Cellular algebra

Definition

Let $R$ be a commutative ring with identity and let $A$ be an associative algebra over $R$. A cell datum for $A$ consists of four pieces of data. They are a finite partially ordered set $\Lambda$, a finite set $M(\lambda)$ for each $\lambda \in \Lambda$, and an $R$-linear map $* \colon A \to A$. The fourth is an injective map $(\lambda, S, T) \mapsto C^{\lambda}_{S,T} \in A$, defined for $\lambda \in \Lambda$ and $S, T \in M(\lambda)$. The data must satisfy:

1. The elements $C^{\lambda}_{S,T}$, for $\lambda \in \Lambda$ and $S, T \in M(\lambda)$, form an $R$-basis of $A$.
2. The map $*$ is an anti-involution, that is $(ab)^* = b^* a^*$ and $(a^*)^* = a$ for all $a, b \in A$, and $(C^{\lambda}_{S,T})^* = C^{\lambda}_{T,S}$.
3. For all $a \in A$, $\lambda \in \Lambda$ and $S, T \in M(\lambda)$, $$a\, C^{\lambda}_{S,T} \equiv \sum_{S' \in M(\lambda)} r_a(S', S)\, C^{\lambda}_{S',T} \pmod{A(< \lambda)},$$ where $r_a(S', S) \in R$ does not depend on $T$, and $A(< \lambda)$ is the $R$-span of the $C^{\mu}_{U,V}$ with $\mu < \lambda$.

The algebra $A$ is cellular if it has a cell datum, and the basis in clause 1 is then a cellular basis.