Plethystic exponential calculus and characteristic polynomials of permutations
Abstract
We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of symmetric products of real/algebraic tori. The proof uses formal power series expansions of plethystic exponentials, and has been motivated by some recent applications of these combinatorial tools in supersymmetric gauge and string theories. Since the methods are elementary, we tried to be self-contained, and relate to other topics such as the q-binoomial theorem, and the cycle index and Molien series for the symmetric group.
Keywords
Cite
@article{arxiv.2105.13049,
title = {Plethystic exponential calculus and characteristic polynomials of permutations},
author = {Carlos A. A. Florentino},
journal= {arXiv preprint arXiv:2105.13049},
year = {2021}
}
Comments
Some statements and treatment simplified; relation with cycle index better explored. New examples and citations added