Jack multiple zeta function

Source abstract

For conjugate shapes $λ$ and $λ'$, Stanley's duality relates the Jack symmetric functions $J_λ(x;α)$ and $J_{λ'}(x;1/α)$. Through Stanley's combinatorial formula, it is an identity between the sums of the weights of the semi-standard Young tableaux giving the same monomial. We introduce the Jack multiple zeta functions, which keep these weights separate, and prove that Stanley's duality lifts to them, including the case of skew shapes. Thus the duality holds before the weights are summed up, and in fact step by step along chains of partitions. Since the duality is an identity among formal symbols, the same duality holds when the multiple zeta functions are replaced by other functions, and replacing them with the monomial quasi-symmetric functions recovers Stanley's duality. Hence our duality is a refinement of Stanley's duality which contains it as a special case. In particular, we obtain a duality for the Schur multiple zeta functions and the classical relation between the multiple zeta and zeta-star functions. We next show that any expansion of a Jack symmetric function in terms of Schur functions can be lifted to Jack multiple zeta functions, yielding an analog of a formula conjectured by Stanley and proved by Koike. We also show that the Jack multiple zeta functions of one-row shape give a one-parameter family interpolating the multiple zeta and zeta-star functions, and we discuss their values at $\{2\}^r$ and at $\{2\}^a,3,\{2\}^b$.