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#Schur multiple zeta functions

The Schur multiple zeta functions were introduced in [NPY18] in order to interpolate between previous generalizations of the Riemann zeta function. They are defined as \[\zeta_\lambda(\svec) \coloneqq \sum_{T \in \SSYT(\lambda)} \prod_{(i,j) \in \lambda} T_{ij}^{-s_{ij}}.\] The product is taken over all boxes in the diagram of \(\lambda.\) Note that this is not a symmetric function and not a polynomial in \(\svec.\) In fact, when \(\lambda = (1),\) \(\zeta_\lambda(\svec)\) coincides with the classical Riemann zeta function.

Example

For the one-box partition, \[\zeta_{(1)}(s) = \sum_{m\geq 1} m^{-s},\] which is the classical Riemann zeta function.

When \(\lambda\) is a single row or a single column, we recover the functions previously introduced by Hoffman [Hof92] and Zaiger [Zag94].

If we set all variables equal, then \[\zeta_\lambda(s,s,s,\dotsc,s) = \schurS_\lambda(1^{-s},2^{-s},3^{-s},\dotsc).\]

Jacobi–Trudi identities and Giambelli formulas are proved in [NPY18]. More general determinant formulas of the same type as Lascoux–Pragacz and Hamel–Goulden are proved in [BC20].

Sum formulas, see [BKSY+23].

Pieri formulas and a Littlewood–Richardson type rule are proved in [Nak23]. For connection with quasisymmetric functions, see [Hof08]. For connections with Chern numbers and hyper-Kähler and Calabi–Yau manifolds, see [Li21].

A refinement of the Littlewood–Richardson rule is available in [Han26].

#Schur multiple zeta P and Q

In [NT22] the authors introduce the multiple zeta analogs of Schur P and Schur Q functions.

They also define multiple zeta functions analogous to the Orthogonal Schur polynomials and the Symplectic Schur polynomials.

#Multiple Schur series

J. Yu introduces a connected, commutative graded Hopf algebra on Young tableaux whose linearized quotient is a quasi-shuffle algebra [Yu26]. Its characters give multiple Schur series , a common framework for Schur multiple zeta values, Schur multiple Eisenstein series, and their \(q\)-analogues. The construction has hook and Jacobi–Trudi formulas.

H. Bachmann and J. Yu further define Schur Eisenstein series and Schur MacMahon series indexed by partitions [BY26]. They relate the two families through the Faà di Bruno Hopf algebra and an \(\mathfrak{sl}_2\)-action. The Schur Eisenstein series indexed by partitions with parts at most three form a basis of the algebra of quasimodular forms; the corresponding integral spanning statement for Schur MacMahon series is conjectural.

Bibliography

  1. [BC20]Henrik Bachmann and Steven Charlton. Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values. European Journal of Combinatorics, 87:103133, June 2020.
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  doi = {10.1016/j.ejc.2020.103133},
  url2 = {https://doi.org/10.1016/j.ejc.2020.103133},
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  month = jun,
  publisher = {Elsevier {BV}},
  volume = {87},
  pages = {103133},
  author = {Henrik Bachmann and Steven Charlton},
  title = {Generalized {J}acobi--{T}rudi determinants and evaluations of {S}chur multiple zeta values},
  journal = {European Journal of Combinatorics}
}
  2. [BKSY+23]Henrik Bachmann, Shin-ya Kadota, Yuta Suzuki, Shuji Yamamoto and Yoshinori Yamasaki. Sum formulas for Schur multiple zeta values. arXiv:2302.03187, 2023.
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Year = {2023},
Eprint = {2302.03187},
  url = {https://arxiv.org/abs/2302.03187},
journal = {arXiv e-prints}
}
  3. [BY26]Henrik Bachmann and Jinbo Yu. Schur Eisenstein series and Schur MacMahon series. arXiv:2607.27702v1, 2026.
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    @article{BachmannYu2026x,
  author = {Henrik Bachmann and Jinbo Yu},
  title = {Schur {E}isenstein series and {S}chur {M}ac{M}ahon series},
  year = {2026},
  eprint = {2607.27702v1},
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  journal = {arXiv e-prints}
}
  4. [Han26]Hikari Hanaki. The Littlewood–Richardson rule for Schur multiple zeta functions. arXiv:2601.03970, 2026.
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    @article{Hanaki2026x,
Author = {Hikari Hanaki},
Title = {The {L}ittlewood--{R}ichardson rule for {S}chur multiple zeta functions},
Year = {2026},
Eprint = {2601.03970},
  url = {https://arxiv.org/abs/2601.03970},
journal = {arXiv e-prints}
}
  5. [Hof92]Michael E. Hoffman. Multiple harmonic series. Pacific Journal of Mathematics, 152(2):275–290, 1992.
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number = {2},
publisher = {Pacific Journal of Mathematics, A Non-profit Corporation},
pages = {275--290},
year = {1992}
}
  6. [Hof08]Michael E. Hoffman. A character on the quasi-symmetric functions coming from multiple zeta values. The Electronic Journal of Combinatorics, 15(1), July 2008.
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    @article{Hoffman2008,
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  url2 = {https://doi.org/10.37236/821},
  year = {2008},
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  publisher = {The Electronic Journal of Combinatorics},
  volume = {15},
  number = {1},
  author = {Michael E. Hoffman},
  title = {A Character on the Quasi-Symmetric Functions coming from Multiple Zeta Values},
  journal = {The Electronic Journal of Combinatorics}
}
  7. [Li21]Ping Li. The complex genera, symmetric functions and multiple zeta values. arXiv:2112.01192, 2021.
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Author = {Ping Li},
Title = {The complex genera, symmetric functions and multiple zeta values},
Year = {2021},
Eprint = {2112.01192},
  url = {https://arxiv.org/abs/2112.01192},
journal = {arXiv e-prints}
}
  8. [Nak23]Shutaro Nakaoka. The Pieri formulas and the Littlewood–Richarson rule for Schur multiple zeta functions. arXiv:2305.19975, 2023.
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    @article{Nakaoka2023x,
Author = {Shutaro Nakaoka},
Title = {The {P}ieri formulas and the {L}ittlewood--{R}icharson rule for {S}chur multiple zeta functions},
Year = {2023},
Eprint = {2305.19975},
  url = {https://arxiv.org/abs/2305.19975},
journal = {arXiv e-prints}
}
  9. [NPY18]Maki Nakasuji, Ouamporn Phuksuwan and Yoshinori Yamasaki. On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions. Advances in Mathematics, 333:570–619, July 2018.
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  volume = {333},
  pages = {570--619},
  author = {Maki Nakasuji and Ouamporn Phuksuwan and Yoshinori Yamasaki},
  title = {On {S}chur multiple zeta functions: {A} combinatoric generalization of multiple zeta functions},
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}
  10. [NT22]Maki Nakasuji and Wataru Takeda. Symmetric Schur multiple zeta functions. arXiv:2208.11909, 2022.
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    @article{NakasujiTakeda2022x,
Author = {Maki Nakasuji and Wataru Takeda},
Title = {Symmetric {S}chur multiple zeta functions},
Year = {2022},
Eprint = {2208.11909},
  url = {https://arxiv.org/abs/2208.11909},
journal = {arXiv e-prints}
}
  11. [Yu26]Jinbo Yu. The Young Tableaux Hopf algebra and multiple Schur series. arXiv:2607.09157v1, 2026.
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  author = {Jinbo Yu},
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  year = {2026},
  eprint = {2607.09157v1},
  url = {https://arxiv.org/abs/2607.09157v1},
  journal = {arXiv e-prints}
}
  12. [Zag94]Don Zagier. Values of zeta functions and their applications. First european congress of mathematics paris, july 6–10, 1992:497–512, 1994.
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  pages = {497--512},
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  title = {Values of Zeta Functions and Their Applications},
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}

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