How the norm one positive definite functions determine a finite group. (English) Zbl 1486.43005
Summary: We study a finite (discrete) group \(G\) through the information we obtain from
\[
P_1(G)=\{\langle\pi(\cdot)\xi,\xi\rangle: \ \pi\colon G\to\mathcal{U}(H)\text{ is unitary}, \ \xi\in H, \ \|\xi\|=1\}
\]
of norm one positive definite functions of \(G\), arising from matrix coefficients of any unitary representation \(\pi\). Below we summarize our results.
- (a)
- Knowing \(P_1(G)\) as a set of functions, when \(G\) is a finite abelian group we can determine \(G\cong\prod_j \mathbb{Z}_{p_j^{r_j}}\) as a direct product of its cyclic subgroups of prime power orders.
- (b)
- Knowing \(P_1(G)\) 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 \(P_1( G')\cong P_1(G)\) as multiplicative semigroups, we show that \(G'\cong G\) as groups.
- (c)
- Knowing \(P_1(G)\) as a compact convex set, we can construct the group von Neumann algebra \(\operatorname{vN}(G)\) of \(G\). Consequently, when \(G^\prime\) is another finite group with \(P_1(G) \cong P_1( G')\) as convex sets, we show that \(\mathrm{vN}(G)\cong \mathrm{vN}( G')\) as von Neumann algebras. In particular, we can tell if \(G\) is abelian.
MSC:
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 43A70 | Analysis on specific locally compact and other abelian groups |
| 20D99 | Abstract finite groups |
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