Cellular algebra

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 ${\displaystyle \lbrace C_{\mathfrak {s}}\mid {\mathfrak {s}}\in M(\lambda )\rbrace }$ 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

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

for all indices ${\displaystyle s,t,u,v\in M(\lambda )}$.

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

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

for all ${\displaystyle a\in A}$,${\displaystyle x,y\in W(\lambda )}$.

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:

Let ${\displaystyle \Lambda _{0}:=\lbrace \lambda \in \Lambda \mid \phi _{\lambda }\neq 0\rbrace }$ and define ${\displaystyle L(\lambda ):=W(\lambda )/\operatorname {rad} (\phi _{\lambda })}$ for all ${\displaystyle \lambda \in \Lambda _{0}}$. Then all ${\displaystyle L(\lambda )}$ are absolute simple ${\displaystyle A}$-modules and every simple ${\displaystyle A}$-module is one of these.

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

• Tensor products of finitely many cellular ${\displaystyle R}$-algebras are cellular.
• A ${\displaystyle R}$-algebra ${\displaystyle A}$ is cellular if and only if its opposite algebra ${\displaystyle A^{\text{op}}}$ is.
• If ${\displaystyle A}$ is cellular with cell-datum ${\displaystyle (\Lambda ,i,M,C)}$ and ${\displaystyle \Phi \subseteq \Lambda }$ is an ideal (a downward closed subset) of the poset ${\displaystyle \Lambda }$ then ${\displaystyle A(\Phi ):=\sum RC_{\mathfrak {st}}^{\lambda }}$ (where the sum runs over ${\displaystyle \lambda \in \Lambda }$ and ${\displaystyle s,t\in M(\lambda )}$) is a two-sided, ${\displaystyle i}$-invariant ideal of ${\displaystyle A}$ and the quotient ${\displaystyle A/A(\Phi )}$ is cellular with cell datum ${\displaystyle (\Lambda \setminus \Phi ,i,M,C)}$ (where i denotes the induced involution and M, C denote the restricted mappings).
• If ${\displaystyle A}$ is a cellular ${\displaystyle R}$-algebra and ${\displaystyle R\to S}$ is a unitary homomorphism of commutative rings, then the extension of scalars ${\displaystyle S\otimes _{R}A}$ is a cellular ${\displaystyle S}$-algebra.
• Direct products of finitely many cellular ${\displaystyle R}$-algebras are cellular.

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

• If ${\displaystyle (A,i)}$ is a finite-dimensional ${\displaystyle R}$-algebra with an involution and ${\displaystyle A=A_{1}\oplus A_{2}}$ a decomposition in two-sided, ${\displaystyle i}$-invariant ideals, then the following are equivalent:

1. ${\displaystyle (A,i)}$ is cellular.
2. ${\displaystyle (A_{1},i)}$ and ${\displaystyle (A_{2},i)}$ are cellular.

• Since in particular all blocks of ${\displaystyle A}$ are ${\displaystyle i}$-invariant if ${\displaystyle (A,i)}$ is cellular, an immediate corollary is that a finite-dimensional ${\displaystyle R}$-algebra is cellular w.r.t. ${\displaystyle i}$ if and only if all blocks are ${\displaystyle i}$-invariant and cellular w.r.t. ${\displaystyle i}$.
• Tits' deformation theorem for cellular algebras: Let ${\displaystyle A}$ be a cellular ${\displaystyle R}$-algebra. Also let ${\displaystyle R\to k}$ be a unitary homomorphism into a field ${\displaystyle k}$ and ${\displaystyle K:=\operatorname {Quot} (R)}$ the quotient field of ${\displaystyle R}$. Then the following holds: If ${\displaystyle kA}$ is semisimple, then ${\displaystyle KA}$ is also semisimple.

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

• If ${\displaystyle A}$ is cellular w.r.t. ${\displaystyle i}$ and ${\displaystyle e\in A}$ is an idempotent such that ${\displaystyle i(e)=e}$, then the algebra ${\displaystyle eAe}$ is cellular.

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. the involution ${\displaystyle i}$. Then the following hold

• ${\displaystyle A}$ is split, i.e. all simple modules are absolutely irreducible.
• The following are equivalent:^[1]

1. ${\displaystyle A}$ is semisimple.
2. ${\displaystyle A}$ is split semisimple.
3. ${\displaystyle \forall \lambda \in \Lambda :W(\lambda )}$ is simple.
4. ${\displaystyle \forall \lambda \in \Lambda$ :\phi _{\lambda }} is nondegenerate.

• The Cartan matrix ${\displaystyle C_{A}}$ of ${\displaystyle A}$ is symmetric and positive definite.
• The following are equivalent:^[7]

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. ${\displaystyle \det(C_{A})=1}$.

• If ${\displaystyle A}$ is Morita equivalent to ${\displaystyle B}$ and the characteristic of ${\displaystyle R}$ is not two, then ${\displaystyle B}$ is also cellular w.r.t. a suitable involution. In particular ${\displaystyle A}$ is cellular (to some involution) if and only if its basic algebra is.^[8]
• Every idempotent ${\displaystyle e\in A}$ is equivalent to ${\displaystyle i(e)}$, i.e. ${\displaystyle Ae\cong Ai(e)}$. If ${\displaystyle \operatorname {char} (R)\neq 2}$ then in fact every equivalence class contains an ${\displaystyle i}$-invariant idempotent.^[5]