対称群上の帯球関数とリース行列式 (表現論と非可換調和解析をめぐる諸問題)

Abstract

A function on the symmetric group mathfrak{S}_{n} is called here a zonal spherical function (with respect to mathfrak{S}{$mu$}) if it is biinvariant with respect to the subgroup mathfrak{S}{$mu$}, the Young subgroup for a partition $mu$vdash n. In the article, we show that the values of such functions are expressed in terms of a certain matrix function called the wreath determinant, which is relative invariant with respect to the left translation, when the Young diagram of $mu$ is rectangular. Further, as an application, we give a simple alternative proof of the Alon-Tarsi conjecture on Latin squares for a certain known specific case.