From Cauchy's determinant formula to bosonic and fermionic immanant identities

Cauchy's determinant formula (1841) involving $\det ((1-u_i v_j)^{-1})$ is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius [J. reine angew. Math. 1882] involving a sum of two geometric series in $u_i v_j$. This theme also resurfaced in a matrix analysis setting in a paper by Horn [Trans. Amer. Math. Soc. 1969] - where the computations are attributed to Loewner - and in recent works by Belton-Guillot-Khare-Putinar [Adv. Math. 2016] and Khare-Tao [Amer. J. Math. 2021]. These formulas were recently unified and extended in [Trans. Amer. Math. Soc. 2022] to arbitrary power series, with commuting/bosonic variables $u_i, v_j$. In this note we formulate analogous permanent identities, and in fact, explain how all of these results are a special case of a more general identity, for any character - in fact, any complex class function - of any finite group that acts on the bosonic variables $u_i$ and on the $v_j$ via signed permutations. (We explain why larger linear groups do not work, via a - perhaps novel - "symmetric function" characterization of signed permutation matrices that holds over any integral domain.) We then provide fermionic analogues of these formulas, as well as of the closely related Cauchy product identities.