Cellular algebra

In abstract algebra, a cellular algebra is a finite-dimensional associative algebra "A" with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of "A".
History.
The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer. However, the terminology had previously been used by Weisfeiler and Lehman in the Soviet Union in the 1960s, to describe what are also known as association schemes.
Definitions.
Let formula_1 be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also formula_2 be a formula_1-algebra.
The concrete definition.
A "cell datum" for formula_2 is a tuple formula_5 consisting of
and satisfying the following conditions:
This definition was originally given by Graham and Lehrer who invented cellular algebras.
The more abstract definition.
Let formula_8 be an anti automorphism of formula_1-algebras with formula_34 (just called "involution" from now on).
A "cell ideal" of formula_2 w.r.t. formula_36 is a two-sided ideal formula_37 such that the following conditions hold:
A "cell chain" for formula_2 w.r.t. formula_36 is defined as a direct decomposition
into free formula_1-submodules such that
Now formula_56 is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent. Every basis gives rise to cell chains (one for each topological ordering of formula_6) and choosing a basis of every left ideal formula_58 one can construct a corresponding cell basis for formula_2.
Examples.
Polynomial examples.
formula_60 is cellular. A cell datum is given by formula_61 and
A cell-chain in the sense of the second, abstract definition is given by
Matrix examples.
formula_66 is cellular. A cell datum is given by formula_67 and
A cell-chain (and in fact the only cell chain) is given by
In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset formula_6.
Further examples.
Modulo minor technicalities all Iwahori–Hecke algebras of finite type are cellular w.r.t. to the involution that maps the standard basis as formula_74. This includes for example the integral group algebra of the symmetric groups as well as all other finite Weyl groups.
A basic Brauer tree algebra over a field is cellular if and only if the Brauer tree is a straight line (with arbitrary number of exceptional vertices).
Further examples include q-Schur algebras, the Brauer algebra, the Temperley–Lieb algebra, the Birman–Murakami–Wenzl algebra, the blocks of the Bernstein–Gelfand–Gelfand category formula_75 of a semisimple Lie algebra.
Representations.
Cell modules and the invariant bilinear form.
Assume formula_2 is cellular and formula_5 is a cell datum for formula_2. Then one defines the "cell module" formula_79 as the free formula_1-module with basis formula_81 and multiplication
where the coefficients formula_83 are the same as above. Then formula_79 becomes an formula_2-left module.
These modules generalize the Specht modules for the symmetric group and the Hecke-algebras of type A.
There is a canonical bilinear form formula_86 which satisfies
for all indices formula_88.
One can check that formula_89 is symmetric in the sense that
for all formula_91 and also formula_2-invariant in the sense that
for all formula_22,formula_91.
Simple modules.
Assume for the rest of this section that the ring formula_1 is a field. With the information contained in the invariant bilinear forms one can easily list all simple formula_2-modules:
Let formula_98 and define formula_99 for all formula_100. Then all formula_101 are absolute simple formula_2-modules and every simple formula_2-module is one of these.
These theorems appear already in the original paper by Graham and Lehrer.
Properties of cellular algebras.
Persistence properties.
If formula_1 is an integral domain then there is a converse to this last point:
If one further assumes formula_1 to be a local domain, then additionally the following holds:
Other properties.
Assuming that formula_1 is a field (though a lot of this can be generalized to arbitrary rings, integral domains, local rings or at least discrete valuation rings) and formula_2 is cellular w.r.t. to the involution formula_36. Then the following hold 