Cellular algebra

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{{#invoke:Hatnote|hatnote}} 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.


The cellular algebras discussed in this article were introduced in a 1996 paper of Graham and Lehrer.[1] 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. [2][3]


Let be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also be a -algebra.

The concrete definition

A cell datum for is a tuple consisting of

The images under this map are notated with an upper index and two lower indices so that the typical element of the image is written as .

and satisfying the following conditions:

  1. The image of is a -basis of .
  2. for all elements of the basis.
  3. For every , and every the equation
with coefficients depending only on , and but not on . Here denotes the -span of all basis elements with upper index strictly smaller than .

This definition was originally given by Graham and Lehrer who invented cellular algebras.[1]

The more abstract definition

Let be an anti automorphism of -algebras with (just called "involution" from now on).

A cell ideal of w.r.t. is a two-sided ideal such that the following conditions hold:

  1. .
  2. There is a left ideal that is free as a -module and an isomorphism
of --bimodules such that and are compatible in the sense that

A cell chain for w.r.t. is defined as a direct decomposition

into free -submodules such that

  1. is a two-sided ideal of
  2. is a cell ideal of w.r.t. to the induced involution.

Now is called a cellular algebra if it has a cell chain. One can show that the two definitions are equivalent.[4] Every basis gives rise to cell chains (one for each topological ordering of ) and choosing a basis of every left ideal one can construct a corresponding cell basis for .


Polynomial examples

is cellular. A cell datum is given by and

A cell-chain in the sense of the second, abstract definition is given by

Matrix examples

is cellular. A cell datum is given by 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 .

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 .[5] 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).[4]

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 of a semisimple Lie algebra.[4]


Cell modules and the invariant bilinear form

Assume is cellular and is a cell datum for . Then one defines the cell module as the free -module with basis and multiplication

where the coefficients are the same as above. Then becomes an -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 which satisfies

for all indices .

One can check that is symmetric in the sense that

for all and also -invariant in the sense that

for all ,.

Simple modules

Assume for the rest of this section that the ring is a field. With the information contained in the invariant bilinear forms one can easily list all simple -modules:

Let and define for all . Then all are absolute simple -modules and every simple -module is one of these.

These theorems appear already in the original paper by Graham and Lehrer.[1]

Properties of cellular algebras

Persistence properties

If is an integral domain then there is a converse to this last point:

  1. is cellular.
  2. and are cellular.

If one further assumes to be a local domain, then additionally the following holds:

Other properties

Assuming that 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 is cellular w.r.t. to the involution . Then the following hold

  1. is semisimple.
  2. is split semisimple.
  3. is simple.
  4. is nondegenerate.
  1. is quasi-hereditary (i.e. its module category is a highest-weight category).
  2. .
  3. All cell chains of have the same length.
  4. All cell chains of have the same length where is an arbitrary involution w.r.t. which is cellular.
  5. .


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