Affine Data and Affine Substitutions of Noncommutative Polynomials

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n,r\in\mathbb{N}$.

1. (Affine data)¶ An affine datum from $m$ to $n$ variables is a pair $T=(A,c)$ of maps $A:[n]\times[m]\to\mathbb{R}$ and $c:[n]\to\mathbb{R}$, whose values are written $A_{ij}$ and $c_{i}$.

2. (Affine substitution)¶ The tuple of $T$ is the $n$-tuple $a^{T}=(a^{T}_{1},\dots,a^{T}_{n})$ in $\mathcal{P}_{m}$ given by

the finite sum in the complex vector space $\mathcal{P}_{m}$. ¶The affine substitution of $T$ is the substitution $\sigma_{T}=\sigma_{a^{T}}:\mathcal{P}_{n}\to\mathcal{P}_{m}$.

3. (Norm)¶ The norm $\lVert T\rVert$ of $T$ is the nonnegative square root of the real number $\sum_{i=1}^{n}\sum_{j=1}^{m}A_{ij}^{2}\ge0$.

4. (Composites and identities)¶ If $S=(B,b)$ is an affine datum from $n$ to $r$ variables, the composite $S\circ T$ is the affine datum $(BA,\,Bc+b)$ from $m$ to $r$ variables given by

¶The identity datum $\mathrm{id}_{m}$ is the affine datum $(E,0)$ from $m$ to $m$ variables with $E_{ij}=1$ if $i=j$ and $E_{ij}=0$ if $i\neq j$, where $0$ denotes the map with value $0$.

5. (Coordinate data)¶ Let $d\in\mathbb{N}$ and write $2d=d+d$. The marginal data $\mathrm{pr}^{1}=(P^{1},0)$ and $\mathrm{pr}^{2}=(P^{2},0)$ and the difference datum $D=(P^{1}-P^{2},0)$ are the affine data from $2d$ to $d$ variables given, for $i\in[d]$ and $j\in[2d]$, by

The diagonal datum $\mathrm{diag}=(Q,0)$ is the affine datum from $d$ to $2d$ variables with $Q_{ij}=1$ if $i=j$ or $i=d+j$, and $Q_{ij}=0$ otherwise ($i\in[2d]$, $j\in[d]$). These data depend on $d$, which is fixed by the context.