Canonical comparisons between standard graph tensor products

Setting. Keep $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$ and the standard graph bimodules $R_w$ of Standard graph bimodules, support filtrations and characters: for $w\in S_n$ the graded $(R,R)$-bimodule $R_w$ equals $R$ as a graded $\mathbb{Q}$-vector space with $f\cdot x=fx$ and $x\cdot g=w(g)x$, where $w(g)(\lambda )=g(w^{-1}\lambda )$, generated by the element $1$ of degree $0$. Recall the degree-zero isomorphism of graded $(R,R)$-bimodules ${\mu }_{w,v}:R_w{\otimes }_R{R}_v⟶R_{wv},a\otimes b⟼aw(b),$ for $w,v\in S_n$, with inverse $1\mapsto 1\otimes 1$ (the composition remark of that item, where the assignment is checked to be balanced, $R$-linear on both sides and surjective between free rank-one left $R$-modules). Here the internal shifts are those of Associative graded algebras, bimodules, and internal shifts: $\mu$ is homogeneous of degree zero.

Word comparisons. Let $t=(x_1,\dots ,x_r)$ and $u=(y_1,\dots ,y_s)$ be finite words in $S_n$ (the empty word allowed) with the same product $x_1\cdots x_r=y_1\cdots y_s=w\mathrm{.}$ Fix the left-to-right iteration of $\mu$ and write ${\mu }_t:R_{x_1}{\otimes }_R\cdots {\otimes }_R{R}_{x_r}⟶R_w,a_1\otimes \cdots \otimes a_r⟼a_1{x}_1(a_2)(x_1{x}_2)(a_3)\cdots (x_1\cdots x_{r-1})(a_r),$ the composite of the binary multiplications ${\mu }_{x_1,x_2},{\mu }_{x_1{x}_2,x_3},\dots$; for $r=0$ the source is the unit bimodule $R=R_e$ and ${\mu }_t={\mathrm{i}\mathrm{d}}_R$, and for $r=1$ it is ${\mathrm{i}\mathrm{d}}_{R_{x_1}}$. Each ${\mu }_t$ is a degree-zero isomorphism of graded $(R,R)$-bimodules with inverse $1\mapsto 1\otimes 1\otimes \cdots \otimes 1$. The canonical comparison between the two word tensors is $c_{t,u}:={\mu }_u^{-1}\circ {\mu }_t:R_{x_1}{\otimes }_R\cdots {\otimes }_R{R}_{x_r}⟶R_{y_1}{\otimes }_R\cdots {\otimes }_R{R}_{y_s}\mathrm{.}$

Basic properties. The comparisons form a transitive system: $c_{t,t}=\mathrm{i}\mathrm{d}$ for every word $t$, and for three words $t,u,v$ with the same product $c_{u,v}\circ c_{t,u}=c_{t,v}$, because ${\mu }_v^{-1}{\mu }_u{\mu }_u^{-1}{\mu }_t={\mu }_v^{-1}{\mu }_t$. In particular $c_{t,u}$ is inverse to $c_{u,t}$. When one of the words has length one the comparison is the corresponding display: for $r=1$, $t=(x_1)$, one has $c_{t,u}={\mu }_u^{-1}$; for $s=1$, $c_{t,u}={\mu }_t$. The identifications are associative in the sense that for a threefold product the two iterated comparisons built from the binary $\mu$'s coincide, this being the equality of the explicit formula above under any rebracketing.

Homogeneity of the Hom spaces. For $w,w^{\mathrm{\prime }}\in S_n$, ${\underset{}{\mathrm{Hom}}}_{R^e}(R_w,R_{w^{\mathrm{\prime }}})=\{\begin{array}{ll}\cong R,& w=w^{\mathrm{\prime }},\\ 0,& w\mathrm{\ne }{w}^{\mathrm{\prime }},\end{array}$ where the underlined Hom is the direct sum over all homogeneous internal degrees. For $w=w^{\mathrm{\prime }}$ the identification sends a homogeneous map $\phi$ to $\phi (1)\in R$ and the internal degree of $\phi$ is the degree of $\phi (1)$; hence the internal degree-zero part is one-dimensional over $\mathbb{Q}$, spanned by ${\mathrm{i}\mathrm{d}}_{R_w}$. The vanishing for $w\mathrm{\ne }{w}^{\mathrm{\prime }}$ is the statement that the supports $Gr(w)\mathrm{\ne }Gr(w^{\mathrm{\prime }})$ are distinct graphs, and the description for $w=w^{\mathrm{\prime }}$ is the map remark of Standard graph bimodules, support filtrations and characters.

Small cases. If $r=s=0$ then $w=e$ and $c_{t,u}={\mathrm{i}\mathrm{d}}_R$; if $t$ and $u$ are both words for $w$, the comparison $c_{t,u}$ is the isomorphism used to compare the two word models of the same standard bimodule $R_w$. No choice is made: ${\mu }_t$ is the unique degree-zero bimodule isomorphism with ${\mu }_t(1\otimes \cdots \otimes 1)=1$, and $c_{t,u}$ is determined by $t$ and $u$ alone.