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

Abstract

Cauchy’s determinant formula (1841) involving det((1−uivj)−1) is a fundamental result in symmetric function theory. It has been extended in several directions, including a determinantal extension by Frobenius (1882) involving a sum of two geometric series in uivj. This theme also resurfaced in a matrix analysis setting in a paper by Horn (1969) – where the computations are attributed to Loewner – and in recent works by Belton et al. (2016) and Khare and Tao (2021). These formulas were recently unified and extended in Khare (2022) to arbitrary power series, with commuting/bosonic variables ui,vj.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 ui and on the vj 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.