BOREL TYPE SUBALGEBRAS OF THE q-SCHUR^{m} ALGEBRA

Jie Du and Hebing Rui

School of Mathematics, University of New South Wales Sydney, 2052, Australia and Department of Mathematics, University of Shanghai for Science and Technology, Shanghai, 200093, China

In [G1], J. A. Green investigated certain subalgebras, called Borel subalgebras, of the Schur algebra associated with the Borel subgroups of the general linear group. Besides their combinatorial definition, these algebras are quasi-hereditary and give rise to a triangular decomposition of the Schur algebra with which Weyl and co-Weyl modules can be described as induced modules by using tensor and hom functors (see [Sa]). In [PW], part of Green’s work has been generalised to the q-Schur algebra. Recently, a new class of quasi-hereditary algebras, called the q-Schur^{2} algebras, associated with the Hecke algebra of the Weyl group of type B has been introduced by Du and Scott [DS] (see [DJM1] for a Morita equivalent version). Associated with Ariki-Koike Hecke algebras, a more general class of quasi-hereditary algebras, called cyclotomic q-Schur algebras, has been introduced by Dipper, James and Mathas in [DJM2]. Since these algebras do not occur naturally in the context of Lie theory or quantum groups (cf. [DS1]), it would be interesting to find possible connections between the representation theories of these algebras and relevant quantum groups. As an attempt to this problem, we will investigate in this paper some Lie-theoretic structure of these algebras, and especially seek the existence of their Borel type subalgebras.

We shall aim at the q-Schur^{2} algebra first, since the construction for the q-Schur^{2} algebra in [DS] is almost parallel to that for the q-Schur algebra. Especially, the existence of both a natural basis (i.e., the counterpart for the centralizer algebra of a permutation module) and a Green-Murphy type (or cellular) basis for a q-Schur^{2} algebra guarantees that we can mimic Green’s construction in this case. Surprisingly, the work for Borel type subalgebras in the m = 2 case can be easily generalized to the cyclotomic q-Schur algebras, though there is no natural basis available in

Typeset by AMS-TEX

Both authors gratefully acknowledge support under ARC Large Grant A69530243. The second author is partially supported by the National Natural Science Foundation, grant no. 19501016.

He also wishes to thank the University of New South Wales for its hospitality during the writing of this paper.

the work [DJM2] for m > 2. In this generalisation, we first aim at a subclass of cyclotomic q-Schur algebras, called q-Schur^{m} algebras indexed by bidegree (n, r) as for the q-Schur algebras, and will prove that Borel type subalgebras exist in a q-Schur^{m} algebra. It is also interesting to note that a Borel type subalgebra of the q-Schur^{m} algebra of degree (n, r) is isomorphic to a Borel subalgebra of the q-Schur algebra of degree (N, r) for some N = N(m, n, r). We will also explain how we can easily get the Borel type subalgebras for an arbitrary cyclotomic q-Schur algebra.

It is worth pointing out that the notion of Borel subalgebras for an arbitrary quasi-hereditary algebra has been introduced by Scott [Sc]. It would be interesting to know if the Borel type subalgebras of q-Schur^{m} algebras fit the definition given in [Sc]. If it was the case, it would imply that the higher derived functors vanished in the case discussed in (4.10) and (5.16(f)). Thus, we would have an analogue of Borel-Bott-Weil theorem for q-Schur^{m} algebras.

We organise the paper as follows. Section one collects results on q-Schur^{2} algebras and related combinatorics. Candidates S_{R}^{2,≽} and S_{R}^{2,≼} for Borel type subalgebras are introduced as subspaces. In section two, we prove that these subspaces are subalgebras, where we discover an important connection with the subalgebra structure on the Borel subalgebras S_{R}^{1,≽} and S_{R}^{1,≼} of the q-Schur algebra S_{R}^{1}. This important observation indicates that a somewhat easy generalization exists. In section three, we will prove that a q-Schur^{2} algebra is a product of the Borel type subalgebras, and hence, we obtain a triangular decomposition of the q-Schur^{2} algebra. The representation theory is investigated in section four, where the quasi-heredity of S_{R}^{2,≽} and S_{R}^{2,≼} is obtained by using the criterion established in [DR] and is used to determine the PIMs and some induced standard and costandard modules. In section five, we shall define the Borel type subalgebras S_{R}^{m,≽} and S_{R}^{m,≼} for the q-Schur^{m} algebra S_{R}^{m} and show how all results in §§2-4 for the m = 2 case are generalized to S_{R}^{m} for arbitrary m. Finally, we determine the tilting modules and the Ringel duals of these Borel type algebras. We will see that S_{R}^{m,≽} and S_{R}^{m,≼} are Ringel dual to each other.

Throughout, unless specified, R denotes a commutative ring with 1.

1. The q-Schur^{2} algebra. Let W be the Weyl group of type B_{r}. As a Coxeter group, we denote the set of Coxeter generators of W by S = {s_{0}, s_{1}, ⋯, s_{r−1}} with relations described in the following Coxeter diagram

∘^{0}=∘^{1}−∘^{2} ⋯ ∘− ∘^{r−1}

In section five, W will be identified with the wreath product (ℤ/mℤ) ≀ 𝔖_{r} for m = 2, where 𝔖_{r} = 𝔖_{{1,2,··· ,r}} is the symmetric group on r letters.

Let Z = ℤ[q_{0}, q_{0}^{−1}, q, q^{−1}] be the Laurent polynomial ring in the indeterminates q_{0}, q and let q_{s_{0}} = q_{0} and q_{s_{i}} = q for 1 ⩽ i ⩽ r − 1. The generic Hecke algebra H associated to W is an associative algebra over Z with a Z-basis {T_{w} | w ∈ W } and multiplication defined by

\begin{matrix}(T_{s} − q_{s})(T_{s} + 1) = 0, if s ∈ S, \\ T_{x}T_{y} = T_{xy}, if l(xy) = l(x) + l(y). \end{matrix}

Here l is the length function on W. For a commutative ring R which is a Z-algebra, let H_{R} = H ⊗_{Z} R be the Hecke algebra over R. For simplicity, we shall continue to use T_{w} for T_{w} ⊗ 1 and q_{s} for q_{s} ⊗ 1.

We need the notion of multi-compositions. Let ℤ^{+} be the set of nonnegative integers. Fix n, r ∈ ℤ^{+} with n > 0. A composition λ of r with n parts is a sequence (λ_{1}, ⋯, λ_{n}) such that λ_{i} ∈ ℤ^{+} and |λ| = ∑_{i} λ_{i} = r, and λ is called a partition if the sequence is decreasing. For any positive integer m, an m-composition λ of r is defined to be a sequence of compositions λ = (λ^{(1)}, ⋯, λ^{(m)}) such that r = ∑_{i=1}^{m} |λ^{(i)}| and λ is called an m-partition if each λ^{(i)} is a partition. Here the number of parts in each λ^{(i)} may be different. Denote by Λ_{m}(r)^{+} the set of all m-partitions of r. Putting a_{0} = 0 and a_{i} = a_{i−1} + |λ^{(i)}| for all i ⩾ 1, the sequence a = (a_{0}, a_{1}, ⋯, a_{m}) is called the cumulative norm sequence (or simply, c.n.s.) of λ. Let ⊴ be the dominance order on m-compositions. Thus, λ ⊴ μ means that, for every i, 1 ⩽ i ⩽ m,

∑_{j=1}^{i−1} a_{i} + ∑_{t=1}^{k} λ_{t}^{(i)} ⩽ ∑_{j=1}^{i−1} b_{i} + ∑_{t=1}^{k} μ_{t}^{(i)}, ∀k,

where a = (a_{0}, a_{1}, ⋯, a_{m}) resp. b = (b_{0}, b_{1}, ⋯, b_{m}) is the c.n.s. of λ resp. μ.

Let Λ(n, r) be the set of all compositions of r with n parts, and let, for m > 0, (1.1)

\{ \begin{matrix}Λ_{m}(n, r) = ⋃_{λ∈Λ(m,r)} Λ(max_{n,r}, λ_{1}) × ⋯ × Λ(max_{n,r}, λ_{m−1}) × Λ(n, λ_{m}) \\ Λ_{m}(n, r)^{+} = Λ_{m}(n, r) ∩ Λ_{m}(r)^{+}, \end{matrix}

where max_{n,r} is the maximum of n, r. Note that Λ(n, r) = Λ_{1}(n, r) and that Λ_{2}(n, r) is denoted Π(n, r) in [DS]. Note also that Λ_{m}(n, r)^{+} is a coideal of the poset Λ_{m}(r)^{+} = Λ_{m}(r, r)^{+}. An m-composition λ ∈ Λ_{m}(n, r) will sometimes be viewed as a single composition by concatenating λ^{(1)}, λ^{(2)}, ⋯, λ^{(m)}. To indicate the difference, the latter will be denoted by λ̄ ∈ Λ(N, r), where N is defined by

(1.2)  N = N(m, n, r) = (m − 1)max_{n,r} + n.

For example, λ̄ = (30 ⋯ 0201) ∈ Λ(9, 6) if λ = ((30 ⋯ 0), (201)) ∈ Λ_{2}(3, 6). Clearly, the map λ ↦ λ̄ defines a bijection from Λ_{m}(n, r) to Λ(N, r). Note that, for λ, μ ∈ Λ_{m}(n, r), we have λ ⊴ μ if and only if λ̄ ⊴ μ̄.

The subgroup of W generated by {s_{1}, ⋯, s_{r−1}} will be identified with 𝔖_{r}. For a (1-)composition λ of r, let

(1.3)  𝔖_{λ} = 𝔖_{{1,··· ,λ_{1}}} × ⋯ × 𝔖_{{λ_{1}+···+λ_{n−1}+1,··· ,r}}

be the Young subgroup of 𝔖_{r} corresponding to λ, and D_{λ} the set of distinguished representatives of right 𝔖_{λ}-cosets. If μ is another composition of r, then D_{λμ} = D_{λ} ∩ D_{μ}^{−1} is the set of distinguished representatives of double 𝔖_{λ}-𝔖_{μ} cosets, and, for d ∈ D_{λμ}, d^{−1}𝔖_{λ}d ∩ 𝔖_{μ} is a Young (or parabolic) subgroup. For convenience, we will use in the sequel the following notation:

(1.4)  𝔖_{λd∩μ} = d^{−1}𝔖_{λ}d ∩ 𝔖_{μ}, 𝔖_{λ∩dμ} = 𝔖_{λ} ∩ d𝔖_{μ}d^{−1}.

To any 2-composition λ = (λ^{(1)}, λ^{(2)}) ∈ Λ_{2}(n, r), we associate a so-called quasiparabolic subgroup W_{λ} of W (see [DS]). By definition, we have W_{λ} = C_{λ}𝔖λ̄, where C_{λ} is the subgroup of W generated by t_{i} = s_{i−1} ⋯ s_{1}s_{0}s_{1} ⋯ s_{i−1} for 1 ⩽ i ⩽ |λ^{(1)}|. Let λ̂ = (|λ^{(1)}|, λ^{(2)}). Then Wλ̂ is the minimal parabolic subgroup of W containing W_{λ}. For quasi-parabolic subgroups, the distinguished coset representatives are introduced in [DS, §2]. Let D_{λ^{(1)}} (resp. Dλ̂) be the set of distinguished representatives in the right coset 𝔖_{λ^{(1)}}\𝔖_{a} with a = |λ^{(1)}| (resp. Wλ̂\W). Then D_{λ} = D_{λ^{(1)}}Dλ̂ (resp. D_{λμ} = D_{λ} ∩ D_{μ}^{−1}) is the set of distinguished representatives in the right W_{λ}-cosets (resp. double W_{λ}-W_{μ}-cosets) in the sense of [DS, (2.2.5)]. The reader should not confuse the notation D_{λμ} for type A with that for type B.

For λ = (λ^{(1)}, λ^{(2)}) ∈ Λ_{2}(n, r), let x_{λ} = xλ̄π_{λ} where π_{λ} = ∏_{i=1}^{|λ^{(1)}|}(q^{i−1} + T_{t_{i}}), and xλ̄ = ∑_{w∈𝔖λ̄} T_{w}. The element x_{λ} serves as the generator of the trivial representation for W_{λ}. Following [DS, §3], we introduce the endomorphism algebras

(1.5)

S^{2} = S_{q}^{2}(n, r) = End_{H}( ⨁_{λ∈Λ_{2}(n,r)} x_{λ}H)
S_{R}^{2} = S_{q}^{2}(n, r; R) = End_{H_{R}}( ⨁_{λ∈Λ_{2}(n,r)} x_{λ}H_{R}).

These endomorphism algebras are called the q-Schur^{2} algebras of degree (n, r) (see also [DJM1] for a Morita equivalent version).

For any d ∈ D_{λμ}, the conjugate intersection W_{λ}^{d} ∩ W_{μ} = d^{−1}W_{λ}d ∩ W_{μ} is a subgroup of W, which will be denoted by W_{ν} with ν = λd ∩ μ (cf. [DS, (2.2.8)]). Similarly, write W d̂ ; λ̂ ∩ Wμ̂ = Wν̂. Then Wν̂ is a parabolic subgroup of W. Let π_{μ\ν̂} ∈ H_{R} be the element obtained by deleting the product πν̂ from π_{μ}. So π_{μ} = πν̂π_{μ\ν̂}. For any λ, μ ∈ Λ_{2}(n, r) and d ∈ D_{λμ}, there is a unique element φ_{λμ}^{d} in S_{R}^{2} such that

(1.6)  φ_{λμ}^{d}(x_{ν}h) = δ_{μ,ν}x_{λ}T_{ud̂}π_{μ\ν̂}T_{v}T_{D_{λ̄d∩μ̄}∩Wμ̄}h,

where d = ud̂v is the right distinguished decomposition of d (see [DS, §2.3]), i.e., d̂ ∈ D_{λ̂μ̂}, u ∈ D_{λ^{(1)}} and d̂v^{−1} ∈ Dλ̂, and T_{X} = ∑_{w∈X} T_{w} for any subset X ⊂ W. Moreover, the set

(1.7)  {φ_{λμ}^{d} | λ, μ ∈ Λ_{2}(n, r), d ∈ D_{λμ}}

forms a basis for S_{R}^{2}. We shall call it the natural basis for S_{R}^{2}.

Write X_{W_{λ}dW_{μ}} = φ_{λμ}^{d}(x_{μ}). Let ι be the anti-automorphism of H_{R} sending T_{w} to T_{w^{−1}}. By [DS, (4.2.2.2)]

(1.8)  X_{W_{λ}dW_{μ}}^{ι} = X_{W_{μ}d^{−1}W_{λ}}.

So ι induces an anti-involution

(1.9)  ι : S_{R}^{2} → S_{R}^{2} such that (φ_{λμ}^{d})^{ι} = φ_{μλ}^{d^{−1}}.

Now we mimic Green’s construction in [G1] to introduce certain subspaces of S_{R}^{2} via its natural basis given in (1.7). First, we need some combinatorics.

Let I(n, r) = {i = (i_{1}, i_{2}, ⋯, i_{r}) | 1 ⩽ i_{j} ⩽ n for 1 ⩽ j ⩽ r}, where n, r are positive integers. Then, the symmetric group 𝔖_{r} acts on I(n, r) by place permutation: iw = (i_{w(1)}, ⋯, i_{w(r)}) for any i ∈ I(n, r) and w ∈ 𝔖_{r}. Following [G1, §3], I(n, r) is a poset with the partial order ≼ defined by setting i ≼ j if i_{k} ⩽ j_{k} for all k with 1 ⩽ k ⩽ r. The weight wt(i) of i is a composition (λ_{1}, ⋯, λ_{n}) of r, where λ_{j} = #{i_{k} ∈ i | i_{k} = j}. Obviously, we have wt(i) ⊵ wt(j) if i ≼ j. For any

λ ∈ Λ(n, r), let i_{λ} = (\underbrace{1 ⋯ 1}_{λ_{1}}, ⋯, \underbrace{n ⋯ n}_{λ_{n}}). If λ = (λ^{(1)}, ⋯, λ^{(m)}) ∈ Λ_{m}(n, r), then

we define i_{λ} = iλ̄ ∈ I(N, r), where N is defined in (1.2).

(1.10) Lemma. Let λ, μ ∈ Λ_{m}(n, r) be of c.n.s. a, b respectively. Then i_{λ}w ≽ i_{μ} if and only if, for every i with 1 ⩽ i ⩽ m and every k, w(j) ⩽ b_{i−1} + ∑_{t=1}^{k} μ_{t}^{(i)} for all j with a_{i−1} < j ⩽ a_{i−1} + ∑_{t=1}^{k} λ_{t}^{(i)}.

Proof. Since i_{λ} = iλ̄ for any λ ∈ Λ_{m}(n, r), we have i_{λ}w ≽ i_{μ} if and only if iλ̄w ≽ iμ̄. So we may assume λ = λ̄ = (λ_{1}, ⋯, λ_{N}) and μ = μ̄ = (μ_{1}, ⋯, μ_{N}). Thus, i_{λ}w ≽ i_{μ} if and only if

i_{w(1)}, ⋯, i_{w(μ_{1})} ⩾ 1,
i_{w(μ_{1}+1)}, ⋯, i_{w(μ_{1}+μ_{2})} ⩾ 2,
⋯,
i_{w(μ_{1}+···+μ_{N−1}+1)}, ⋯, i_{w(r)} ⩾ N,

which are equivalent to

w(1), ⋯, w(λ_{1}) ⩽ μ_{1},
w(λ_{1} + 1), ⋯, w(λ_{1} + λ_{2}) ⩽ μ_{1} + μ_{2},
⋯,
w(λ_{1} + ⋯ + λ_{N−1} + 1), ⋯, w(r) ⩽ r,

as required. □

(1.11) Definition. For any μ ∈ Λ_{m}(n, r), let

Ω_{m}^{≽}(μ) = {(λ, d) | λ ∈ Λ_{m}(n, r), d ∈ Dλ̄μ̄ and i_{λ}d ≽ i_{μ}}
Ω_{m}^{≼}(μ) = {(λ, d) | λ ∈ Λ_{m}(n, r), d ∈ Dμ̄λ̄ and i_{μ}d ≼ i_{λ}}.

and

S_{R}^{2,≽} = S_{R}^{2,≽}(n, r) = R-span{φ_{λμ}^{d} | (λ, d) ∈ Ω_{2}^{≽}(μ), μ ∈ Λ_{2}(n, r)}
S_{R}^{2,≼} = S_{R}^{2,≼}(n, r) = R-span{φ_{μλ}^{d} | (λ, d) ∈ Ω_{2}^{≼}(μ), μ ∈ Λ_{2}(n, r)}.

Clearly, (λ, d) ∈ Ω_{m}^{≽}(μ) if and only if (λ, d^{−1}) ∈ Ω_{m}^{≼}(μ), and therefore, we have, by (1.9), S_{R}^{2,≼} = (S_{R}^{2,≽})^{ι}.

2. The subalgebra structure on S_{R}^{2,≽} and S_{R}^{2,≼}. In this section, we will prove that the vector spaces S_{R}^{2,≽} and S_{R}^{2,≼} are actually subalgebras of S_{R}^{2}. We shall see that the subalgebra structure of S_{R}^{2,≽} and S_{R}^{2,≼} are closely related to the subalgebra structure of the Borel subalgebras S_{R}^{1,≽} and S_{R}^{1,≼} for the corresponding q-Schur algebra. Let us first look at the q-Schur algebra case.

For a commutative ring R, let R[M_{n}(q)] be the associative algebra over R generated by X_{ij} with 1 ⩽ i, j ⩽ n such that

(2.1)  \{ \begin{matrix}(1) X_{ij}X_{ik} = qX_{ik}X_{ij}, & if j > k, \\ (2) X_{ji}X_{ki} = X_{ki}X_{ji}, & if j > k, \\ (3) X_{ij}X_{rs} = q^{−1}X_{rs}X_{ij}, & if i > r, j < s, \\ (4) X_{ij}X_{rs} − X_{rs}X_{ij} = (q^{−1} − 1)X_{is}X_{rj}, & if i < r, j < s. \end{matrix}

As an R-module, R[M_{n}(q)] has a basis {∏_{ij} X_{ij}^{t_{ij}} | t_{ij} ∈ ℤ^{+}}, where the products are formed with respect to any fixed order of the X_{ij}’s (see, for example, [DPW, (1.1)] with α = q, β = 1 there). Let A_{q}(n, r) be the r-th homogeneous component of R[M_{q}(n)]. Then A_{q}(n, r) has a basis

{X_{λμ}^{d} = X_{i_{λ}d,i_{μ}} | λ, μ ∈ Λ(n, r), d ∈ D_{λμ}},

where D_{λμ} denotes the set of distinguished representatives for the double cosets 𝔖_{λ}\𝔖_{r}/𝔖_{μ}, and X_{ij} = X_{i_{1},j_{1}}X_{i_{2},j_{2}} ⋯ X_{i_{r},j_{r}} if i = (i_{1}, ⋯, i_{n}) and j = (j_{1}, ⋯, j_{n}). Denote by A_{q}(n, r)^{∗} the linear dual of A_{q}(n, r). Then, by [DPW, (5.5)],

A_{q}(n, r)^{∗} ≅ End_{H_{R}}(⊕_{λ∈Λ(n,r)}x_{λ}H_{R})

where H_{R} is the Hecke algebra associated to the symmetric group 𝔖_{r} over the commutative ring R.

(2.2) Definitions and Results. The algebra

S_{R}^{1} = S_{q}(n, r; R) = End_{H_{R}}(⊕_{λ∈Λ(n,r)}x_{λ}H_{R})

is called the q-Schur algebra of degree (n, r) (cf. [DJ2]). The natural basis for a q-Schur algebra is given as follows: For λ, μ ∈ Λ(n, r), d ∈ D_{λμ}, let ψ_{λμ}^{d} ∈ S_{R}^{1} be defined by

ψ_{λμ}^{d}(x_{ν}h) = δ_{μν} ∑_{w∈𝔖_{λ}d𝔖_{μ}} T_{w}h = δ_{μν}T_{𝔖_{λ}d𝔖_{μ}}h.

Then S_{R}^{1} has an R-basis {ψ_{λμ}^{d} | λ, μ ∈ Λ(n, r), d ∈ D_{λμ}}. Moreover, if we identify ψ_{λμ}^{d} with its image under the isomorphism above, the basis {ψ_{λμ}^{d} | λ, μ ∈ Λ(n, r), d ∈ D_{λμ}} is the dual of the basis {X_{λ,μ}^{d} | λ, μ ∈ Λ(n, r), d ∈ D_{λμ}} for A_{q}(n, r). Thus, we have ψ_{λμ}^{d}(X_{ρν}^{d_{1}}) = δ_{λρ}δ_{μν}δ_{d,d_{1}} (see [DPW, (5.7)]).

The following theorem was first obtained by Parshall and Wang (see [PW, §11.2]). The proof below is different.

(2.3) Theorem. Suppose λ, μ, ν ∈ Λ(n, r) and d_{1} ∈ D_{λμ}, d_{2} ∈ D_{μν}.

(a) If i_{λ}d_{1} ≽ i_{μ} and i_{μ}d_{2} ≽ i_{ν}, then ψ_{λμ}^{d_{1}}ψ_{μν}^{d_{2}} = ∑\begin{matrix}d∈D_{λν} \\ i_{λ}d≽i_{ν} \end{matrix} a_{d}ψ_{λν}^{d} for some a_{d} ∈ R.

(b) If i_{λ}d_{1} ≼ i_{μ} and i_{μ}d_{2} ≼ i_{ν}, then ψ_{λμ}^{d_{1}}ψ_{μν}^{d_{2}} = ∑\begin{matrix}d∈D_{λν} \\ i_{λ}d≼i_{ν} \end{matrix} a_{d}ψ_{λν}^{d} for some a_{d} ∈ R.

(c) Let S_{R}^{1,≽} = S_{R}^{1,≽}(n, r) (resp. S_{R}^{1,≼} = S_{R}^{1,≼}(n, r) ) be the free R-submodule of S_{R}^{1} spanned by {ψ_{λμ}^{d} | (λ, d) ∈ Ω_{1}^{≽}(μ), μ ∈ Λ_{1}(n, r)} (resp. {ψ_{μλ}^{d} | (λ, d) ∈ Ω_{1}^{≼}(μ), μ ∈ Λ_{1}(n, r)}). Then the subspaces S_{R}^{1,≽} and S_{R}^{1,≼} are subalgebras for the q-Schur algebra S_{R}^{1}.

Proof. By embedding S_{R}^{1} into S_{R}^{2} via ψ_{λμ}^{d} ↦ φ_{(λ,0)(μ,0)}^{d} (or a direct construction), we obtain an anti-automorphism ι on S_{R}^{1} of order 2 which turns (b) into (a). The statement (c) follows easily from (a) and (b). So it remains to prove (b).

Suppose λ, μ ∈ Λ(n, r), d ∈ D_{λμ} with i_{λ}d ≼ i_{μ}. We claim that ψ_{λμ}^{d}(X_{ik}) ≠ 0 implies i ≼ k. Indeed, we have, by the hypothesis and (2.2), k = i_{μ}w. If l(w) = 0, i.e., w = e, then i_{μ} = k. So i = i_{λ}d ≼ i_{μ} = k. Assume now l(w) > 0. Write w = w′t with t = (a, a + 1) and l(w) = l(w′) + 1. Then k_{a} > k_{a+1}. If i_{a} ⩽ i_{a+1}, then, by (2.1(3)), X_{ik} = qX_{it,kt} = qX_{it,i_{μ}w′}. By induction, it ≼ i_{μ}w′ = kt, which implies i ≼ k. If i_{a} > i_{a+1}, then by (2.1(4)(3)),

\begin{aligned}X_{i_{a}k_{a}}X_{i_{a+1}k_{a+1}} & = X_{i_{a+1}k_{a+1}}X_{i_{a}k_{a}} − (q^{−1} − 1)X_{i_{a+1}k_{a}}X_{i_{a}k_{a+1}} \\ & = X_{i_{a+1}k_{a+1}}X_{i_{a}k_{a}} − (1 − q)X_{i_{a}k_{a+1}}X_{i_{a+1}k_{a}}. \end{aligned}

Thus, X_{ik} = X_{it,kt} −(1−q)X_{i,kt}. Since ψ_{λμ}^{d}(X_{ik}) ≠ 0, we have either ψ_{λμ}^{d}(X_{it,kt}) ≠ 0 or ψ_{λμ}^{d}(X_{i,kt}) ≠ 0. By induction, either it ≼ kt or i ≼ kt. If it ≼ kt, then i ≼ k. Assume i ≼ kt. By definition, i_{a+1} ⩽ k_{a} and i_{a} ⩽ k_{a+1}, which implies i_{a+1} < i_{a} ⩽ k_{a+1} < k_{a}. Hence, i ≼ k. This completes the proof of our claim.

Let ψ_{λμ}^{d_{1}}ψ_{μν}^{d_{2}} = ∑_{d∈D_{λν}} a_{d}ψ_{λν}^{d}. We hope to prove that if i_{λ}d_{1} ≼ i_{μ} and i_{μ}d_{2} ≼ i_{ν}, then a_{d} ≠ 0 implies i_{λ}d ≼ i_{ν}. Suppose now a_{d} ≠ 0. Then ψ_{λμ}^{d_{1}}ψ_{μν}^{d_{2}}(X_{ij}) ≠ 0, where (i, j) = (i_{λ}d, i_{ν}). If we denote by Δ the comultiplication on A_{q}(n, r), then,

ψ_{λμ}^{d_{1}}ψ_{μν}^{d_{2}}(X_{ij}) = mult ∘ (ψ_{λμ}^{d_{1}} ⊗ ψ_{μν}^{d_{2}}) ∘ Δ(X_{ij}) = ∑_{k∈I(n,r)} ψ_{λμ}^{d_{1}}(X_{ik})ψ_{μν}^{d_{2}}(X_{kj}) ≠ 0

So there is a k ∈ I(n, r) such that ψ_{λμ}^{d_{1}}(X_{ik}) ≠ 0 and ψ_{μν}^{d_{2}}(X_{kj}) ≠ 0. Thus, i ≼ k and k ≼ j by the above claim, and hence, i ≼ j, or i_{λ}d ≼ i_{ν}. □

In the rest of this section, we shall try to find the relation between φ_{λμ}^{d} and ψ_{λ̄μ̄}^{d} for λ, μ ∈ Λ_{2}(n, r) and d ∈ Dλ̄μ̄ and then we prove that S_{R}^{2,≽} and S_{R}^{2,≼} are subalgebras via (2.3). First, we recall a result on the Weyl group of type B_{r}.

For λ = (λ^{(1)}, λ^{(2)}) ∈ Λ_{2}(n, r), let C_{λ} be the subgroup of W generated by t_{i} with 1 ⩽ i ⩽ |λ^{(1)}| (as given in §1). Recall from (1.6) that ν̂ ∈ Λ_{2}(n, r) is defined by W d̂ ; λ̂ ∩ Wμ̂ = Wν̂ where λ, μ ∈ Λ_{2}(n, r) and d̂ is distinguished uniquely determined by d. In particular, we have (cf. [DS, §4.2]):

(2.4)  Cν̂ = Cd̂ ; λ̂ ∩ Cμ̂ = Cd̂ ; λ ∩ C_{μ} = Z_{C_{λ}∩C_{μ}}(d̂)

where Z_{C_{λ}∩C_{μ}}(d̂) is the centerlizer of d̂ in C_{λ} ∩ C_{μ}.

(2.5) Theorem. The subspaces S_{R}^{2,≽} and S_{R}^{2,≼} are subalgebras of S_{R}^{2}.

Proof. We claim first that, for any λ, μ ∈ Λ_{2}(n, r), d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ},

(2.6)  φ_{λμ}^{d}(x_{μ}) = x_{λ}T_{μd̂}π_{μ\ν̂}T_{v}h_{λ̄d∩μ̄} = T_{𝔖λ̄d𝔖μ̄}π_{μ} = ψ_{λ̄μ̄}^{d}(xμ̄)π_{μ},

where h_{λ̄d∩μ̄} = T_{D_{λ̄d∩μ̄}∩𝔖μ̄} and d = ud̂v is the right distinguished decomposition of d (see (1.6)). Indeed, since i_{λ}d ≽ i_{μ} implies d(j) ⩽ |μ^{(1)}| for all j ⩽ |λ^{(1)}| (see (1.10)), we have d^{−1}C_{λ}d ⊆ C_{μ}. Let d = ud̂v be the right distinguished decomposition of d with u ∈ D_{λ^{(1)}}, v^{−1} ∈ D_{μ^{(1)}} and d̂∈ D_{λ̂μ̂}. Then, d̂^{−1}C_{λ}d̂⊆ C_{μ}, since C_{λ} is a normal subgroup of W (B_{a}) with a = |λ^{(1)}|. For any t_{j} ∈ C_{λ}, write t_{j}d̂ = d̂(d̂^{−1}t_{j}d̂). We have l(t_{j})+l(d̂) = l(d̂)+l(d̂^{−1}t_{j}d̂) as d̂^{−1}t_{j}d̂∈ C_{μ}. Consequently, l(d̂^{−1}t_{j}d̂) = l(t_{j}). Comparing the numbers of s_{0} occurring in d̂^{−1}t_{j}d̂ and t_{j}, we have d̂^{−1}t_{j}d̂= t_{j}, and hence, C_{λ} = d̂^{−1}C_{λ}d̂. Thus, C_{λ} is a subgroup of C_{μ}, and hence, Cν̂ = C_{λ} and π_{λ\ν̂} = 1. Let d^{−1} = u_{1}d̂_{1}v_{1} be the right distinguished decomposition for d^{−1}. Then, by (1.6),

X_{W_{μ}d^{−1}W_{λ}} = φ_{μλ}^{d^{−1}}(x_{λ}) = x_{μ}T_{u_{1}d̂_{1}}π_{λ\ν̂}T_{v_{1}}h_{λ̄d_{1}∩μ̄} = π_{μ}T_{𝔖μ̄d^{−1}𝔖λ̄}.

Thus, φ_{λμ}^{d}(x_{μ}) = X_{W_{λ}dW_{μ}} = (X_{W_{μ}d^{−1}W_{λ}})^{ι} = T_{𝔖λ̄d𝔖μ̄}π_{μ}. The last equality in (2.6) follows from the definition of ψ_{λ̄μ̄}^{d}.

If φ_{λμ}^{d_{1}}, φ_{νρ}^{d_{2}} ∈ S_{R}^{2,≽}, then φ_{λμ}^{d_{1}}φ_{νρ}^{d_{2}} = 0 unless μ = ν. Let d_{2} = ud̂_{2}v be the right distinguished decomposition of d_{2}. By repeatedly applying (2.6), we have

(2.7)  \begin{matrix}φ_{λμ}^{d_{1}}φ_{μρ}^{d_{2}}(x_{ρ}) = & φ_{λμ}^{d_{1}}(x_{μ}T_{ud̂_{2}}π_{ρ\μ̂}T_{v}h_{λ̄d_{2}∩μ̄}) \\ = & T_{𝔖λ̄d_{1}𝔖μ̄}π_{μ}T_{ud̂_{2}}π_{ρ\μ̂}T_{v}h_{λ̄d_{2}∩μ̄} \\ = & T_{𝔖λ̄∩D_{λ̄∩d_{1}μ̄}}T_{d_{1}}T_{𝔖μ̄d_{2}𝔖ρ̄}π_{ρ} \\ = & ψ_{λ̄μ̄}^{d_{1}}ψ_{μ̄ρ̄}^{d_{2}}(xρ̄)π_{ρ} \\ = & ∑_{d∈D_{λ̄ρ̄},iλ̄d≽iρ̄} a_{d}ψ_{λ̄ρ̄}^{d}(xρ̄)π_{ρ}, \end{matrix}

where the notation λ̄ ∩ d_{1}μ̄ is given in (1.4), and the last equality follows from (2.3(a)). Noting i_{λ} = iλ̄, we obtain

(2.8)  φ_{λμ}^{d_{1}}φ_{μρ}^{d_{2}} = ∑_{d∈D_{λ̄ρ̄},i_{λ}d≽i_{ρ}} a_{d}φ_{λρ}^{d} ∈ S_{R}^{2,≽}

for some a_{d} ∈ R. Hence the subspace S_{R}^{2,≽} is a subalgebra of S_{R}^{2}. Applying (1.9), we see that S_{R}^{2,≼} is also a subalgebra of S_{R}^{2}. □

(2.9) Corollary. We have the following isomorphisms of R-algebras S_{R}^{2,≽}(n, r) ≅ S_{R}^{1,≽}(max_{n,r} + n, r), S_{R}^{2,≼}(n, r) ≅ S_{R}^{1,≼}(max_{n,r} + n, r).

Proof. Immediately follow from (2.7) and (2.8). □

3. The triangular decomposition of S_{R}^{2}. In [G1], Green proved that a Schur algebra is a product of its two (opposite) Borel subalgebras. In this section, we are going to generalize this result to q-Schur^{2} algebras. Let S_{R}^{2,≽}S_{R}^{2,≼} denote the submodule of S_{R}^{2} generated by all products ab with a ∈ S_{R}^{2,≽} and b ∈ S_{R}^{2,≼}. We shall prove that S_{R}^{2} = S_{R}^{2,≽}S_{R}^{2,≼}, and call this the triangular decomposition of S_{R}^{2}. Using the embedding of a q-Schur algebra into a q-Schur^{2} algebra, we will reobtain the triangular decomposition of a q-Schur algebra, which was first established by Parshall and Wang [PW] in the context of quantum linear groups. Our proof below is in fact a simple application of the Green-Murphy type basis for the q-Schur^{2} algebra given in [DS]. These basis elements have double indices of semi-standard tableaux. Let us briefly recall some combinatorics first.

As usual, we identify each μ ∈ Λ(n, r) with its Young diagram, which consists of boxes arranged in a manner as illustrated by the example μ = (4021) ∈ Λ(4, 7) for which we have

μ = ^{− − −}

A μ-tableau t is obtained by replacing each box by one of the numbers 1, 2, ⋯, r, and t is called to be regular if the set of entries in t is equal to {1, ⋯, r}. A regular μ-tableau t is row-standard if its entries are increasing along each row. Let t^{μ} be the regular μ-tableau in which the numbers 1, 2, ⋯, r appear in order along successive rows. For example, for μ = (4021)

t^{μ} = \begin{matrix}1 & 2 & 3 & 4 \\ − & − & − \\ 5 & 6 \\ 7 \end{matrix}

The symmetric group 𝔖_{r} acts on the regular tableaux by permuting its entries. The set of distinguished representatives D_{μ} can be characterised as

D_{μ} = {d ∈ 𝔖_{r} | t^{μ}d is row standard}.

A μ-tableau t is said to be of type λ if the number of entries i in t is equal to λ_{i}. For any μ ∈ Λ(n, r)^{+}, a μ-tableau is said to be semi-standard if its entries are weakly increasing along each row and strictly increasing down each column. Let 𝔗(μ, λ) (resp. 𝔗^{ss}(μ, λ) ) be the set of all μ-tableaux (resp. semi-standard μ-tableaux ) of type λ.

For w ∈ 𝔖_{r} and 𝔰 ∈ 𝔗^{ss}(μ, λ), let δ(w, 𝔰) ∈ D_{λ} be defined by the row-standard λ-tableau t^{λ}δ(w, 𝔰) for which i belongs to row a if the place occupied by i in t^{μ}w is occupied by a in 𝔰. Thus, we obtain a map δ(∗, ∗) : 𝔖_{r} × 𝔗(μ, λ) → D_{λ}. In particular, the map δ(1, ∗) gives a bijection between 𝔗(μ, λ) and D_{λ}, and 𝔰 has non-decreasing rows (i.e., 𝔰 is weakly row-standard) if and only if δ(1, 𝔰) ∈ D_{λμ} ([DJ1, (1.7)]).

We now introduce appropriate terminology for multi-compositions. For μ = (μ^{(1)}, ⋯, μ^{(m)}) ∈ Λ_{m}(n, r), a multi-tableau 𝔰 = (𝔰_{1}, 𝔰_{2}, ⋯, 𝔰_{m}) is called a μtableau if

𝔰̄ = 𝔰_{1} ; ⋮ ; 𝔰_{m}

is a μ̄-tableau. Clearly, each 𝔰_{i} is a μ^{(i)}-tableau. The multi-tableau 𝔰 is said to be regular (resp. row-standard) if 𝔰̄ is regular (resp. row standard). For μ, λ ∈ Λ_{m}(n, r), a μ-tableau 𝔰 is said to be of type λ if the μ̄-tableau 𝔰̄ is of type λ̄. Let t^{μ} be the regular tableau such that t̄^{μ} = tμ̄. For example,

t^{μ} = (t^{μ^{(1)}}, ⋯, t^{μ^{(m)}}) = 1 2 3 5 6 9 4 7 8

if μ = ((310 ⋯ 0), (220 ⋯ 0), (10)) ∈ Λ_{3}(2, 9). Here the rows in each single tableau corresponding to 0 parts at the end of each λ^{(i)} are omitted for visual clarity.

The symmetric group 𝔖_{r} acts on regular μ-tableaux diagonally, i.e., 𝔰w = (𝔰_{1}w, ⋯, 𝔰_{m}w). Note that 𝔰̄w = \overline{𝔰w}.

The notion of semi-standard multi-tableaux has been introduced by Du and Scott in [DS] for m = 2 (see also a version given in [DJM1]), and by Dipper, James and Mathas for arbitrary m in [DJM2]. The following definition is a generalized version of [DS, (1.2.2)].

(3.1) Definition. For μ ∈ Λ_{m}(n, r)^{+} and λ ∈ Λ_{m}(n, r), a μ-tableau 𝔰 of type λ is said to be semi-standard if

(a) 𝔰_{1}, ⋯, 𝔰_{m} have non-decreasing rows and strictly increasing columns,

(b) all entries in 𝔰_{j} are strictly bigger than (j − 1)max_{n,r} for 2 ⩽ j ⩽ m.

Note that, if we write every entry in 𝔰 as i + (j − 1)max_{n,r} with 1 ⩽ i < max_{n,r} and replace it by the symbol i_{j}, the definition above is turned into the definition given in [DJM2, (4.4)].

Let 𝔗_{m}^{ss}(μ, λ) be the set of all semi-standard μ-tableaux of type λ. For any 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), let

(3.2)  δ(𝔰) = δ(1, 𝔰̄).

Since 𝔰 is a semi-standard μ-tableau of type λ, 𝔰̄ is a weakly row-standard μ̄-tableau of type λ̄, and hence, δ(𝔰) ∈ Dλ̄μ̄.

(3.3) Lemma. For any 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), we have i_{λ}δ(𝔰) ≽ i_{μ}.

Proof. Let a resp. b be the c.n.s. of λ resp. μ. For a tableau t, write row_{t}(a) = i, if a is an entry in t whose row index is i. By (1.10), it suffices to prove that for every i with 1 ⩽ i ⩽ m and every k, δ(𝔰)(j) ⩽ b_{i−1} + ∑_{t=1}^{k} μ_{t}^{(i)} for all j with a_{i−1} < j ⩽ a_{i−1} + ∑_{t=1}^{k} λ_{t}^{(i)}. Suppose there are i_{0}, j_{0}, k_{0} with a_{i_{0}−1} < j_{0} ⩽ a_{i_{0}−1} + ∑_{t=1}^{k_{0}} λ_{t}^{(i)} such that δ(𝔰)(j_{0}) > b_{i_{0}−1} + ∑_{t=1}^{k_{0}} μ_{t}^{(i)}. Then rowt̄^{μ}(δ(𝔰)(j_{0})) ⩾ (i_{0} − 1)max_{n,r} + (k_{0} + 1). Since 𝔰 is a semi-standard μ-tableau of type λ, we have by (3.1(a)) that a ⩾ (i_{0} −1)max_{n,r} +(k_{0} +1), where a is the entry of 𝔰̄ in the place which is occupied by δ(𝔰)(j_{0}) in t̄^{μ}. By definition, a = row_{t̄^{λ}δ(𝔰)}(δ(𝔰)(j_{0})) ⩾ (i_{0} − 1)max_{n,r} + (k_{0} + 1). However, a = row_{t̄^{λ}δ(𝔰)}(δ(𝔰)(j_{0})) = rowt̄^{λ}(j_{0}), since the symmetric group 𝔖_{r} acts on t̄^{λ} by entry permutation. Therefore, a ⩽ (i_{0}−1)max_{n,r}+k_{0}, a contradiction. □

The first part of the following theorem guarantees the existence of the expected triangular decomposition of S_{R}^{2}.

(3.4) Theorem. ( [DS, §6]) (a) The set

{φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}} | μ ∈ Λ_{2}(n, r)^{+}, λ, ν ∈ Λ_{2}(n, r), 𝔰 ∈ 𝔗_{2}^{ss}(μ, λ), 𝔱 ∈ 𝔗_{2}^{ss}(μ, ν)}

is an R-basis of S_{R}^{2}.

(b) For any commutative Noetherian ring R, the algebra S_{R}^{2} is quasi-hereditary.

We shall generalize this result to the q-Schur^{m} algebra S_{R}^{m} in order to get its triangular decomposition.

(3.5) Theorem. Let S_{R}^{2,≽} and S_{R}^{2,≼} be the subalgebras of S_{R}^{2} as defined in (1.11). Then S_{R}^{2} has the following triangular decomposition

S_{R}^{2} = S_{R}^{2,≽}S_{R}^{2,≼} = ∑_{μ∈Λ_{2}(n,r)^{+}} S_{R}^{2,≽}φ_{μμ}^{1}S_{R}^{2,≼}.

Proof. Applying (3.3) for m = 2, we have φ_{λμ}^{δ(𝔰)} ∈ S_{R}^{2,≽} for any 𝔰 ∈ 𝔗_{2}^{ss}(μ, λ). Let ι

be the anti-involution on H_{R} as in (1.8). By (1.9), φ_{μν}^{δ(𝔱)^{−1}} = (φ_{νμ}^{δ(𝔱)})^{ι} ∈ (S_{R}^{2,≽})^{ι} = S_{R}^{2,≼} for every 𝔱 ∈ 𝔗_{2}^{ss}(μ, ν). So the basis given in (3.4) is contained in S_{R}^{2,≽}S_{R}^{2,≼}, and therefore, S_{R}^{2} = S_{R}^{2,≽}S_{R}^{2,≼}. Finally, for any 𝔰 ∈ 𝔗_{2}^{ss}(μ, λ) and 𝔱 ∈ 𝔗_{2}^{ss}(μ, ν),

φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}} = φ_{λμ}^{δ(𝔰)}φ_{μμ}^{1}φ_{μν}^{δ(𝔱)^{−1}} ∈ S_{R}^{2,≽}φ_{μμ}^{1}S_{R}^{2,≼}.

So the last equality follows. □

4. The quasi-heredity of S_{R}^{2,≼} and S_{R}^{2,≽}. In this section, we will discuss the representation theory of the Borel type subalgebras S_{R}^{2,≼} and S_{R}^{2,≽} of a q-Schur^{2} algebra S_{R}^{2}. We will show that both S_{R}^{2,≼} and S_{R}^{2,≽} are quasi-hereditary. The proof is based on the notion of a standardly based algebra introduced by the authors in [DR]. Then, using the triangular decomposition, we will also prove that the standard modules and costandard modules for S_{R}^{2} can be induced from a left (resp. right) standard modules for S_{R}^{2,≼} (resp. S_{R}^{2,≽}). We should point out that, by (2.9), the quasi-heredity (4.5) of S_{R}^{2,≼} and S_{R}^{2,≽} follows from that of the Borel subalgebras of q-Schur algebras (see [DR, (5.6.1)]). For completeness, we include a direct proof below. First, we recall the definition of standardly based algebras.

(4.1) Definition. Assume that R is a commutative ring with 1. Let A be an R-algebra and (Λ, ⩽) a poset. A is called a standardly based algebra on Λ (or to be standardly based) if the following conditions hold.

(a) For any λ ∈ Λ, there are index sets I(λ), J(λ) and subsets

A^{λ} = {a_{i,j}^{λ} | (i, j) ∈ I(λ) × J(λ)}

of A such that the union A = ∪_{λ∈Λ}A^{λ} is disjoint and forms an R-basis for A. (b) For any a ∈ A, a_{i,j}^{λ} ∈ A, we have

\begin{matrix}a · a_{i,j}^{λ} ≡ ∑_{i′∈I(λ)} f_{i′,λ}(a, i)a_{i′,j}^{λ} mod A(>λ) \\ a_{i,j}^{λ} · a ≡ ∑_{j′∈J(λ)} f_{λ,j′}(j, a)a_{i,j′}^{λ} mod A(>λ) \end{matrix}

where A(> λ) is the R-submodule of A spanned by A^{μ} with μ > λ, and f_{i′,λ}(a, i), f_{λ,j′}(j, a) ∈ R are independent of j and i, respectively. Such a base A is called a standard base for the algebra A.

Note that a cellular algebra [GL] must be standardly based. However, the converse may not be true. We shall prove that S_{R}^{2,≽} and S_{R}^{2,≼} are standardly based algebras whose standard bases are not cellular bases.

(4.2) Definition. For any λ ∈ Λ, let f_{λ} : J(λ) × I(λ) → R be a function, whose value f_{λ}(j, i′) at (j, i′) ∈ J(λ) × I(λ) is defined by

a_{ij}^{λ}a_{i′j′}^{λ} ≡ f_{λ}(j, i′)a_{ij′}^{λ} mod A(>λ).

The function f_{λ} induces a bilinear form β_{λ} : A^{λ} ×A^{λ} → R such that β_{λ}(a_{ij}^{λ}, a_{i′j′}^{λ}) = f_{λ}(j′, i), where A^{λ} is the free R-submodule of A spanned by a_{ij}^{λ} for all (i, j) ∈ I(λ) × J(λ). We say A is a standardly full-based algebra if im (β_{λ}) = R for all λ ∈ Λ. (compare [DR, (1.3.1)]).

The following result has been proved by the authors in [DR, (3.2.1), (4.2.7)].

(4.3) Theorem. (a) Let R be a commutative Noetherian ring. If A is a standardly full-based algebra, then A is a quasi-hereditary algebra over R in the sense of [CPS2].

(b) If R is a commutative local Noetherian ring, then A is split quasi-hereditary if and only if A is a standardly full-based algebra.

If A is quasi-hereditary with poset Λ, then we use Δ(A, λ) and ∇(A, λ) (λ ∈ Λ) to denote the standard and costandard (left) A-modules. Since A^{op} is also quasihereditary, the left A^{op}-modules Δ(A^{op}, λ) and ∇(A^{op}, λ) are naturally right Amodules by shifting the left action of A^{op} to the right action of A. We shall denote these right modules by Δ^{op}(A, λ) and ∇^{op}(A, λ).

Let A be a standardly based algebra on the poset Λ with a standard base as in (4.1). For each λ ∈ Λ and (i, j) ∈ I(λ) × J(λ), let Δ(λ, j) (resp. Δ(i, λ)) be the left (resp. right) A-submodule generated by the elements a_{ij}^{λ} + A(>λ) with j fixed (resp. with i fixed). The following result (see [DR, (3.2.2)]) identifies these modules with the standard and costandard modules under the quasi-heredity structure of A.

(4.4) Theorem. If R is a commutative local Noetherian ring and A is standardly full-based, then the modules Δ(λ, j) (resp. Δ(i, λ)) is isomorphic to the left standard module Δ(A, λ) (resp. right standard modules Δ^{op}(A, λ)) of A under its quasiheredity structure as considered in (4.3).

We are now ready to prove the main results of this section.

(4.5) Theorem. (a) The subalgebras S_{R}^{2,≽} and S_{R}^{2,≼} are standardly full-based, and hence are quasi-hereditary over a commutative Noetherian ring R.

(b) If R is a field, then all the costandard modules ∇(S_{R}^{2,≽}, λ) (resp. standard modules Δ(S_{R}^{2,≼}, λ)) for S_{R}^{2,≽} (resp. S_{R}^{2,≼}) are simple and of dimension one.

Proof. For any μ ∈ Λ_{2}(n, r). Let ⩽ be the dominance order ⊴ on Λ = Λ_{2}(n, r). Let I(μ) = Ω_{2}^{≽}(μ) and J(μ) = {i_{μ}} (see (1.11) for the definition of Ω_{2}^{≽}(μ)). For any φ_{λ,μ}^{d} ∈ S_{R}^{2,≽}, write φ_{λ,μ}^{d} = φ_{i_{λ}d,i_{μ}}^{μ}. It is easy to see that (4.1(a)) holds. Now,

(2.8) implies (4.1)(b). So S_{R}^{2,≽} is standardly based. Moreover, it is full in the sense of (4.2) since φ_{μ,μ}^{1}φ_{μ,μ}^{1} = φ_{μ,μ}^{1} for every μ ∈ Λ_{2}(n, r). So S_{R}^{2,≽} is quasi-hereditary over R by (4.3). Similarly, we have S_{R}^{2,≼} is quasi-hereditary. Thus (a) follows. The statement (b) follows immediately from (a) and (4.4), easily. □

(4.6) Proposition. Suppose that the base ring R is a field.

(a) Let V_{λ} = S_{R}^{2,≽}φ_{λλ}^{1} for each λ ∈ Λ_{2}(n, r). Then {V_{λ} | λ ∈ Λ_{2}(n, r)} is a complete set of non-isomorphic principal indecomposable S_{R}^{2,≽}-modules. Moreover,

dim_{R}V_{λ} = ∏_{i=1}^{max_{n,r}+n} ( \begin{matrix}λ_{i} + i − 1 \\ i − 1 \end{matrix} ), where λ_{i} is the i-th component of λ̄, i.e.,

λ̄ = (λ_{1}, ⋯, λ_{max_{n,r}+n}).

(b) Let rad V_{λ} be the radical of V_{λ}. Then rad V_{λ} is spanned by φ_{μλ}^{d} ∈ S_{R}^{2,≽} with φ_{μλ}^{d} ≠ φ_{λλ}^{1}. Thus, the dimension of any irreducible S_{R}^{2,≽}-module is one.

Proof. (a) Since any element in φ_{λλ}^{1}S_{R}^{2,≽}φ_{λλ}^{1} is a R-linear combination of φ_{λλ}^{d} with i_{λ}d ≽ i_{λ} and d ∈ D_{λλ}, we must have d = 1. Thus, φ_{λλ}^{1} is a primitive idempotent element, since φ_{λλ}^{1}S^{2,≽}φ_{λλ}^{1} = Rφ_{λλ}^{1}. On the other hand, the identity element has the decomposition 1 = ∑_{λ∈Λ_{2}(n,r)} φ_{λλ}^{1}. So {V_{λ} | λ ∈ Λ_{2}(n, r)} is a complete set of non-isomorphic principal indecomposable S_{R}^{2,≽}-modules. The last assertion follows from [Sa, (6.1)] by counting the elements in Ω_{2}^{≼}(λ).

(b) The first assertion follows from a straightforward computation. The last assertion follows from the fact that {V_{λ}/rad V (λ) | λ ∈ Λ_{2}(n, r)} is the complete set of non-isomorphic simple S_{R}^{2,≽}-modules. □

We are now going to look at the relationship between the standard modules and costandard modules for q-Schur^{2} algebras (see (3.4)) and their Borel type subalgebras.

Let R be a commutative ring. For any λ ∈ Λ_{2}(n, r), let χ_{λ} : S_{R}^{2,≽} → R be an R-linear map such that

(4.7)  χ_{λ}(φ_{μν}^{d}) = \{ \begin{matrix}1, & if μ = ν = λ and d = 1 \\ 0, & otherwise. \end{matrix}

Clearly, χ_{λ} is an algebra homomorphism, and via χ_{λ}, R can be made into a rank-one (left or right) S_{R}^{2,≽}-module R_{λ}. One may define similarly algebra homomorphism χ_{λ} : S_{R}^{2,≼} → R and (left or right) S_{R}^{2,≼}-module R̂_{λ}. The following result can be proved easily.

(4.8) Proposition. If R is a commutative local Noetherian ring, then R̂_{λ} (resp. R_{λ}) is isomorphic to the left standard module Δ(S_{R}^{2,≼}, λ) (resp. right standard module Δ^{op}(S_{R}^{2,≽}, λ)).

Proof. Obviously, Δ(S_{R}^{2,≼}, λ) is a free R-module spanned by φ̄_{λλ}^{1} = φ_{λλ}^{1}+S_{R}^{2,≼}(>λ), where S_{R}^{2,≼}(> λ) is the submodule of S_{R}^{2,≼} spanned by φ_{μν}^{d} ∈ S_{R}^{2,≼} with μ > λ. Obviously, φφ̄_{λλ}^{1} = χ_{λ}(φ)φ̄_{λλ}^{1} for φ ∈ S_{R}^{2,≼}. Thus Δ(S_{R}^{2,≼}, λ) ≅ R̂_{λ}. One can prove Δ^{op}(S_{R}^{2,≽}, λ) ≅ R_{λ} similarly. □

(4.9) Theorem. Let R be a commutative local Noetherian ring and λ ∈ Λ_{2}(n, r). Then we have

(a) S_{R}^{2} ⊗_{S_{R}^{2,≼}} Δ(S_{R}^{2,≼}, λ) ≅ \{ \begin{matrix}Δ(S_{R}^{2}, λ), & if λ ∈ Λ_{2}(n, r)^{+} \\ 0, & otherwise, \end{matrix} , and

(b) Δ^{op}(S_{R}^{2,≽}, λ) ⊗_{S_{R}^{2,≽}} S_{R}^{2} ≅ \{ \begin{matrix}Δ^{op}(S_{R}^{2}, λ), & if λ ∈ Λ_{2}(n, r)^{+} \\ 0, & otherwise. \end{matrix} .

Proof. We prove (a). The proof of (b) is similar. By (4.8), it suffices to prove S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} = 0 for λ ∉ Λ_{2}(n, r)^{+}. By (3.4), S_{R}^{2} has an R-basis φ_{ρμ}^{d(𝔰)}φ_{μν}^{d(𝔱)^{−1}}, where

𝔰 ∈ 𝔗_{2}^{ss}(μ, ρ) and 𝔱 ∈ 𝔗_{2}^{ss}(μ, ν). By (3.3), φ_{μν}^{δ(𝔱)^{−1}} ∈ S_{R}^{2,≼}. If λ ∈ Λ_{2}(n, r)\Λ_{2}(n, r)^{+}, then φ_{μν}^{δ(𝔱)^{−1}} ⊗_{S_{R}^{2,≼}} 1 = 1 ⊗ χ_{λ}(φ_{μν}^{δ(𝔱)^{−1}})1 = 0. Thus, S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} = 0.

Suppose now λ ∈ Λ_{2}(n, r)^{+}. By (4.8), it is equivalent to prove S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} ≅ Δ(S_{R}^{2}, λ). The proof is similar to that given by J. A. Green in [G2]. Recall from [DS, §6], that the Green-Murphy basis element φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}} is denoted by Φ_{𝔰𝔱}^{μ}, where 𝔰 ∈ 𝔗_{2}^{ss}(μ, λ), 𝔱 ∈ 𝔗_{2}^{ss}(μ, ν) and μ ∈ Λ_{2}(n, r)^{+}, λ, ν ∈ Λ_{2}(n, r). Let S_{R}^{2}(> μ) be the submodule of S_{R}^{2} spanned by Φ_{𝔰′𝔱′}^{ρ} with ρ > μ, where ⩽ is the dominance order ⊴ on Λ_{2}(n, r). Then the standard module Δ(S_{R}^{2}, λ), λ ∈ Λ_{2}(n, r)^{+}, for the q-Schur^{2} algebra S_{R}^{2} satisfies Δ(S_{R}^{2}, λ) ≅ S_{R}^{2}φ̄_{λλ}^{1}, where φ̄_{λλ}^{1} = φ_{λλ}^{1} + S_{R}^{2}(> λ). Write S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} = S_{R}^{2}(1 ⊗ 1). It is enough to find two S_{R}^{2}-homomorphisms

f : S_{R}^{2}φ̄_{λλ}^{1} → S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} and g : S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} → S_{R}^{2}φ̄_{λλ}^{1} such that f(φ̄_{λλ}^{1}) = 1 ⊗ 1 and g(1 ⊗ 1) = φ̄_{λλ}^{1}.

We first define f. Consider the S_{R}^{2}-homomorphism f̃: S_{R}^{2}φ_{λλ}^{1} → S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ}

satisfying f(φ_{λλ}^{1}) = 1 ⊗ 1. For φ_{ρν}^{d(𝔰)}φ_{νμ}^{d(𝔱)^{−1}} ∈ S_{R}^{2}(> λ), φ_{ρν}^{d(𝔰)}φ_{νμ}^{d(𝔱)^{−1}} ⊗ 1 = 0, since ν > λ. It follows that f̃(S_{R}^{2}(>λ)) = 0. So f̃ induces an S_{R}^{2}-homomorphism f from S_{R}^{2}φ̄_{λλ}^{1} to S_{R}^{2} ⊗_{S_{R}^{2,≼}} R̂_{λ} such that f(φ̄_{λλ}^{1}) = 1 ⊗ 1.

To define g, we note that, for any φ_{ρ,μ}^{d} ∈ S_{R}^{2,≼}, if ρ ≠ λ, then φ_{ρ,μ}^{d} ∈ S_{R}^{2}(> λ) by [DS, (6.1.4)]. Consequently, we have φφ̄_{λλ}^{1} = χ_{λ}(φ)φ̄_{λλ}^{1} for all φ ∈ S_{R}^{2,≼}. Now, using universal property of tensor product, g is defined. □

For λ ∈ Λ_{2}(n, r), let H^{p}(λ) = Ext _{S_{R}^{2,≽}}^{p}(_{S_{R}^{2,≽}}S_{R}^{2}, R_{λ}). Then, H^{0}(λ) may be re-

garded as a module induced from a Borel subalgebra. The following result generalises the classical Borel-Weil theorem (compare the remarks after [G1, (5.9)]).

(4.10) Corollary. Let ∇(S_{R}^{2}, μ) be the (left) costandard module for S_{R}^{2} associated to μ ∈ Λ_{2}(n, r)^{+}, and λ ∈ λ_{2}(n, r). Then

H^{0}(λ) ≅ \{ \begin{matrix}∇(S_{R}^{2}, λ), & if λ ∈ Λ_{2}(n, r)^{+} \\ 0, & otherwise. \end{matrix}

Proof. By the adjoint isomorphism (see, e.g., [CR, 2.19]), we see that

H^{0}(λ) ≅ Hom_{R}(Δ^{op}(S_{R}^{2,≽}, λ) ⊗_{S_{R}^{2,≽}} S_{R}^{2}, R).

Now, the result follows from (4.9). □

(4.11) Remark. As mentioned in the introduction, it would be interesting to know if H^{p}(λ) = 0 for all λ ∈ Λ_{2}(n, r) and p > 0.

5. The general case. In [AK], Ariki and Koike introduced certain algebras associated to the complex reflection groups (ℤ/mℤ) ≀ 𝔖_{r}, which are now known as Ariki-Koike Hecke algebras. When m = 1 and 2, these algebras are isomorphic to the Hecke algebras of types A_{r−1} and B_{r}, respectively. In [DJM2], Dipper, James and Mathas introduced some associated endomorphism algebras, called cyclotomic q-Schur algebras. In this section, we generalize the results in §§2-4 to cyclotomic q-Schur algebras. For simplicity, we will mainly work on a subclass of cyclotomic q-Schur algebras, called q-Schur^{m} algebras. We will see that results for other cyclotomic q-Schur algebras can be obtained easily from that for q-Schur^{m} algebras.

Though the natural basis like (1.7) is not available for a q-Schur^{m} algebra S_{R}^{m} with m > 2, having a close look at the work in previous sections, we notice that only part of the basis elements in (1.7), which are involved in the definition of Green-Murphy type basis (3.4), have been used. Note also that a Murphy type basis for S_{R}^{m} is introduced in [DJM2]. So, if we can prove this basis is a Green type basis (cf. [G2]), i.e., is of the from (3.4), then our generalization to arbitrary m will be almost straightforward. So we aim to construct those elements φ_{λμ}^{d} involved in (1.11). To do this, it suffices to generalize the relation (2.6). The next combinatorial lemma will lead a generalization of (2.6). Recall the c.n.s. of a multi-composition in §1.

(5.1) Lemma. If λ, μ ∈ Λ_{m}(n, r) and d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ}, then d = w_{m} ⋯ w_{1} with l(d) = ∑_{i=1}^{m} l(w_{i}), where w_{j} ∈ 𝔖_{{a_{j−1}+1,··· ,b_{j}}} for all 1 ⩽ j ⩽ m, where a = (a_{0}, a_{1}, ⋯, a_{m}) resp. b = (b_{0}, b_{1}, ⋯, b_{m}) is the c.n.s. of λ resp. μ.

Proof. We first construct inductively a sequence of elements x_{m}, ⋯, x_{1} with x_{j} ∈ 𝔖_{{a_{j−1}+1,··· ,b_{j}}} such that d^{−1} = x_{1} ⋯ x_{m}. Applying (1.10) to d, we have, for every i with 1 ⩽ i ⩽ m, d(j) ⩽ b_{i−1} for all j ⩽ a_{i−1}. Thus, for i = m, we obtain {d(1), d(2), ⋯, d(a_{m−1})} ⊆ {1, 2, ⋯, b_{m−1}}. Taking complements, we have

{b_{m−1} + 1, ⋯, r} ⊆ {d(j) | a_{m−1} + 1 ⩽ j ⩽ r}.

So we may choose x_{m} ∈ 𝔖_{{a_{m−1}+1,··· ,b_{m}}} such that x_{m}d(j) = j for all b_{m−1} + 1 ⩽ j ⩽ b_{m} = r and x_{m}d(j) = d(j) for all j ⩽ a_{m−1}. Now, since d(j) ⩽ b_{m−2} for all j ⩽ a_{m−2}, it follows that

{d(1), d(2), ⋯, d(a_{m−2})} ⊆ {1, 2, ⋯, b_{m−2}},

and hence,

\begin{aligned}\{b_{m−2} + 1, ⋯, b_{m−1}, b_{m−1} + 1, ⋯, b_{m}\} & ⊆ \{d(j) | a_{m−2} + 1 ⩽ j ⩽ r\} \\ & = \{x_{m}d(j) | a_{m−2} + 1 ⩽ j ⩽ r\}. \end{aligned}

Therefore,

{b_{m−2} + 1, ⋯, b_{m−1}} ⊆ {x_{m}d(j) | a_{m−2} + 1 ⩽ j ⩽ b_{m−1}}.

Thus, there is an x_{m−1} ∈ 𝔖_{{a_{m−2}+1,··· ,b_{m−1}}} such that x_{m−1}x_{m}d(j) = j for all j with b_{m−2} + 1 ⩽ j ⩽ r, and x_{m−1}x_{m}d(j) = d(j) for j ⩽ a_{m−2}. Continue this process. After finitely many steps, we will find x_{2}, ⋯, x_{m} with x_{i} ∈ 𝔖_{{a_{i−1}+1,··· ,b_{i}}}, such that x_{2} ⋯ x_{m}d(j) = j for all j ⩾ b_{1} +1 and x_{2} ⋯ x_{m}d(j) = d(j) for all j ⩽ a_{1}. Finally, choose x_{1} ∈ 𝔖_{{1,··· ,b_{1}}} such that x_{1} ⋯ x_{m}d(j) = j for all j ⩽ b_{1}. Since x_{1}(j) = j for all b ⩾ b_{1} + 1, we have x_{1} ⋯ x_{m}d(j) = j for all j ⩾ b_{1} + 1. Thus, x_{1} ⋯ x_{m}d = 1, and d^{−1} = x_{1} ⋯ x_{m}.

Next, applying induction on m and noting the exchange condition for Coxeter groups, one sees easily that there are y_{j} ∈ 𝔖_{{a_{j−1}+1,··· ,b_{j}}} such that d^{−1} = y_{1}y_{2} ⋯ y_{m} and l(d^{−1}) = ∑_{i=1}^{m} l(y_{i}). Putting w_{i} = y_{i}^{−1}, the result follows. □

(5.2) Remark. This lemma can be used to give a weak version of the middle equality in (2.6). That is, we want to prove that, for λ, μ ∈ Λ_{2}(n, r) and d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ}, T_{𝔖λ̄d𝔖μ̄}π_{μ} ∈ x_{λ}H_{R}.

Write λ = (λ^{(1)}, λ^{(2)}), μ = (μ^{(1)}, μ^{(2)}) and put a=|λ^{(1)}| and b = |μ^{(1)}|. Then the c.n.s. of λ and μ are (0, a, r) and (0, b, r), respectively, and a ⩽ b. Since i_{λ}d ≽ i_{μ}, by (5.1), we can write d = w_{2}w_{1} with w_{1} ∈ 𝔖_{{1,··· ,b}}, and w_{2} ∈ 𝔖_{{a+1,··· ,r}}. Writing T_{𝔖λ̄d𝔖μ̄} = xλ̄T_{d}h_{1} for some h_{1} ∈ 𝔖μ̄ and noting that π_{μ} is in the center of H_{R}(Wμ̂), we have

T_{𝔖λ̄d𝔖μ̄}π_{μ} = xλ̄T_{d}h_{1}π_{μ} = xλ̄T_{w_{2}}T_{w_{1}}π_{μ}h_{1}
= xλ̄T_{w_{2}}π_{μ}T_{w_{1}}h_{1} = xλ̄T_{w_{2}}π_{λ}π_{μ\λ}T_{w_{1}}h_{1}
= xλ̄π_{λ}T_{w_{2}}π_{μ\λ}T_{w_{1}}h_{1} ∈ x_{λ}H_{R}.

Note that, with (5.2), φ_{λμ}^{d} can be defined easily for those λ, μ ∈ Λ_{2}(n, r) and d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ} (compare the general definition in [DS, (4.2.6)]). We are now ready to generalise (5.2) to the Ariki-Koike Hecke algebras.

In the rest of the paper, let H_{R} be the Ariki-Koike Hecke algebra associated to the complex reflection group (ℤ/mℤ)≀𝔖_{r}. Then H_{R} is an associative algebra over a commutative ring R containing q, q^{−1}, u_{1}, ⋯, u_{r} with generators T_{i}, 0 ⩽ i ⩽ r − 1, and relations

T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1} for 1 ⩽ i ⩽ r − 2
T_{i}T_{j} = T_{j}T_{i}, if |i − j| ⩾ 2
(T_{i} − q)(T_{i} + 1) = 0, if i ≠ 0
(T_{0} − u_{1}) ⋯ (T_{0} − u_{m}) = 0.

Note that the subalgebra generated by T_{i} with 1 ⩽ i ⩽ r − 1 is the Hecke algebra H_{R}(𝔖_{r}) of type A_{r−1}, and, when m = 2 and u_{1} = q_{0} and u_{2} = −1, H_{R} is the Hecke algebra of type B_{r} as defined at the beginning of §1. We will use the notation H_{R}(𝔖_{λ}) for the subalgebra of H_{R} defined by a parabolic subgroup 𝔖_{λ} of 𝔖_{r},

Ariki and Koike proved that the set

(5.3)  {L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} | w ∈ 𝔖_{r}, and 0 ⩽ c_{i} ⩽ m − 1, i = 1, 2, ⋯, r},

is a R-basis of H_{R}, where L_{i} = q^{1−i}T_{i−1} ⋯ T_{1}T_{0}T_{1} ⋯ T_{i−1}. Moreover, we have the following results (see [AK, (2.1)-(2.2)]).

(5.4)  \{ \begin{matrix}(1) & L_{i} and L_{j} commute for all 1 ⩽ i, j ⩽ r \\ (2) & If a ∈ R and i ≠ k, then T_{i} commutes with ∏_{i=1}^{k}(L_{i} − a) \end{matrix}

Suppose that λ ∈ Λ_{m}(n, r) is of c.n.s. a = (a_{0}, ⋯, a_{m}). Following [DJM2], we denote

(5.5)  π_{λ} = ∏_{k=2}^{m} ∏_{i=1}^{a_{k−1}}(L_{i} − u_{k}),  x_{λ} = π_{λ}xλ̄.

(5.6) Lemma. Let λ, μ ∈ Λ_{m}(n, r) be of c.n.s. a and b, respectively. If d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ}, then T_{𝔖λ̄d𝔖μ̄}π_{μ} ∈ x_{λ}H_{R}.

Proof. We first note that, if T_{i} ∈ H_{R}(𝔖μ̄), then i ≠ b_{j} for all j. By (5.4(2)) and (5.5), all the elements in H_{R}(𝔖μ̄) commute with π_{μ}. Since T_{𝔖λ̄d𝔖μ̄} = xλ̄T_{d}h for some h ∈ H_{R}(𝔖μ̄), we need only prove T_{d}π_{μ} ∈ π_{λ}H_{R}. For simplicity, we put π_{b_{k}} = ∏_{i=1}^{b_{k}}(L_{i} − u_{k+1}). Then, π_{μ} = π_{b_{1}} ⋯ π_{b_{m−1}}. Since a_{k} ⩽ b_{k}, the product π_{a_{k}} is part of π_{b_{k}}. Let π_{b_{k}\a_{k}} denote the product obtained by deleting π_{a_{k}} from π_{b_{k}}.

By (5.1), T_{d} = T_{w_{m}} ⋯ T_{w_{2}}T_{w_{1}} for some w_{j} ∈ 𝔖_{{a_{j−1}+1,··· ,b_{j}}}, 1 ⩽ j ⩽ m. Thus, for every 1 ⩽ j ⩽ m, (5.4(2)) implies that T_{w_{j}} commutes with the product π_{a_{j−1}} ∏_{k=j}^{m−1} π_{b_{k}}, and T_{w_{m}} ⋯ T_{w_{j+1}} commutes with π_{a_{l}} for all l < j. Taking j = 1, we obtain T_{w_{1}}π_{μ} = π_{μ}T_{w_{1}}, and, for j = 2, we have by (5.4(1))

T_{w_{2}}T_{w_{1}}π_{μ} = T_{w_{2}}π_{μ}T_{w_{1}}
= T_{w_{2}}π_{a_{1}}(π_{b_{2}} ⋯ π_{b_{m−1}})π_{b_{1}\a_{1}}T_{w_{1}}
= π_{a_{1}}(π_{b_{2}} ⋯ π_{b_{m−1}})T_{w_{2}}π_{b_{1}\a_{1}}T_{w_{1}}.

Now, if we rewrite the product in the parentheses as a product of π_{a_{2}}(π_{b_{3}} ⋯ π_{b_{m−1}}) and π_{b_{2}\a_{2}}, then the former commutes with T_{w_{3}}. Continue this process. We finally obtain

T_{d}π_{μ} = π_{λ}T_{w_{m}}π_{b_{m−1}\a_{m−1}}T_{w_{m−1}} ⋯ π_{b_{1}\a_{1}}T_{w_{1}} ∈ π_{λ}H_{R},

as desired. □

Note that, for d = δ(𝔰), (5.6) holds by (3.3). This case was proved in [DJM2, (4.10)].

Cyclotomic q-Schur algebras S_{R}(Λ) associated to H_{R} are introduced in [DJM2] for a finite set Λ of m-compositions of r such that Λ∩Λ_{m}(r)^{+} is a coideal of Λ_{m}(r)^{+}. We are interested in those cyclotomic q-Schur algebras defined by Λ_{m}(n, r).

(5.7) Definition. The q-Schur^{m} algebra S_{R}^{m} of degree (n, r) is the cyclotomic q-Schur algebra associated to the poset Λ_{m}(n, r). In other words, we define

S_{R}^{m} = S_{R}^{m}(n, r) = End_{H_{R}}(⊕_{λ∈Λ_{m}(n,r)}x_{λ}H_{R}).

The q-Schur^{m} algebra is a natural generalization of the q-Schur algebra and the q-Schur^{2} algebra. When m = 2 and u_{1} = q_{0}, u_{2} = −1, S_{R}^{2} is isomorphic to S_{R}^{2}. Many nice properties of q-Schur^{2} algebras such as the quasi-heredity (over fields), a cellular basis, etc. have also been established in [DJM2] for a cyclotomic q-Schur algebra. In particular, as in the m = 2 case, we have, for any m-compositions λ, μ, the isomorphism ([DJM2, (5.17(ii))]) Hom_{H_{R}}(x_{μ}H_{R}, x_{λ}H_{R}) ≅ x_{λ}H_{R} ∩ H_{R}x_{μ}, and a Murphy type basis for x_{λ}H_{R} ∩ H_{R}x_{μ} ([DJM2, (6.3)]). This leads to a nice basis, i.e., a cellular basis, for any cyclotomic q-Schur algebra (see [DJM2, (6.6)]).

Let us give a little more details about this construction. For any standard μtableau s, let 𝔣_{λ}(s) be the μ-tableau of type λ obtained from s by replacing each entry a in s by i_{k} if a is in the row i of the k-th component of t^{λ}. Take a semistandard μ-tableau 𝔰 of type λ, and consider the inverse image T_{𝔰,λ} = 𝔣_{λ}^{−1}(𝔰). Thus, T_{𝔰,λ} is the set of all standard μ-tableaux whose image under the map 𝔣_{λ} is 𝔰. Let ⊴ be the partial order on the set of standard μ-tableaux (see [DJM2, (3.1.1)]). Thus, for standard μ-tableaux s, t, s ⊴ t means s ↓ k ⊴ t ↓ k for 1 ⩽ k ⩽ r, where s ↓ k is a multi-composition determined by the entries 1, 2, ⋯, k in s. For example, if

s = 1 2 8 6 7 3 5 9 4

then s ↓ 6 = ((221)(10)) and s ↓ 8 = ((321)(20)). Among the elements in the set T_{𝔰,λ}, there is a unique standard μ-tableau, which is maximal with respect to ⊴. Such a tableau was denoted by first(𝔰) in [DJM2]. Let d(𝔰) be a distinguished coset representative in D_{μ} defined by t^{μ}d(𝔰) = first(𝔰). Then, if we define, for 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), 𝔱 ∈ 𝔗_{m}^{ss}(μ, λ), Φ_{𝔰𝔱} ∈ S_{R}^{m} such that

(5.8)  Φ_{𝔰𝔱}^{μ}(x_{ρ}) = δ_{νρ}h_{1}T_{d(𝔰)^{−1}}x_{μ}T_{d(𝔱)}h_{2},

where h_{1} = ∑_{x∈𝔖λ̄∩D_{μ̄d(𝔰)∩λ̄}} T_{x^{−1}} and h_{2} = ∑_{x∈𝔖ν̄∩D_{μ̄d(𝔱)∩ν̄}} T_{x}, then, by [DJM2, (6.6)], the set

(5.9)  {Φ_{𝔰𝔱} | 𝔰 ∈ 𝔗^{ss}(ν, λ), 𝔱 ∈ 𝔗^{ss}(ν, μ), λ, μ ∈ Λ_{m}(n, r), ν ∈ Λ_{m}(n, r)^{+}}

forms a basis for the q-Schur^{m} algebra S_{R}^{m}(n, r). Note that the element Φ_{𝔰𝔱}^{μ}(x_{ν}) is in fact a sum of Murphy’s basis elements, which is denoted m_{ST} in [DJM2, (6.2)] (see the proof of (4.10) there). Note also that we adopt the Φ-notation used in [DS] instead of the φ-notation in [DJM2] as we have already used φ for natural basis elements.

As a consequence of (5.9), we obtain the quasi-heredity of the q-Schur^{m} algebra. For a field R, the following result is given in [DJM2, (6.18)]. The general case follows from [DR, (3.2.1)] (see (4.3a)).

(5.10) Theorem. Let R be a commutative Noetherian ring. Then the algebra S_{R}^{m} is quasi-hereditary.

We are now going to define the Borel type subalgebras of a q-Schur^{m} algebra S_{R}^{m}(n, r). For λ, μ ∈ Λ_{m}(n, r) and d ∈ Dλ̄μ̄ with i_{λ}d ≽ i_{μ}, let φ_{λμ}^{d} ∈ S_{R}^{m} be defined by

(5.11)  φ_{λμ}^{d}(x_{ν}h) = δ_{μν}T_{𝔖λ̄d𝔖μ̄}π_{μ}h,  ν ∈ Λ_{m}(n, r), h ∈ H_{R}.

This is well-defined by (5.6). Let ι be the R-linear anti-automorphism on H_{R} sending T_{i} to T_{i} for 0 ⩽ i ⩽ r − 1 (see [GL, (5.5)]). Then (π_{λ})^{ι} = π_{λ} for any λ ∈ Λ_{m}(n, r) (see (5.5) for the definition of π_{λ}). It implies that (π_{λ}T_{𝔖λ̄d𝔖μ̄})^{ι} = T_{𝔖μ̄d^{−1}𝔖λ̄}π_{λ} for any λ, μ ∈ Λ_{m}(n, r). Thus we can define φ_{λμ}^{d} = (φ_{μλ}^{d^{−1}})^{ι} if i_{λ}d ≼ i_{μ}.

(5.12) Lemma. For any μ ∈ Λ_{m}(n, r), let Ω_{m}^{≽}(μ) and Ω_{m}^{≼}(μ) be defined in (1.11). Then

(a) the set {φ_{λμ}^{d} | (λ, d) ∈ Ω_{m}^{≽}(μ), μ ∈ Λ_{m}(n, r)} is R-linearly independent.

(b) the set {φ_{μλ}^{d} | (λ, d) ∈ Ω_{m}^{≼}(μ), μ ∈ Λ_{m}(n, r)} is R-linearly independent.

Proof. Suppose ∑\begin{matrix}μ∈Λ_{m}(n,r) \\ (λ,d)∈Ω_{m}^{≽}(μ) \end{matrix} a_{λ,μ}^{d}φ_{λμ}^{d} = 0 with a_{λμ}^{d} ∈ R. Applying the left hand

side to x_{μ}, we have ∑_{(λ,d)∈Ω_{m}^{≽}(μ)} a_{λμ}^{d}T_{𝔖λ̄d𝔖μ̄}π_{μ} = 0. Since π_{μ} is a monic polynomial in L_{i}, by (5.3), we have ∑_{(λ,d)∈Ω_{m}^{≽}(μ)} a_{λμ}^{d}T_{𝔖λ̄d𝔖μ̄}L_{1}^{a_{1}} ⋯ L_{r}^{a_{r}} = 0, where L_{1}^{a_{1}} ⋯ L_{r}^{a_{r}} is the highest term of π_{μ}. This implies a_{λμ}^{d} = 0, for all (λ, d) ∈ Ω_{m}^{≽}(μ), by (5.3) again. Thus, (a) follows. The statement (b) follows from (a). □

(5.13) Theorem. Maintain the notation above. Let S_{R}^{m,≽} = S_{R}^{m,≽}(n, r) (resp. S_{R}^{m,≼} = S_{R}^{m,≼}(n, r)) be the free R-submodule of S_{R}^{m} spanned by φ_{λμ}^{d} with (λ, d) ∈ Ω_{m}^{≽}(μ) (resp. φ_{μλ}^{d} with (λ, d) ∈ Ω_{m}^{≼}(μ) ) for all μ ∈ Λ_{m}(n, r). Then S_{R}^{m,≽} and S_{R}^{m,≼} are subalgebras of the q-Schur^{m} algebra S_{R}^{m}, and we have, for N = N(m, n, r) as in (1.2), the following isomorphisms of R-algebras:

S_{R}^{m,≽}(n, r) ≅ S_{R}^{1,≽}(N, r), S_{R}^{m,≼}(n, r) ≅ S_{R}^{1,≼}(N, r).

Proof. By definition (5.11), we have φ_{λμ}^{d}(x_{μ}) = T_{𝔖λ̄d𝔖μ̄}π_{μ} = ψ_{λ̄μ̄}^{d}(xμ̄)π_{μ} for (λ, d) ∈ Ω_{m}^{≽}(μ) and μ ∈ Λ_{m}(n, r). Now, the multiplicative closure condition and the isomorphisms can be proved similarly as in (2.7) and (2.8) □

These two subalgebras are called the Borel type subalgebras of S_{R}^{m}.

Before generalizing (3.4) to S_{R}^{m} via (5.9), we observe from (3.3) that φ_{λμ}^{δ(𝔰)} ∈ S_{R}^{m,≽} if 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ). So we must find some relation between this element δ(𝔰) and the element d(𝔰) used in the definition (5.8).

We first recall from (3.2) the definition of δ(𝔰). Let 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ). Then δ(𝔰) is the distinguished coset representative defined by row standard tableau tλ̄δ(𝔰) for which i belongs to row a if the place occupied by i in tμ̄ is occupied by a in 𝔰̄. For example, if μ = ((3210 ⋯ 0), (210 ⋯ 0), (111)) ∈ Λ_{3}(3, 12)^{+} and λ = ((2110 ⋯ 0), (220 ⋯ 0), (112)) ∈ Λ_{3}(3, 12), we take

𝔰 = 1 1 14 13 13 25 2 27  14  26 3  27

Then

t^{μ} = 1 2 3 7 8 10 4 5  9  11 6  12 and t^{λ}δ(𝔰) = 1 2 7 8 10 4  3 9 11 5  5 12

Here, again rows corresponds those 0-parts are omitted. Note that the standard μ-tableau t^{μ}δ(𝔰)^{−1} is obtained by replacing all the numbers i in 𝔰 by the sequence obtained by reading the i-th row in tλ̄, the replacements in 𝔰 are made from left to right, down successive rows (compare [DS, (1.2.2)]). Thus, in the previous example, we have

t^{μ}δ(𝔰)^{−1} = 1 2 7 5 6 9 3 11  8  10 4  12

Clearly, the standard μ-tableau t^{μ}δ(𝔰)^{−1} is the maximal element first(𝔰) in T_{𝔰,λ} under the partial order ⊴ on standard tableaux. Therefore, we have the following result.

(5.14) For any 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), t^{μ}δ(𝔰)^{−1} = first(𝔰). Therefore, d(𝔰) = δ(𝔰)^{−1}.

This result implies immediately the following.

(5.15) Theorem. Maintain the notation in (5.8) and (5.11). For any μ ∈ Λ_{m}(n, r)^{+}, λ, ν ∈ Λ_{m}(n, r), and 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), 𝔱 ∈ 𝔗_{m}^{ss}(μ, ν), we have φ_{λ,μ}^{δ(𝔰)} ∈ S_{R}^{m,≽}, φ_{μ,ν}^{δ(𝔱)^{−1}} ∈ S_{R}^{m,≼} and Φ_{𝔰𝔱} = φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}}. Thus, the set

{φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}} | λ, ν ∈ Λ_{m}(n, r), μ ∈ Λ_{m}(n, r)^{+}, 𝔰 ∈ 𝔗_{m}^{ss}(μ, λ), 𝔱 ∈ 𝔗_{m}^{ss}(μ, ν)}

forms a basis for S_{R}^{m}.

Proof. We have seen that part of the first assertion follows from (3.3). Let 𝔱^{μ} be the unique element in 𝔗_{m}^{ss}(μ, μ). By (5.14) and (5.8), one checks easily that φ_{λμ}^{δ(𝔰)} = Φ_{𝔰𝔱^{μ}}, φ_{μν}^{δ(𝔱)^{−1}} = Φ_{𝔱^{μ}𝔱} and Φ_{𝔰𝔱^{μ}}Φ_{𝔱^{μ}𝔱} = Φ_{𝔰𝔱}. Hence we have Φ_{𝔰𝔱} = φ_{λμ}^{δ(𝔰)}φ_{μν}^{δ(𝔱)^{−1}}. The last assertion follows from (5.9). □

As in (4.7), we have an algebra homomorphism χ_{λ} from S_{R}^{m,≽} to R. Let R_{λ} be the S_{R}^{m,≽}-module induced by χ_{λ}.

(5.16) Theorem. Let S_{R}^{m,≽} (resp. S_{R}^{m,≼}) be the Borel type subalgebras of S_{R}^{m}. (a) The q-Schur^{m} algebra S_{R}^{m} has a triangular decomposition

S_{R}^{m} = S_{R}^{m,≽}S_{R}^{m,≼} = ∑_{λ∈Λ_{m}(n,r)^{+}} S_{R}^{m,≽}φ_{λλ}^{1}S_{R}^{m,≼}.

(b) The Borel type subalgebras S_{R}^{m,≽} and S_{R}^{m,≼} are quasi-hereditary with simple costandard modules and simple standard modules, respectively, if R is a commutative local Noetherian ring.

(c) The set {φ_{λλ}^{1} | λ ∈ Λ_{m}(n, r)} is the complete set of primitive idempotents in S_{R}^{m,≽} and S_{R}^{m,≼} if R is a field.

(d) Suppose R is a field, and let V_{λ} = S_{R}^{m,≽}φ_{λλ}^{1} for λ ∈ Λ_{m}(n, r). Then V_{λ} = Δ(S_{R}^{m,≽}, λ) and {V_{λ} | λ ∈ Λ_{m}(n, r)} is a complete set of all principal indecomposable S_{R}^{m,≽}-modules. Moreover, if we write λ̄ = (λ_{1}, ⋯, λ_{N}), where N is defined in (1.2), then

dim_{R}V_{λ} = ∏_{i=1}^{N} ( \begin{matrix}λ_{i} + i − 1 \\ i − 1 \end{matrix} ) .

(e) Suppose R is a field. Then {V_{λ}/radV_{λ} | λ ∈ Λ_{m}(n, r)} is the complete set of non-isomorphic simple S_{R}^{m,≽}-modules and any simple module is of dimension one.

(f) Suppose that R is a commutative local Noetherian ring. Let Δ(S_{R}^{m}, λ) (resp. ∇(S_{R}^{m}, λ) ) be the left standard module (resp. costandard module) of S_{R}^{m} with respect to λ ∈ Λ_{m}(n, r)^{+}. Let Δ(S_{R}^{m,≼}, λ) (resp. Δ^{op}(S_{R}^{m,≽}, λ)) be the left (resp. right) standard module for S_{R}^{m,≼} (resp. for S_{R}^{m,≽}). Then

S_{R}^{m} ⊗_{S_{R}^{m,≼}} Δ(S_{R}^{m,≼}, λ) ≅ \{ \begin{matrix}Δ(S_{R}^{m}, λ), & if λ ∈ Λ_{m}(n, r)^{+}, \\ 0, & otherwise, \end{matrix}

and

Hom_{S_{R}^{m,≽}}(_{S_{R}^{m,≽}}S_{R}^{m}, R_{λ}) ≅ \{ \begin{matrix}∇(S_{R}^{m}, λ), & if λ ∈ Λ_{m}(n, r)^{+}, \\ 0, & otherwise. \end{matrix}

Proof. All statements can be proved formally by using arguments similar to those given in §4. □

(5.17) Remark. (1) We remark that an arbitrary cyclotomic q-Schur algebra S_{R}(Λ) defined in [DJM2] is Morita equivalent to a centralizer subalgebra of a q-Schur^{m} algebra defined by a coideal of Λ_{m}(n, r)^{+} for some n. This is because, by [DJM2, (3.9i)], S_{R}(Λ) is Morita equivalent to S_{R}(Λ^{+}), where Λ^{+} = Λ∩Λ_{m}(r)^{+} and hence, is Morita equivalent to the subalgebra eS_{R}^{m}(n, r)e, where e = ∑_{λ∈Λ′} φ_{λλ}^{1} for any subset Λ′ of Λ_{m}(n, r) whose intersection with Λ_{m}(r)^{+} is Λ^{+}. Moreover, the Borel type subalgebras of eS_{R}^{m}(n, r)e are S_{R}^{m,≽} ∩ eS_{R}^{m}e and S_{R}^{m,≼} ∩ eS_{R}^{m}e.

(2) It would be nice to prove that the Borel type subalgebra S_{R}^{m,≼} is a Borel subalgebra of the quasi-hereditary algebra S_{R}^{m} in the sense of [Sc], and to establish the Borel-Bott-Weil theorem in this generality as described in (4.11).

6. Tilting modules and Ringel duals. In this section, we discuss the tilting modules and Ringel duals for Borel subalgebras S_{R}^{m,≽} and S_{R}^{m,≼} over a field R.

Let A be a quasi-hereditary algebra over a field R. Then the category of left A-modules _{A}C is a highest weight category on a poset (Λ, ⩽) in the following sense (see [CPS1]).

(a) For each λ ∈ Λ, there is a simple A-module L(λ) such that {L(λ) | λ ∈ Λ} is the set of the non-isomorphic left A-modules.

(b) For each λ ∈ Λ, the standard module Δ(λ) has simple head L(λ) and all other composition factors L(μ) satisfy μ < λ.

(c) The projective cover P (λ) of L(λ) has a filtration 0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{t} = P (λ) such that any section F_{i}/F_{i−1} ≅ Δ(μ) with μ > λ if i ≠ t and μ = λ if i = t. The section F_{t}/F_{t−1} is called the top section.

The conditions (b) and (c) can be replaced by (b′) and (c′) as follows.

(b′) The costandard module ∇(λ) has socle L(λ) and all other composition factors L(μ) satisfy μ < λ;

(c′) The injective envelope I(λ) has a filtration 0 = F′_{0} ⊂ F′_{1} ⊂ ⋯ ⊂ F′_{t′} = I(λ) such that any section F_{i}/F_{i−1} ≅ Δ(μ) with μ > λ if i ≠ 1 and μ = λ if i = 1. The section F_{1}/F_{0} is called the bottom section.

We shall say a left A-module M has a Δ-filtration (resp. ∇-filtration) if M has a filtration whose sections are of forms Δ(λ) (resp. ∇(λ) ) for some λ ∈ Λ. A module M is called a tilting module if it has Δ-filtration and ∇-filtration. Let _{A}C(tilt) be the subcategory of tilting modules. In [R], Ringel proved that, for each λ ∈ Λ, there is a unique indecomposable X(λ) ∈ _{A}C(tilt), called partial tilting module with respect to λ, such that X(λ) has a Δ-filtration with bottom section Δ(λ) and a ∇-filtration with top section ∇(λ). A module X ∈ _{A}C(tilt) is called a full tilting module if X has a decomposition X = ⊕_{λ∈Λ}X(λ)^{⊕m_{λ}(X)} such that m_{λ}(X) > 0 for all λ ∈ Λ. For full tilting modules X and Y, Ringel proved that the endomorphism algebras End_{A}(X) and End_{A}(Y) are Morita equivalent. Such an endomorphism algebra is called a Ringel dual of the quasi-hereditary algebra A. It is known that the Ringel dual is a quasi-hereditary algebra.

(6.1) Theorem. Let R be a field. The Borel type subalgebras S_{R}^{m,≽} and S_{R}^{m,≼}, viewed as regular modules, are full tilting modules, and therefore, they are Ringel dual to each other.

Proof. By (5.16)(d), V_{λ} = S_{R}^{m,≽}φ_{λλ}^{1} is a principal indecomposable S_{R}^{m,≽}-module, and is isomorphic to the standard module Δ(λ) = Δ(S_{R}^{m,≽}, λ). Hence 0 ⊂ V_{λ} is a Δ-filtration of V_{λ} with bottom section Δ(λ). On the other hand, V_{λ} has a basis φ_{μ,λ}^{d} with (μ, d) ∈ Ω_{m}^{≽}(λ). If we order these basis elements φ_{μ,λ}^{d} as φ_{1}, φ_{2}, ⋯, where φ_{i} = φ_{μ_{(i)},λ}^{d}, such that μ_{(i)} ⊴ μ_{(j)} implies i < j, and define M_{i} as the

R-space spanned by φ_{j} with j ⩽ i, then M_{i} is an S_{R}^{m,≽}-module, and M_{t} = V_{λ} where t = #Ω_{m}^{≽}(λ). Consider the filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = V_{λ} of V_{λ}. Obviously, its section M_{i}/M_{i−1} is of dimensional one for all 1 ⩽ i ⩽ t. Suppose that M_{i}/M_{i−1} is spanned by φ̄_{μλ}^{d}. Then, for any φ_{ρν}^{d′} ∈ S_{R}^{m,≽}, we have φ_{ρν}^{d′}φ̄_{μλ}^{d} = 0 if ρ ≠ μ. If ρ = μ, then ν = μ, d′ = 1, and φ_{ρν}^{d′}φ̄_{μλ}^{d} = φ̄_{μλ}^{d}. Thus, φ_{ρν}^{d′}φ̄_{μλ}^{d} = χ_{μ}(φ_{ρν}^{d′})φ̄_{μλ}^{d}. So M_{i}/M_{i−1} is isomorphic to R_{μ} defined by χ_{μ}. On the other hand, by definition, ∇(μ) ≅ Hom_{R}(Δ^{op}(μ), R), where Δ^{op}(μ) is the right standard S_{R}^{m,≽}-module defined by χ_{μ}, too (see (4.8)). Therefore, M_{i}/M_{i−1} ≅ ∇(μ). In particular, M_{t}/M_{t−1} ≅ ∇(λ). So V_{λ} is the partial tilting module corresponding to λ. Since 1 = ∑_{λ∈Λ_{m}(n,r)} φ_{λλ}^{1}, we have that S_{R}^{m,≽} = ⊕_{λ∈Λ_{m}(n,r)}V_{λ}, and so, S_{R}^{m,≽} is a full tilting S_{R}^{m,≽}-module. Consequently, the Ringel dual End_{S_{R}^{m,≽}}(S_{R}^{m,≽})

(≅ (S_{R}^{m,≽})^{op}) of S_{R}^{m,≽} is isomorphic to S_{R}^{m,≼}. □

Acknowledgement. The authors would like to thank L. Scott for his helpful comments during the writing of this paper. They also wish to thank A. Mathas for his comments on an earlier version of this paper, and to thank the referee for some useful comments.

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E-mail: jied@maths.unsw.edu.au  hbruik@online.sh.cn