When is a cellular algebra quasi–hereditary ?

by

Steffen König and Changchang Xi

1 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 or quasi– hereditary. 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 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 [8]). 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’).

The subject of this note are the following problems arising from this situation.

Problem 1. How to characterize among the cellular algebras the quasi–hereditary ones by a structural property?

1991 Mathematics Subject Classification: Primary 16E10, 16G10, 18G20. Secondary 17B10, 19A31, 20C05, 20G05, 81R05.

Both authors have obtained support from Volkswagen Foundation. C.C.Xi also obtained support from NSFC (Grant No. 19831070).

Problem 2. How to characterize among the cellular algebras the quasi–hereditary ones by a numerical property?

Problem 3. Given a cellular algebra with a cell chain of ideals, how to decide whether it is quasi–hereditary?

Problem 4. Determine precisely for which choice of parameters a Brauer algebra or a partition algebra is quasi–hereditary.

We give full answers to these problems. More precisely, problem 1 is answered by the equivalence of (a) and (b) in 1.1 below, problem 2 is answered by the equivalence of (a) and (c), problem 3 is answered by the equivalence of non–(a) and non–(e) in the more technical theorem 3.1 (which extends theorem 1.1 by adding another two equivalent statements). Problem 4 is answered by theorems 1.3 and 1.4, which are applications of 3.1.

Theorem 1.1 Let k be a field and A a cellular k–algebra (with respect to an involution i). Then the following are equivalent:

(a) The algebra A is a quasi–hereditary algebra.

(b) A has finite global dimension.

(c) The Cartan matrix of A has determinant one.

Given a k–algebra A, it may be difficult to show that A is not cellular since the definition involves the choice of a basis. Our proof of theorem 1.1 provides us with an invariant of cellular algebras as follows:

Proposition 1.2 Let k be a field and A a cellular k–algebra. Then the determinant of the Cartan matrix of A is a positive integer.

A more detailed version of proposition 1.2 is proposition 3.2 below.

We note that our result is in accordance with the socalled ’Cartan determinant conjecture’ which states that a finite dimensional algebra of finite global dimension should have Cartan determinant one (not just plus or minus one, as Eilenberg had shown).

Applying the equivalence of (a) and (b) in the theorem, we can solve problem 4, namely we give a precise description for which parameters the Brauer algebras and partition algebras are quasi–hereditary.

Theorem 1.3 Let k be any field, fix δ ∈ k and denote by B(r, δ) the Brauer algebra on 2r vertices and with parameter δ.

Then B(r, δ) is quasi–hereditary if and only if

(1) δ is not zero or r is odd; and

(2) the characteristic of k is either zero or strictly bigger than r.

This extends previous results by Graham and Lehrer; they proved the ’if’–part in [6], 4.16. and 4.17.

For partition algebras we have

Theorem 1.4 Let k be any field, fix δ ∈ k and denote by P (r, δ) the partition algebra on 2r vertices and with parameter δ.

Then P (r, δ) is quasi–hereditary if and only if δ is not zero and the characteristic of k is either zero or strictly bigger than r.

Martin [11] had shown this in case of characteristic zero and δ ≠ 0. In [15], the ”if part” of 1.4 is proved.

In section two we recall the definitions of cellular and quasi–hereditary algebras and then we collect a few auxiliary statements about such algebras.

In section three we prove theorem 1.1, or actually the more detailed theorem 3.1, and also proposition 1.2.

In section four we deal with applications. In particular we recall the relevant definitions and prove theorems 1.3 and 1.4 and a few other results on related classes of algebras like Temperley–Lieb algebras and Jones’ annular algebras.

2 Background

First we recall the two equivalent definitions of cellular and the definition of quasi– hereditary. Then we collect several facts to be used in the proofs later on.

For simplicity we stick to the ground ring being an (arbitrary) field k. By algebra we always mean a finite dimensional associative algebra with unit.

Definition 2.1 (Graham and Lehrer, [6]) An associative k–algebra A is called a 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 λ ∈ I there is a finite set M(λ). The algebra A has a k–basis C_{S,T}^{λ} where (S, T) runs through all elements of M(λ) × M(λ) for all λ ∈ I.

(C2) The map i is a k–linear anti–automorphism of A with i^{2} = id which sends C_{S,T}^{λ} to C_{T,S}^{λ}.

(C3) For each λ ∈ I and S, T ∈ M(λ) and each a ∈ A the product aC_{S,T}^{λ} can be written as (∑_{U∈M(λ)} r_{a}(U, S)C_{U,T}^{λ})+r′ where r′ is a linear combination of basis elements with upper index μ strictly smaller than λ, and where the coefficients r_{a}(U, S) ∈ k do not depend on T.

In the following we shall call a k–linear anti–automorphism i of A with i^{2} = id an involution of A. In [7] it has been shown that this definition is equivalent to the following one.

Definition 2.2 Let A be a k–algebra. Assume there is an antiautomorphism i on A with i^{2} = id. A two–sided ideal J in A is called a cell ideal if and only if i(J) = J and there exists a left ideal Δ ⊂ J such that Δ has finite k–dimension and that there is an isomorphism of A–bimodules α : J ≃ Δ ⊗_{k} i(Δ) (where i(Δ) ⊂ J is the i–image of Δ) making the following diagram commutative:

[figure: J; α; Δ ⊗_{k} i(Δ); J; i; α; Δ ⊗_{k} i(Δ); x ⊗ y ↦ i(y) ⊗ i(x)]

The algebra A (with the involution i) is called cellular if and only if there is a vector space decomposition A = J′_{1} ⊕ J′_{2} ⊕ … ⊕ J′_{n} (for some n) with i(J′_{j}) = J′_{j} for each j and such that setting J_{j} = ⊕_{l=1}^{j}J′_{l} gives a chain of two sided ideals of A: 0 = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ … ⊂ J_{n} = A (each of them fixed by i) and for each j (j = 1, …, 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}.

The modules Δ(j), 1 ≤ j ≤ n, are called standard modules of the cellular algebra A, and the above chain in A is called a cell chain. (Standard modules are called cell modules in [6]).

Let us also recall the definition of quasi-hereditary algebras introduced in [3].

Definition 2.3 (Cline, Parshall and Scott [3]) Let A be a k-algebra. An ideal J in A is called a heredity ideal if J is idempotent, J(rad(A))J = 0 and J is a projective left (or, right) A-module. The algebra A is called quasi-hereditary provided there is a finite chain 0 = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ … ⊂ 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.

We also need the notion of a Cartan matrix in the following abstract sense (which coincides with the one used in group theory if A is the group algebra of a finite group over a splitting field). Denote the simple A–modules by L(1), …, L(m) and their projective covers by P (1), …, P (m). The entries c_{j,h} of the Cartan matrix C(A) are the composition multiplicities [P (j) : L(h)]. The determinant of C(A) is called the Cartan determinant. In general this can be any integer.

Now we collect a number of auxiliary statements for later use.

For the first three assertions we fix a cellular algebra A with involution i and cell chain 0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A.

Lemma 2.1 Let A be a cellular algebra with involution i and cell chain 0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A. Then:

(1) There is a natural bijection between isomorphism classes of simple A–modules and indices l ∈ {1, …, n} such that J_{l}^{2} ⊄ J_{l−1}. The inverse of this bijection is given by sending such an l to Δ(l)/rad(Δ(l)) (which in this case is simple).

In the following we index the simple modules in this way by a subset of the set {1, …, n}.

(2) If l is the index of a simple module L(l) as in (1), then the composition factors L(j) of the standard module Δ(l) satisfy j ≥ l and j = l occurs with multiplicity one (and this factor is the unique simple quotient Δ(l)/rad(Δ(l)).

(3) The given cell chain of A is a heredity chain (making A into a quasi–hereditary algebra) if and only if all J_{l} satisfy J_{l}^{2} ⊄ J_{l−1} if and only if n equals the number of isomorphism classes of simple modules.

Proof. (1) is implicit in theorem 3.4 in [6] and it is given another proof in proposition 4.1 in [7]. (2) is proposition 3.6 in [6] and it also follows from proposition 4.1 of [7]. (3) is implicit in remark 3.10 in [6] and stated as corollary 4.2 in [7].

Note that we will have to prove the stronger statement that if one given cell chain is not a heredity chain then there is no heredity chain at all.

We also will make use of the following three statements.

(4) A quasi–hereditary algebra has finite global dimension (Parshall and Scott [13]), which actually is bounded by 2n − 2 where n is the length of the heredity chain (Dlab and Ringel [4]).

(5) A quasi–hereditary algebra has Cartan determinant one. This has been shown by Burgess and Fuller [2]. It also follows from (a slight modification of) the computation of the Cartan determinant of a cellular algebra which we give in the next section.

(6) If an algebra A has finite global dimension, then its Cartan determinant has absolute value one. This has been observed by Eilenberg [5]. It also follows directly from the observation that the Grothendieck group of A–mod is free abelian of finite rank, both the simple modules and the projective modules form a basis, and the Cartan matrix transforms one of these bases into the other.

A key ingredient in the proof below is the following property of symmetric matrices with real entries. The identity matrix is denoted by I. For a matrix X, we denote by X^{tr} the transposed matrix of X.

Proposition 2.2 Let X be a positive definite matrix, Y a positive semidefinite matrix and Z a ’square root’ of X, that is, Z^{2} = X and Z = Z^{tr} and Z is positive definite. Then the matrix U = Z^{−1}Y Z^{−1} is positive semidefinite and has the same eigenvalues as the matrix V = X^{−1}Y.

Proof. The existence of Z is an easy exercise in linear algebra.

Under our assumptions, λX − Y equals (for any λ ∈ k) the product Z(λI − Z^{−1}Y Z^{−1})Z. Hence a vector x satisfies (λX − Y)x = 0 if and only if the vector u = Zx satisfies (λI − U)u = 0. Moreover the eigenvalue λ of V which corresponds to the eigenvector v can be written as λ = \frac{v^{tr}Y v}{v^{tr}Xv}, hence it is a non–negative real number.

3 The criterion

A more detailed version of theorem 1.1 is the following.

Theorem 3.1 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) A has finite global dimension.

(c) The Cartan matrix of A has determinant one.

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

We remark that the equivalence of non–(a) and non–(e) answers problem 3 from the introduction.

We also note that the length of a cell chain in general is not an invariant of the algebra (see [9] for an example). This forces us to formulate condition (a’) as depending on the choice of a cell chain.

The key to the proof is the following refinement of proposition 1.2.

Proposition 3.2 Let k be a field and A a cellular k–algebra. Denote by m the number of isomorphism classes of simple A–modules.

Then the determinant of the Cartan matrix of A is a positive integer. It equals one if and only A has a cell chain of length m.

Proof of proposition 3.2.

We fix a base field k and a cellular algebra A with an involution i and a cell chain 0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A. Each subquotient J_{l}/J_{l−1} in the chain has the form Δ(l) ⊗_{k} i(Δ(l)). Denote the number of isomorphism classes of simple A–modules by m. As in [6], 3.5, we form the ’decomposition matrix’ D of A (with respect to the given cell chain). It is an n × m–matrix with integer entries d_{a,b} = [Δ(a) : L(b)], the composition multiplicity of L(b) in Δ(a). By assertion (1) in lemma 2.1, the indices of simple modules can be naturally identified with the indices of those ideals J_{l} satisfying J_{l}^{2} ⊄ J_{l−1}. Hence we may assume, possibly after rearranging the rows of D, that D is

of the form ( \begin{matrix}D_{1} \\ D_{2} \end{matrix} ), where both D_{1} and D_{2} are integer matrices and D_{2} (whose rows

correspond to those indices l such that J_{l}^{2} ⊄ J_{l−1}) is a square matrix. In case n = m we consider D_{1} as a matrix of size 0 × 0. We note that D equals D_{2} if and only if n equals m. By assertion (3) of lemma 2.1, this happens if and only if the given cell chain is a heredity chain.

By assertion (2) of lemma 2.1, D_{2} actually is lower triangular with all diagonal entries equal to one.

The Cartan matrix C of A satisfies C = D^{tr}D (see [6]). In fact, the composition multiplicity [Δ(j) : L(h)] equals dim_{k}(e(h)Δ(j)) if e(h) is a primitive idempotent generating a projective cover Ae(h) of L(h). This projective module is filtered by J_{0}e(h) ⊂ J_{1}e(h) ⊂ … ⊂ J_{n}e(h) = Ae(h). The subquotients in this filtration are of the form Δ(l) ⊗_{k} i(Δ(l))e(h). Hence Δ(j) occurs in this filtration with multplicity dim_{k}(i(Δ(j))e(h)) which equals dim_{k}(e(h)Δ(j)), which is the desired claim.

Next we are going to compute the determinant of C, which by definition of C must

be an integer. We have C = (D_{1}^{tr}, D_{2}^{tr}) ( \begin{matrix}D_{1} \\ D_{2} \end{matrix} ) = (D_{1}^{tr}D_{1} + D_{2}^{tr}D_{2}). Denote by C_{1}

the product D_{2}^{tr}D_{2}. The unitriangularity of D_{2} gives det(C_{1}) = 1. Hence det(C) = det(C_{1}^{−1}C) = det(I + C_{1}^{−1}D_{1}^{tr}D_{1}) where I denotes the identity matrix. Clearly C_{1} is positive definite and D_{1}^{tr}D_{1} is positive semidefinite. Of course, D_{1}^{tr}D_{1} is zero if and only if n = m.

The positive definite symmetric matrix C_{1} can be written as C_{2}^{2} for some symmetric C_{2}. Proposition 2.2 implies that the matrix C_{3} = C_{2}^{−1}D_{1}^{tr}D_{1}C_{2}^{−1} has the same eigenvalues as the matrix C_{1}^{−1}D_{1}^{tr}D_{1}. By construction, C_{3} is symmetric and its eigenvalues are non–negative real numbers. Moreover, all eigenvalues are zero precisely in the case n = m. Going back to the matrix I + C_{1}^{−1}D_{1}^{tr}D_{1} we conclude that all its eigenvalues are of the form 1 + λ, where λ are eigenvalues of the matrix C_{1}^{−1}D_{1}^{tr}D_{1} and therefore non–negative real numbers, and all λ equal zero if and only n = m. We note that this implies statement (5) mentioned after lemma 2.1 (under the assumption that the algebra is cellular – for quasi–hereditary algebras without an involution i one has to slightly modify the argument).

Altogether we have shown: det(C) is a positive integer and it is equal to one if and only if for the given cell chain we have n = m.

Proof of theorem 3.1.

We keep the notations from proposition 3.2 and its proof.

The equivalence of (a) and (a’) follows from assertion (3) in lemma 2.1.

From proposition 3.2 we get that det(C) is equal to one if and only if for the given cell chain we have n = m. But of course, det(C) does not depend on the choice of the cell chain. Hence we have shown: if the length n of some cell chain satisfies n = m, then the same must be true for any cell chain. This proves the equivalence of (a) (or (a’)) (c) and (d).

Assertion (4) (made after lemma 2.1) shows that (a) implies (b).

Assertion (6) finally tells us that (by proposition 1.2, which has been proved already) we can conclude (c) from (b).

We remark that the computation of det(C) goes through in a more general situation:

Proposition 3.3 Let A be a finite dimensional k–algebra with a chain 0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A of two–sided ideals whose subquotients are of the form Δ(l)⊗∇(l) such that for all primitive idempotents e ∈ A there is an isomorphism of vector spaces eΔ(l) ≃ ∇(l)e. Then the Cartan determinant of A is a positive integer.

4 Applications

First we discuss Brauer algebras. They have been introduced by Brauer [1] in order to extend classical Schur–Weyl duality to semisimple algebraic groups of types B and C. More precisely, Schur–Weyl duality means the following double centraliser property: Fix a field k and two natural numbers n and r. Then the general linear group GL_{n}(k) acts (diagonally) from the left on the vector space (k^{n})^{⊗r} whereas the symmetric group Σ_{r} acts from the right by permuting places of tensors. These two actions centralise each other. Replacing in this setup the algebraic group GL_{n}(k) by a subgroup of type B or C, one has to replace the symmetric group by the Brauer algebra if one wants to keep the double centraliser property.

For the definition of the Brauer algebra, we fix a field k, an indeterminate x, and a natural number r. Then B(r, x) has a basis consisting of all diagrams, which consist of 2r vertices, divided into 2 ordered sets, the r top vertices and the r bottom vertices, and r edges such that each edge belongs to exactly 2 vertices and each vertex belongs to exactly one edge. Multiplication of basis elements is defined by concatenating diagrams: Assume we are given two basis elements, say a and b. First, draw an edge from bottom vertex i of a to top vertex i of b (for each i = 1, …, r). This produces a diagram which is almost of the desired form except that there may be cycles not attached to any of the (new) top and bottom vertices. Denote the number of these cycles by d. Then delete all cycles; the result is a basis element, say c. Now the product ab is by definition x^{d}c.

Of course, for a field element, say δ, the Brauer algebra B(r, δ) is defined by using δ instead of x, that is, by forming the quotient of B(r, x) modulo x − δ. (For Brauer’s application to orthogonal or symplectic groups one has to choose δ to be an integer.)

Let us illustrate this definition by an example. We multiply two elements in B(4, δ):

[figure: •; •; •; •; •; •; •; •; ·; •; •; •; •; •; •; •; •; =; δ^{1}·; •; •; •; •; •; •; •; •]

Proof of theorem 1.3.

It has been shown in [6] (and later on in a different way also in [10]) that Brauer algebras are cellular for any choice of r and δ. In the following we are going to use the notation of [8], where cell structures are written as ’inflations’, that is, by data of linear algebra. The cell chain exhibited both in [6] and in [10] shows that as a free k–module, A is equal to

kΣ_{r} ⊕ (V_{r−2} ⊗ V_{r−2} ⊗ kΣ_{r−2}) ⊕ (V_{r−4} ⊗ V_{r−4} ⊗ kΣ_{r−4}) ⊕ …,

and the iterated inflation starts with kΣ_{r}, inflates it along V_{r−2} ⊗ V_{r−2} ⊗ kΣ_{r−2} and so on, ending with an inflation of k = kΣ_{1} or k = kΣ_{0} as bottom layer (depending on whether r is odd or even).

Now we have to distinguish several cases.

If r is even and δ equals zero, then the above cell chain contains a nilpotent ideal, namely the layer V_{0} ⊗ V_{0} ⊗ k, whose elements are the scalar multiples of the diagram d

[figure: d :=; •; •; •; •; …; •; •; •; •; •; •; …; •; •]

whose square is zero. Hence this chain is not a heredity chain. By theorem 1.1, the algebra cannot be quasi–hereditary.

For any choice of parameters, the Brauer algebra has a quotient kΣ_{r}, the group algebra of the symmetric group on r letters, and this quotient actually arises as B(r, δ)/J for some ideal in the cell chain. In particular, factoring out J of the cell chain of B(r, δ) yields a cell chain of kΣ_{r}. If the characteristic of k divides the order of Σ_{r}, that is, if 2 ≤ char(k) ≤ r, the group algebra kΣ_{r} has infinite global dimension. By theorem 1.1, the length of the cell chain of kΣ_{r} and hence also that of B(r, δ) must be strictly bigger than the number of simple modules. Therefore, again by theorem 1.1, the algebra B(r, δ) cannot be quasi–hereditary.

If the characteristic of k is zero or bigger than r, then all the above group algebras kΣ_{r−2l} are semisimple and the subquotients in the cell chain of B(r, δ) are inflations of the simple components of these algebras. The multiplication in such a subquotient is non–zero if and only if there is an idempotent in this layer (see e.g. proposition 4.1 of [7]). Hence the given cell chain is already a heredity chain (and B(r, δ) is quasi– hereditary), provided we can show that the multiplication in each layer is non–zero. To do this, we fix a layer in the filtration, say coming from V_{r−2l} ⊗ V_{r−2l} ⊗ kΣ_{r−2l} for some l. Denote V_{r−2l} by V. By the construction in [6] or [10], the chosen layer has the form V ⊗ V ⊗ J, where the k–space V has a basis consisting of configurations formed by l connecting 2l out of 2r vertices. And J is some subquotient in a cell chain of kΣ_{r−2l}, hence by our present assumption, J is just a full matrix algebra over k. Multiplication in this layer (i.e. neglecting terms in lower layers of the cell chain) is of the form (a ⊗ b ⊗ x)(c ⊗ d ⊗ y) = a ⊗ d ⊗ xφ(b, c)y, where a, b, c, d are in V, x, y are in J and the bilinear form φ : V × V → J assigns to a pair b, c of configurations a scalar multiple of a some element in J. If r is strictly bigger than 2l, than we choose:

[figure: b :=; c :=; •; •; •; •; …; •; •; •; •; …; •; •; •; •; •; …; •; •; •; …]

Then (see e.g. [10], proof of lemma 5.3) φ(b, c) is a non–zero element in J and we can choose elements x and y in the simple algebra J such that the whole product (a ⊗ b ⊗ x)(c ⊗ d ⊗ y) = a ⊗ d ⊗ xφ(b, c)y becomes non–zero. This gives us a non–zero structure constant in case r > 2l. In the remaining case r = 2l we have to use the assumption that δ is not zero. Then choosing the above diagram d for both b and c we still get a non–zero structure constant. This finishes the proof.

Now we turn to the partition algebras defined in [11]. These algebras are of interest in statistical mechanics [12]. Let us recall some definitions from [11].

Let M be a finite set. We denote by E_{M} the set of all equivalent relations on, or equivalently all partitions of the set M:

\begin{aligned}E_{M} := & \{ρ = ((M_{1})(M_{2}) ⋯ (M_{i}) ⋯) | ∅ ≠ M_{i} ⊂ M, ∪_{i}M_{i} = M, \\ & M_{i} ∩ M_{j} = ∅ (i ≠ j)\} \end{aligned}

For example, we take M = {1, 2, 3}, then

E_{M} = {(123), (1)(23), (12)(3), (13)(2), (1)(2)(3)}.

if μ ∈ E_{M} and ν ∈ E_{N}, then we define μ · ν ∈ E_{M∪N} to be the smallest ρ in E_{M∪N} such that ρ contains both μ and ν.

To definine the partition algebras, we put

M = {1, 2, ⋯, n, 1′, 2′, …, n′}, M′ = {1′, 2′, ⋯, n′, 1′′, 2′′, …, n′′}.

Let f : E_{M} × E_{M} ⟶ Z be such that f(μ, ν) is the number of parts of μ · ν ∈ E_{M∪M′} (note that |M ∪ M′| = 3n) containing exclusively elements j′ with a single prime.

For example, in case n = 3, ((123)(1′2′)(3′)) · ((1′)(2′3′)(1′′)(2′′)(3′′)) = ((123)(1′2′3′)(1′′)(2′′)(3′′)) and f(μ, ν) = 1.

Let C : E_{M} × E_{M} ⟶ E_{M} be such that C(μ, ν) is obtained by deleting all single primed elements of μ · ν (discarding the f(μ, ν) empty brackets so produced), and replacing all double primed elements with single primed ones.

The partition algebra P (n, q) is defined as follows.

Let k be a field, q ∈ k, and let n be a natural number. We define a product on E_{M}:

E_{M} × E_{M} ⟶ E_{M} (μ, ν) ⟼ μν = q^{f(μ,ν)}C(μ, ν)

This product is associative. Let P (n, q) denote the vector space over k with the basis E_{M}. Then, by linearly extending the product on E_{M}, the vector space P (n, q) becomes a finite dimensional algebra over k with the above product. P (n, q) is called the partition algebra.

If we take B_{M} = {ρ ∈ E_{M} | each part of ρ has exactly two elements of M} and define the product of two elements in B_{M} in the same way as in P (n, q), then the subspace B(n, q) of P (n, q) with the basis B_{M} becomes a finite dimensional algebra. This is just the Brauer algebra B(n, q). Similarly, if we take P_{M} = {ρ ∈ B_{M} | ρ is planar}, then we get the Temperley–Lieb algebra A_{n}(q) with the basis P_{M}.

The proof of theorem 1.4 is quite similar to that of theorem 1.3. The partition algebras have been shown to be cellular in [15], and the proof given there contains all the ingredients for adapting the above proof.

Along similar lines (using again [6] or adapting [10]) one also can show the following extensions of results of Westbury [14] and of Graham and Lehrer [6].

Proposition 4.1 Let A_{n}(δ) be a Temperley–Lieb algebra of type A. Then A_{n}(δ) is quasi–hereditary if and only if δ ≠ 0 or n odd.

Proposition 4.2 Let J_{n}(δ) be the Jones’ annular algebra. Then J_{n}(δ) is quasi– hereditary if and only if δ ≠ 0.

Finally we give an example of an application of proposition 1.2.

Proposition 5.3 of [7] states that a Brauer tree algebra is cellular if and only if the Brauer tree is a straight line. For example, associated with a straight line consisting of three edges there is an algebra A which as left module over itself looks as follows:

AA = \begin{matrix}1 & & & 2 & & & 3 \\ 2 & ⊕ & 1 & & 3 & ⊕ & 2 \\ 1 & & & 2 & & & 3 \end{matrix}

Slightly changing (not: deforming) the structure constants we get another algebra, say B, which looks as follows:

BB = \begin{matrix}1 & & & 2 & & & 3 \\ 2 & ⊕ & 1 & & 3 & ⊕ & 2 \\ 3 & & & 2 & & & 1 \end{matrix}

Both algebras A and B are self–injective. But A is cellular as mentioned before, whereas B has Cartan determinant zero, hence by 1.2 it cannot be cellular.

References

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[6] J.J.Graham and G.I.Lehrer, Cellular algebras. Invent. Math. 123, 1–34 (1996).

[7] S.König and C.C.Xi, On the structure of cellular algebras. In: I.Reiten, S.Smalø and Ø.Solberg (Eds.): Algebras and Modules II. Canadian Mathematical Society Conference Proceedings Vol. 24 (1998), 365–386.

[8] S.König and C.C.Xi, Cellular algebras: inflations and Morita equivalences. Preprint 97–078, Bielefeld, 1997. To appear in Journal of the London Math.Soc.

[9] S.König and C.C.Xi, On the number of cells of a cellular algebra. Preprint 97–126, Bielefeld, 1997. To appear in Communications in Algebra.

[10] S.König and C.C.Xi, A characteristic free approach to Brauer algebras. Preprint 98–005, Bielefeld, 1998.

[11] P.P.Martin, The structure of the partition algebras. J. Algebra 183, 319–358 (1996).

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[15] C.C.Xi, Partition algebras are cellular. To appear in Compos. Math.

Steffen König, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany,

   koenig@mathematik.uni-bielefeld.de

Changchang Xi, Department of Mathematics, Beijing Normal University, 100875 Beijing, P.R. China,

   xicc@bnu.edu.cn