# Cellular algebra

{{#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.

## History

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]

## Definitions

Let ${\displaystyle R}$ be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also ${\displaystyle A}$ be a ${\displaystyle R}$-algebra.

### The concrete definition

A cell datum for ${\displaystyle A}$ is a tuple ${\displaystyle (\Lambda ,i,M,C)}$ consisting of

${\displaystyle C:{\dot {\bigcup }}_{\lambda \in \Lambda }M(\lambda )\times M(\lambda )\to A}$
The images under this map are notated with an upper index ${\displaystyle \lambda \in \Lambda }$ and two lower indices ${\displaystyle {\mathfrak {s}},{\mathfrak {t}}\in M(\lambda )}$ so that the typical element of the image is written as ${\displaystyle C_{\mathfrak {st}}^{\lambda }}$.

and satisfying the following conditions:

${\displaystyle aC_{\mathfrak {st}}^{\lambda }\equiv \sum _{{\mathfrak {u}}\in M(\lambda )}r_{a}({\mathfrak {u}},{\mathfrak {s}})C_{\mathfrak {ut}}^{\lambda }\mod A(<\lambda )}$
with coefficients ${\displaystyle r_{a}({\mathfrak {u}},{\mathfrak {s}})\in R}$ depending only on ${\displaystyle a}$,${\displaystyle {\mathfrak {u}}}$ and ${\displaystyle {\mathfrak {s}}}$ but not on ${\displaystyle {\mathfrak {t}}}$. Here ${\displaystyle A(<\lambda )}$ denotes the ${\displaystyle R}$-span of all basis elements with upper index strictly smaller than ${\displaystyle \lambda }$.

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

### The more abstract definition

Let ${\displaystyle i:A\to A}$ be an anti automorphism of ${\displaystyle R}$-algebras with ${\displaystyle i^{2}=id}$ (just called "involution" from now on).

A cell ideal of ${\displaystyle A}$ w.r.t. ${\displaystyle i}$ is a two-sided ideal ${\displaystyle J\subseteq A}$ such that the following conditions hold:

1. ${\displaystyle i(J)=J}$.
2. There is a left ideal ${\displaystyle \Delta \subseteq J}$ that is free as a ${\displaystyle R}$-module and an isomorphism
${\displaystyle \alpha :\Delta \otimes _{R}i(\Delta )\to J}$
of ${\displaystyle A}$-${\displaystyle A}$-bimodules such that ${\displaystyle \alpha }$ and ${\displaystyle i}$ are compatible in the sense that
${\displaystyle \forall x,y\in \Delta :i(\alpha (x\otimes i(y)))=\alpha (y\otimes i(x))}$

A cell chain for ${\displaystyle A}$ w.r.t. ${\displaystyle i}$ is defined as a direct decomposition

${\displaystyle A=\bigoplus _{k=1}^{m}U_{k}}$

into free ${\displaystyle R}$-submodules such that

Now ${\displaystyle (A,i)}$ 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 ${\displaystyle \Lambda }$) and choosing a basis of every left ideal ${\displaystyle \Delta /J_{k-1}\subseteq J_{k}/J_{k-1}}$ one can construct a corresponding cell basis for ${\displaystyle A}$.

## Examples

### Polynomial examples

${\displaystyle R[x]/(x^{n})}$ is cellular. A cell datum is given by ${\displaystyle i=id}$ and

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

${\displaystyle 0\subseteq (x^{n-1})\subseteq (x^{n-2})\subseteq \ldots \subseteq (x^{1})\subseteq (x^{0})=R}$

### Matrix examples

${\displaystyle R^{d\times d}}$ is cellular. A cell datum is given by ${\displaystyle i(A)=A^{T}}$ and

A cell-chain (and in fact the only cell chain) is given by

${\displaystyle 0\subseteq R^{d\times d}}$

In some sense all cellular algebras "interpolate" between these two extremes by arranging matrix-algebra-like pieces according to the poset ${\displaystyle \Lambda }$.

### 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 ${\displaystyle T_{w}\mapsto T_{w^{-1}}}$.[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 ${\displaystyle {\mathcal {O}}}$ of a semisimple Lie algebra.[4]

## Representations

### Cell modules and the invariant bilinear form

Assume ${\displaystyle A}$ is cellular and ${\displaystyle (\Lambda ,i,M,C)}$ is a cell datum for ${\displaystyle A}$. Then one defines the cell module ${\displaystyle W(\lambda )}$ as the free ${\displaystyle R}$-module with basis ${\displaystyle \lbrace C_{\mathfrak {s}}|{\mathfrak {s}}\in M(\lambda )\rbrace }$ and multiplication

${\displaystyle aC_{\mathfrak {s}}:=\sum _{\mathfrak {u}}r_{a}({\mathfrak {u}},{\mathfrak {s}})C_{\mathfrak {u}}}$

where the coefficients ${\displaystyle r_{a}({\mathfrak {u}},{\mathfrak {s}})}$ are the same as above. Then ${\displaystyle W(\lambda )}$ becomes an ${\displaystyle A}$-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 ${\displaystyle \phi _{\lambda }:W(\lambda )\times W(\lambda )\to R}$ which satisfies

${\displaystyle C_{\mathfrak {st}}^{\lambda }C_{\mathfrak {uv}}^{\lambda }\equiv \phi _{\lambda }(C_{\mathfrak {t}},C_{\mathfrak {u}})C_{\mathfrak {sv}}^{\lambda }\mod A(<\lambda )}$

One can check that ${\displaystyle \phi _{\lambda }}$ is symmetric in the sense that

${\displaystyle \phi _{\lambda }(x,y)=\phi _{\lambda }(y,x)}$

for all ${\displaystyle x,y\in W(\lambda )}$ and also ${\displaystyle A}$-invariant in the sense that

${\displaystyle \phi _{\lambda }(i(a)x,y)=\phi _{\lambda }(x,ay)}$

### Simple modules

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

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

## Properties of cellular algebras

### Persistence properties

If ${\displaystyle R}$ is an integral domain then there is a converse to this last point:

If one further assumes ${\displaystyle R}$ to be a local domain, then additionally the following holds:

### Other properties

Assuming that ${\displaystyle R}$ 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 ${\displaystyle A}$ is cellular w.r.t. to the involution ${\displaystyle i}$. Then the following hold

1. ${\displaystyle A}$ is quasi-hereditary (i.e. its module category is a highest-weight category).
2. ${\displaystyle \Lambda =\Lambda _{0}}$.
3. All cell chains of ${\displaystyle (A,i)}$ have the same length.
4. All cell chains of ${\displaystyle (A,j)}$ have the same length where ${\displaystyle j:A\to A}$ is an arbitrary involution w.r.t. which ${\displaystyle A}$ is cellular.
5. ${\displaystyle \det(C_{A})=1}$.

## References

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