Proof of The Chain Rule for Free Difference Quotients under an Affine Substitution

Each result cited is universally quantified over the data in its own statement.

Throughout, $a=a^{T}=(a_{1},\dots,a_{n})$ is the tuple of $T$, so that $\sigma_{T}=\sigma_{a}$ by Affine Data and Affine Substitutions of Noncommutative Polynomials §substitution, and for $w\in W_{n}$ the polynomial $a_{w}\in\mathcal{P}_{m}$ is the product along $w$; in particular $a_{\varnothing}=1$ and $a_{(i)}=a_{i}$ for the one-letter word $(i)$, $i\in[n]$. For a word $v$ (in either alphabet) we write $|v|$ for its length, with $|\varnothing|=0$, and for $l\in[|v|]$ the words $v_{<l}$ and $v_{>l}$ are those of the preamble of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law, as is the symbol $\delta$. A sum over an empty range of indices is $0$. We fix $k\in[m]$ for the whole proof.

Step 1 (A two-sided difference quotient). For $f,g\in\mathcal{P}_{m}$ and $v\in W_{m}$ put

$$E(v;f,g)=\sum_{l=1}^{|v|}\delta_{v_{l}k}\,\gamma\bigl(f\,x_{v_{<l}}\bigr)\,\gamma\bigl(x_{v_{>l}}\,g\bigr)\in\mathbb{C},$$

which is $0$ when $v=\varnothing$. By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, part (a), for fixed $f,g$ there is exactly one linear map $D(\cdot\,;f,g):\mathcal{P}_{m}\to\mathbb{C}$ with $D(x_{v};f,g)=E(v;f,g)$ for every $v\in W_{m}$, and it is given by $D(0;f,g)=0$ and $D(q;f,g)=\sum_{v\in\operatorname{supp}q}q(v)\,E(v;f,g)$ for $q\neq0$. We record three properties.

(i) By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, $1x_{u}=x_{u}=x_{u}1$ for every $u\in W_{m}$, so for $|v|\ge1$ the number $E(v;1,1)$ is the value at $x_{v}$ prescribed for $\partial^{\gamma}_{k}$ in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient (with $d=m$, $\lambda=\gamma$, $j=k$), while $E(\varnothing;1,1)=0=\partial^{\gamma}_{k}(1)$ and $1=x_{\varnothing}$. Thus $D(\cdot\,;1,1)$ and $\partial^{\gamma}_{k}$ are linear maps that agree on every monomial, and by the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, part (a), $\partial^{\gamma}_{k}(q)=D(q;1,1)$ for every $q\in\mathcal{P}_{m}$.

(ii) For fixed $q\in\mathcal{P}_{m}$ the number $D(q;f,g)$ is linear in $f$ for fixed $g$ and linear in $g$ for fixed $f$: each $E(v;f,g)$ is, because the product of $\mathcal{P}_{m}$ is bilinear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and $\gamma$ is linear, and $D(q;f,g)$ is either $0$ or a linear combination, with coefficients $q(v)$ not depending on $f,g$, of the numbers $E(v;f,g)$.

(iii) $D(1;f,g)=E(\varnothing;f,g)=0$. For $l'\in[m]$ the variable $x_{l'}$ is the monomial of the one-letter word $(l')$, whose only position is $l=1$ with $(l')_{<1}=(l')_{>1}=\varnothing$; hence, using $f1=f$ and $1g=g$, $D(x_{l'};f,g)=\delta_{l'k}\,\gamma(f)\,\gamma(g)$.

Step 2 (Leibniz rule). We claim that for all $q,r,f,g\in\mathcal{P}_{m}$

$$D(qr;f,g)=D(q;f,rg)+D(r;fq,g).\qquad\text{(L)}$$

First let $q=x_{u}$ and $r=x_{v}$ with $u,v\in W_{m}$. Then $qr=x_{uv}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. The word $uv$ has length $|u|+|v|$, and its positions split into two ranges: for $l\in[|u|]$ one has $(uv)_{l}=u_{l}$, $(uv)_{<l}=u_{<l}$ and $(uv)_{>l}=u_{>l}v$; for $l=|u|+l'$ with $l'\in[|v|]$ one has $(uv)_{l}=v_{l'}$, $(uv)_{<l}=u\,v_{<l'}$ and $(uv)_{>l}=v_{>l'}$ (either range is empty when $u$, respectively $v$, is empty). By Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and the associativity in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, $x_{u_{>l}v}\,g=x_{u_{>l}}\,(x_{v}g)$ and $f\,x_{u v_{<l'}}=(f x_{u})\,x_{v_{<l'}}$. Splitting the sum defining $E(uv;f,g)$ along the two ranges therefore gives

$$E(uv;f,g)=E(u;f,x_{v}g)+E(v;fx_{u},g),$$

which is (L) for $q=x_{u}$, $r=x_{v}$. Now fix $f,g$ and $v\in W_{m}$, and put $r=x_{v}$. Both sides of (L) are linear in $q$: the left side because $q\mapsto qx_{v}$ is linear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra and $D(\cdot\,;f,g)$ is linear; the right side because $D(\cdot\,;f,x_{v}g)$ is linear and $q\mapsto D(x_{v};fq,g)$ is linear by (ii) and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra. They agree on monomials $q$, so they agree for every $q\in\mathcal{P}_{m}$ by the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, part (a). Finally fix an arbitrary $q\in\mathcal{P}_{m}$. Both sides of (L) are linear in $r$: the left side as before; on the right, $r\mapsto D(q;f,rg)$ is linear by (ii) and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, and $r\mapsto D(r;fq,g)$ is linear by definition. They agree on every monomial $r=x_{v}$ by what was just shown, so (L) holds for all $r$ by the same uniqueness.

Step 3 (The entries of the affine tuple). Let $i\in[n]$. By Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, $a_{i}=c_{i}1+\sum_{l'=1}^{m}A_{il'}x_{l'}$, so by linearity of $D(\cdot\,;f,g)$ and (iii), for all $f,g\in\mathcal{P}_{m}$,

$$D(a_{i};f,g)=c_{i}\cdot0+\sum_{l'=1}^{m}A_{il'}\,\delta_{l'k}\,\gamma(f)\,\gamma(g)=A_{ik}\,\gamma(f)\,\gamma(g).\qquad\text{(A)}$$

Step 4 (Products along words). We prove by induction on $L\in\{0\}\cup\mathbb{N}$ the statement $S(L)$: for every $w\in W_{n}$ of length $L$ and all $f,g\in\mathcal{P}_{m}$,

$$D(a_{w};f,g)=\sum_{l=1}^{L}A_{w_{l}k}\,\gamma\bigl(f\,a_{w_{<l}}\bigr)\,\gamma\bigl(a_{w_{>l}}\,g\bigr).\qquad\text{(W)}$$

For $L=0$ we have $w=\varnothing$, $a_{\varnothing}=1$, and $D(1;f,g)=0$ by (iii), which is the empty sum. Assume $S(L)$ and let $w\in W_{n}$ have length $L+1$. Let $u=(w_{1},\dots,w_{L})$ (so $u=\varnothing$ if $L=0$) and $i=w_{L+1}$, so that $w=u(i)$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $a_{w}=a_{u}a_{(i)}=a_{u}a_{i}$. By (L) with $q=a_{u}$ and $r=a_{i}$,

$$D(a_{w};f,g)=D(a_{u};f,a_{i}g)+D(a_{i};fa_{u},g).$$

By $S(L)$ applied to $u$ with $a_{i}g$ in place of $g$, the first term equals $\sum_{l=1}^{L}A_{u_{l}k}\,\gamma(f a_{u_{<l}})\,\gamma(a_{u_{>l}}a_{i}g)$, and by (A) the second term equals $A_{ik}\,\gamma(fa_{u})\,\gamma(g)$. For $l\in[L]$ we have $w_{l}=u_{l}$, $w_{<l}=u_{<l}$ and $w_{>l}=u_{>l}(i)$, so $a_{w_{>l}}=a_{u_{>l}}a_{i}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values and $a_{u_{>l}}a_{i}g=a_{w_{>l}}g$ by associativity; for $l=L+1$ we have $w_{l}=i$, $w_{<l}=u$, $w_{>l}=\varnothing$ and $a_{\varnothing}g=g$. Hence the two terms add up to the right side of (W) for $w$, which proves $S(L+1)$.

Step 5 (Conclusion). For every $u\in W_{n}$, $\sigma_{T}(x_{u})=a_{u}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, so $\gamma(a_{u})=\gamma(\sigma_{T}(x_{u}))=\mu(x_{u})$. Let $w\in W_{n}$. By (i), $\sigma_{T}(x_{w})=a_{w}$, (W) with $f=g=1$, and $1a_{u}=a_{u}=a_{u}1$,

$$\partial^{\gamma}_{k}\bigl(\sigma_{T}(x_{w})\bigr)=D(a_{w};1,1)=\sum_{l=1}^{|w|}A_{w_{l}k}\,\mu(x_{w_{<l}})\,\mu(x_{w_{>l}})=\sum_{i=1}^{n}A_{ik}\sum_{l=1}^{|w|}\delta_{w_{l}i}\,\mu(x_{w_{<l}})\,\mu(x_{w_{>l}}),$$

the last step because, for each $l$, the only index $i\in[n]$ with $\delta_{w_{l}i}\neq0$ is $i=w_{l}$. If $|w|\ge1$, the inner sum is $\partial^{\mu}_{i}(x_{w})$ by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient (with $d=n$ and $\lambda=\mu\in\Sigma_{n}$); if $w=\varnothing$, both sides are $0$, since the sum over $l$ is empty and $\partial^{\mu}_{i}(1)=0$. Thus the identity of the lemma holds for $p=x_{w}$, for every $w\in W_{n}$. Both $p\mapsto\partial^{\gamma}_{k}(\sigma_{T}(p))$ and $p\mapsto\sum_{i=1}^{n}A_{ik}\,\partial^{\mu}_{i}(p)$ are linear maps $\mathcal{P}_{n}\to\mathbb{C}$ (a composite of linear maps, and a linear combination of linear maps), and they agree on every monomial; by the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension, part (a), they are equal, which is the asserted identity for every $p\in\mathcal{P}_{n}$.