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\begin{document}
\title{Cellular algebras and quasi-hereditary algebras: a comparison}

\author{Steffen K\"onig}
\address{Fakult\"{a}t f\"{u}r Mathematik,
Universit\"{a}t Bielefeld,                             
Postfach 100131,                                   
D-33501 Bielefeld,                                 
Germany}
\email{koenig@mathematik.uni-bielefeld.de}

\author{Changchang Xi}
\address{Department of Mathematics,
Beijing Normal University, 
100875 Beijing, P. R. China}
\email{xicc@bnu.edu.cn}
\thanks{The second author was partially supported by NSF of China
(No. 19831070).}

\issueinfo{5}{10}{}{1999}
\dateposted{June 24, 1999}
\pagespan{71}{75}
\PII{S 1079-6762(99)00063-3}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 16D80, 16G30, 20C30, 20G05; 
Secondary 16D25, 18G15, 20F36, 57M25, 81R05}

\date{March 15, 1999} 

\revdate{April 22, 1999}

\commby{Dave Benson}

\begin{abstract}
Cellular algebras have been defined in a computational way
by the existence of a special kind of basis. We compare them with 
quasi-hereditary algebras, which are known to carry much homological and
categorical structure. Among the properties to be discussed here are
characterizations of quasi-hereditary algebras inside the 
class of cellular algebras in terms of vanishing of cohomology and in
terms of positivity of the Cartan determinant.
\end{abstract}
\maketitle

\section{Introduction}

To a large extent, algebraic representation theory of Lie algebras,
algebraic groups and related finite groups deals with finite-dimensional
algebras which are cellular \cite{GL} or quasi-hereditary \cite{CPS}. 
Group algebras of symmetric groups and their
Hecke algebras are known to be cellular as well
as various generalizations (e.g. Brauer algebras, cyclotomic Hecke algebras,
Temperley--Lieb algebras, partition algebras). Several of these algebras
also have been used in other contexts like topology (invariants of knots or
manifolds) or statistical mechanics. Schur algebras associated with
semisimple algebraic groups in any characteristic and blocks of the
Bernstein--Gelfand--Gelfand category $\mathcal{O}$ 
associated with semisimple complex Lie
algebras are cellular as well, but they also satisfy the stronger condition
to be quasi-hereditary. A quasi-hereditary structure comes both with 
desirable numerical properties (decomposition matrices
are square matrices, the number of simple modules can be read off from a
defining chain of ideals) and with homological structure 
(finite global dimension,
vanishing results on certain cohomology groups, stratification of
derived module categories, 
existence of `tilting modules' and derived equivalences, possibility to define
`Kazhdan--Lusztig' theory), and also there is a categorical definition (which
cannot exist for cellular algebras; see below). Many cellular algebras,
in particular Brauer algebras and partition algebras, are known to be
quasi-hereditary for some choice of parameters and not quasi-hereditary
for some other choice (typically `at zero'). 

In contrast to quasi-hereditary algebras, whose definition 
already comes with a lot of structure, cellular algebras have been defined 
first in a purely computational way, by requiring the existence of a basis
with nice multiplicative properties. However, recently a theory has emerged
which discusses homological and categorical 
structures in this class of algebras. In particular,
this theory clarifies the relation with quasi-hereditary algebras
both from the abstract point of view and from that of checking
examples in practice. The aim of this note is to survey this development.

\section{Definitions}

From now on, by an algebra we always mean an associative algebra, which is
finite dimensional over a field $k$. 

The original definition of cellular algebras is as follows.

\begin{Def}[Graham and Lehrer, \cite{GL}]
An associative
$k$-algebra $A$ is called a \textit{cellular algebra} with cell datum $(I,M,C,i)$
if the following conditions are satisfied:

(C1) The finite set $I$ is partially ordered. 
Associated with each $\lambda \in I$ there is a finite
set $M(\lambda)$. The algebra $A$ has a $k$-basis $C^{\lambda}_{S,T}$,
where $(S,T)$ runs through
all elements of $M(\lambda) \times M(\lambda)$ for all $\lambda \in I$.

(C2) The map $i$ is a $k$-linear anti-automorphism of $A$ with $i^2 =
\operatorname{id}$
which sends $C^{\lambda}_{S,T}$ to
$C^{\lambda}_{T,S}$.

(C3) For each $\lambda \in I$ and $S,T \in M(\lambda)$ and each $a \in A$
the product $a C^{\lambda}_{S,T}$ can be written as 
$(\sum_{U \in M(\lambda)} r_a(U,S) C^{\lambda}_{U,T}) + r'$,
where $r'$ is a linear combination of basis
elements with upper index $\mu$ strictly less than $\lambda$, and 
where the coefficients $r_a(U,S) \in k$ do not depend on $T$.
\end{Def}

In the following we shall call a $k$-linear anti-automorphism $i$ of $A$
with $i^2 = \operatorname{id}$ an involution of $A$.
In \cite{KX1} it has been shown that this definition is equivalent to 
the following one.

\begin{Def}[\cite{KX1}] Let $A$ be a $k$-algebra.
Assume there is an anti-automorphism $i$ on $A$ with $i^2 = 
\operatorname{id}$. A two-sided
ideal $J$ in $A$ is called a \textit{cell ideal} if and only if 
$i(J) = J$ and there exists \label{ourdef} a
left ideal $\Delta \subset J$ such 
that $\Delta$ has finite $k$-dimension and that there is an isomorphism of 
$A$-bimodules $\alpha: J \simeq \Delta \otimes_k i(\Delta)$ (where 
$i(\Delta) \subset J$ is the $i$-image of $\Delta$) making the following
diagram commutative:
\begin{equation*}
\xymatrix{
J \ar[r]^-{\alpha} \ar[d]_{i} & \Delta \otimes_k i(\Delta) \ar[d]^{x \otimes y
\mapsto i(y) \otimes i(x)} \\
J\ar[r]^-{\alpha} &
\Delta \otimes_k i(\Delta)}
\end{equation*}
The algebra $A$ (with the involution $i$) is called \textit{cellular} if and
only if there is a vector space 
decomposition $A=J_1' \oplus J_2' \oplus \dots \oplus J_n'$
(for some $n$) with $i(J_j')=J_j'$ for each $j$ and such that 
setting $J_j = \bigoplus_{l=1}^j J_l'$ gives a
chain of two-sided ideals of $A$: $0=J_0 \subset J_1 \subset J_2 \subset \dots
\subset J_n = A$ (each of them fixed by $i$)
and for each $j$ ($j = 1, \dots, n$),
the quotient $J_j'=J_j/J_{j-1}$ is a cell
ideal (with respect to the involution induced by $i$ on the quotient) of
$A/J_{j-1}$.
\end{Def}

The first definition can be used to check concrete examples.
The second definition, however, is often more handy for theoretical
and structural purposes.


Typical examples of cellular algebras are the following: 
Group algebras of symmetric groups, or
more general Hecke algebras of type $A$ or even of Ariki--Koike type (i.e.
cyclotomic Hecke algebras) \cite{GL}, 
Brauer algebras of types $B$ and $C$ \cite{GL} (see also 
\cite{KX4} for another proof), partition algebras \cite{X} 
and various kinds of Temperley--Lieb algebras \cite{GL}.


Let us also recall the definition of quasi-hereditary algebras
introduced in \cite{CPS}.

\begin{Def}[Cline, Parshall, and Scott \cite{CPS}] Let $A$ be a $k$-algebra.
An ideal $J$ in $A$ is called a \textit{heredity ideal} if $J$ is idempotent,
$J(\operatorname{rad}(A))J = 0$ and $J$ is a projective left (or right) $A$-module. The
algebra $A$ is called \textit{quasi-hereditary} provided there is a finite
chain $0=J_0\subset J_1\subset J_2\subset \dots \subset J_n = A$ of ideals in
$A$ such that $J_j/J_{j-1}$ is a heredity ideal in $A/J_{j-1}$ for all $j$.
Such a chain is then called a heredity chain of the quasi-hereditary
algebra $A$.
\end{Def}

Examples of quasi-hereditary algebras are blocks of category $\mathcal{O}$
\cite{BGG} and Schur algebras \cite{PS,D}. The precise relation to
highest weight categories is described in \cite{CPS}.

\section{Results}

Being interested in structural results, the first question one has to ask
is that of Morita invariance. In the case of quasi-hereditary algebras,
Cline, Parshall, and Scott \cite{CPS} proved the equivalence of the definition
of quasi-hereditary algebras given above with another one, which is in terms 
of `highest weight
categories' and hence categorical. Thus Morita invariance follows
immediately. For cellular algebras the situation is more delicate.

\begin{Theo}[\cite{KX2}] Let $k$ be a field of characteristic different from 
two. Then
the notion of `cellular algebra' over $k$ is Morita invariant. This is not 
true over fields of characteristic two.
\end{Theo}

Often, cellular structures can be defined in a characteristic free way, 
e.g. for integral group rings of symmetric groups. The 
second part of the theorem says that,
in general, it is impossible to transfer a cellular structure to a Morita
equivalent algebra (unless two is invertible in the ground ring). In 
particular, there cannot exist a purely categorical definition of cellular
algebras. 


For a quasi-hereditary algebra $A$, the length of a longest heredity chain 
equals the 
number of isomorphism classes of simple $A$-modules. Conversely, a cellular
algebra with a cell chain of this length must be quasi-hereditary. However,
if the algebra is cellular, but not quasi-hereditary, then the length of
a cell chain is not an invariant of the algebra any more. An example (of a
local algebra of dimension fourteen) is given in \cite{KX3}.


In the representation theory of finite groups or related topics one often
uses the following hierarchy of finite-dimensional algebras:
\[\begin{array}{c}
\{ semisimple \} \\ \cap \\ \{symmetric \} \\ \cap \\
\{ weakly {\phantom x} symmetric \} \\
\cap \\ \{ quasi\text{-}Frobenius = self\text{-}injective \}\end{array}
\] 


Here self-injective means
that each projective module is injective as well, whereas weakly symmetric says
that the projective cover of any given simple module is the injective
envelope of the same simple module. That is, the permutation 
$\operatorname{top}(P)
\mapsto \operatorname{soc}(P)$ (for $P$ indecomposable projective--injective) 
is the identity. It is well known that 
in general all these inclusions are proper. Within the class of cellular
algebras, however, one inclusion is the identity.

\begin{Theo}[\cite{KX6}]
Let $A$ be a cellular algebra. If $A$ is
self-injective, then it is weakly symmetric.
\end{Theo}

The main results we have obtained deal with the question when a cellular
algebra is quasi-hereditary. This question splits into the following
problems.

%Problem 1
\begin{problem} How to characterize the
quasi-hereditary algebras among the cellular ones by a structural property?
\end{problem}
%Problem 2
\begin{problem} How to characterize the
quasi-hereditary algebras among the cellular ones by a numerical property?
\end{problem}
%Problem 3
\begin{problem}\label{prb:3} Given a cellular algebra with a cell chain of ideals,
how to decide whether it is quasi-hereditary?
\end{problem}

%To these problems we shall give globally a homological and numerical answer.
%Problems 1, 2 and 3 are answered by the next result. 
The equivalence of (a) and (c) in the following theorem 
answers the first problem. Problem 2 is solved by the
equivalence between (a) and (d). And the equivalence between non-(a) and
non-(b) tells us how to prove that a cellular algebra is not
quasi-hereditary, i.e. how to solve Problem \ref{prb:3}.

\begin{Theo}[\cite{KX5}] Let $k$ be a field and $A$ a cellular
$k$-algebra (with respect to an involution $i$). 
Then the following are equivalent:

(a) Some cell chain of $A$ (with respect to some involution, possibly
different from $i$) is a heredity chain as well, i.e. it makes $A$ 
into a quasi-hereditary algebra.

(a$'$) There is a cell chain of $A$ (with respect to some involution, possibly
different from $i$) whose length equals the number of 
isomorphism classes of simple $A$-modules.

(b) Any cell chain of $A$ (with respect to any involution) is a heredity chain.

(c) The algebra $A$ has finite global dimension; i.e., there exists an 
$N \in {\mathbb N}$ such that $Ext^i_A(X,Y)=0$ for all $i \geq N$ and for
all $A$-modules $X$ and $Y$.

(d) The Cartan matrix (recording the composition multiplicities of 
simple modules in indecomposable projective modules) of $A$ has 
determinant one. \label{criterion}
\end{Theo}

The determinant of the Cartan matrix of a cellular algebra always is a
positive integer \cite{KX5}.


The main open cases of cellular algebras to be checked for quasi-heredity
were Brauer algebras \cite{We,GL} and partition algebras \cite{M}.

%Problem 4
\begin{problem}
 Determine precisely for which choice of parameters a 
Brauer algebra or a partition algebra is quasi-hereditary.
\end{problem}

Problem 4 is answered by the next two results which apply the equivalence
of (a) and (b) in Theorem \ref{criterion}.

\begin{Theo}[\cite{KX5}]
Let $k$ be any field, fix $\delta \in k$, and denote by $B(r,\delta)$ the
Brauer algebra on $2r$ vertices and with parameter $\delta$. 

Then $B(r,\delta)$ is quasi-hereditary if and only if 

(1) $\delta$ is not zero or $r$ is odd; and

(2) the characteristic of $k$ is either zero or strictly greater than
$r$.
\end{Theo} 

This extends previous results by Graham and Lehrer \cite{GL};
they proved the `if'-part in \cite{GL}, 4.16. and 4.17.  


For partition algebras we have

\begin{Theo}[\cite{KX5}] \label{partition}
Let $k$ be any field, fix $\delta \in k$, and denote by $P(r,\delta)$ the
partition algebra on $2r$ vertices and with parameter $\delta$. 

Then $P(r,\delta)$ is quasi-hereditary if and only if $\delta$ is not
zero and the characteristic of $k$ is either zero or strictly greater than
$r$.
\end{Theo}

Martin \cite{M} had shown this in case of characteristic
zero and $\delta \neq 0$. In \cite{X}, the `if'-part of Theorem \ref{partition}
is proved.



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\end{document}