The symmetric functions catalog

An overview of symmetric functions and related topics

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 [NT22b] 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