On the $ω$-circular $*$-distribution for complex unimodular $ω$

Source abstract

We consider a notion of $ω$-circular $*$-distribution $μ_ω$, defined in terms of a parameter $ω\in \mathbb{C}$ with $| ω|=1$. This $*$-distribution appears as limit law in an algebraic central limit theorem (CLT) for $ω$-commuting unitaries, and interpolates between the Fermionic setting (at $ω= -1$) and the Bosonic one (at $ω= 1$). On a combinatorial level, $μ_ω$ is described in terms of pair-partitions of sets $\{1, ... ,2n\}$, where one has a total order among the $n$ pairs of the partition and where every pair is assigned a direction, to the effect that one can define the sign of a crossing between two pairs. The main goal of the paper is to study a significant subspace, with "Fock space" flavour, of the $L^2$-space associated to a natural multi-variate version of $μ_ω$. The inner product on this subspace is defined by a sesquilinear form $\langle \cdot \, , \, \cdot \rangle_ω$ on non-commutative polynomials. We explicitly describe $\langle \cdot \, , \, \cdot \rangle_ω$ in terms of some tractable non-negative definite functions on symmetric groups, and we prove that it is non-degenerate if and only if $ω$ is not a root of unity.