Character polynomials for two rows and hook partitions

2018/12/21 by Ashraf, Ahmed Umer
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.09377

Abstract

Representation theory of the symmetric group \mathfrakS_n has a very distinctive combinatorial flavor. The conjugacy classes as well as the irreducible characters are indexed by integer partitions λ\vdash n. We introduce class functions on \mathfrakS_n that count the number of certain tilings of Young diagrams. The counting interpretation gives a uniform expression of these class functions in the ring of character polynomials, as defined by \citemurnaghanfirst. A modern treatment of character polynomials is given in \citeorellana-zabrocki. We prove a relation between these combinatorial class functions in the (virtual) character ring. From this relation, we were able to prove Goupil's generating function identity \citegoupil, which can then be used to derive Rosas' formula \citerosas for Kronecker coefficients of hook shape partitions and two row partitions.

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