Finite de Finetti theorems for free easy quantum groups

We prove various finite de Finetti theorems for non-commutative distributions which are invariant under the free easy quantum group actions. This complements the free de Finetti theorems by Banica, Curran and Speicher, which mostly focus on infinite sequences. We also discuss some refined results for the infinite setting.

💡 Research Summary

This paper establishes finite de Finetti theorems for non‑commutative distributions that are invariant under the actions of free easy quantum groups (FEQGs). While earlier work by Banica, Curran, and Speicher focused on infinite sequences, the authors fill the gap by providing a complete characterization for finite families of random variables.

The authors begin by recalling the combinatorial description of FEQGs via categories of non‑crossing partitions C. For each m > 0 they denote C(m)=C(m,0). They then review free probability basics: a *‑probability space (A,φ), free cumulants κ_m, the moment‑cumulant relation, and the Möbius inversion on the lattice of partitions. The quantum group G_n is introduced as a compact matrix quantum group with fundamental corepresentation u, together with its coaction α_n on A and the Haar state h. The Haar moments are expressed through the Weingarten matrix Wg_{C,m,n}, which is the inverse of the Gram matrix G_{C,m,n}(π,σ)=n^{#(π∨σ)}.

The central result (Theorem 1) states that for a family (x_1,…,x_n) in a *‑probability space, the following are equivalent:

(i) G_n‑invariance: (φ⊗id)∘α_n(x)=φ(x)·1 for all x∈A.

(ii) There exists a unital sub‑algebra B⊂A and a φ‑preserving conditional expectation E:A→B such that for every m and every multi‑index i∈