Cellular bases for algebras with a Jones basic construction

We define a method which produces explicit cellular bases for algebras obtained via a Jones basic construction. For the class of algebras in question, our method gives formulas for generic Murphy--type cellular bases indexed by paths on branching diagrams and compatible with restriction and induction on cell modules. The construction given here allows for a uniform combinatorial treatment of cellular bases and representations of the Brauer, Birman-Murakami-Wenzl, Temperley-Lieb, and partition algebras, among others.


Introduction
The notion of cellularity was introduced by Graham and Lehrer [GL] as a tool for studying non-semisimple representations of Hecke algebras and other algebras with geometric connections. Cellular algebras are defined by the existence of a cellular basis with combinatorial properties that reflect the "Robinson-Schensted correspondence" in the Iwahori-Hecke algebra of the symmetric group. Important examples of cellular algebras include the Iwahori-Hecke algebras of the symmetric group, Brauer algebras, Birman-Murakami-Wenzl algebras, Temperley-Lieb algebras and partition algebras (see [GL], [Mu], [Xi], [Xi1]).
In this paper, we consider a tower of unital algebras R = A 0 ⊆ A 1 ⊆ A 2 ⊆ · · · (1.1) over an integral domain R which are obtained by a repeated Jones basic construction on a tower of cellular algebras with cellular bases that are well behaved with respect to restriction and induction on cell modules. For such pairs of towers, we demonstrate: (1) An explicit filtration of each cell module for A i+1 by cell modules for A i and; (2) An inductive construction by which a cellular basis for A i+1 is explicitly defined in terms of a cellular basis for A i .
Having established an inductive construction of cellular bases in the above general setting, we apply our method obtain explicit cellular bases for the Brauer algebras, Birman-Murakami-Wenzl (BMW) algebras, Temperley-Lieb algebras, and partition algebras. In each of the aforementioned examples, we write down explicit formulae for cellular bases that are indexed by paths on an appropriate branching diagram. The given bases are compatible with restriction and induction on cell modules, and are analogues of Murphy's cellular bases for the Iwahori-Hecke algebra of the symmetric group [Mu]. In the particular cases of the Brauer and BMW algebras, our results recover the construction given in [En1]. Hence this work may be regarded as a generalisation of [En1].
The hypotheses which allow for an inductive construction of Murphy-type bases in this paper differ only slightly from the framework for cellularity of algebras related to the Jones basic construction given by Goodman and Graber in [GG] and, in each of the examples we consider, the existence of a Murphy-type cellular basis which is compatible with induction and restriction has already been established non-explicitly in [GG].
Given that our bases are compatible with restriction and induction on cell modules, the Jucys-Murphy elements in each algebra under consideration here will act triangularly relative to the Murphy-type bases given here (see §3 of [GG1] or §3 of [Mat1]). Thus, in the generic setting, and after a unitriangular transformation, the Murphy-type cellular bases will give seminormal bases of each of the algebras under consideration here. In [En2], the Murphy-type bases given in §4 are used to explicit combinatorial formulae for the seminormal representations of the partition algebras.
Using the definition of the partition algebras as diagram algebras, Xi [Xi] has given cellular bases for the partition algebras (see also [DW], [Wi]). The bases for partition algebras given in [Xi] are obtained by adjoining certain tangle diagrams to basis elements for the the group algebra of the symmetric group, a process which formally corresponds to König and Xi's method of constructing cellular algebras by inflation [KX]. The method established in [Xi] has also been used to give prove cellularity of bases for the Brauer and BMW algebras which are indexed by tangle diagrams (see [Xi1], [En], [Wi]). The significance of the approach taken in this paper is that by indexing the basis elements by paths in suitably constructed branching diagrams, rather than by tangles, we are able to obtain cellular structures which admit explicit cell module filtrations under induction and restriction.
Finally, we note that the results of Ariki and Mathas [AM] and Mathas [Mat2] on restriction and induction on cell modules of the cyclotomic hecke algebras, imply that the construction of cellular bases given here applies equally well to the cyclotomic BMW algebras with admissible parameters. In this setting, our construction would recover the generalisation of [En1] to the cyclotomic case given by Rui and Xu in [RX].
In §2 we recall the definition of cellularity from [GL] and [GG]. We define the branching diagram of a tower of cellular algebras in terms of restriction and induction on cell modules and formulate a set of hypotheses on the towers (1.1) and (1.2), analogous to the framework for cellularity given by Goodman and Graber [GG]. In §3 we construct explicit filtration which show that restriction and induction on certain modules in the tower (1.1) are controlled by restriction and induction on cell modules in the tower (1.2). In §4 we show that (1.1) is a tower of cellular algebras with Murphy type bases that are well behaved with respect to restriction and induction on cell modules. In §5 we apply the preceding construction to particular examples.

Preliminiaries
Cellular algebras were defined by Graham and Lehrer [GL]. The construction in this paper will use a slightly weaker version of cellularity which is due to Goodman and Graber [GG].
Definition 2.1. Let R be an integral domain. A cellular algebra is a tuple (A, * ,Â, , A ) where (1) A is a unital R-algebra and * : A → A is an algebra anti-automorphism of A; (2) (Â, ) is an ordered set, andÂ λ , for λ ∈Â, is an indexing set; (3) The set A = c λ st | λ ∈Â and s, t ∈Â λ , is an R-basis for A, for which the following conditions hold: (a) Given λ ∈Â, t ∈Â λ , and a ∈ A, there exist r v , for v ∈Â λ , such that, for all s ∈Â λ , where A ✄λ is the R-module generated by If A is an algebra with cell datum (A, * ,Â, , A ) we will frequently omit reference to the cell datum for A and simply refer to A as a cellular algebra.
Let A be a cellular algebra, λ ∈Â and s ∈Â λ . The cell module A λ is the R-submodule of A/A ✄λ generated by c λ st + A ✄λ | t ∈Â λ with right A-action given by (2.1). By the condition (a) in Definition 2.1, A λ is independent of the choice of s.
If A is a cellular algebra over R, λ ∈Â, and N ⊆ M is an inclusion of right A-modules, write If M is a right A-module, an order preserving A cell-module composition series for M is a filtration by right A-modules, such that λ (s) ✄ λ (t) inÂ whenever t > s. If A ⊆ B is an inclusion of algebras over R, define the induced module Definition 2.2. Let R be an integral domain. A tower of cellular algebras with a branching diagram is a sequence of cellular algebras over R such that H 0 = R, and for i = 0, 1, . . . , the following conditions hold: (1) H i ⊆ H i+1 and 1 H i = 1 R .
(4) If λ ∈Ĥ i and µ ∈Ĥ i+1 , then H µ i+1 appears as a subquotient H µ i+1 = N j /N j−1 for some j = 1, . . . , p, A prototypical example of a tower of cellular algebras with a branching diagram is the tower of Iwahori-Hecke algebras of the symmetric group (see §5.2).
For the remainder of this section, we assume the following axioms which may be regarded as an adaptation to our setting of the framework for cellularity given by Goodman and Graber in §2.4 of [GG].
Let R be an integral domain and be two towers of R-algebras each with a common multiplicative identity. We assume that: In the context of hypotheses (A)-(H) above, the reader may find it useful to bear in mind the tower of Brauer algebras R = B 0 ⊆ B 1 ⊆ · · · where B k , for k ∈ Z 2 , is obtained by adjoining an element e k−1 to the group algebra of the symmetric group on k letters (see §5.3).

Define the two sided ideal
For brevity, we will continue to write

Propositions 2.3 and 2.4 reformulate the definition of the module
defines an isomorphism of right A k−1 -modules.
Proof. The statement follows from the assumptions that e k−1 commutes with A k−2 and the map A k−1 → e k−1 A k−1 given by a → e k−1 a is injective.

Induction and Restriction
In this section, we continue to assume that the pair (2.4) satisfy the hypotheses (A)-(H). and Proof. The inclusion (3.1) follows from the assumption that The proof of (3.3) is given in the following steps.
Step 1. Since x Step 2. The assumption that commutes and there is an isomorphism We establish the following notation.
(1) If λ ∈Ĥ i , then we have an H i+1 cell-module composition series (2) If µ ∈Ĥ i+1 , then we have an H i cell-module composition series In the above setting, fix Similarly, let In the next statement we make explicit the A k -module filtrations given in Proposition 3.1.

Proof. By Proposition 3.1 there is an inclusion of
Define an R-module homomorphism ϕ ℓ,k−1 : and note that ϕ ℓ,k−1 (a) = a for all a ∈ A k−2ℓ−1 . In light of (3.6), the isomorphism (3.7) is realised by the A k -module homomorphism which maps as an A k−2ℓ−2 -module, using the isomorphism (3.8) and the assumptions on the structure of Res for j = 1, . . . , r.

Proposition 3.1 has shown that
and note that ϑ ℓ,k−1 (a) = a for all a ∈ A k−2ℓ+1 . Since k−1 A k = U r , the isomorphism (3.9) is realised by the A k -module map A k−2ℓ−1 H λ k−2ℓ−1 as an A k−2ℓ -module, using the map (3.10) and the assumptions on the structure of Ind This completes the proof of the theorem.

The Murphy Basis
In this section we continue to assume that the pair (2.4) satisfy the hypotheses (A)-(H) and use Theorem 3.3 to construct an explicit Murphy-type cellular basis for each algebra in the tower R = A 0 ⊆ A 1 ⊆ A 2 ⊆ · · · .
The author is is grateful to Arun Ram for pointing out that the inductive procedure used to construct bases in the proof of Theorem 4.1 in fact yields the closed expression for the operators a is a filtration by right A i−1 -modules and, for j = 1, . . . , p, the R-linear map for u ∈Â (λ (j) ,ℓ j ) i−1 and t ∈Â (µ,m) i such that t| i−1 = u, is an isomorphism of right A i−1 -modules.

Applications
If the pair (2.4) satisfy the assumptions (A)-(H), then by Theorem 4.1, a cellular basis for A i , for i 0, of the form (4.3) is determined explicitly by the following data: (1) The branching diagramĤ whose vertices on level i, for i 0, consist of the elements of the partially ordered setĤ i .
In the examples below, we use Theorem 4.1 and the bases for the Iwahori-Hecke algebra of the symmetric group given by Murphy [Mu] to produce explicit cellular bases, in the form of the data (5.1)-(5.4), for important examples of algebras obtained by a Jones basic construction. By Theorems 3.2 and 3.3 our construction will yield cellular bases that are compatible with induction and restriction on cell modules. Note that in each example considered below, the bases obtained are cellular in the strict sense of [GL]. We first establish some notation. 5.1. Combinatorics. Let k denote a non-negative integer and S k be the symmetric group acting on {1, . . . , k} on the right. For i an integer, 1 i < k, let s i denote the transposition (i, i + 1). Then S k is presented as a Coxeter group by generators s 1 , s 2 , . . . , s k−1 , with the relations An product w = s i 1 s i 2 · · · s i j in which j is minimal is called a reduced expression for w and j = ℓ(w) is the length of w. If i, j = 1, . . . , k, define If k > 0, a partition of k is a non-increasing sequence λ = (λ 1 , λ 2 , . . . ) of integers, λ i 0, such that i 1 λ i = k; otherwise, if k = 0, write λ = ∅ for the empty partition. The fact that λ is a partition of k will be denoted by λ ⊢ k. If λ is a partition, we will also write |λ| = i 1 λ i . The integers {λ i | for i 1} are the parts of λ. If λ ⊢ k, the Young diagram of λ is the set The elements of [λ] are the nodes of λ and more generally a node is a pair (i, j) ∈ N × N. The diagram [λ] is traditionally represented as an array of boxes with λ i boxes on the i-th row. For example, if λ = (3, 2), then [λ] = . Let [λ] be the diagram of a partition. Usually, we will identify the partition λ with its diagram and write λ in place of [λ].

5.2.
Iwahori-Hecke algebras of the symmetric group. Let R be an integral domain and q be a unit in R. Let H k = H k (q 2 ) denote the Iwahori-Hecke algebra of the symmetric group which is presented by the generators T 1 , . . . , T k−1 , and the relations The branching diagram of the tower R = H 0 ⊆ H 1 ⊆ · · · is the graphĤ with: (1) vertices on level i:Ĥ i = {λ | λ ⊢ i}, ordered by the dominance order on partitions.
The first few levels of the graphĤ are given in (5.7).
The generic branching rules encoded in the graphĤ can be made explicit using the cellular basis for H k given by Murphy [Mu]. The Murphy basis is constructed as follows.
Proposition 5.2. Let λ ∈Ĥ i and µ (1) , . . . , µ (p) be an ordering of Proposition 5.3. Let µ ∈Ĥ i+1 and λ (1) , . . . , λ (r) be an ordering of {λ | λ ∈Ĥ i and λ → µ} such that λ (s) ✄ λ (t) whenever t > s. If is a filtration by H i -modules where, for j = 1, . . . , r, 5.3. Brauer algebras. The Brauer algebras B k (z) were defined by Brauer [Br]. Wenzl [We] 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 [GL]. A Murphy type cellular basis for the Brauer algebras has been given in [En1].
Remark 5.5. The basis (5.9) coincides with the Murphy-type basis for B i (z) given in [En1].

Birman-Murakami-Wenzl algebras.
The BMW algebras B k (q, z) were defined by Birman and Wenzl [BW] and Murakami [Mur] to give an algebraic realisation of the Kauffman link invariant [Ka]. Wenzl [We] showed that the BMW algebras are obtained from the Iwahori-Hecke algebras of the symmetric group by a Jones basic construction, and that the BMW algebras over a field of characteristic zero are generically semisimple. Cellularity of the BMW algebras was proved by Xi [Xi1] and a Murphy-type cellular basis for the BMW algebras was given in [En1].
, where q, z are indeterminants over Z. Following §3 of [We] or §2.3 of [DRV], the BMW algebra W k = W k (q, z) is the unital R-algebra presented by the generators g 1 , . . . , g k−1 , which are assumed to be invertible, and relations g i g j = g j g i , i = j, j + 1, where e i is defined by The above relations also imply that i+1 e i = z ±1 e i , e i e j = e j = e j e i , if i = j + 1.
Remark 5.7. The basis (5.10) differs from the Murphy-type basis for W i (q, z) given in [En1] by a unitriangular transformation.

5.5.
Temperley-Lieb algebras. The Temperley-Lieb algebras were defined by Jones [Jo], who used them to define link invariants in [Jo1]. The cellularity of Temperley-Lieb algebras was established by Graham and Lehrer [GL]. Härterich [Hä] has given Murphy bases for generalised Temperley-Lieb algebras, and Goodman and Graber [GG1] have shown that the Temperley-Lieb algebras form a strongly coherent tower of cellular algebras. Let z be an indeterminant and R = Z[z]. The Temperley-Lieb algebra A k = A k (z) is the unital R-algebra presented by the generators e 1 , . . . , e k−1 and the relations e i e j = e j e i , i = j + 1.