---
abstract: |
    We show that an adaptation of Landrock’s Lemma for symmetric algebras also holds for cellular algebras and BGG algebras. This is a result relating the radical layers of any two projective modules. The reason it holds in our setting is that there is a duality between injective hulls and projective covers. As a corollary we deduce that BGG reciprocity respects Loewy structure.

    We then specialise to $(q)$-Schur algebras, in this setting there are applications to tilting modules. The results of Jantzen then give us a reciprocity for the Loewy structure of tilting modules from the region $(p-1)\rho+X_1$ in terms of the Loewy structure of Weyl modules from lower down in the alcove geometry.

    [**[Key words:]{}**]{} cellular algebras, BGG algebras, Loewy structure.
address:
- 'Corpus Christi College, Cambridge, CB2 1RH, England, UK'
- 'Magdalene College, Cambridge, CB3 0AG, England, UK'
author:
- 'C. Bowman'
- 'S.  Martin'
date: 30th January 2010
title: A Reciprocity Result for Projective Indecomposable Modules of Cellular Algebras and BGG Algebras
---

Introduction
============

Cellular algebras were introduced in [@GL] as the setting in which to study representations of algebraic groups, Lie algebras and the related finite dimensional algebras. The many-fold generalisations of Hecke algebras (for example cyclotomic Hecke algebras, Brauer algebras, affine Hecke algebras, partition algebras) have been shown to be cellular, as have the $(q)$-Schur algebras. As such, the structure of projective indecomposable modules of cellular algebras is of great interest.

The notion of a BGG algebra is defined in [@CPSdual] and in [@dual]. Such algebras are defined as quasi-hereditary algebras with the additional structure of a duality functor which fixes simple modules. The subcategory of such modules is of particular interest within the category of quasi-hereditary algebras and includes category $\mathcal{O}$ and $(q)$-Schur algebras as examples.

In [@qschur], [@Zeit2] projective-injective modules of $(q)$-Schur algebras are studied and shown to be precisely the projective tilting modules for the algebra. The bijection between the natural indexing sets for the two families of modules is shown to be a generalisation of the Mullineux map.

In [@and] the Loewy structure of tilting modules is studied in both the modular case, and for quantum groups. The projective tilting modules for the quantum algebraic group are shown to be rigid, and several results on rigidity of modular tilting modules from the first $p^2$-alcove are formulated.

In Corollary \[tilt\] we will see that Landrock’s lemma for Schur algebras will allow us to deduce the complete Loewy structure of certain tilting modules from that of the lighter Weyl modules.

We belatedly define rigidity and fix some notation. The socle of an $A$-module $M$ is defined to be the largest semisimple submodule of $M$, and denoted ${\operatorname{soc}}M$. Now consider the socle of $M/{\operatorname{soc}}M$, and let ${\operatorname{soc}}^2 M$ denote the submodule of $M$ containing ${\operatorname{soc}}M$ such that ${\operatorname{soc}}^2 M /{\operatorname{soc}}M$ is the socle of $M/{\operatorname{soc}}M$. We then inductively define the socle series ${\operatorname{soc}}^i M$. The socle layers are the semisimple quotients ${\operatorname{soc}}_i M={\operatorname{soc}}^{i}M/{\operatorname{soc}}^{i-1}M$. The radical of of an $A$-module $M$ is defined to be the smallest module with a semisimple quotient, and denote ${\operatorname{rad}}M$. We then let ${\operatorname{rad}}^2 M = {\operatorname{rad}}({\operatorname{rad}}M)$ and inductively define the radical series, ${\operatorname{rad}}^i M$, of $M$. The radical layers are then the semisimple quotients ${\operatorname{rad}}_i M = {\operatorname{rad}}^{i}M/{\operatorname{rad}}^{i-1}M$.

We let ${\operatorname{head}}M$ denote the first radical layer. The length of the radical series coincides with the length of the socle series. For a given module, $M$, this common number is called the Loewy length denoted $ll(M)$. An $A$-module is *rigid* if the radical and socle series coincide, i.e. if ${\operatorname{rad}}_iM={\operatorname{soc}}_{ll(M)-i}M$.

Cellular Algebras and BGG Algebras
==================================

We recall the original definition of a cellular algebra.

An associative $k$-algebra $A$ is called a cellular algebra with cell datum $(\pi;M;C; i)$ if the following conditions are satisfied:

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

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

(C3) For each $\lambda \in \pi$ and $S,T \in M(\lambda)$ and each $a \in A$ the product $aC^\lambda_{S,T}$ 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 strictly less than $\lambda$, and where coefficients $r_a(U, S) \in k$ do not depend on $T$.

The more abstract definition which follows has been shown in [@xi] to be equivalent to the one above.

Let $A$ be a $k$-algebra. Assume there is an anti-automomorphism $i$ on $A$ with $i^2= \text{id}$. A two-sided ideal $I$ in $A$ is called a *cell ideal* if and only if $i(I)=I$ and there exists a left ideal $L \subset I$ such that $L$ has finite $k$-dimension and that there is an isomorphism of $A$-bimodules $\alpha:I \cong L \otimes_k i(L)$ making the following diagram commutative: $$\xymatrix@=6pt{
&I \ar@{>}[rrrr]^{\alpha}\ar@{>}[dddd]^{i}  &&&&L \otimes_k i(L)\ar@{>}[dddd]^{x \otimes y \mapsto i(y) \otimes i(x)}\\
&\\
&\\
&\\
&I \ar@{>}[rrrr]^{\alpha}           &&&&L \otimes_k i(L)    
}$$ The algebra $A$ is called *cellular* if and only if there is a vector space decomposition $A=I'_1 \oplus \ldots \oplus I'_n$ with $i(I'_j)=I'_j$ for each $j$ and such that setting $I_j=\oplus_{k=1}^jI'_l$ gives a chain of two-sided ideals of $A$ and for each $j$ the quotient $I'_j=I_j/I_{j-1}$ is a cell ideal of $A/I_{j-1}$.

 {#projectivereciprocity}

There are several equivalent definitions of a quasi-hereditary algebra, we use the definition from [@dual] as it allows us to fix convenient notation in the process. Let $A$ be an associative finite dimensional $k$-algebra, for $k$ an algebraically closed field. Let $L(\lambda)$ constitute a full set of simple $A$-modules as $\lambda$ varies over an indexing poset $\pi$. Let $P(\lambda)$ denote the projective cover and let $I(\lambda)$ denote the injective hull of $L(\lambda)$. By $\Delta(\lambda)$ we denote the maximal factor module of $P(\lambda)$ such that all composition factors are of the form $L(\mu)$ for $\mu \leq \lambda$ in $\pi$. The modules $\Delta(\lambda)$ are called standard modules. Costandard modules are defined in a dual manner.

Let $A$ be an algebra with standard modules $\Delta$. The algebra is called quasi-hereditary if

1.  ${\operatorname{End}}_A(\Delta(\lambda))\cong k $ for all $\lambda \in \pi$,

2.  every projective module has a filtration by standard modules.

A duality on $A$-mod is a contravariant, exact, additive functor $\delta$ from $A$-mod to itself such that $\delta \circ \delta$ is naturally equivalent to the identity functor on $A$-mod and $\delta$ induces a $k$-linear map on the vector spaces ${\operatorname{Hom}}_A(M,N)$ for all $M, N \in A$-mod.

Let $A$ be a quasi-hereditary algebra. If there exists a duality functor $\delta$ on $A$-mod such that $\delta(L(\lambda))=L(\lambda)$ for all $\lambda$, then $A$ is said to be a BGG-algebra.

BGG algebras were defined in [@CPSdual] in order to generalise the famous BGG-reciprocity principle for category $\mathcal{O}$ to the general setting of highest weight categories. They show that for $\lambda, \mu \in \pi$ we have that $[P(\mu):\Delta(\lambda)]=[\Delta(\lambda):L(\mu)]$, and so calculating the characters of projective modules is equivalent to calculating the characters of the smaller standard modules.

 {#section-1}

We have that the anti-involution $i$ defines a duality functor, $\sharp$, on a cellular algebra in the following way. Let $V$ be a left $A$-module, then $V^\sharp={\operatorname{Hom}}(V,k)$ is a left $A$-module with the action: $$\begin{aligned}
(a \circ f)(v)=(f(i(a)v)     \ \ \ (f \in V^*, a \in A, v \in V).\end{aligned}$$

We extend $\sharp$ to a duality on BGG algebras by letting $\sharp$ act as $\delta$ on a BGG algebra. We have for BGG algebras (see [@dual]) and cellular algebras (see [@Cao]) that the duality $\sharp$ fixes simple modules, and interchanges their projective and injective hulls.

The Result
==========

The following proposition is an adaptation of Landrock’s Lemma (see [@al2] Lemma 1.9.10) for symmetric algebras to the setting of cellular algebras and BGG algebras.

\[thm\] Let $A$ be a cellular algebra or a BGG algebra. For $\lambda, \mu \in \pi$ we have the following reciprocity: $$\tag{$\dagger$}
 [ {\operatorname{rad}}_iP(\mu): L(\lambda)] =[{\operatorname{rad}}_iP(\lambda):L(\mu)]. \\$$

Let $A$ denote a BGG algebra with poset $\pi$. For weights $\lambda, \mu \in \pi$ we have that $$\begin{aligned}
[ {\operatorname{rad}}_iP(\mu):{\operatorname{head}}\Delta(\lambda)]&=[ {\operatorname{rad}}_i\Delta(\lambda):L(\mu)],\tag{$\dagger\dagger$}\end{aligned}$$ and so BGG reciprocity respects Loewy structure.

Note that the $\dagger$ reciprocity relates the occurrences of simple modules in projective modules. The $\dagger \dagger$ reciprocity relates how the Weyl filtrations sit within the Loewy structure of the projective modules. Thus by ${\operatorname{head}}\Delta(\lambda)$ we mean the head of the Weyl module, not a simple module isomorphic to the head.

Let $A$ denote a cellular algebra or a BGG algebra. Let $\bar{A}=A / {\operatorname{rad}}^iA$, $\bar{P}(\lambda)=P(\lambda)/{\operatorname{rad}}^iP(\lambda)$. Then $\bar{A}$ is a finite dimensional algebra with a complete list of simple modules given by $L(\lambda)$ as $\lambda$ ranges over $\pi$, each with corresponding prinicipal indecomposable modules $\bar{P}(\lambda)$. Denote the Cartan matrix by $\bar{C}_{ij}$. Then the Cartan matrix $\bar{C}_{ij}$ is symmetric.

\[land\] Let $\beta_1, \ldots, \beta_{m_s}$ be a basis of a complement to ${\operatorname{Hom}}_{A}(P(\nu)/{\operatorname{rad}}^{s-1}P(\nu),I(\mu))$ in ${\operatorname{Hom}}_{A}(P(\nu)/{\operatorname{rad}}^{s}P(\nu),I(\mu))$. Then let $$\begin{aligned}
V=\sum_{i=1}^{m_s}{\operatorname{im}}(\beta_i) \subseteq I(\mu), \ \ \ \ \ W / {\operatorname{rad}}^{s-1}P(\nu)=\bigcap_{i=1}^{m_s} \ker (\beta_i).\end{aligned}$$ Then we have that $$\begin{aligned}
m_s= \dim_k({\operatorname{Hom}}_{A} (L(\mu),P(\nu) / W)) &= \dim_k ({\operatorname{Hom}}_{A}(V, L(\nu))), \\&=\dim_k ({\operatorname{Hom}}_{A}((L(\nu), V^\sharp)).\end{aligned}$$

This follows as any $\beta=\sum b_i \beta_i$ is of Loewy length $s$ and if $\eta$ is the canonical homomorphism $V \to V/{\operatorname{rad}}V$ then $\beta_1\eta, \ldots , \beta_{m_s}\eta$ are linearly independent and so $V/{\operatorname{rad}}V \cong L(\nu)^{m_s}$, which gives the first equality. The second equality follows from properties of the duality.

We define: $$\begin{aligned}
a_\nu &=\dim({\operatorname{Hom}}_{A}(L(\nu), {\operatorname{rad}}^{s-1}P(\mu)/{\operatorname{rad}}^sP(\mu))), \\
\text{and } a_\mu &=\dim({\operatorname{Hom}}_{A}(L(\mu), {\operatorname{rad}}^{s-1}P(\nu)/{\operatorname{rad}}^sP(\nu))).  \end{aligned}$$ By symmetry we need only prove that $a_\mu \leq a_\nu$. We have that ${\operatorname{soc}}(P/W)= L(\mu)^{a_\mu}$ because any simple module not isomorphic to $L(\mu)$, or from a previous radical layer, is in $\bigcap_{i=1}^{m_s} \ker (\beta_i)$, and so by lemma \[land\] we have that: $$\begin{aligned}
a_\mu = m_s &=  \dim_k ({\operatorname{Hom}}_{A}(L(\mu),P(\nu) / W)),\\
 &= \dim_k ({\operatorname{Hom}}_{A}(V, L(\nu))), \\&=\dim_k ({\operatorname{Hom}}_{A}(L(\nu), V^\sharp)    ). \end{aligned}$$ Now, $V^\sharp$ has simple head isomorphic to $L(\mu)$ and is therefore a homomorphic image of $P(\mu)$. Also $V^\sharp$ has Loewy length $s$ and ${\operatorname{rad}}^{s-1}V^\sharp=L(\nu)^{m_s}=L(\nu)^{a_\mu}$ and therefore we have that $a_\mu \leq a_\nu$. This proves $\dagger$.

In order to prove $\dagger \dagger$ we proceed by induction on the partial ordering of $A$. If $\pi$ is a singleton, then the algebra is semisimple and we are done. If not, let $\lambda$ be a maximal element of $\pi$, then by induction the result holds for $A/\langle e_\lambda\rangle$ where $\langle e_\lambda\rangle$ is the ideal generated by the primitive central idempotent corresponding to the weight $\lambda$. This quotient is then a BGG algebra with respect to the indexing set $\pi \backslash \{ \lambda\}$. By our inductive assumption we only need show that the reciprocity holds for pairs $\mu\in \pi\backslash \{\lambda\}$ and $\lambda$. We have that $\lambda$ is maximal and therefore by \[projectivereciprocity\] we have that $P(\lambda)\cong\Delta(\lambda)$. Now by applying $\dagger$ to $\mu \in \pi\backslash \{\lambda\}$ and $\lambda$ we have that: $$\begin{aligned}
[{\operatorname{rad}}_i\Delta(\lambda):\text{L}(\mu)] &= [{\operatorname{rad}}_iP(\lambda):\text{L}(\mu)],\\ &=[{\operatorname{rad}}_iP(\mu):L(\lambda)].\end{aligned}$$ Now as projective modules have a Weyl filtration, and there is no heavier weight than $\lambda$ in $\pi$, we have that $\dagger \dagger$ follows.

Algebraic Groups
================

 {#section-2}

Let $G$ be a semisimple, simply connected linear algebraic group over an algebraically closed field $k$ of positive characteristic $p$. We fix a Borel subgroup $B$ and a maximal torus $T$ with $T \subset B
\subset G$ and we let $B$ determine the negative roots. We write $X =
X(T)$ for the character group of $T$, and $Y=Y(T)$ denote the co-character group. We let $X^+$ denote the set of dominant weights, and let $X_1$ denote the set of restricted weights, that is: $$X_1 := \{ \nu \in X^+ \mid \langle \alpha^\vee, \nu \rangle {\leqslant}p \text{ for all simple roots } \alpha \}.$$

By $G$-module we always mean a rational $G$-module, i.e. a $K[G]$-comodule, where $K[G]$ is the coordinate algebra of $G$. For each $\lambda \in X^+$ we have the following (see [@Jantzen]) finite dimensional $G$-modules: $$\begin{tabular}{ll}
${\mathrm{L}}(\lambda)$ & simple module of highest weight $\lambda$;\\
$\Delta(\lambda)$ & Weyl module of highest weight $\lambda$;\\
$\nabla(\lambda)$ & $= {\operatorname{ind}}_B^G k_\lambda$; dual Weyl module of
 highest weight $\lambda$; \\
${\mathrm{T}}(\lambda)$ & indecomposable tilting module of highest weight
$\lambda$
\end{tabular}$$ where $k_\lambda$ is the 1-dimensional $B$-module upon which $T$ acts by the character $\lambda$ with the unipotent radical of $B$ acting trivially. The simple modules ${\mathrm{L}}(\lambda)$ are contravariantly self-dual. The module $\nabla(\lambda)$ has simple socle isomorphic to ${\mathrm{L}}(\lambda)$; the module $\Delta(\lambda)$ is isomorphic to the contravariant dual of $\nabla(\lambda)$, hence has simple head isomorphic to ${\mathrm{L}}(\lambda)$.

Schur Algebras
==============

Let $S(\pi)$ denote the generalised (quantum) Schur algebra corresponding to a finite saturated set $\pi$ of dominant weights of the semisimple, complex finite dimensional Lie algebra $\frak{g}$, in the sense of [@don1]. Such an algebra is a BGG-algebra with standard modules $\Delta$ and costandard modules $\nabla$. We remark that in this case the duality $V^\sharp$ is denoted by $V^\circ$ and is called contravariant duality.

 {#Jantzen}

Assume that the field $k$ is of characteristic $p\geq 2h-2$ . By results of Jantzen in [@jan] we have, for $\lambda \in X_1$, that the $S(\pi)$-tilting module $T(2(p-1)\rho+\omega_0\lambda)$ is isomorphic to $P(\lambda)$ the projective module for the generalised Schur algebra $S(\pi)$ where $\pi=\{r \in X : r\leq 2(p-1)\rho\}$.

Applications to Schur algebras and Algebraic Groups
===================================================

Following Andersen and Kaneda [@and] we consider the Loewy structure of tilting modules for algebraic groups.

\[tilt\] Let $G$ be a semisimple, simply connected algebraic group and let $p\geq 2h-2$. We have the following description of the radical layers of the tilting modules with highest weight from the region $(p-1)\rho+ X_1$, $$\begin{aligned}
[ {\operatorname{rad}}_iT(2(p-1)\rho+\omega_0\mu):{\operatorname{head}}\Delta(\lambda)]= [{\operatorname{rad}}_i\Delta(\lambda):L(\mu)].\end{aligned}$$

We have that \[Jantzen\] implies that as $S(\pi)$-modules, $T(2(p-1)\rho+\omega_0\lambda)\cong P(\lambda)$, substituting this into $\dagger\dagger$ our corollary follows.

Therefore, determining the radical series of the $kG$-tilting modules from $(p-1)\rho+ X_1$ is equivalent to determining the Loewy structure of the Weyl modules from lower down in the linkage ordering. This is illustrated in \[examp\]. In Corollary 4.4 of [@and] we see that these modules are rigid for $p\geq3h-3$.

Corollary \[tilt\] follows from the bijection in \[Jantzen\] between the specified tilting modules and the projective modules for the Schur algebra. The question of when tilting modules are projective is addressed in [@qschur] and [@Zeit2] and a generalised Mullineux bijection is described which generalises corollary \[tilt\].

The condition that $p\geq2h-2$ in Corollary \[tilt\] is necessary. A counter example of Corollary \[tilt\] for small primes is given in Ringel’s appendix to [@DM]. This is because \[Jantzen\] fails.

An example {#examp}
----------

Let $G=\text{SL}_3(k)$ for $k$ an algebraically closed field of characteristic $p \geq 2h-2=4$. We study the tilting modules for $\text{SL}_3(k)$ by considering the structure of the corresponding generalised Schur algebra $S(\pi)$ where $\pi=\{r \in X: r \leq 2(p-1)\rho \}$. Take $T$ to be the subgroup of diagonal matrices and $B$ to be the group of lower triangular matrices. We let $W$ denote the Weyl group, which is isomorphic to the symmetric group on three letters.

Let $X(T) \cong \mathbb{Z}^2$ denote the character group of $G$ and $Y \cong \text{Hom}_\mathbb{Z}(X,\mathbb{Z})$ denote the co-character group. Setting $\alpha_1=(2,-1)$ and $\alpha_2=(-1,2)$ we have that $R=\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}$ are the roots of $G$ and that $S=\{\alpha_1,\alpha_2\}$ are the simple roots. For $\gamma \in R$ we let $\check{\gamma} \in Y$ denote the dual root.

We enumerate the alcoves in the dominant chamber of $A_2$ as follows:

![image](triangles.jpg)

The structure of the Weyl modules for $p\geq 5$ is well known (see for example [@DS] and [@ron]), and easily calculated by hyperalgebra calculations (which in this case are well within the range of GAP). We have that: $$\begin{aligned}
\Delta(1)=[1], \ \ \ \Delta(2)=[2,1], \ \ \ \  \ \Delta(3)=[3,2], \end{aligned}$$ $$\begin{aligned}
\Delta(4) =\begin{minipage}{34mm}
  \def\objectstyle{\scriptstyle}
\xymatrix@=6pt{
  &4 \ar@{-}[dl] \ar@{-}[dr] & \\
3\ar@{-}[dr] &1 \ar@{-}[d] \ar@{-}[u] &3'\ar@{-}[dl] \\
  & 2&
} \; \end{minipage}, \ \ \ \ 
\Delta(5) =\begin{minipage}{34mm}
\def\objectstyle{\scriptstyle}
\xymatrix@=6pt{
&5\ar@{-}[dl] \ar@{-}[dr] \\
4\ar@{-}[drr] \ar@{-}[d] \ar@{-}[dr] &      &4' \ar@{-}[d] \ar@{-}[dl] \ar@{-}[dll] \\
3\ar@{-}[dr] &1 \ar@{-}[d] \ar@{-}[ul] &3'\ar@{-}[dl] \\
  & 2&
}  \end{minipage}.\end{aligned}$$ By \[Jantzen\] we have that $P(2)=T(5)$. Now, 2 appears in the first layer of $\Delta(2)$, second layer of $\Delta(3)$ and $\Delta(3')$, the third layer of $\Delta(4)$ and $\Delta(4')$, and fourth layer of $\Delta(5)$. Therefore the $kG$-tilting module $T(5)$ has the following Loewy structure:

@=6pt[ & & & &2@[-]{}\[d\] & & &\
& & &3@[-]{}\[d\] &1 &3’@[-]{}\[d\] &\
& &4@[-]{}\[d\]@[-]{}\[dl\]@[-]{}\[dr\] &2 & &2 &4’@[-]{}\[d\]@[-]{}\[dr\]@[-]{}\[dl\]\
&3 &1 &3’ &5@[-]{}\[dr\]@[-]{}\[dl\] &3 &1 &3’\
& &2@[-]{}\[u\]@[-]{}\[ur\]@[-]{}\[ul\] &4@[-]{}\[d\]@[-]{}\[dr\]@[-]{}\[drr\] & &4’ @[-]{}\[d\]@[-]{}\[dl\]@[-]{}\[dll\] &2@[-]{}\[u\]@[-]{}\[ur\]@[-]{}\[ul\] &\
& & &3 &1 &3’ & &\
& & & &2@[-]{}\[u\]@[-]{}\[ur\]@[-]{}\[ul\] & & &\
]{}

note that we have highlighted where the Weyl filtration lies within the Loewy layers.

We would like to thank Steve Doty for many useful discussions and comments on previous versions of this manuscript.

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