The Littlewood-Richardson rule for Schur multiple zeta functions

The Schur multiple zeta function was defined as a multivariable function by Nakasuji-Phuksuwan-Yamasaki. Inspired by the product formula of Schur functions, the products of Schur multiple zeta functions have been studied. While the product of two Schur functions expands as a linear combination of Schur functions, it is known that a similar expansion for the product of Schur multiple zeta functions can be obtained by symmetrizing, i.e., by taking the summation over all permutations of the variables. In this paper, we present a more refined formula by restricting the summation from the full symmetric group to its specific subgroup.

💡 Research Summary

The paper studies products of Schur multiple zeta functions, a multivariable generalisation of both Euler–Zagier multiple zeta values and Schur functions. For a partition δ, the Schur multiple zeta function ζ_δ(s) is defined as a sum over all semi‑standard Young tableaux M of shape δ, with each entry m_{ij} raised to the negative power s_{ij}. The series converges in a domain where the real parts of the variables satisfy certain inequalities (≥1 on corner positions, >1 elsewhere).

Classically, the product of two ordinary Schur functions expands as a linear combination of Schur functions with Littlewood–Richardson coefficients c^λ_{µν}. Nakao showed that an analogous expansion holds for Schur multiple zeta functions if one symmetrises over the full symmetric group acting on the combined set of variables s∪t, i.e. over Sym(s∗t) where s∗t is the tableau obtained by placing the µ‑shape tableau s below an empty rectangle of width µ₁ and the ν‑shape tableau t to its right (the “concatenated” shape).

The main contribution of this work is a refinement of Nakao’s symmetrisation. Instead of averaging over the whole symmetric group, the authors restrict the sum to a specific subgroup Sym(B(s∗t)), where B(s∗t) – called the “body” – is obtained from the concatenated tableau by deleting contiguous boxes from the bottom of the first column and from the right of the first row. Theorem 1.1 states that, under mild real‑part conditions on the variables (variables in certain rows/columns need only be ≥1, the rest >1), the following identity holds:

\