Cellular Bases for Algebras with a Jones Basic Construction

We construct analogues for the Brauer, BMW, partition, and Jones–Temperley–Lieb algebras of the Murphy basis of the Hecke algebra of the symmetric group. The bases are cellular bases indexed by paths on branching diagrams, and compatible with restriction of cell modules. The Jucys–Murphy elements for each class of algebras act by triangular matrices on the Murphy basis.


Introduction
This paper develops analogues for the Brauer, BMW, partition, and Jones-Temperley-Lieb algebras of the Murphy basis [33] of the Iwahori-Hecke algebras H n (q 2 ) of the symmetric groups S n . The Murphy basis of H n (q 2 ) has many remarkable properties. First, it is a cellular basis in the sense of Graham and Lehrer [16]. For cellular algebras (i.e. algebras with a cellular basis) in general, one can define a family of modules known as cell modules. Then for any specialization of the cellular algebra over a field, all simple modules appear as quotients of the cell modules, and the algebra is semisimple exactly when all the cell modules are simple. The Murphy basis has a number of additional special properties. The Jucys-Murphy elements of the Hecke algebra act in a triangular fashion on the Murphy basis; this action allows the construction of the seminormal representations and the classification of simple modules and blocks, see [28], Chapter 3. Moreover, the Murphy basis is well adapted to the tower of Hecke algebras (H n (q 2 )) n≥0 , as will be explained below.
Several papers in the literature have aimed at generalizations or axiomatizations of the Murphy basis, the seminormal basis, and the set of Jucys-Murphy elements, for example [9,14,29,37]. The present paper is also a contribution to this theme.
In order to develop analogues of the Murphy basis for the Brauer algebras, etc., we first develop a new interpretation of the Murphy basis of the Hecke algebras H n (q 2 ). We begin with observing that the Hecke algebras H n = H n (q 2 ) defined over the generic ground ring Z[q ±1 ], with q an indeterminant, have the following properties: • The Hecke algebras H n are cyclic cellular algebras. This means that each cell module is a cyclic H n -module. • The cellular structures on the tower of algebras H n are coherent. This means that a cell module of H n , restricted to the H n−1 or induced to H n+1 has a "cell filtration", that is, a filtration with cell modules as subquotients.
We therefore begin by studying coherent towers (A n ) n≥0 of cyclic cellular algebras in general. We first obtain some rather simple general results about cellular bases in such towers, in Section 3. We observe that there exists a system of "branching factors" associated to each edge of the generic branching diagram for the tower. In fact, there is a system of "d-branching factors" related to the cell filtration of restricted cell modules, and a system of "u-branching factors" related to the cell filtration of induced modules. Then we note that an ordered product of d-branching factors along paths on the generic branching diagram determine bases of each cell module of each A n as well as a cellular basis of each A n .
In Section 4, we consider the tower of Hecke algebras H n (q 2 ). Recalling that this is a coherent tower of cyclic cellular algebras, we compute branching factors for reduced and induced cell modules. We show that the bases obtained via ordered products of d-branching factors coincide with the Murphy bases.
This construction could be regarded as a cellular analogue of the constructions in [34][35][36]38]. Each of the examples that we want to study is a tower (A n ) n≥0 of algebras that generically is obtained from another tower (Q n ) n≥0 by repeated Jones basic constructions. For the Brauer, BMW, or partition algebras, the tower (Q n ) n≥0 is a tower of Hecke algebras or symmetric group algebras. For the Jones-Temperley-Lieb algebras defined over a ring R, the tower (Q n ) n≥0 is just the constant sequence R. Cellularity of such a tower (A n ) n≥0 was previously studied in [13,14]. Here we augment the framework of those papers by the assumption that the algebras Q n are cyclic cellular. It follows easily from the previous work in [13,14] that the tower (A n ) n≥0 is a coherent tower of cyclic cellular algebras. We show here that branching factors, and therefore cellular bases for the tower (A n ) n≥0 can be obtained by explicit formulas from branching factors for the tower (Q n ) n≥0 . These are our analogues of Murphy's bases.
Finally, in Section 5, we apply our results to the Brauer algebras, Birman-Murakami-Wenzl (BMW) algebras, Jones-Temperley-Lieb algebras, and partition algebras, and obtain explicit Murphy bases for each of these families of algebras.
A complication in our approach to the Murphy bases is that the results of [13,14] do not apply to Jones the basic construction algebras defined over their generic ground ring, say R 0 , but only to the algebras defined over R 0 [δ −1 ], where δ is the "loop parameter"; see Section 5.1, where a mistake in [13,14] is discussed and corrected. Therefore, the Murphy bases appear a priori to be bases only for the algebras defined over R 0 [δ −1 ]. However, as the bases are explicit, we can check for each of our examples that the Murphy basis is actually a basis for the algebras defined over the generic ground ring R 0 . A considerable portion of the work in Section 5 is devoted to this final step. For the Jones-Temperley-Lieb algebras, our Murphy basis turns out to be none other than the usual diagram basis.
Our Murphy bases are "families of path bases", in the sense of Definition 3.8. It is known that for each of our examples, the tower (A n ) n≥0 has a family of Jucys-Murphy elements in the sense of [14]. It follows from [14,Propositions 3.6 and 3.7] that the Jucys-Murphy elements act triangularly on our Murphy bases. Hence, Mathas' theory of cellular algebras with Jucys-Murphy elements and seminormal representations [29] can be applied. It is shown in [1] that triangularity actually holds with respect to dominance order, strengthening the triangularity statements of [13].
The Murphy bases for the partition algebras given in Theorem 6.26 have been used to obtain an analogue of the Young seminormal form for partition algebras in [10]. In the cases of the Brauer and BMW algebras, our results recover the Murphy type bases obtained in [9]; however, the construction here is simpler, and does not involve computations in the braid group. Rui and Si [37] used the path bases from [9] to compute Gram determinants for cell modules of the BMW algebras, and to obtain definitive semisimplicity results.
The Murphy bases of the (abstract) Brauer algebras are used in [2] to produce integral cellular bases of Brauer's centralizer algebras acting on symplectic or orthogonal tensor space. The analogous result holds also for the walled Brauer algebras acting on mixed tensor space [2,38].
Murphy bases are used in [3] to construct skew cell modules of diagram algebras, which are analogous the the skew Specht modules of the symmetric groups.
There are many other examples of coherent towers of cellular algebras obtained by repeated Jones basic constructions, see [13]. Our method of obtaining explicit Murphy type bases as ordered products of branching factors has been applied to the walled Brauer algebras in [2] and should also apply to such examples as the cyclotomic BMW algebras.

Cellular Algebras
Cellular algebras were defined by Graham and Lehrer [16]. In this paper we use a slightly weaker version of cellularity which was introduced in [12,13]. (1) A is a unital R-algebra and * : A → A is an algebra involution, that is an R-linear anti-automorphism of A such that (x * ) * = x for x ∈ A; (2) ( A, ) is a finite partially ordered set, and A λ , for λ ∈ A, is a finite indexing set; (3) The set A = c λ st λ ∈ A and s, t ∈ A λ , is an R-basis for A, for which the following conditions hold: (a) Given λ ∈ A, t ∈ A λ , and a ∈ A, there exist coefficients r v (t, a) ∈ R, for v ∈ A λ , such that, for all s ∈ A λ , c λ st a ≡ v∈ A λ r v (t, a)c λ sv mod A λ , (2.1) where A λ is the R-module generated by c μ st μ ∈ A, s, t ∈ A μ and μ λ . (b) If λ ∈ A and s, t ∈ A λ , then (c λ st ) * ≡ (c λ ts ) mod A λ . The tuple (A,  * , A, , A ) is a cell datum for A.
If A is an algebra with cell datum (A, * , A, , A ) we will frequently omit reference to the cell datum for A and simply refer to A as a cellular algebra. The basis A is called a cellular basis of A.
From points 3(a) and 3(b) of the definition of cellularity, we have for a ∈ A and s, t ∈ A λ , An order ideal ⊂ A is a subset with the property that if λ ∈ and μ λ, then μ ∈ . It follows from the axioms of a cellular algebra that for any order ideal in A, A = span c λ st λ ∈ , s, t ∈ A λ is a two sided ideal of A. In particular A λ and A λ = span c μ st μ ∈ A, s, t ∈ A μ and μ λ are two sided ideals.

Definition 2.2 Let
A be a cellular algebra, and λ ∈ A. The cell module λ A is the right A-module defined as follows. As an R-module, λ A is free with basis indexed by A λ , say {c λ t | t ∈ A λ }. The right A-action is given by where the coefficients r v (t, a) are those of Eq. (2.1).
Thus, for any s ∈ A λ , the map c λ t → c λ st + A λ is an injective A-module homomorphism of the cell module λ A into A λ /A λ . If A is an R-algebra with involution * , then * induces functors M → M * interchanging left and right A-modules, and taking A-A bimodules to A-A bimodules. We identify M * * with M via x * * → x and for modules A M and N A we have (M ⊗ R N) * ∼ = N * ⊗ R M * , as A-A bimodules, with the isomorphism determined by (m ⊗ n) * → n * ⊗ m * . For a right Amodule M A , using both of these isomorphisms, we identify (M * ⊗ M) * with M * ⊗ M * * = M * ⊗ M, via (x * ⊗ y) * → y * ⊗ x. Now we apply these observations with A a cellular algebra and λ A a cell module. The assignment determines an A-A bimodule isomorphism from A λ / A λ to ( λ A ) * ⊗ R λ A . Moreover, we have * • α λ = α λ • * , which reflects the cellular algebra axiom (c λ st ) * ≡ c λ ts mod A λ . When it is necessary to identify the algebra we are working with, we will write α A λ instead of α λ . The importance of the maps α λ for the structure of cellular algebras was stressed by König and Xi [23,24].

Generic Ground Rings
The most important examples of cellular algebras are actually families A S of algebras defined over various integral ground rings S, possibly containing distinguished elements (parameters) which enter into the definition of the algebras. The prototypical example is the Iwahori-Hecke algebra of the symmetric group H k = H k (q 2 ), which can be defined over any integral domain S with a distinguished invertible element q; see Section 4.2 for the detailed description.
Again in the most important examples, there is a "generic ground ring" R for A with the following properties: (1) For any integral ground ring S there is a ring homomorphism from R to S, and the algebra over S is the specialization of the algebra over R, that is (2) R has characteristic zero, and if F denotes the field of fractions of R, then A F is split semisimple; and the cell modules λ A F are the simple A F modules.
For example, the generic ground ring for the Iwahori-Hecke algebra is Z[q, q −1 ], where q is an indeterminant over Z. Indeed, the entire point of the theory of cellular algebras is to provide a setting for a modular representation theory of important classes of algebras such as the Iwahori-Hecke algebras, Brauer algebras, Birman Murakami Wenzl algebras, etc. The cell modules of A R are integrally defined versions of the simple modules of A F which specialize to A k -modules for any field k (with appropriate parameters). The simple A k modules are found as quotients of the cell modules λ A k . See [16,28] for details.

Equivalent Cellular Bases
A cellular algebra A with cell datum (A, * , A, , A ) always admits different cellular bases B. In fact, any choice of an R-basis in each cell module of A can be globalized to a cellular basis of A, see Lemma 2.3. We say that a cellular basis is a cellular basis of A equivalent to the original cellular basis A .

Extensions of Cellular Algebras
Definition 2.4 Suppose A is a unital R-algebra with involution * , and J is an * -invariant ideal. Let us say that J is a cellular ideal in A if it satisfies the axioms for a cellular algebra (except for being unital) with cellular basis {c λ st | λ ∈ J and s, t ∈ J λ } ⊆ J and we have, as in point (3a) of the definition of cellularity, For μ ∈ J , the cell modules μ A and μ J can be identified; this is because of condition (2.2) in the definition of cellular ideals. We have A μ = J μ ⊆ J , and similarly for A μ .

Cellular Algebras with Cyclic Cell Modules
Definition 2.7 A cellular algebra is said to be cyclic cellular if every cell module is cyclic.

Remark 2.8
For examples of cyclic cellular algebras, see Section 6. Cyclic cellularity was also introduced in [11], and some additional examples, beyond those studied here are presented in that paper. (1) A is cyclic cellular.
(2) For each λ ∈ A, there exists an element c λ ∈ A λ with the properties: For a cyclic cellular algebra A and λ ∈ A, we let δ λ A denote a generator of the cell module λ A . The element c λ in Lemma 2.9 can be taken to be any lifting to A λ of α −1 ((δ λ A ) * ⊗δ λ A ). When it is necessary to identify the algebra we are working in, we write c A λ . We record a version of Lemma 2.3 that is adapted to the context of cyclic cellular algebras: , so this follows immediately from Lemma 2.3.

Remark 2.11 (Extensions of cyclic cellular algebras) Let
A be an algebra with involution over R, let J be a cellular ideal in A and suppose that H = A/J is cellular. If both J and H are cyclic cellular, then so is A. This is evident from Lemma 2.5 and Remark 2.6. Let

Bases in Towers of Cellular Algebras
In this section we obtain some elementary results on bases in towers of cellular algebras. Then one can associate to each edge λ → μ in the branching diagram H for the tower (H F N ) n≥0 of split semisimple algebras a "branching factor" d λ→μ . The ordered product of these branching factors along paths in H determines a basis of each cell module of each algebra H n . The collection of these bases is a "family of path bases," which means that the bases behave well with respect to restriction to smaller algebras in the tower, see Definition 3.8. The existence of these special bases of the cell modules depends on the existence of cell filtrations for the restricted modules Res

Coherence Conditions for Towers of Cellular Algebras
If A is a cellular algebra over R, λ ∈ A, and N ⊆ M is an inclusion of right A-modules, with subquotients isomorphic to cell modules. Say that the filtration is order preserving if Observe that all the modules occurring in a cell filtration are necessarily free as Rmodules. Evidently, a given cell module occurs at most once as a subquotient in an order preserving cell filtration.
Here and in the remainder of the paper, we will consider increasing sequences of cellular algebras over an integral domain R. Whenever we have such a sequence of algebras, we assume that all the inclusions are unital and that the involutions are consistent; that is the involution on H i+1 , restricted to H i , agrees with the involution on H i .   The tower is called strongly coherent if the cell filtrations can be chosen to be order preserving.
In the examples of interest to us, we will also have uniqueness of the multiplicities of the cell modules appearing as subquotients of the cell filtrations, and Frobenius reciprocity connecting the multiplicities in the two types of filtrations, see Corollary 3.5.
Only the filtrations of restricted modules Res

Inclusions Matrices, Branching Diagrams, and Cell Filtrations
We recall the notion of an inclusion matrix for an inclusion of split semisimple algebras over a field. Suppose A ⊆ B are finite dimensional split semisimple algebras over a field F (with the same identity element). Let {V λ | λ ∈ A}, be the set of isomorphism classes of simple A-modules and {W μ | μ ∈ B} the set of isomorphism classes of simple B-modules. We associate a B × A inclusion matrix ω to the inclusion A ⊆ B, as follows.
For each μ ∈ B and λ ∈ A, define ω(μ, λ) to be the multiplicity of V λ in a direct sum decomposition of Res B A (W μ ). Say that the inclusion A ⊆ B is multiplicity-free if the inclusion matrix has entries in {0, 1}. Now consider an increasing sequence (B n ) n≥0 of split semisimple algebras over a field F . Suppose that all the inclusions B n ⊆ B n+1 are multiplicity-free. (This suffices for the examples we want to treat). The branching diagram B of the sequence (B n ) n≥0 is a graph with vertex set n≥0 B n , where B n indexes the set of isomorphism classes simple B nmodules. Fix n ≥ 0 and let ω n denote the inclusion matrix of B n ⊆ B n+1 . For λ ∈ B n and μ ∈ B n+1 , the branching diagram has a unique edge connecting λ and μ if ω n (μ, λ) = 0. In this case, we write λ → μ. In our examples we have B 0 = F , so that B 0 is a singleton.

Path Bases and Cell Filtrations
We consider an increasing sequence (H n ) n≥0 of cellular algebras over an integral domain R with field of fractions F . We assume the following conditions are satisfied: (1) H 0 = R, and H F n is split semisimple for all n. (2) The branching diagram H for the tower (H F n ) n≥0 is multiplicity free. We let (H n , * , , H n , H n ) denote a cell datum for H n . Denote the unique element of H 0 by ∅.
Since for all n ≥ 0 and all μ ∈ H n , the rank of the cell module μ H n equals the dimension over F of ( μ H n ) F , and the latter is the number of paths on the branching diagram H from ∅ to μ, we can take H μ n to be the set of such paths. We define a total order on paths on H as follows: Definition 3.7 Let s = (λ (l) , λ (l+1) , . . . , λ (m) ) and t = (μ (l) , μ (l+1) , . . . , μ (m) ) be two paths from H l to H m . Say that s precedes t in reverse lexicographic order (denoted s t) if s = t, or if for the last index j such that λ (j ) = μ (j ) , we have λ (j ) < μ (j ) in H j .   Proof The first statement is proved in [14, Proposition 2.18]; we will give more concrete construction of path bases in Section 3.5, in the case that all the algebras H n are cyclic cellular.
For the converse, suppose we are given a family of path bases {b t | t ∈ H μ n } for n ≥ 0 and for μ ∈ H n . For n ≥ 1 and μ ∈ H n , let λ (1) , . . . , λ (s) be a list of {λ ∈ H n−1 | λ → μ}, ordered so that i < j if λ (i) λ (j ) . Let It follows from the definition of a path basis that N j is an H n−1 submodule of Res H n H n−1 ( μ H n ), and that N j /N j −1 ∼ = λ (j ) H n−1 . If {λ ∈ H n−1 | λ → μ} is totally ordered, then clearly this cell filtration is order preserving.

Cyclic Cellularity and Branching Factors
Suppose that A ⊆ B are cyclic cellular algebras over an integral domain R. We have the following observations regarding cell filtrations of restricted and induced modules: (1) Let μ ∈ B and suppose that Res B A ( μ B ) has a cell filtration: (2) Let λ ∈ A and suppose Ind B A ( λ A ) has a cell filtration: Let δ λ A be a generator of the A-module λ A ; then δ λ A ⊗1 is a generator of the B-module is a cyclic B-module, there exists an element We call the elements d B λ→μ and u B λ→μ branching factors. They are not canonical, but in the examples in Sections 4 and 6, it will be possible to make natural choices for these elements.
Proof Evidently, statement (1) implies statement (2). We prove both statements by induction on n, the case n = 1 being evident. Fix n > 1 and suppose the statements hold for cell modules of H k for 1 ≤ k ≤ n − 1. For each i we have an isomorphism of H n−1 -modules is a basis of M j . But this basis is equal to δ μ H n d t t ∈ H μ n and t(n − 1) ∈ λ (1) , λ (2) , . . . , λ (j ) .

Example: The Iwahori-Hecke Algebra of the Symmetric Groups
In this section, we apply the theory of Section 3 to the Iwahori-Hecke algebra of the symmetric groups. In particular, we recall that the sequence of Hecke algebras is a coherent tower of cyclic cellular algebras, and we compute the branching factors for reduced and induced cell modules. We show that the path bases obtained via ordered products of branching factors coincide with the Murphy bases.

Combinatorics
Let n denote a non-negative integer and S n be the symmetric group acting on {1, . . . , n} on the right. For i an integer, 1 ≤ i < n, let s i denote the transposition (i, i + 1). Then S n is presented as a Coxeter group by generators s 1 , s 2 , . . . , s n−1 , with the relations We will assume familiarity with the usual combinatorics associated with the symmetric groups: compositions and partitions, and their diagrams, tableaux, dominance order, etc. We will follow the terminology and notations of [28], especially Section 3.1. Our convention regarding diagrams is illustrated by the example: for the partition λ = (3, 2), its diagram is [λ] = The notation λ n indicates that λ is a partition of n. The diagram of a partition is commonly called a Young diagram. We denote the set of Young diagrams of size n by Y n For a composition λ of size n, let T (λ) denote the set of all λ-tableaux (possibly with repeated entries) and T 0 (λ) the set of λ-tableaux in which each number 1, 2, . . . , n appears exactly once. For a partition λ, write T Std (λ) for the set of standard λ-tableaux. If t ∈ T 0 (λ) and 1 ≤ k ≤ n, we write node t (k) for the node in λ containing the entry k, row t (k) for the row coordinate of k in t and col t (k) for the column coordinate of k in t.
The symmetric group S n acts freely and transitively on T 0 (λ), on the right, by acting on the integer labels of the nodes of [λ]. For example, Let t λ denote the standard λ-tableau in which 1, 2, . . . , n are entered in increasing order from left to right along the rows of [λ]. Thus in the previous example where n = 6 and λ = (3, 2, 1), For each t ∈ T 0 (λ), let w(t) denote the unique permutation such that t = t λ w(t). The Young subgroup S λ is defined to be the row stabilizer of t λ in S n . For instance, when n = 6 and λ = (3, 2, 1), as in (4.1), then S λ = s 1 , s 2 , s 4 .
Let λ n and let t ∈ T 0 (λ). Let α be an addable node of λ. Then we write t ∪ α for the tableau of shape λ ∪ α which agrees with t on the nodes of λ and which has the entry n + 1 in node α. If t is a standard λ-tableau, then the node of t containing the entry n is a removable node β of λ. Write t = t ↓ n−1 for the standard tableau of shape λ \ β obtained by removing the node β.

Iwahori-Hecke Algebras of the Symmetric Group
Let R be an integral domain and q be a unit in R. Let H n = H n (q 2 ) denote the Iwahori-Hecke algebra of the symmetric group, which is presented by the generators T 1 , . . . , T n−1 , and the relations If we need to refer explicitly to the ground ring R, we write H n (R; If R is a field of characteristic zero and q is not a proper root of unity, then it is known that each of the algebras H n is split semisimple with simple modules labeled by the set Y n of Young diagrams of size n; moreover the branching diagram H of the tower (H n ) n≥0 is Young's lattice; namely for Young diagrams λ and μ with |μ| = |λ| + 1, we have λ → μ if and only if μ is obtained from λ by adjoining one node.
If μ ∈ H n , define H μ n to be the set of paths (μ (0) = ∅, μ (1) , . . . , μ (n) = μ) on Young's lattice H from ∅ to μ. There is an evident bijection between the set of such paths and standard tableaux of shape λ.
If μ ∈ H n , let In the following statement, recall that for λ ∈ H i and t ∈ T Std (λ), w(t) denotes the unique permutation in S i such that t λ w(t) = t.  We let {m λ t | t ∈ T Std (λ)} denote the basis of the cell module λ H n derived from the Murphy basis. Then we have m λ t = m λ t λ T w(t) . In particular, we see that the Hecke algebra is a cyclic cellular algebra, with λ H n generated by m λ t λ . The bimodule isomorphism , so plays the role of the element c λ in Section 2.5. We record this as a corollary:

Cell Filtrations and Branching Factors
Our next task is to recall that the sequence of Hecke algebras (H n ) n≥0 is a strongly coherent tower of cellular algebras, and to determine the branching factors d (n) μ→λ and u (n) μ→λ when μ → λ. First we discuss the cell filtrations of restrictions of cell modules and the branching factors d (n) μ→λ .
Theorem 4.4 (Jost, Murphy) Let n ≥ 1 and λ ∈ H n . Let λ H n be the corresponding cell module of H n . Then Res H n H n−1 ( λ H n ) has an order preserving filtration by cell modules of H n−1 .
Jost [22] has shown, using the Dipper-James description of Specht modules of the Hecke algebras [5], that the restriction of a Specht module has a filtration by Specht modules. Together with Murphy's result that the cell modules of the Hecke algebras can be identified with the Specht modules [33,Theorem 5.3], this shows that the restriction of a cell module has a cell filtration. A direct proof of Theorem 4.4 using Murphy's description of the cellular structure is given in [15].
We now give a more precise description of the cell filtration in Theorem 4.4. Let α 1 , . . . , α p be the list of removable nodes of λ, listed from bottom to top and let μ (j ) = λ \ α j . Thus i ≤ j if and only if μ (i) μ (j ) . Let N 0 = (0) and for 1 ≤ j ≤ p, let N j be the R-submodule of λ H n spanned by by the basis elements m λ t such that the node containing n in t is one of α 1 , . . . , α j . Then we have The explicit form of the assertion of Theorem 4.4 is that the N j are H n−1 -submodules of We can now determine the branching factors d More explicitly, let a(α) be the entry of t λ in the node α. Then Proof Under the isomorphism This means that we can chose d (n) μ→λ = T w(t μ ∪α) . Now it is straightforward to check that w(t μ ∪ α) = (n, n − 1, . . . , a(α)), so that d (n) μ→λ = T (n,n−1,...,a(α)) = T a(α),n .
Let λ ∈ H n and let t be a standard λ-tableau. We identify t with a path on the branching diagram (1) .
(4.6) Lemma 4.6 Let λ be a partition of n, let α be a removable node of λ, and let μ = λ \ α. Let a(α) be the entry of t λ in the node α. Let s ∈ T 0 (μ) be a μ-tableau. Then Proof We have

Lemma 4.7 Let λ be a partition of n and let t be a standard λ-tableau. Then
Proof Let α be the node of λ containing the entry n in t and let μ = λ \ α. Let t be the standard μ tableau obtained from t by removing the node α. Let a(α) be the entry of t λ in the node α. Then t = t ∪ α, so by the previous lemma and Corollary 4.5, μ→λ T w(t ) . By induction, we obtain the desired formula T w(t) = d t .

Corollary 4.8 The bases of the cell modules and the cellular basis of the Hecke algebra H n given in Proposition 3.10 and Corollary 3.11 coincide with the Murphy bases:
Next we turn to the cell filtration of induced cell modules and the branching factors u (n) μ→ν .  Let α = α 1 , α 2 , . . . , α p = ω be the list of addable nodes of μ, listed from top to bottom. Let ν (i) = μ ∪ α i . Note that i ≤ j if and only if ν (i) ν (j ) . The cell modules of H n+1 occurring as subquotients in the cell filtration of Ind One proof of Theorem 4.9 is obtained by combining [5,Sect. 7] with [33,Theorem 5.3]. A different proof was recently given by Mathas [30]; this proof is based on Murphy's Theorem 4.12 on the existence of a cell filtration of permutation modules of H n . We are going to sketch Mathas' proof in order to point out how the branching factors u (n+1) μ→ν can be extracted from it.

Definition 4.11
Let λ, μ n and T : λ → N be a λ-tableau. Then: (2) T is semistandard if the entries of T are weakly increasing along each row from left to right and strictly increasing along each column from top to bottom.
Let λ, μ n and t ∈ T Std (λ). Define μ(t) to be the tableau obtained from t by replacing each entry j in t with the row index of the entry j in t μ . If T SStd (1) M μ is free as an R-module, with basis Since H n+1 is free of rank n+1 as a left H n -module, it follows that the induction functor Ind We will write Ind for this functor in the following discussion. Because of exactness, we have In particular Mathas' proof of Theorem 4.9 proceeds by exhibiting a cell filtration of Another consequence of the freeness of H n+1 as a left H n -module is the following: if M is a right ideal in H n , then We will simply identify Ind(M) with MH n+1 . Recall that ω denotes the lowest addable node of μ, and note that m μ = m μ∪ω . Hence, (4.14) To proceed, we need to relate semistandard tableaux of size n and type μ and semistandard tableaux of size n + 1 and type μ ∪ ω. Let l denote the number of non-zero parts of μ, so that ω = (l + 1, 1). If S is a semistandard tableau of shape λ and type μ, and β is an addable node of λ, then we define the semistandard tableau S ∪ β of shape λ ∪ β and type and S(β) = l + 1. We write T SStd μ∪ω (S) for the set of semistandard tableaux S ∪ β as β ranges over addable nodes of λ. It is easy to see that every U ∈ T SStd μ∪ω ( H n+1 ) is obtained as S ∪ β for some S and some β. Recall that S 1 , . . . , S m = T μ is the list of all semistandard tableaux of size n and type μ, listed so that Shape(S i ) Shape(S j ) implies i ≤ j . Mathas defines the following Rsubmodules of M μ∪ω :

Remark 4.16
The proof of this result in the published version of [30] has a gap, but this was repaired in the version posted to the arxiv.
By applying Theorem 4.12 to M μ∪ω , we see that M μ∪ω /N m−1 is free with basis (4.16) We can now exhibit an order preserving cell filtration of M μ∪ω /N m−1 ∼ = Ind( μ H n ). In the following, we write N = N m−1 . Recall that α = α 1 , α 2 , . . . , α p = ω is the list of addable nodes of μ listed from top to bottom and ν (j ) = μ ∪ α j . Let J 0 = (0) and for 1 ≤ i ≤ p, define J i ⊆ M μ∪ω /N by  (4.17) and This completes the sketch of Mathas' proof of Theorem 4.9. It remains to see how the cell filtration (4.17) carries over to Ind( μ H n ), and to identify the branching factors u Since m T μ t μ = m μ , the composite isomorphism is given by We need to examine how this isomorphism acts on the basis (4.16) of M μ∪ω /N . Let β be an addable node of μ and let ν = μ ∪ β. Suppose that β is in row r, and let In particular, D(ω) = 1.
The following lemma can be extracted from [30].

Cellularity and the Jones Basic Construction: A Correction
In [13,14], Goodman and Graber developed a theory of cellularity for algebras with a Jones basic construction. Examples of such algebras include the Birman-Murakami-Wenzl, Brauer, partition, and Jones-Temperley-Lieb algebras, among others. There was, however, a mistake in the proof in [13] that these algebras constitute coherent towers of cellular algebras. In this section, we will review the setting of [13,14], describe the error, and explain what needs to be done to correct it. The setting in [13], as modified in [14] is the following. First recall that an essential idempotent in an algebra A over a ring R is an element e such that e 2 = δe for some nonzero δ ∈ R. Let R be an integral domain with field of fractions F and consider two towers of algebras with common multiplicative identity, It is assumed that the two towers satisfy the following list of axioms: (1) There is an algebra involution * on ∪ n A n such that (A n ) * = A n , and likewise, there is an algebra involution * on ∪ n H n such that (H n ) * = H n . (2) A 0 = H 0 = R and A 1 = H 1 (as algebras with involution).
(3) For n ≥ 2, A n contains an essential idempotent e n−1 such that e * n−1 = e n−1 and A n /(A n e n−1 A n ) ∼ = H n as algebras with involution. (4) For n ≥ 1, e n commutes with A n−1 and e n A n e n ⊆ A n−1 e n . (5) For n ≥ 1, A n+1 e n = A n e n , and the map x → xe n is injective from A n to A n e n . (6) For n ≥ 2, e n−1 ∈ A n+1 e n A n+1 . n≥0 is a strongly coherent tower of cellular algebras.
Under these hypotheses, it is claimed in [13,14] that (A n ) n≥0 is a strongly coherent tower of cellular algebras. The strategy of the proof is to show by induction that the following statements hold for all n ≥ 0: • A n is a cellular algebra.
• For 2 ≤ n, J n = A n e n−1 A n is a cellular ideal in A n .
• For 2 ≤ n, the cell modules of J n are of the form = ⊗ A n−2 e n−1 A n , where is a cell module of A n−2 .
• The finite tower (A k ) 0≤k≤n is strongly coherent.
For n ≤ 1 these statements are evident. Assuming the statements hold for some fixed n ≥ 1, one first proves that J n+1 is a cellular ideal in A n+1 with cell modules of the form = ⊗ A n−1 e n A n+1 , where is a cell module of A n−1 . It follows from Lemma 2.5 that A n+1 is cellular.
It then remains to show that for each cell module of A n+1 , the restriction of to A n has an order preserving cell filtration, and that for each cell module of A n , the induction of to A n+1 has an order preserving cell filtration. In fact, we will go over the details of the proof of these last two steps below in Theorem 5.6. For now, we note that in the proof of the statement about induced modules, it was falsely claimed in [13], in the last paragraph on page 335, that if is a cell module of J n then J n = . In fact, this does not follow from the axioms (1)-(8) listed above, so it is necessary to add an additional axiom to our framework, as follows: (9) For n ≥ 2, e n−1 A n e n−1 A n = e n−1 A n .
From this, it follows that for a cell module = ⊗ A n−2 e n−1 A n of J n , we have J n = , and the proof in [13] can proceed as before. Let us now consider the applicability of the augmented framework axioms (1)-(9) to the principal examples considered in [13,14]. In fact, in each example, a stronger version of axiom (6) holds, namely e n−1 e n e n−1 = e n−1 and e n e n−1 e n = e n for n ≥ 2.
Thus for n ≥ 3, e n−1 A n e n−1 A n ⊇ e n−1 e n−2 e n 1 A n = e n−1 A n . Therefore, Axiom (6) reduces to the statement When A n is the n-th BMW, Brauer, partition, or Jones-Temperley-Lieb algebra defined over an integral ground ring R, we have A 1 = H 1 = R. Let δ be the non-zero element of R such that e 2 1 = δe 1 . Then we have e 1 A 2 e 1 A 2 = e 1 A 1 e 1 A 2 = e 2 1 A 2 = δe 1 A 2 , where we have used e 1 A 2 = e 1 A 1 = Re 1 . In each of these examples, e 1 A 2 is free as an Rmodule, and hence Axiom (9) holds if and only if δ is invertible in R. It follows that Axiom (9) does not hold when R is the generic ground ring, but it does hold when R is the generic ground ring with δ −1 adjoined.
In fact, for these algebras, it is false that (A n ) n≥0 is a coherent tower of cellular algebras, over the generic ground ring, but, by [13], as corrected above, it is true over the generic ground ring with δ −1 adjoined. This is illustrated by the example of the Jones-Temperley-Lieb algebras in the following section.

An example: The Jones-Temperley-Lieb Algebras
We first state an elementary result about the commutativity of specialization and induction.
Let A be an algebra over an integral domain R and let ϕ : R → k be a ring homomorphism from R to a field k. Write A k for A ⊗ R k, and for a right A-module M, write M k for the right A k -module M ⊗ R k. The algebra A 2 has two cell modules, each of rank 1. They are 0 = e 1 A 2 = Re 1 is three dimensional, as one sees by examining the generic branching diagram for the tower (A k n ) n≥0 . However, when k = Q and δ = 0, k . It follows from this and Corollary 5.2 that at least one of Ind( 0 ) or Ind( 1 ) fails to be free as an R-module, and in particular one of these induced modules does not have a cell filtration. (R 0 ; δ)) n≥0 over the generic ground ring R 0 = Z[δ] is not a coherent tower of cellular algebras.

Standing Assumptions
For the remainder of the paper we will work in the setting described by axioms (1)-(9) of Section 5.1, and assume in addition that (10) Each H n is a cyclic cellular algebra.

Cellularity of the Algebras A n
Next we review some of the consequence of our axioms that were obtained in [13,14], as corrected above in Section 5.1. In the following let (H i , * , H i , , H i ) denote the cell datum for H i .
(1) Each A n is a cellular algebra. In fact, this is demonstrated by showing that J n = A n e n−1 A n is a cellular ideal of A n . Since the quotient algebra H n = A n /J n is assumed to be cellular, it follows from Lemma 2.5 that A n is cellular. (2) The partially ordered set A n in the cell datum for A n can be realized as

Remark 5.4
The parameterization of A n given here differs from that used in [13,14].
Taking Axiom (10) into account, we obtain: The tower (A n ) n≥0 is a strongly coherent tower of cyclic cellular algebras.
Proof From [13,14], with the correction noted in Section 5.1, we have that the tower is a strongly coherent tower of cellular algebras. It remains to show that each A n is cyclic cellular. We prove this by induction on n. The statement is known for n = 0 and n = 1, since A 0 = R and A 1 = H 1 . Fix n ≥ 0 and assume the algebras A k for k ≤ n are cyclic cellular. The cell modules (λ,0) A n+1 are cell modules of H n+1 , so cyclic by axiom (9). For l > 0, we can take

(λ,l) A n
We suppose that generators δ λ H n of λ H n have been chosen for all n ≥ 0 and for all λ ∈ H n . We suppose also that H n -H n bimodule isomorphisms α λ : H (2) select generators δ (λ,l) A n for each cell module; (3) choose A n -A n bimodule isomorphisms (4) and finally choose elements c (λ,l) ∈ A (λ,l) such that When l = 0, we identify (λ,0) A n with λ H n , and we proceed according to the prescription of Remark 2.6 and Remark 2.11. Namely, δ (λ,0) ; and c (λ,0) is any element of π −1 n (c λ ). We continue by induction on n. For n ≤ 1 there is nothing to do, since A 0 = R and A 1 = H 1 . Fix n ≥ 2 and suppose that all the desired data has been chosen for all k ≤ n and all (μ, m) ∈ A k . We have to consider (λ, l) ∈ A n+1 with l > 0. As a model of the cell module (λ,l) A n+1 we can take (λ,l−1) A n−1 ⊗ A n−1 e n A n+1 , and for the generator of the cell module we can take δ (λ,l) ⊗ A n−1 e n . Next we define α (λ,l) . According to [13,Sect. 4], as A n+1 -A n+1 bimodules, with the isomorphism determined by a 1 xe n a 2 → a 1 e n ⊗ x ⊗ e n a 2 . Similarly We identify (e n A n+1 ) * with A n+1 e n (as A n+1 -A n−1 bimodules). Thus Thus Note that c (λ,l−1) e n ∈ A Let us restate this last observation, replacing n+1 by n. We have shown that if (λ, l) ∈ A n and l > 0, then (we can take) c (λ,l) = c (λ,l−1) e n−1 (5.2) By induction, we have c (λ,l) = c (λ,0) e n−2l+1 e n−2l+3 · · · e n−1 .
Expressions of this form will appear again, so we establish the notation

Branching Factors
We continue to work with a pair of towers of algebras (5.1) satisfying the standing assumptions of Section 5.3. We know already that both of the towers (H n ) n≥0 and (A n ) n≥0 are strongly coherent towers of cyclic cellular algebras, with H F n and A F n split semisimple for all n. Therefore, the analysis of Section 3.5, concerning branching factors and path bases, is applicable to both towers. We will show that the branching factors and path bases for the tower (A n ) n≥0 can be computed by explicit formulas from those for the tower (H n ) n≥0 .
We suppose that we have chosen once and for all the following data for the tower (H n ) n≥0 , following observations (1) and (2) and  (λ,l)→(μ,m) in A n+1 associated to each edge (λ, l) → (μ, m) in the branching diagram A, with properties analogous to those listed above.
Neither the cell filtrations nor the branching factors are canonical. However, it was shown in [13] that cell filtrations of the induced and restricted modules for the tower (A n ) can be obtained recursively, based on the cell filtrations of induced and restricted modules for the tower (H n ). We will show here that the branching factors for the tower (A n ) can also be chosen to satisfy recursive relations, so that they are determined completely by the liftings u (n+1) λ→μ andd (n+1) λ→μ of the branching factors for the tower (H n ). Each of the statements in the following theorem should be interpreted as applying whenever they make sense. For example, in statement (2), the branching factor d (n+1) (λ,l)→(μ,m+1) makes sense when n ≥ 1, (λ, l) ∈ A n , (μ, m + 1) ∈ A n+1 , and (λ, l) → (μ, m + 1) in the branching diagram A. This implies that (μ, m) ∈ A n−1 and (μ, m) → (λ, l) in A, so that the branching factor u (n) (μ,m)→(λ,l) also makes sense.

Theorem 5.6 The branching factors for the tower (A n ) n≥0 can be chosen to satisfy:
Proof To prove this result, we have to look into, and add some detail to, the proof in [13,14] that the tower (A n ) is strongly coherent.
First we consider branching factors for reduced modules. The argument is an elaboration of the proof of [ , as well as branching factors u Pulling all this data back via ϕ, we have a cell filtration of Res This means that we can choose u (n) (μ,m)→(λ (j ) ,l j ) for d (n+1) (λ (j ) ,l j )→(μ,m+1) . This proves point (2). Next we turn to the branching factors for induced modules. Statement (3) is evident when n = 0 since A 0 = H 0 = R and A 1 = H 1 . Statement (4) only makes sense when n ≥ 1, so it remains to verify both statements (3) and (4) for n ≥ 1. The argument is an elaboration of the proof of Proposition 4.14 in [13].
Let n ≥ 1 and let be a cell module of A n . According to [13,Proposition 4.14], ⊗ A n J n+1 embeds in Ind    (λ,l)→(μ,m+1) , we have to construct a particular cell filtration of ⊗ A n J n+1 . By axiom (7) and [13, Corollary 4.6], we have J n+1 = A n e n A n ∼ = A n e n ⊗ A n−1 e n A n , as A n -A n+1 bimodules, the isomorphism being given by a 1 e n a 2 → a 1 e n ⊗ A n−1 e n a 2 . We have A n e n ∼ = A n as an A n -A n−1 bimodule, so ⊗ A n J n+1 ∼ = ⊗ A n A n e n ⊗ A n−1 e n A n ∼ = ⊗ A n A n ⊗ A n−1 e n A n ∼ = Res( ) ⊗ A n−1 e n A n .
The composite isomorphism ϕ : ⊗ A n J n+1 → Res( ) ⊗ A n−1 e n A n is given by ϕ(x ⊗ A n a 1 e n a 2 ) = xa 1 ⊗ A n−1 e n a 2 . In particular, ϕ(x ⊗ A n e n ) = x ⊗ A n−1 e n . We assume that we have a chosen cell filtration of Res( ), and we have chosen branching fac- . By [13,Lemma 4.12], M j −1 ⊗ A n−1 e n A n embeds in M j ⊗ A n−1 e n A n for each j , and the quotient is isomorphic to Writing M j = M j ⊗ A n−1 e n A n , we obtain a cell filtration of Res( ) ⊗ A n−1 e n A n , . Pulling back this data via the isomorphism ϕ : ⊗ A n J n+1 → Res( ) ⊗ A n−1 e n A n , we get a cell filtration of ⊗ A n J n+1 , . We conclude that we can take

Applications
We will apply our results to the following examples: the BMW algebras, the Brauer algebras, the partition algebras, and the Jones-Temperley-Lieb algebras. For each example, let R 0 denote the generic ground ring and let R = R 0 [δ −1 ], where e 2 1 = δe 1 . We show that our results apply to the algebras defined over R, and we give explicit Murphy bases for the algebras.
We are then able to check, by a computation specific to each algebra, that the Murphy bases are, in fact, bases for the algebras defined over the generic ground ring R 0 . = (a i , 1) and i = (a i , 0).

Several of our examples involve tangle diagrams in the rectangle
Recall that a knot diagram means a collection of piecewise smooth closed curves in the plane which may have intersections and self-intersections, but only simple transverse intersections. At each intersection or crossing, one of the two strands (curves) which intersect is indicated as crossing over the other.
An (n, n)-tangle diagram is a piece of a knot diagram in R consisting of exactly n topological intervals and possibly some number of closed curves, such that: (1) the endpoints of the intervals are the points 1, . . . , n, 1, . . . , n, and these are the only points of intersection of the family of curves with the boundary of the rectangle, and (2) each interval intersects the boundary of the rectangle transversally.
An (n, n)-Brauer diagram is a "tangle" diagram containing no closed curves, in which information about over and under crossings is ignored. Two Brauer diagrams are identified if the pairs of boundary points joined by curves is the same in the two diagrams. By convention, there is a unique (0, 0)-Brauer diagram, the empty diagram with no curves. For n ≥ 1, the number of (n, n)-Brauer diagrams is (2n − 1)!! = (2n − 1)(2n − 3) · · · (3) (1).
For any of these types of diagrams, we call P = {1, . . . , n, 1

Birman-Murakami-Wenzl Algebras
The Birman-Murakami-Wenzl (BMW) algebras were introduced by Birman and Wenzl [1] and independently by Murakami [32] . The version of the presentation given here follows [31]. Cellularity of the BMW algebras was established in [8,9,43].  Morton and Wassermann [31] give a realization of the BMW algebra as an algebra of (n, n)-tangle diagrams modulo regular isotopy and the following Kauffman skein relations: (1) Crossing relation: (2) Untwisting relation: In the tangle picture, the generators g i and e i are represented by the diagrams There is evidently a unital algebra homomorphism from W n to W n+1 taking generators to generators; from the tangle realization, one can see that this homomorphism is injective, so W n is a subalgebra of W n+1 . The symmetry of the defining relations for W n ensures that the assignments g * i = g i , e * i = e i . determine an involutory algebra anti-automorphism of W n . In the tangle picture, the involution * acts on tangles by flipping them over a horizontal line.
If v ∈ S n and v = s i 1 s i 2 · · · s i j is a reduced expression then the element g v = g i 1 g i 2 · · · g i j depends only on v. For i, j = 1, 2, . . . ,let Let J n denote the ideal W n e n−1 W n ; in the tangle picture, this is the ideal spanned by tangle diagrams with at least one horizontal strand. The map W n /J n → H n = H n (S, q 2 ) determined by g v + J n → T v , for v ∈ S n , is an algebra isomorphism.

The Murphy Basis
The generic ground ring for the BMW algebras is where z, q, and δ are indeterminants over Z. R 0 is an integral domain whose field of , and write W n (R) for W n (R; z, q, δ) and H n (R) for H n (R; q 2 ). It is observed in [13], Section 5.4, that the pair of towers (W n (R)) n≥0 and (H n (R)) n≥0 satisfy the framework axioms (1)-(7) of Section 5.1. Axiom (8) holds by Corollary 4.10. Axiom (9) hold for W n (R), by the remarks at the end of Section 5.1. Finally, Axiom (10) holds by Corollary 4.3. Therefore, by Theorem 5.5, the tower of algebras (W n (R)) n≥0 is a strongly coherent tower of cyclic cellular algebras. By the discussion in Section 5.4, the partially ordered set W n in the cell datum for W n (R) can be realized as  (R; z, q, δ) denote the BMW algebra over R. For n ≥ 0, the set is an R-basis for W n (R), and (W n (R), * , W n , , W n ) is a cell datum for W n (R).
In the remainder of this section, we will show that the Murphy bases W n are bases of the BMW algebras defined over the generic ground ring R 0 . First note that the elements d * s c (λ,l) d t are actually defined over R 0 and are linearly independent. The issue is to show that W n spans the BMW algebra over R 0 . To do this, we examine the transition matrix between a Morton-Wassermann basis of the BMW algebra and W n .

Morton-Wassermann Tangle Bases
We begin by describing the Morton-Wassermann tangle bases of the BMW algebras. We identify the BMW algebras with their tangle realizations, following [31].
To each (n, n)-tangle diagram T , associate a Brauer diagram conn(T ) by deleting the closed strands in T and forgetting information about over and under crossings. Thus conn(T ) has a strand connecting two vertices if and only if T has a strand connecting the same two vertices.
Order the vertices of a tangle or Brauer diagram by 1 < 2 · · · < n < n < · · · < 1, that is, in clockwise order around the boundary of R. The length (D) of a Brauer diagram D is the minimal number of crossings of strands in a physical drawing of the diagram, that is, the number of 4-tuples of vertices (a, b, c, d) such that a < b < c < d and (a, c) and (b, d) are strands of D. Definition 6.3 Say that an (n, n)-tangle diagram T is layered with respect to some total ordering (t 1 , t 2 , . . . , t k ) of its strands, if (1) whenever i < j, every crossing of t i with t j is an over crossing, and (2) each individual strand of T is unknotted, i.e. ambient isotopic to a strand with no self-crossings. Say that T is layered if it is layered with respect to some total ordering of its strands. Say that a layered tangle diagram is simple if it has no closed strands and no strand has self-crossings.
Note that any simple layered tangle diagram T is ambient isotopic to a simple layered tangle diagram in which any two distinct strands have at most one crossing; the number of crossings in such a representative of T is the length of conn(T ). Proof Assume without loss of generality that the number of crossings of T and of S is the length of D. Suppose that S is layered with respect to an ordering (t 1 , t 2 , . . . , t n ) of its strands and T is layered with respect to an ordering (t π(1) , t π(2) , . . . , t π(n) ) for some permutation π of {1, 2, . . . , n}. For brevity, say that T is layered with respect to π. The permutation π may not be unique, so assume that π has been chosen with minimal length for the given tangle diagram T .
If π is the identity permutation, then T and S are ambient isotopic, so represent the same element of W n . Assume that π is not the identity and assume inductively that the assertion holds when T is replaced by a simple layered tangle diagram T with conn(T ) = D, whenever T is layered with respect to a permutation π with (π ) < (π).
Since π is not the identity permutation, there exists i such that π(i) > π(i + 1). If the strands t π(i) and t π(i+1) do not cross, then T is also layered with respect to the shorter permutation π = (i, i + 1) • π, contradicting the choice of π as having minimal length. Therefore t π(i) and t π(i+1) have a (unique) crossing, with t π(i) crossing over t π(i+1) . Because T is layered with respect to π there is no third strand t = t π(k) such that t π(i) has an over crossing with t and t has an over crossing with t π(i+1) . Let U be the tangle diagram obtained by changing the crossing of t π(i) and t π(i+1) , and let T 0 and T ∞ be the two tangle diagrams obtained by smoothing this crossing. It follows that all three of these tangle diagrams are simple and layered, T 0 and T ∞ have fewer than (D) crossings, and by the Kauffman skein relation, Since U is layered with respect to π = (i, i + 1) • π, with (π ) = (π ) − 1, the conclusion follows from the induction hypothesis. Proof We can assume that the number of crossings of T is (D), where D = conn(T ). We proceed by induction on the number of crossings. If T has no crossings, then T is an element of B, because up to ambient isotopy, there is a unique simple layered tangle diagram with underlying Brauer diagram D. Assume that (D) is positive and that the statement holds for all simple layered tangle diagrams with fewer than (D) crossings. There is a simple layered tangle diagram S in B with conn(S) = D. By the previous lemma, T − S is a Z[q − q −1 ]linear combination of simple layered tangle diagrams with fewer than (D) crossings, and thus the result follows from the induction hypothesis.

Lemma 6.8 Let B be a Morton-Wassermann tangle basis of W n (R 0 ). The matrix with respect to B of left or right multiplication by
Proof Let T be an element of B; assume without loss of generality that the number of crossings of T is (D) where D denotes conn(T ). We have to show that T g i is in the We proceed by induction on the number of crossings of T . If T has no crossings, then T g i is simple and layered, so the assertion follows from Proposition 6.6. Assume that (D) > 0 and that the assertion holds when T is replaced by an element of B with fewer crossings. If the vertices i and i + 1 of T are connected by a strand, then T g i = z −1 T , so we are done. Otherwise, let s and t denote the distinct strands of T incident on the vertices i and i + 1. Let S be a simple layered tangle diagram such that conn(S) = D, S has (D) crossings, and S is layered with respect to an ordering (t, s, . . . ) of the strands. Then Sg i is simple and layered, so is in the Z[q − q −1 ]-span of B, by Proposition 6.6. Moreover (T − S)g i is in the Z[z ±1 , (q − q −1 )]-span of B, by combining Lemma 6.5, Proposition 6.6, and the induction hypothesis.
The proof for right multiplication by g −1 i or by left multiplication by g ±1 i is similar.

The Transition Matrix from a Tangle Basis to the Murphy Basis
We examine the coefficients of the expansion of an element Proof There are two cases to consider. CASE 1, l = m and λ ⊂ μ. Then for some a ≤ k − 2m,

The Murphy Basis and the Generic Ground Ring
Let B denote the matrix of expansion coefficients of the elements of W n with respect to some Morton-Wassermann tangle basis B of W n (R 0 ) (and some choice of ordering of W n and of B). By Proposition 6.12, we know that the matrix B has entries in Z[z ±1 , q ±1 ] ⊂ R 0 . On the other hand, since W n is a basis of the BMW algebra over R = R 0 [δ −1 ], it follows that B is invertible over R. We are going to show that B is invertible over Z[z ±1 , q ±1 ] and therefore W n is a basis of W n over R 0 .
The Brauer algebra B n over Z[δ] is the specialization of W n (R 0 ) at q = 1 and z = 1. (See the following Section 6.3 for details). Under the specialization, the Morton-Wassermann basis of W n (R 0 ) specializes to the usual diagram basis of the Brauer algebra, and W n specializes to the corresponding collection of elements of the Brauer algebra, denoted B n . Moreover, the evaluation of B at q = 1 and z = 1, which we denote by B Z , is the matrix of expansion coefficients of the elements of B n with respect to the diagram basis of the Brauer algebra. Let d denote the determinant of B andd the determinant of B Z , which is the evaluation of d at q = 1 and z = 1. Since B is a matrix over Z[z ±1 , q ±1 ], it follows that B Z is a matrix over Z, and henced is an integer.
Butd is an integer, so it follows thatd = ±1 and thus B Z is invertible over Z. Lemma 6.14 B is invertible over R 0 .
Proof Since B is invertible over R, d = det(B) is a unit in R. We can regard R as a subring of g for some integers a, b and some natural numbers c, e, f, g. But the specialization of d at q = 1 and z = 1 is equal to ±1 and therefore we must have c = e = f = g = 0. Thus d = ±q a z b is a unit in R 0 , so B is invertible over R 0 .
The invertibility of B over R 0 together with Proposition 6.2 implies the following theorem: Proof For k ≥ 0, (λ, l) ∈ W k , and t ∈ W } is the basis of the cell module (λ,l) W k derived from the cellular bases W . The collection of these bases, as k and (λ, l) vary, is a family of path bases, because the path basis condition holds over R = R 0 [δ −1 ], according to Lemma 3.12, and therefore it holds over R 0 as well. It follows from Lemma 3.9 that restrictions of cell modules have an order preserving cell filtration.

Brauer Algebras
The Brauer algebras were defined by Brauer [4]. Wenzl [40] showed that the Brauer algebras are obtained from the group algebra of the symmetric group by the Jones basic construction, and that the Brauer algebras over a field of characteristic zero are generically semisimple. Cellularity of the Brauer algebras was established by Graham and Lehrer [16].
Let S be an integral domain with a distinguished element δ. The Brauer algebra B n = B n (S; δ) is the free S-module with basis the set of (n, n)-Brauer diagrams. The product of two Brauer diagrams is obtained by stacking them and then replacing each closed loop by a factor of δ; see [4] or [40] for details. The involution * on (n, n)-Brauer diagrams which reflects a diagram in the axis y = 1/2 extends linearly to an algebra involution of B n (S; δ). Note that the Brauer diagrams with only vertical strands are in bijection with permutations of {1, . . . , n}, and that the multiplication of two such diagrams coincides with the multiplication of permutations. Thus the Brauer algebra contains the group algebra SS n of the permutation group S n as a unital subalgebra. The identity element of the Brauer algebra is the diagram corresponding to the trivial permutation. We will note below that SS n is also a quotient of B n (S; δ).
Let s i and e i denote the following (n, n)-Brauer diagrams: It is easy to see that e 1 , . . . , e n−1 and s 1 , . . . , s n−1 generate B n (S; δ) as an algebra. We have e 2 i = δe i , so that e i is an essential idempotent if δ = 0 and nilpotent otherwise. Note that e * i = e i and s * i = s i . The products ab and ba of two Brauer diagrams have at most as many through strands as a. Consequently, the span of diagrams with fewer than n through strands is an ideal J n in B n (S; δ). The ideal J n is generated by e n−1 . We have B n (S; δ)/J n ∼ = SS n , as algebras with involutions; in fact, the isomorphism is determined by v + J n → v, for v ∈ S n .
Morton and Wassermann show [31] that B n (S; δ) is a specialization of the BMW algebra W n (S; q, z, δ) at q = 1 and z = 1. Consequently, B n (S; δ) has a presentation by generators s i and e i (1 ≤ i ≤ n − 1) and relations specializing those of the BMW algebra.

The Murphy Basis
The generic ground ring for the Brauer algebras is R 0 = Z[δ], where δ is an indeterminant. Write R = Z[δ ±1 ], and write B n (R) = B n (R; δ).
For n ≥ 0 write H n = RS n . Specializing the cellular basis for H n (q 2 ) given in Theorem 4.1 at q = 1 gives a cellular basis for H n . As for the Hecke algebras, H n is the set Y n of Young diagrams of size n, and the branching diagram for the tower (H n ) n≥0 of symmetric group algebras is Young's lattice.
It is shown in [13,Sect. 5.2] that the pair of towers (B n (R)) n≥0 and (H n ) n≥0 satisfy the framework axioms (1)-(7) of Section 5.1. Axiom (8) holds by Corollary 4.10, and specialization from the Hecke algebras to the symmetric group algebras. Axiom (9) hold for B n (R), by the remarks at the end of Section 5.1. Moreover, by Corollary 4.3, the symmetric group algebras are cyclic cellular, so Axiom (10) is satisfied as well. Therefore, by Theorem 5.5, the tower of algebras (B n (R)) n≥0 is a strongly coherent tower of cyclic cellular algebras.
By the discussion in Section 5.4, the partially ordered set B n in the cell datum for B n can be realized as The order relation on B n , and the branching rule for the branching diagram B for the tower (B n ) n≥0 are exactly the same as for the BMW algebras discussed in the previous section. Proof The proof is the same as that of Corollary 5.17.

Remark 6.21
The basis (6.9) coincides with the Murphy-type basis for B n (δ) given in [9].

Jones-Temperley-Lieb Algebras
The Jones-Temperley-Lieb algebras were defined by Jones [19], and were used to define the Jones link invariant in [21]. The cellularity of Jones-Temperley-Lieb algebras was established by Graham and Lehrer [16]. Härterich [18] has given Murphy bases for generalized Temperley-Lieb algebras.
Let S be an integral domain and δ ∈ S. The Jones-Temperley-Lieb algebra A n = A n (S; δ) is the unital S-algebra presented by the generators e 1 , . . . , e n−1 and the relations e i e i±1 e i = e i , e i e j = e j e i if |i − j | ≥ 2, and e 2 i = δe i . The Jones-Temperley-Lieb algebra can also be realized as the subalgebra of the Brauer algebra, with parameter δ, spanned by Brauer diagrams without crossings. Because of the symmetry of the relations the assignment e i → e i determines an involution * of A n . The span of diagrams with at least one horizontal strand (that is, all diagrams other than the identity diagram) is an ideal J n ; it is the ideal generated by e n−1 . The map A n /J n → S determined by 1 A n + J n → 1 S is an isomorphism of algebras with involution.
The generic ground ring for the Jones-Temperley-Lieb algebras is (R; δ), and H n = R for n ≥ 0.

The Murphy Basis
It is shown in [13,Sect. 5.3] that the pair of towers (A n (R)) n≥0 and (H n ) n≥0 satisfies the framework axioms (1)-(7) of Section 5.1. Axioms (8) and (10) are evident since H n = R for all n. Axiom (9) hold for A n (R), by the remarks at the end of Section 5.1. Therefore, by Theorem 5.5, the tower of algebras (A n (R)) n≥0 is a strongly coherent tower of cyclic cellular algebras.
For each n ≥ 0, the partially ordered set H n in the cell datum for H n is a singleton which we label as {n}, and the branching diagram for the tower (H n ) n≥0 is ∅ = 0 → 1 → 2 →· · · .
The branching diagram A for the tower (A n ) n≥0 is that obtained by reflections from H . It can be realized as follows: For n ≥ 0, let A n = j 0 ≤ j ≤ n and n − j is even and order A n by writing m l if l ≥ m as integers. The branching diagram A has an edge connecting j on level n and k on level n + 1 if and only if |j − k| = 1.
Evidently, the algebra H n = R has the cellular basis {1}. We can choose the element c n in H n (see Lemma 2.9) to be 1 and also all the branching factors d (n) (n−1)→n and u (n) (n−1)→n to be 1. According to Eq. (5.5), for j ∈ A n , we can take where l = (n − j)/2, and e (l) n−1 is defined in Eq. (5.4). By Theorem 5.7, the branching factors for the tower (A n ) n≥0 can be chosen as follows: If j ∈ A i and k ∈ A i+1 with j → k, we take is an R-basis for A n , and (A n , * , A n , , A n ) is a cell datum for A n .

The Murphy Basis Coincides with the Diagram Basis
Next, we will show that the Murphy type cellular basis A n of A n given in Proposition 6.22 actually coincides with the diagram basis, so is in particular a basis for the Jones-Temperley-Lieb algebra over the generic ground ring Z[δ]. Let S be an integral domain and δ ∈ S. Let k and n be non-negative integers of the same parity. A (k, n)-Temperley-Lieb diagram is a planar diagram with k upper vertices and n lower vertices connected in pairs with no crossings. The product of a (k, n)-TL diagram and an (n, m)-TL diagram is defined by the same rule as the product of two ordinary TL diagrams of the same size; the result is a power of δ times a (k, m)-TL diagram. The Temperley-Lieb category is category whose objects are non-negative integers; if n−k is odd, then Hom(k, n) = 0, and if n − k is even then Hom(k, n) is the free S-module on the basis of (k, n)-TL diagrams. Composition of morphisms is the bilinear extension of the product of diagrams described above. There is a map * from (k, n)-TL diagrams to (n, k)-TL diagrams defined by reflection in a horizontal line. The linear extension of * is a contravariant functor from the TL category to itself with * • * = id. The rank of a (m, n)-TL diagram is the number of its vertical strands.
Fix n ≥ 0. A TL n-dangle of rank k is a (k, n)-TL diagram with k vertical strands and (n − k)/2 horizontal strands. Any (n, n)-TL diagram T of rank k can be written uniquely as T = y * x, where x and y are n-dangles of rank k. A Dyck sequence of length n and rank k is a sequence (a 1 , . . . , a n ) such that a i ∈ {±1}, each partial sum j i=1 a i is non-negative, and n i=1 a i = k. There is a bijection between Dyck sequences of length n and rank k, and n-dangles of rank k, given as follows. Given a Dyck sequence (a i ) of length n and rank k, there is a unique n-dangle x of rank k with the following property: a vertex j is the right endpoint of a horizontal strand of x if and only if a j = −1. Conversely, given an n-dangle x of rank k, label the right endpoint of each horizontal strand with −1 and all other bottom vertices with +1. Then the resulting sequence of labels in {±1}, read from left to right, is a Dyck sequence of rank k. The two maps, from Dyck sequences to dangles and from dangles to Dyck sequences, are inverses.
There is a bijection between paths on the generic branching diagram for the Temperley-Lieb algebras, of length n, from ∅ to k, and Dyck sequences of length n and rank k. A path is given by a sequence (0 = b 0 , 1 = b 1 , b 2 is a Dyck sequence of length n and rank k. Conversely, given a Dyck sequence of length n and rank k, its sequence of partial sums defines a path on the branching diagram, of length n, from ∅ to k. Evidently, the two maps, from paths to Dyck sequences and from Dyck sequence to paths, are inverses. Composing the two bijections described above, we have a bijection between paths on the branching diagram and dangles. For a path t on the branching diagram, let x(t) denote the corresponding dangle. Theorem 6.23 Fix n and k ≤ n with n − k even. Let s and t be elements of A k n . Then We do this by induction on n, the case n = 1 being evident. Assume that the assertion holds for some fixed n, for all k with k ≤ n and n − k even, and for all t ∈ A k n . Let s ∈ A j n+1 for some j , s = (k 0 , k 1 , . . . , k n = k, k n+1 = j), and let t be the truncation of s of length n, t = (k 0 , k 1 , . . . , k n = k).

Thus the Murphy type basis
Write l = (n − k)/2 and l = (n + 1 − j)/2. There are two cases: Case 1. j = k + 1, l = l. In this case, x(s) is obtained from x(t) by adding a vertical strand at the new vertex n + 1. On the other hand, using the induction hypothesis at the last step. Multiplication of an n-dangle of rank k on the left by x Case 2. j = k − 1, l = l + 1. In this case, x(s) is obtained from x(t) by "closing" the rightmost vertical strand; that is, if j is the vertex adjacent to this strand, the strand is replaced by a horizontal strand joining j and n + 1. On the other hand, by the same computation as in the previous case. But multiplication of an n-dangle of rank k on the left by x

Partition Algebras
The partition algebras A n (k), for k, n ∈ Z ≥0 ,are a family of algebras defined in the work of Martin and Jones in [20,[25][26][27] in connection with the Potts model and higher dimensional statistical mechanics. Jones [20] showed that the even partition algebra A 2n (k) is in Schur-Weyl duality with the symmetric group S k acting diagonally on the n-fold tensor product V ⊗n of its k-dimensional permutation representation V . In [25], Martin defined the odd partition algebra A 2n+1 (k) as the centralizer of the subgroup S k−1 ⊆ S k acting on V ⊗n . Including the algebras A 2n+1 (k) in the tower allowed for the simultaneous analysis of the whole tower of algebras (6.11) using the Jones basic construction, by Martin [25] and Halverson and Ram [17]. Cellularity of the partition algebras was proved in [6,41,42]. can be represented by the diagrams: If ρ 1 , ρ 2 ∈ P 2n , then the composition ρ 1 • ρ 2 is the partition obtained by placing ρ 1 above ρ 2 and identifying each vertex in the bottom row of ρ 1 with the corresponding vertex in the top row of ρ 2 and deleting any components of the resulting diagram which contains only elements from the middle row.
The Brauer algebra B n (S; δ) embeds as a subalgebra of A 2n (S; δ), spanned by partitions with each block having two elements. In particular, A 2n (S; δ) has a subalgebra isomorphic to the symmetric group algebra SS n , spanned by permutation diagrams. The permutation subalgebra is generated by the transpositions The multiplicative identity of A 2n (S; δ) is the trivial permutation. It is not hard to see that the partition algebra A 2n (S; δ) is generated by the transpositions s i (1 ≤ i ≤ n − 1) and elements e j (1 ≤ j ≤ 2n − 1), where Halverson and Ram [17,Theorem 1.11] and East [7,Theorem 36] give a presentation for A 2n in terms of the generators e j and s i . The algebras A 2n (S; δ) and A 2n−1 (δ) have an algebra involution * which acts on diagrams by flipping them over the horizontal line y = 1/2. The generators s i and e j are * -invariant A 2n−1 (S; δ) is defined as a subalgebra of A 2n (S; δ), and A 2n (S; δ) embeds in A 2n+1 (S; δ) as follows: define a map ι : P 2n → P 2n+1 by adding an additional block {n + 1, n + 1}. The linear extension of ι is a monomorphism of algebras with involution.
Let d ∈ P 2n . Call a block of d a through block if the block has non-empty intersection with both [n] and [n]. The number of through blocks of d is called the propagating number of d, denoted pn(d). Clearly, pn(d) ≤ n for all d ∈ P 2n . The only d ∈ P 2n with propagating number equal to n are the permutation diagrams. If x, y ∈ P 2n and xy = δ r z, then pn(z) ≤ min{pn(x), pn(y)}. Hence the span of the set of d ∈ P 2n with pn(d) < n is an ideal J 2n ⊂ A 2n (S; δ). Moreover, J 2n−1 := J 2n ∩ A 2n−1 is the span of d ∈ P 2n−1 with pn(d) < n. One can check that for k ≥ 2, J k is the ideal of A k (S; δ) generated by e k−1 . The ideal J k is *invariant, and the span of permutation diagrams in A k is a * -invariant linear complement for J k . It follows that A 2n (S; δ)/J 2n ∼ = SS n and A 2n−1 (S; δ)/J 2n−1 ∼ = SS n−1 as algebras with involution; the isomorphisms are determined by v + J k → v, where v is a permutation diagram.

The Murphy Basis
The generic ground ring for the partition algebras is R 0 = Z[δ], where δ is an indeterminant. Write R = Z[δ ±1 ], and let F = Q(δ) denote the field of fractions of R. Write A n for A n (R; δ) and write H 2i = H 2i+1 = RS i for i ≥ 0. The tower (H n ) n≥0 is a strongly coherent tower of cyclic cellular algebras, and H F n is split semisimple. The branching diagram of the tower (H n ) n≥0 is the graph H with (1) H 2i = H 2i+1 = the set Y i of Young diagrams of size i.
It is shown in [13,Sect. 5.7] that the pair of towers (A n ) n≥0 and (H n ) n≥0 satisfy the framework axioms (1)-(7) of Section 5.1. Axiom (8) holds by Corollary 4.10. Axiom (9) holds for the partition algebras, by the remarks at the end of Section 5.1. Finally, Axiom (10) holds by Corollary 4.3. Therefore, by Theorem 5.5, the tower of algebras (A n ) n≥0 is a strongly coherent tower of cyclic cellular algebras. The first few levels of A are given in Fig. 5. Next, we determine the branching coefficients for the two towers (H n ) n≥0 and (A n ) n≥0 . Let λ ∈ H 2i−1 and μ ∈ H 2i with λ → μ in H . If μ = λ ∪ {(r, μ r )}, let a = r j =1 μ j . Then the branching factors for the inclusion H 2i−1 ⊆ H 2i in the tower (H i ) i≥0 are given by 2k−1 is equal to a single partition of even type (k, l), and that no factor of δ arises in the computation of the product, as shown in Fig. 6. If m = l + 1, then μ ⊂ l and (λ,l)→(μ,l+1) is equal to a sum of distinct partitions, each of even type (k, l), and again no factor of δ appears in the computation of the product, as shown in Fig. 7.
Assume now that is of even type (k, m) and (λ, l) → (μ, m) is an edge from level 2k − 1 to level 2k in A. Thus the lower vertices k − m + 1, . . . , k each constitute a block of . If l = m, then λ ⊂ μ and 2k−2 is a single partition, of odd type (k, l), and no power of δ occurs in the computation of the product. The diagram for this case is similar to Fig. 8, except that the lower vertex k − l of is now an singleton block of . Proof If n = 2k + 1 is odd, then c (λ,0) e (l) n−1 is a sum of partitions of odd type (k + 1, l). If n = 2k is even, then c (λ,0) e (l) n−1 is a sum of partitions of even type (k, l). The argument proceeds as in the proof of Proposition 6.12, with Lemma 6.28 taking the place of Lemma 6.11.

Theorem 6.30
The set A n = {d * s c (λ,l) d t | s, t ∈ A (λ,l) n , (λ, l) ∈ A n },is a basis for the partition algebra A n (R 0 ; δ) over the generic ground ring R 0 = Z[δ].
Proof The transition matrix B between the diagram basis of the partition algebra and the set A n has integer entries, according to Proposition 6.29, and in particular d = det(B) is an integer. Since A n is a basis for the partition algebra over R = Z[δ ±1 ], it follow that B is invertible over R, so the integer d is a unit in R. It follows that d = ±1 and hence B is invertible over Z. Hence A n is a basis of the partition algebra over R 0 . Corollary 6.31 Let A n denote the partition algebra over the generic ground ring R 0 = Z[δ]. For n ≥ 0 and for a cell module of A n+1 , the restricted module Res A n+1 A n ( ) has an order preserving cell filtration.
Proof The proof is the same as that of Corollary 6.17.
Acknowledgments The first author is grateful to Arun Ram and Andrew Mathas for many stimulating conversations during the course of this research and for detailed comments on previous versions of this paper. The second author thanks Andrew Mathas for many helpful conversations and the School of Mathematics and Statistics at the University of Sydney for its hospitality. We thank Steffen König and the Institute for Algebra and Number Theory at the University of Stuttgart for hospitality. This research was supported by the Australian Research Council (grant ARC DP-0986774) at the University of Melbourne, (grants ARC DP-0986349, ARC DP-110103451) at the University of Sydney and EPSRC (grant EP/L01078X/1) at City University London.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Proof This follows from Eq. (4.7) and Lemma A.1 and Lemma A.2.
Our next goal is to obtain similar formulas for the Murphy type bases of the various algebras treated in Section 6.
Proposition A.4 Let A n denote the n-th BMW, Brauer, partition or Jones-Temperley-Lieb algebra. The Murphy type basis of A n established in Section 6 can be written in the form u s d t λ ∈ A n ands, t ∈ A λ n .
Sketch of proof. We need to show that if x ∈ A n and y ∈ A n+1 with x → y in the branching diagram A, then where the elements c x ∈ A n and c y , u (n+1) x→y , d (n+1) x→y ∈ A n+1 are as specified in Section 6. The result will then follow from Lemma A.1. For the Temperley-Lieb algebras, Eq. (A.4) is evident from the formulas in Section 6.4 for the elements c x and for the branching factors.
For the BMW, Brauer and partition algebras, (A.4) can be established in two steps. The first step is to show that (A.4) holds when x = (λ, 0) ∈ A n and y = (μ, 0) ∈ A n+1 . For the Brauer and partition algebras this special case of (A.4) follows from Lemma 4.18, part (1), as all the elements involved lie in a copy of the symmetric group algebra contained in A n+1 . For the BMW algebras, it is necessary to establish an analogue of Lemma 4.18, part (1) which is valid in the algebra of the braid group.
The second step in the proof of (A.4) is to establish the general case from the special case. This involves a straightforward computation using the formulas of Theorem 5.7.