Cauchy identity

Well known form of Cauchy identity is

(1)

where is a Young diagram, are Schur functions and are cycle type permutation centralizer sizes. It can be found even on wiki. Classic references are [FH91; Mac95].

Maybe in the references above the following identity

(2)

is missing, but you can easily recognize in this statement exponent of an infinite sum expansion. Also you can find it in eq. (7.20) of [Sta23].

References

[Mac95]

I. G. Macdonald, Symmetric functions and hall polynomials, 2nd ed. Oxford University Press, USA, 1995.

[FH91]

W. Fulton and J. Harris, Representation theory: A first course. Springer, 1991.

[Sta23]

R. P. Stanley, Enumerative combinatorics 2, Second. Cambridge University Press, 2023.