Cellular bases and cell modules

Let $R$ be a commutative ring with $1$ and $A$ a unital $R$-algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms) that is free as an $R$-module on a finite set $C=\{c_{st}^{\lambda }:\lambda \in \mathrm{\Lambda },s,t\in T(\lambda )\}$, where $(\mathrm{\Lambda },⊵)$ is a finite poset (Partial order and partially ordered set; its strict order is written $⊳$) and each $T(\lambda )$ is a finite set. For $\lambda \in \mathrm{\Lambda }$ let $A^{⊳\lambda }$ be the $R$-submodule of $A$ spanned by all $c_{uv}^{\mu }$ with $\mu ⊳\lambda$.

Then $(C,\mathrm{\Lambda })$ is a cellular basis of $A$, and $A$ is a cellular algebra, if:

(i) the $R$-linear map $i:A\to A$ with $i(c_{st}^{\lambda })=c_{ts}^{\lambda }$ for all $\lambda ,s,t$ is an algebra anti-automorphism of $A$, i.e. $i(ab)=i(b)i(a)$ and $i(1)=1$ (equivalently $i$ is an isomorphism of $A$ with the opposite algebra; the displayed formula forces $i^2=\mathrm{i}\mathrm{d}$); and

(ii) for every $\lambda \in \mathrm{\Lambda }$, every $t\in T(\lambda )$ and every $a\in A$ there exist coefficients $r_v\in R$ ($v\in T(\lambda )$), independent of $s$, such that for every $s\in T(\lambda )$ $c_{st}^{\lambda }a\equiv {\sum }_{v\in T(\lambda )}{r}_v{c}_{sv}^{\lambda }(\mathrm{m}\mathrm{o}\mathrm{d}{A}^{⊳\lambda })\mathrm{.}$

Cell modules. For $\lambda \in \mathrm{\Lambda }$ let $C^{\lambda }$ be the free $R$-module with basis $\{c_t^{\lambda }:=c_{tt}^{\lambda }:t\in T(\lambda )\}$; condition (ii) makes $C^{\lambda }$ a right $A$-module by $c_t^{\lambda }a:={\sum }_v{r}_v{c}_v^{\lambda }$ when $c_{tt}^{\lambda }a\equiv {\sum }_v{r}_v{c}_{tv}^{\lambda }(\mathrm{m}\mathrm{o}\mathrm{d}{A}^{⊳\lambda })$, and it makes $C^{\lambda }$ a left $A$-module via $i$, the left action being $a\cdot c_t^{\lambda }:=c_t^{\lambda }i(a)$. Both actions are well defined on the freely spanning basis elements by (ii): the coefficients $r_v$ are unique because $C$ is an $R$-basis of $A$ and the quotient by $A^{⊳\lambda }$ kills exactly the basis elements indexed by strictly larger labels, while applying the anti-automorphism (i) to (ii) gives the same triangularity for left multiplication. The module $C^{\lambda }$ is the (right) cell module of $\lambda$, and $A^{⊵\lambda }\mathrm{/}{A}^{⊳\lambda }\cong C^{\lambda }{\otimes }_R{C}^{\lambda }$ via $c_{st}^{\lambda }+A^{⊳\lambda }\mapsto c_s^{\lambda }\otimes c_t^{\lambda }$, where $A^{⊵\lambda }$ is spanned by all $c_{uv}^{\mu }$ with $\mu ⊵\lambda$. A cellular algebra may have several cellular bases.