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