How the norm one positive definite functions determine a finite group

Abstract

We study a finite (discrete) group G through the information we obtain fromP1(G)={〈π(⋅)ξ,ξ〉:π:G→U(H)is unitary,ξ∈H,‖ξ‖=1} of norm one positive definite functions of G, arising from matrix coefficients of any unitary representation π. Below we summarize our results.(a)Knowing P1(G) as a set of functions, when G is a finite abelian group we can determine G≅∏jZpjrj as a direct product of its cyclic subgroups of prime power orders.(b)Knowing P1(G) as a multiplicative semigroup, we can construct the subgroup lattice L(G) of G. With L(G) in stock, we can tell if G is cyclic, simple, perfect, solvable, supersolvable, or nilpotent. When G′ is a finite simple group with P1(G′)≅P1(G) as multiplicative semigroups, we show that G′≅G as groups.(c)Knowing P1(G) as a compact convex set, we can construct the group von Neumann algebra vN(G) of G. Consequently, when G′ is another finite group with P1(G)≅P1(G′) as convex sets, we show that vN(G)≅vN(G′) as von Neumann algebras. In particular, we can tell if G is abelian.