Schur Multiple Zeta Functions

• Schur multiple zeta functions are defined as Dirichlet-type series over semi-standard Young tableaux, extending classical Euler–Zagier MZVs.
• They satisfy determinantal identities like Jacobi–Trudi and Giambelli, and incorporate duality and product rules analogous to Schur functions.
• Contemporary research explores their analytic continuation, functional equations, and combinatorial generalizations in number theory and representation theory.

A Schur multiple zeta function (SMZF) is a multivariate generalization of Euler–Zagier multiple zeta values (MZVs) indexed by arbitrary Young diagrams or skew shapes, interpolating between MZVs and zeta-star values, and unifying their combinatorics with that of the theory of Schur symmetric functions. Defined as Dirichlet-type series over semi-standard Young tableaux, SMZFs inherit and extend structurally significant properties of both classical MZVs and Schur polynomials, including determinantal identities (Jacobi–Trudi, Giambelli), duality relations, representation-theoretic product rules (Pieri, Littlewood–Richardson), and connections to quasi-symmetric functions and zeta-functions of root systems. The subject encompasses developments such as laced and skew Giambelli formulas, Hurwitz-type extensions, algebraic and functional equations, shuffle product phenomena, and explicit evaluations for checkerboard and ribbon shapes.

1. Definition and Fundamental Properties

Given a partition (possibly skew) $\lambda$, the Schur multiple zeta function attached to $\lambda$ is

$$\zeta_\lambda(s) = \sum_{M \in \mathrm{SSYT}(\lambda)} \prod_{(i,j)\in\lambda} m_{i,j}^{-s_{i,j}}$$

where the sum runs over all semi-standard Young tableaux $M=(m_{i,j})$ of shape $\lambda$ (rows weakly increase, columns strictly increase, entries in $\mathbb{N}_{>0}$), and $s=(s_{i,j})$ is a complex exponent tableau. The domain of absolute convergence is prescribed by

$$\operatorname{Re} s_{i,j} \geq 1 \text{ for } (i,j)\notin \mathrm{Corners}(\lambda),\quad \operatorname{Re} s_{i,j} > 1 \text{ for } (i,j)\in\mathrm{Corners}(\lambda)$$

(Nakasuji et al., 2017, Nakasuji et al., 2021, Matsumoto et al., 18 Sep 2025). For skew shapes $\lambda/\mu$, the same definition applies, summing over SSYTs of that skew shape.

Specializations:

• For $\lambda=(k)$ (single row): $\zeta_\lambda$ reduces to the classical multiple zeta-star function.
• For $\lambda=(1^k)$ (single column): $\zeta_\lambda$ reduces to the ordinary multiple zeta function.

2. Determinant Formulas: Jacobi–Trudi and Giambelli

SMZFs satisfy determinantal identities paralleling those of Schur functions.

Jacobi–Trudi Formulas:

Assume “diagonally constant” exponents $s_{ij} = a_{j-i}$. Then

$$\zeta_\lambda(s) = \det\left[\zeta^\star(a_{-j+1}, \ldots, a_{-j+\lambda_i-i+j}) \right]_{1 \le i,j \le r}$$

for $r$ rows (Nakasuji et al., 2017, Takeda et al., 5 Aug 2025). An analogous E-type formula uses columns and ordinary (strict) MZVs.

Giambelli Formula:

With $\lambda = (p_1,\ldots,p_N\mid q_1,\ldots,q_N)$ in Frobenius notation,

$$\zeta_\lambda(s) = \det\Bigl[\zeta_{(p_i+1,1^{q_j})}(s^{F}_{i,j})\Bigr]_{1 \leq i,j \leq N}$$

where $s^F_{i,j}$ is the restriction of $s$ to the corresponding hook (Matsumoto et al., 2023, Matsumoto et al., 18 Sep 2025). This formula extends to "laced" and "skew" types, with antipode/reflection phenomena in the quasi-symmetric analog.

For laced or winged shapes $[\lambda_*|\lambda|\lambda^*]$, determinant identities relate SMZFs of such shape to those of pasted hooks (Matsumoto et al., 18 Sep 2025). For content-parametrized exponents, a complete reduction to products/determinants of classical MZVs or zeta-star functions is achieved (Matsumoto et al., 2023).

3. Duality, Ohno-Type Relations, and Algebraic Structure

A duality generalizing the classical MZV duality holds for SMZFs: $$\zeta_{\lambda/\mu}(k) = \zeta_{(\lambda/\mu)^{\dagger}}(k^+)$$ where $(\lambda/\mu)^{\dagger}$ is the 180$^{\circ}$-rotation of the skew diagram and $k^+$ is an explicit dual filling (Nakasuji et al., 2021). The Ohno relation (parameter-shifted duality) holds: $O(k; \ell) = O(k^+;\ell)$ for sums in which integer fillings are shifted in all possible ways summing to $\ell$. Recent advances interpolate the Ohno relation to a meromorphic functional identity in a complex parameter $s$, yielding a full analytic continuation framework (Nakasuji et al., 2022).

These dualities and sum relations equip the algebra of Schur MZVs with rich $\mathbb{Q}$-linear structures and suggest analogues of depth-graded, shuffle, and motivic filtrations characteristic of classical MZVs (Nakasuji et al., 2021).

4. Product Structures: Pieri, Littlewood–Richardson, and Shuffle Formulas

Pieri and Littlewood–Richardson:

Multiplication of Schur MZFs admits a combinatorial rule mirroring that of Schur functions. For hook-shaped and general shapes, Pieri formulas utilize explicit pushing/insertion rules and are realized via crystal base combinatorics (Nakasuji et al., 2021, Nakaoka, 2023). The product of Schur MZFs,

$$\zeta_\mu(\mathbf s) \zeta_\nu(\mathbf t) = \sum_{\lambda} c^\lambda_{\mu,\nu}\, \zeta_\lambda(w_\lambda(\mathbf s,\mathbf t))$$

expands as a linear combination of Schur MZFs, where $c^\lambda_{\mu,\nu}$ are Littlewood–Richardson coefficients and $w_\lambda$ are tableau fillings respecting Yamanouchi conditions (Nakaoka, 2023, Hanaki, 7 Jan 2026). For SMZFs, a refined LR formula is available: the required symmetrization can often be restricted to a subgroup that permutes only the "body" variables, fixing "corner" exponents (Hanaki, 7 Jan 2026).

Shuffle Product:

For hook-type SMZVs, explicit shuffle product formulas are derived via combinatorial poset models and extended with Hurwitz-type (shifted) SMZFs to cover nontrivial shuffle phenomena (Nakasuji et al., 2022). Each shuffle product can be decomposed as a finite sum of hook-type SMZVs with combinatorially determined coefficients.

5. Extensions and Special Structures

Hurwitz-type and Factorial Variants:

Allowing variable shifts $x_{ij}\in\mathbb{R}_{\geq 0}$ yields Schur–Hurwitz MZFs

$$\zeta_\lambda(\mathbf{s}\mid\mathbf{x}) = \sum_{M \in \mathrm{SSYT}(\lambda)} \prod_{(i,j)} (m_{ij}+x_{ij})^{-s_{ij}}$$

which accomodate additional analytic continuation and differentiation identities. Determinantal structures (Jacobi–Trudi, Giambelli) as well as connections to zeta-functions of type $A$ root systems generalize verbatim in this setting (Matsumoto et al., 19 Mar 2025).

Quasi-symmetric and Symplectic/Orthogonal Types:

SMZFs admit natural analogues in the context of quasi-symmetric, Schur P/Q, symplectic, and orthogonal functions, with Pfaffian and determinant identities describing symplectic and orthogonal SMZFs under suitable constraints (Nakasuji et al., 2022). The antipode in the Hopf algebra of QSym induces structural isomorphisms relating SMZFs of skew and reflected shapes (Nakasuji et al., 2017, Matsumoto et al., 18 Sep 2025).

Checkerboard and Ribbon Shapes:

Explicit evaluation is available for certain shapes and fillings, notably those with checkerboard patterns (alternating exponents) and ribbons. For example, Schur MZVs with entries alternating between $1$ and $3$ admit explicit expressions as polynomials in odd zeta values, and computational methods based on generalized Jacobi–Trudi determinants have produced characterizations of their behavior in terms of parity and depth (Bachmann et al., 2017, Bachmann et al., 2019, Bachmann et al., 2023).

6. Connection to Root System Zeta Functions and Functional Equations

For anti-hook shapes and beyond, SMZFs can be expressed in terms of zeta-functions associated to $A_r$ root systems. In particular, certain Schur MZVs for anti-hook shapes are linear combinations of Weyl group multiple Dirichlet series and modified type $A$ zeta functions, revealing deep links between the combinatorics of tableaux and the analytic properties of root system zeta functions (Matsumoto et al., 2020, Matsumoto et al., 2023). These links facilitate the derivation of new functional relations among root system zeta-functions, informed by the combinatorics of Young diagram decompositions.

7. Perspectives and Open Directions

Active research directions in the theory of Schur multiple zeta functions include:

• The complete determination of algebraic relations (double shuffle, duality, sum formulas) and structural and dimension-theoretic properties of the “Schur MZV algebra” (Nakasuji et al., 2021).
• The investigation of motivic, Galois, and coaction structures extending those of classical MZVs.
• Analytic and $p$-adic interpolation, and generalizations to $q$- and $t$-deformations (Macdonald-type factorial zetas) (Matsumoto et al., 19 Mar 2025).
• Extension of product and combinatorial rules (Littlewood–Richardson, Pieri, shuffle) to refined settings, symmetric and non-symmetric types, and broader families of diagrams (Hanaki, 7 Jan 2026, Nakaoka, 2023).
• Systematic evaluation and classification of special values, especially for checkerboard, ribbon, and rectangular shapes, including determinant formulas and parity phenomena (Bachmann et al., 2019, Bachmann et al., 2023).

The theory of Schur multiple zeta functions thus forms a nexus between algebraic combinatorics, special values in analytic number theory, and representation theory, offering a template for new zeta-type invariants attached to combinatorial and representation-theoretic data.