The Free Unitary Heat Generator on Unitary Laws

Let $\mathbb{N}$ be the set of natural numbers with its addition and order, carried into the real numbers $\mathbb{R}$ by the natural-number image, and for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$. Let $d\in\mathbb{N}$, let $W_{2d}$, the generators $g(l)$ and signs $\varepsilon(l)$ of letters be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words, with concatenation written $(u,v)\mapsto uv$, let $\mathcal{L}_{d}$ be the set of unitary laws of $d$-tuples, and let $\mathbb{C}\supseteq\mathbb{R}$ be the field of complex numbers. Finite sums are those of Sum over a Finite Index Set.

For natural numbers $a\le b$, $b-a+1$ denotes the natural number $m$ with $a+m=b+1$; for a natural number $q$ with $1<q$, $q-1$ denotes the natural number $m$ with $m+1=q$. Both exist and are unique by claim 7 of Properties of the Order on the Natural Numbers and claim 4 of Arithmetic of Addition on the Natural Numbers, because $a<b+1$ by claims 5 and 1 of Properties of the Order on the Natural Numbers and claim 1 of Arithmetic of Addition on the Natural Numbers.

Let $w\in W_{2d}$ be a word of length $k\in\mathbb{N}$. For $a,b\in[k]$ with $a\le b$, $w_{[a,b]}$ denotes the word of length $b-a+1$ whose $i$-th letter is $w_{a+i-1}$, for $i\in[b-a+1]$; for $a,b\in[k]$ with $b<a$ we put $w_{[a,b]}=\varnothing$.

1. (Pairs, start and end positions)¶ $P(w)$ is the set of pairs $(p,q)\in[k]\times[k]$ with $p<q$ and $g(w_{p})=g(w_{q})$; it is finite by claim 3 of Basic Properties of Finite Sets, as a subset of the set $[k]\times[k]$, which is finite by claim 1 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets, $[k]$ being finite by claim 1 of Basic Properties of Finite Sets. For $p\in[k]$ put $\sigma_{p}=p+1$ if $\varepsilon(w_{p})=1$ and $\sigma_{p}=p$ if $\varepsilon(w_{p})=-1$; for $q\in[k]$ with $1<q$ put $\tau_{q}=q$ if $\varepsilon(w_{q})=1$ and $\tau_{q}=q-1$ if $\varepsilon(w_{q})=-1$. For $(p,q)\in P(w)$ one has $1\le p<q$, so $1<q$ and $\tau_{q}$ is defined, and

$$\sigma_{p}\le p+1\le q\le\tau_{q}+1,\qquad\tau_{q}\le q\le k,$$

so $\sigma_{p},\tau_{q}\in[k]$.

2. (The two pieces of a pair)¶ For $(p,q)\in P(w)$ the inner piece is $y_{pq}(w)=w_{[\sigma_{p},\tau_{q}]}$, and the outer piece is the concatenation

$$x_{pq}(w)=w_{[\tau_{q}+1,k]}\;w_{[1,\sigma_{p}-1]},$$

where the first factor is read as $\varnothing$ if $\tau_{q}=k$ and the second as $\varnothing$ if $\sigma_{p}=1$; otherwise $\tau_{q}<k$, so that $\tau_{q}+1\in[k]$, and $1<\sigma_{p}$, so that $\sigma_{p}-1\in[k]$ is defined as above.

3. (The free unitary heat generator)¶ For $\lambda\in\mathcal{L}_{d}$, $\Theta\lambda:W_{2d}\to\mathbb{C}$ is the map with $\Theta\lambda(\varnothing)=0$ and, for every word $w$ of length $k\in\mathbb{N}$,

$$\Theta\lambda(w)=-\frac{k}{2}\,\lambda(w)-\sum_{(p,q)\in P(w)}\varepsilon(w_{p})\,\varepsilon(w_{q})\;\lambda\bigl(y_{pq}(w)\bigr)\,\lambda\bigl(x_{pq}(w)\bigr),$$

where the sum is read as $0$ if $P(w)$ is empty. The map $\lambda\mapsto\Theta\lambda$ is the free unitary heat generator.

4. (Restriction to cyclically reduced words)¶ For $\lambda\in\mathcal{L}_{d}$, $\widehat{\Theta}\lambda:W^{\circ}_{d}\to\mathbb{C}$ is the restriction of $\Theta\lambda$ to the set $W^{\circ}_{d}$ of cyclically reduced words.