User:AndrewReeves/Cellular algebra (draft article)

In algebraic representation theory, a cellular algebra is an algebra which has a basis with certain combinatorial properties. The existence of such a basis leads to a natural characterisation of the simple modules of the algebra. Cellular algebras are often quasi-hereditary, though this is not always the case.

The first definition of a cellular algebra was given in 1996 by J.J. Graham and G. I. Lehrer. Later, S. König and C.C. Xi gave an alternative (but equivalent) definition. In different applications, either one of the two definitions may turn out to be the more useful; the original definition is more combinatorial, the second is stated more in terms of abstract ring theory.

Examples of cellular algebras include the group algebra of the symmetric group, the Brauer algebra, all Hecke algebras of finite type, the Temperley-Lieb algebra and the Birman-Murakami-Wenzl algebra.

Definition
The original definition given by Graham and Lehrer is as follow:.

Let $$R$$ be a commutative ring and let $$A$$ be an $R$-algebra free over $$R$$. $$A$$ is cellular if there exists a (not necessarily unique) cell datum consisting of such that the image of $$C$$ is an $$R$$-basis $$\{ C_{S,T}^t \}$$ of $$A$$, the map $$C_{S,T}^t \mapsto C_{T,S}^t$$ extends $$R$$-linearly to an antiautomorphism of $$A$$ and for any $$a \in A$$ and every $$S,S' \in M(t)$$ there exist $$r_a(S',S) \in R$$ such that $$a C_{S,T}^t = \sum_{S' \in M(t)} r_a(S',S) C_{S',T}^t + r'$$ where $$r'$$ denotes a linear combination of basis elements $$C_{S,T}^{t'}$$ with $$t' < t$$
 * A poset $$\mathcal{T}$$;
 * For each $$t \in \mathcal{T}$$ a set $$M(t)$$;
 * An injection $$C : \bigsqcup_{t \in T} M(t) \times M(t) \rightarrow A$$;

Cell Modules
Suppose that $$A$$ is a cellular $$R$$-algebra with cell datum ($$\mathcal{T}$$,$$M(t)$$,$$C$$). Using the same notation as above, we can define for any $$t \in \mathcal{T}$$ a cell module $$W(t)$$. This is the module with $$R$$-basis $$\{ C_S | S \in M(t)\}$$ and algebra action defined by $$a \circ C_S = \sum_{S' \in M(t)} r_a(S',S) C_{S'}$$ for all $$a \in A$$ and $$S \in M(t)$$

For any cell module $$W(t)$$ there is an $R$-bilinear form $$\phi_t : W(t) \times W(t) \rightarrow R$$ given by $$\phi_t(C_S,C_T) = r_{C_{S,T}^t}(S,S)$$. Graham and Lehrer showed that this form is symmetric and invariant under the action of $$A$$. Based on this they also showed how to extract all the simple modules of $$A$$ from the cell modules.

Simple Modules
Suppose that $$R$$ is a field and that $$\mathcal{T}$$ is finite (which implies that $$A$$ is a finite dimensional algebra).

Define the radical $$\mathfrak{R}(t) := \{ x \in W(t) \; | \; \phi_t(x,y) = 0 \quad \forall y \in W(t) \}$$. This is a submodule of the cell module $$W(t)$$, so the quotient $$L(t) := W(t)/\mathfrak{R}(t)$$ is well-defined. It is trivial only when $$\phi(t)$$ is identically zero.

Let $$\mathcal{T}^0$$ be the set of all $$t \in \mathcal{T}$$ such that $$\phi_t$$ is not identically zero. Then a theorem of Graham and Lehrer states that
 * $$L(t)$$ is absolutely irreducible for any $$t \in \mathcal{T}^0$$;
 * $$\{L(t) \; | \; t \in \mathcal{T}^0 \}$$ is a complete set of representatives of the distinct isomorphism classes of simple $$A$$-modules;
 * The cell module $$W(t)$$ is simple if and only if $$\phi_t$$ is non-degenerate on $$W(t)$$;
 * The following statements are equivalent:
 * $$A$$ is a semisimple algebra;
 * Every cell module of $$A$$ is absolutely irreducible;
 * The forms $$\phi_t$$ are non-degenerate for every $$t \in \mathcal{T}$$;