Evaluation of Word Polynomials by a Unitary Law

Let $\mathbb{N}$ be the natural numbers, $d\in\mathbb{N}$, $W_{2d}$ as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, and $\mathcal{L}_{d}$ the set of unitary laws of $d$-tuples. Let word polynomials, their support sets and the word polynomials $e_{w}$ be as in Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions, and let $\mathbb{C}$ be the field of complex numbers. Sums over a nonempty finite index set are those of Sum over a Finite Index Set.

(Evaluation)¶ For $\lambda\in\mathcal{L}_{d}$ and a word polynomial $X$ with support set $F$,

$$\lambda(X)=\sum_{w\in F}X(w)\,\lambda(w)\in\mathbb{C}.$$

It does not depend on $F$: for two support sets $F,G$ of $X$, both sums equal the sum over the finite set $F\cup G$ (finite by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §finite-union) by the second part of Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, since $X(w)\lambda(w)=0$ for $w$ in $F\cup G$ outside $F$, respectively outside $G$. In particular $\lambda(e_{w})=\lambda(w)$ for every $w\in W_{2d}$, using the support set $\{w\}$.