation $\lambda/\mu$2, the Giambelli formula takes the hook-determinant form

$\lambda/\mu$3

for content-parametrized parameters in the convergence region (Matsumoto et al., 2023). This reduces arbitrary content-parametrized Schur multiple zeta functions to hook-type building blocks. Matsumoto–Nakasuji then derived explicit hook expansions in terms of Euler–Zagier multiple zeta and multiple zeta-star functions, and also reformulated them in terms of modified zeta-functions of type $\lambda/\mu$4 root systems (Matsumoto et al., 2023).

These determinant formulas are not isolated identities. The 2017 foundational paper identifies diagonally constant truncated SMZFs with suitable specializations of Macdonald’s “ninth variation” Schur functions, so that Jacobi–Trudi, Giambelli, and dual Cauchy become consequences of the corresponding identities in that broader symmetric-function setting (Nakasuji et al., 2017). Later work extended the determinant formalism to “laced” or extended shapes, and also obtained new Giambelli-type formulas for certain skew shapes by combining quasi-symmetric antipode arguments with anti-diagonal transposition (Matsumoto et al., 18 Sep 2025). Taken together, these results show that determinant calculus is not ancillary but intrinsic to the subject.

3. Quasi-symmetric, integral, and duality formalisms

A second organizing layer comes from quasi-symmetric functions and iterated integrals. For a skew shape $\lambda/\mu$5 and a positive-integer filling $\lambda/\mu$6, Nakasuji–Phuksuwan–Yamasaki defined Schur-type quasi-symmetric functions $\lambda/\mu$7 whose zeta-evaluation recovers $\lambda/\mu$8. Under Hoffman’s harmonic-algebra isomorphism and the standard zeta-evaluation map, these functions sit naturally inside $\lambda/\mu$9, and the Hopf-algebra antipode acts by

$D(\lambda/\mu)$0

where $D(\lambda/\mu)$1 is the anti-diagonal transpose of the diagram (Nakasuji et al., 2017). This gives a combinatorial mechanism for transposition-type identities beyond ordinary MZV duality.

For ribbon shapes, the same paper constructed explicit iterated-integral representations generalizing the Euler–Zagier integral formulas. Reversing the integration variables yields ribbon duality: when $D(\lambda/\mu)$2 is again a ribbon, the integral representation transports $D(\lambda/\mu)$3 to $D(\lambda/\mu)$4 (Nakasuji et al., 2017). This ribbon calculus provides the bridge from tableau combinatorics to the analytic methods usually associated with MZVs.

Nakasuji and Ohno then established a genuine Schur duality formula for admissible diagonal integer tableaux. If $D(\lambda/\mu)$5 is such a tableau on $D(\lambda/\mu)$6, $D(\lambda/\mu)$7 is its dual tableau, and $D(\lambda/\mu)$8 is the conjugate skew shape, then

$D(\lambda/\mu)$9

They also proved the Schur analogue of the Ohno relation: for the Schur–Ohno sums

$\mathrm{SSYT}(\lambda/\mu)$0

one has $\mathrm{SSYT}(\lambda/\mu)$1 for all $\mathrm{SSYT}(\lambda/\mu)$2 (Nakasuji et al., 2021).

The generalized duality was then interpolated to a complex parameter by Nakasuji–Ohno–Takeda. They introduced a Mellin–Barnes type function $\mathrm{SSYT}(\lambda/\mu)$3 such that $\mathrm{SSYT}(\lambda/\mu)$4 for $\mathrm{SSYT}(\lambda/\mu)$5, and proved

$\mathrm{SSYT}(\lambda/\mu)$6

thereby extending the Ohno symmetry from integer sums to a meromorphic identity in the complex $\mathrm{SSYT}(\lambda/\mu)$7-plane (Nakasuji et al., 2022). This places Schur Ohno theory within the analytic-continuation paradigm familiar from the Euler–Zagier case.

4. Special shapes, explicit evaluations, and relations to classical zeta values

Several families of SMZFs admit unusually explicit evaluations. Bachmann and Yamasaki studied checkerboard-style fillings, in which the exponents alternate between two values $\mathrm{SSYT}(\lambda/\mu)$8 and $\mathrm{SSYT}(\lambda/\mu)$9 according to the parity of $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$0, with the corner boxes forced to carry $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$1. For checkerboardable shapes this gives a distinguished diagonal-constant SMZF, denoted $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$2 (Bachmann et al., 2017).

Their most striking results occur in the case $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$3. For primitive ribbons $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$4 and $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$5, they proved

$\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$6

so every odd single zeta value of the forms $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$7 and $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$8 acquires a new checkerboard sum representation (Bachmann et al., 2017). Using the Jacobi–Trudi formula, they further showed that certain checkerboard Schur multiple zeta values of staircase type are determinants of Hankel-type matrices whose entries are odd single zeta values. In particular, for suitable parity conditions on $\mathbf{s}=(s_{ij})_{(i,j)\in D(\lambda/\mu)}$9 and $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$0, the staircase value $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$1 is a determinant with entries $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$2 (Bachmann et al., 2017).

A different family of explicit formulas concerns sums over all admissible tableaux of fixed weight. For ribbons with $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$3 corners, Bachmann, Kadota, Suzuki, Yamamoto, and Yamasaki showed that the weight-$$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$4 sum is always a $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$5-linear combination of classical MZVs of depth at most $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$6. For anti-hooks, and more generally certain one-corner configurations, the sum collapses to a rational multiple of a single Riemann zeta value, recovering and extending the classical sum formulas for $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$7 and $$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$8 (Bachmann et al., 2023).

Anti-hook and hook-type SMZFs also connect to zeta-functions of root systems. Matsumoto and Nakasūji expressed anti-hook Schur multiple zeta-functions in terms of modified type-$$\zeta_{\lambda/\mu}(\mathbf{s}) = \sum_{M\in \mathrm{SSYT}(\lambda/\mu)} \prod_{(i,j)\in D(\lambda/\mu)} M_{ij}^{-s_{ij}}.$$9 root-system zeta-functions, obtaining functional relations between Euler–Zagier zetas, Schur multiple zeta-functions, and root-system zetas (Matsumoto et al., 2020). The content-parametrized hook formulas of Matsumoto–Nakasuji provide a related decomposition into Euler–Zagier $\Re s_{ij}\ge 1$0, $\Re s_{ij}\ge 1$1, and modified type-$\Re s_{ij}\ge 1$2 zeta-functions (Matsumoto et al., 2023). These results identify SMZFs as one of the meeting points of tableau combinatorics, multiple Dirichlet-series methods, and Lie-theoretic zeta functions.

5. Product formulas, insertion rules, and Littlewood–Richardson phenomena

Product identities for SMZFs are more subtle than for ordinary Schur functions because the exponent variables are part of the data. For hook-type SMZFs, Nakasuji and Takeda introduced a “pushing rule” for inserting horizontal or vertical strips and proved Pieri-type formulas using extended Jacobi–Trudi identities and nonintersecting-path methods (Nakasuji et al., 2021). In this setting the product with a one-row or one-column SMZF expands into a sum over pushed tableaux of new hook shapes.

Nakaoka generalized this to arbitrary shapes. Working first with truncated SMZFs and identifying them with sums over $\Re s_{ij}\ge 1$3 crystals, he established general Pieri formulas in which multiplication by a one-row or one-column Schur multiple zeta function is expressed through combinatorial “pushing” operations indexed by sets $\Re s_{ij}\ge 1$4 and $\Re s_{ij}\ge 1$5. The same crystal-theoretic framework yields a Littlewood–Richardson rule for products of general SMZFs, recovering the usual Littlewood–Richardson coefficients after symmetrization over the relevant exponent variables (Nakaoka, 2023).

That symmetrization is essential. Hanaki showed that, unlike the classical Schur-function product $\Re s_{ij}\ge 1$6, the Schur multiple zeta product requires summation over permutations of variables to recover an LR-type expansion. He then refined the earlier formula by restricting the required symmetrization from the full symmetric group to a specific subgroup that permutes only the “body” variables of the concatenated shape, leaving certain arm variables fixed (Hanaki, 7 Jan 2026). This refinement makes the combinatorics more faithful to the geometry of the tableaux while preserving the classical coefficients $\Re s_{ij}\ge 1$7.

A parallel multiplicative theory exists for hook-type shuffle products. Using 2-labeled Schur posets and their iterated-integral realizations, Nakasuji and Takeda proved shuffle product formulas for hook-type Schur multiple zeta values and, to accommodate derived terms, introduced elementary factorial Schur multiple zeta functions of hook type. They then established an explicit shuffle formula for these elementary factorial objects, showing that the hook-type and elementary-factorial families are closed under shuffle in the same sense that ordinary MZVs are (Nakasuji et al., 2022).

6. Variants, analogues, and current extensions

The Schur multiple zeta framework has been extended in several orthogonal directions. Nakasuji and Takeda defined Schur $\Re s_{ij}\ge 1$8- and $\Re s_{ij}\ge 1$9-multiple zeta functions by summing over marked shifted tableaux of strict shape, and also introduced symplectic and orthogonal Schur multiple zeta functions. In that setting they proved a pfaffian formula for Schur $\Re s_{ij}>1$0-multiple zeta functions, determinant formulas for the symplectic and orthogonal cases, and parallel quasi-symmetric generalizations. One basic identity is

$\Re s_{ij}>1$1

reflecting the difference between marked-tableau conventions for $\Re s_{ij}>1$2- and $\Re s_{ij}>1$3-objects (Nakasuji et al., 2022). Hanaki later established Littlewood–Richardson-type product expansions for Schur $\Re s_{ij}>1$4- and skew Schur $\Re s_{ij}>1$5-multiple zeta functions, again with symmetrization over exponent variables and with a refined subgroup version in the skew $\Re s_{ij}>1$6 case (Hanaki, 23 Mar 2026).

A second extension is the Hurwitz-type theory. Matsumoto and Nakasuji introduced shifted parameters $\Re s_{ij}>1$7 and defined

$\Re s_{ij}>1$8

In the diagonal-parametrized case $\Re s_{ij}>1$9, $2$0, they obtained Hurwitz analogues of the Jacobi–Trudi and Giambelli formulas, hook decompositions, and meromorphic continuation. They also derived derivative identities with respect to the shift parameters, including

$2$1

which relate shift derivatives to exponent shifts inside the same diagram (Matsumoto et al., 19 Mar 2025).

A third line of development uses Macdonald’s ninth variation more aggressively. Recent work established quadratic relations for ninth-variation Schur functions by means of the Desnanot–Jacobi identity and Plücker relations, then specialized these identities to diagonally constant SMZFs. In rectangular checkerboard cases this yields explicit polynomial evaluations in ordinary zeta values (Takeda et al., 5 Aug 2025). Closely related work proved new Giambelli-type formulas for laced or extended shapes and for certain skew types via the antipode on quasi-symmetric functions, producing further determinant identities among SMZFs (Matsumoto et al., 18 Sep 2025).

Across these developments, the recurring theme is that Schur multiple zeta functions behave simultaneously as Dirichlet series, as tableau generating functions, and as specializations of symmetric or quasi-symmetric objects. The published results show that determinant identities, duality, crystal combinatorics, and root-system methods all survive in this setting, while the full algebraic structure of the space generated by Schur multiple zeta values remains, in the words of the duality literature, far from understood (Nakasuji et al., 2021).