Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words

Let $\mathbb{N}$ be the set of natural numbers and let $d\in\mathbb{N}$. Let $W_{2d}$, the inverse letters, the lengths $|\cdot|$, the adjoint words $w\mapsto w^{*}$, the reduced words and the set $W^{\circ}_{d}$ of cyclically reduced words be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words and Reduced and Cyclically Reduced Words in Unitary Letters, 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 $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples. Let $\mathbb{C}$ be the field of complex numbers with conjugation $z\mapsto\overline{z}$ and modulus $|z|$.

Then the following hold.

1. (Adjoints)¶ For every $\lambda\in\mathcal{L}_{d}$ and every $w\in W_{2d}$, $\lambda(w^{*})=\overline{\lambda(w)}$.

2. (Bound)¶ For every $\lambda\in\mathcal{L}_{d}$ and every $w\in W_{2d}$, $|\lambda(w)|\le1$.

3. (Stripping a reduced word)¶ Let $z\in W_{2d}$ be reduced. Then there are $b\in W_{2d}$ and $u\in W^{\circ}_{d}$ with $z=b\,u\,b^{*}$ and $|z|=2|b|+|u|$.

4. (Conjugation invariance)¶ For all $b,u\in W_{2d}$ and every $\mu\in\mathcal{L}_{d}$, $\mu(b\,u\,b^{*})=\mu(u)$. In particular, whenever a reduced $z\in W_{2d}$ satisfies $z=b\,u\,b^{*}$ with $b\in W_{2d}$ and $u\in W^{\circ}_{d}$, then $\mu(z)=\mu(u)$ for every $\mu\in\mathcal{L}_{d}$.

5. (Reduction)¶ For every $w\in W_{2d}$ there is $u\in W^{\circ}_{d}$ with $|u|\le|w|$ and $\mu(w)=\mu(u)$ for every $\mu\in\mathcal{L}_{d}$.

6. (Determination by cyclically reduced words)¶ If $\mu,\nu\in\mathcal{L}_{d}$ satisfy $\mu(u)=\nu(u)$ for every $u\in W^{\circ}_{d}$, then $\mu=\nu$.

7. (Nonemptiness)¶ The map $W_{2d}\to\mathbb{C}$ with constant value $1$ belongs to $\mathcal{L}_{d}$; in particular $\mathcal{L}_{d}$ is nonempty.