2023-06-01

## Schur multiple zeta functions

The Schur multiple zeta functions where introduced in [NPY18] in order to interpolate between previous generalizations of the Riemann zeta function. We have that $\zeta_\lambda(\svec)$ is 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.

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 [BKSYY23].

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

### 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.

## References

- [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.
- [BKSYY23] Henrik Bachmann, Shin-ya Kadota, Yuta Suzuki, Shuji Yamamoto and Yoshinori Yamasaki. Sum formulas for Schur multiple zeta values. arXiv e-prints, 2023.
- [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.
- [Hof92] Michael E. Hoffman. Multiple harmonic series. Pacific Journal of Mathematics, 152(2):275–290, 1992.
- [Li21] Ping Li. The complex genera, symmetric functions and multiple zeta values. arXiv e-prints, 2021.
- [Nak23] Shutaro Nakaoka. The Pieri formulas and the Littlewood–Richarson rule for Schur multiple zeta functions. arXiv e-prints, 2023.
- [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.
- [NT22] Maki Nakasuji and Wataru Takeda. Symmetric Schur multiple zeta functions. arXiv e-prints, 2022.
- [Zag94] Don Zagier. Values of zeta functions and their applications. In First European Congress of Mathematics Paris, July 6–10, 1992. Birkhäuser Basel, 1994.