A Newton Identity and Finite-Rank Reconstruction for the Queer Lie Superalgebra

Abstract

We establish a Newton-type identity for the queer Lie superalgebra $\mathfrak q_N$, relating Sergeev's odd cyclic central elements to Nazarov's one-row Capelli elements. The identity is obtained by comparing Ivanov's generating function for factorial Schur $Q$-functions with the queer Perelomov-Popov product of Grigoryev and Nazarov. Its coefficient expansion yields a triangular change of generators between the odd cyclic and odd one-row families. In particular, the odd one-row Capelli elements generate the center, while the even one-row elements are redundant. In fixed rank, we derive determinantal relations and a generic reconstruction theorem. The basic cyclic Hankel determinant is identified with a resultant and factored into the failure-of-strong-typicality and shifted-resonance factors. After localization at this determinant, the center is generated by the first $2N$ odd cyclic elements; consequently, generic central characters are determined by their values on these elements.