Laws of d-Tuples of Unitaries

Let $\mathbb{N}$ be the set of natural numbers and let $d\in\mathbb{N}$, and let $[2d]$ be the initial segment determined by $2d=d+d$. Let $W_{2d}$, the letters $l\in[2d]$, their inverse letters $l\mapsto l^{-1}$ and the adjoint words $w\mapsto w^{*}$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, with concatenation written $(u,v)\mapsto uv$; products of several words are unambiguous by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid. Let $\mathbb{C}$ be the field of complex numbers with conjugation $z\mapsto\overline{z}$.

1. (Unitary laws)¶ A unitary law of $d$-tuples is a map $\lambda:W_{2d}\to\mathbb{C}$ with the following four properties.

¶(a) $\lambda(\varnothing)=1$.

¶(b) $\lambda(u\,l\,l^{-1}\,v)=\lambda(uv)$ for all $u,v\in W_{2d}$ and every letter $l\in[2d]$.

¶(c) $\lambda(uv)=\lambda(vu)$ for all $u,v\in W_{2d}$.

¶(d) For every nonempty finite set $F\subseteq W_{2d}$, the map $F\times F\to\mathbb{C}$, $(v,v')\mapsto\lambda(v^{*}v')$, is a positive semidefinite kernel on $F$.

2. (The space of unitary laws)¶ $\mathcal{L}_{d}$ denotes the set of unitary laws of $d$-tuples.