ARIKI-KOIKE ALGEBRAS WITH SEMISIMPLE BOTTOMS

Jie Du and Hebing Rui

Schools of Mathematics, University of New South Wales Sydney, 2052, Australia Department of Mathematics, University of Shanghai for Science & Technology Shanghai, 200093, P.R.China

30 July, 1998

Consider the cyclic group C_{m} of order m and the wreath product W = W_{m}^{r} = C_{m} ≀ 𝔖_{r}, where 𝔖_{r} is the symmetric group on r letters. Then the direct product C = C_{m} × ⋯ × C_{m} of r copies C_{m} is normal in W and sits at the “bottom” of W. Let F be a splitting field of C in which m is not zero. Then the “bottom” F C of F W is semisimple and simple C-modules N_{i} are indexed by the set I(m, r) of all r-tuples i = (i_{1}, ⋯, i_{r}) with 1 ⩽ i_{j} ⩽ m. Moreover, there is a central primitive idempotent decomposition 1 = ∑_{i∈I(m,r)} e_{i} such that N_{i} ≅ e_{i}F C. The symmetric group 𝔖_{r} acts on I(m, r) by place permutation, and the set of W -orbits is identified with the set Λ(m, r) of compositions of r with m parts. Thus i is in the orbit λ ∈ Λ(m, r), denoted wt(i) = λ, if λ_{j} = #{i_{k} : i_{k} = j} for all j. Notice that, if wt(i) = wt(j), then we have isomorphism of induced modules: N_{i} ↑^{W} ≅ N_{j} ↑^{W}. Therefore, putting N_{λ} = N_{i} and e_{λ} = e_{i} if wt(i) = λ, we have right W -module isomorphism

F W ≅ ⨁_{λ∈Λ(m,r)}(N_{λ} ⊗_{F C} F W)^{⊕d_{λ}} ≅ ⨁_{λ∈Λ(m,r)} e_{λ}F W^{⊕d_{λ}}.

Standard results will give a Morita equivalence between the categories of F W - modules and eF W e-modules, where e = ∑_{λ∈Λ(m,r)} e_{λ}. Since e_{λ}F W e_{μ} ≅ δ_{λμ}F 𝔖_{λ}, where 𝔖_{λ} is the Young subgroup corresponding to λ, we have Morita equivalence

(1)  F W -mod ∼^{Morita} (⊕_{λ∈Λ(m,r)}F 𝔖_{λ})-mod.

On the other hand, for the Hecke algebra of type B (i.e., the Hecke algebra associated to the group W_{2}^{r}), Dipper and James established a Morita equivalence analogous to (1) in [DJ2].

Typeset by AMS-TEX

Both authors gratefully acknowledge support received from Australian Research Council under Large ARC Grant A69530243. The second author is partially supported by the National Natural Science Foundation in P.R. China. He wishes to thank the University of New South Wales for its hospitality during his visit.

This paper is going to generalize these results to the Ariki-Koike algebra H = H_{m}^{r}, an Iwahori-Hecke type algebra associated with W_{m}^{r} (see [AK]). A major difficulty here is the non-existence of a subalgebra based on the bottom C, comparing with the classical case, and also, the group W is no long a Coxeter group. However, since the semi-simplicity of F C is simply equivalent to the condition that the order |C| of the bottom C is non-zero in F, our strategy here is to find a q-analogue of the order of C, which is called the Pioncaré polynomial of C, and with the invertibility of such a polynomial, to look for those idempotents e_{λ}. Thus, we eventually establish a q-analogue of the Morita equivalence (1) above. A by-product of our results is the introduction of the Poincaré polynomial d_{W} of the complex reflection group W. We shall see that the semi-simplicity of H over a field F is equivalent to d_{W} ≠ 0 in F.

We organize the paper as follows. In §1, we introduce the poset Λ[m, r], which is isomorphic to Λ(m, r) and discuss some combinatorics related to symmetric groups. In §2, a useful lemma (2.8) related to the poset structure on Λ[m, r] is proved. Candidates of those idempotents e_{λ} are constructed in §3. The main results are presented in §4, where we prove that the invertibility of the ‘Poincaré polynomial’ f_{m,r} is a necessary and sufficient condition for the existence of those idempotents e_{λ}, and the Morita equivalence is established. Finally, in §5, we lift the Morita equivalence to the endomorphism algebra level. Two by-products for H_{F} over a field F in which f_{m,r} is nonzero are the classification of simple modules and the criterion of semisimplicity.

The main results of the paper have been announced by the first author at the “Simposium on Modular representations of finite groups”, Virginia, May 1998, and at the “International Conference on Representation Theory ”, Shanghai, June-July, 1998. At the Vriginia conference, R. Dipper announced some Morita theorems for Ariki-Koike algebras joint with A. Mathas with quite different treatment. For example, our method works over the set Λ(m, r) of compositions with m parts, while, in their method, they first treat the case where compositions have 2 parts. Thus, they annouced a Morita equivalence between an Ariki-Koike algebra and a tensor product of two smaller such algebras. To obtain our result, they have to break two-part compositions further down.

Throughout, R denotes a commutative ring with identity 1.

1. The poset Λ[m, r]. Let r be a non-negative integer. A composition λ of r with m > 0 parts is a sequence (λ_{1}, ⋯, λ_{m}) of nonnegative integers such that |λ| = ∑_{i=1}^{m} λ_{i} = r, and λ is called a partition if the sequence is weakly decreasing. Let Λ(m, r) (resp. Λ(m, r)^{+}) be the set of compositions (resp. partitions) of r with m-parts. With the usual dominance order ⊴, both Λ(m, r) and Λ(m, r)^{+} are posets.

For notational convenience, we shall use another poset Λ[m, r]. For any λ ∈ Λ(m, r), let [λ] = [a_{0}, a_{1}, ⋯, a_{m}] where a_{0} = 0 and a_{i} = λ_{1} + ⋯ + λ_{i} for all i and put Λ[m, r] = {[λ] : λ ∈ Λ(m, r)}. The following results are almost obvious.

(1.1) Lemma. (a) Alternatively, we have

Λ[m, r] = {[a_{0}, a_{1}, ⋯, a_{m}] : 0 = a_{0} ⩽ a_{1} ⩽ ⋯ ⩽ a_{m} = r, a_{i} ∈ ℤ, ∀i}.

(b) For any a, b ∈ Λ[m, r], define a ≼ b by setting a_{i} ⩽ b_{i} for every i with 1 ⩽ i ⩽ m. Then Λ[m, r] is a poset with partial ordering ≼. Moreover, the map Ξ : Λ(m, r) → Λ[m, r] defined by Ξ(λ) = [λ] is an isomorphism between posets (Λ(m, r), ⊴) and (Λ[m, r], ≼). In particular, we have for all λ, μ ∈ Λ(m, r)

λ ⊴ μ if and only if [λ] ≼ [μ].

(c) If Θ = Ξ^{−1} is the inverse map of Ξ, then, for a = [a_{i}] ∈ Λ[m, r],

Θ(a) = (a_{1} − a_{0}, a_{2} − a_{1}, ⋯, a_{m} − a_{m−1}).

If a ≼ b and a ≠ b, we write a ≺ b.

(1.2) Notation. For any a = [a_{0}, ⋯, a_{m}] ∈ Λ[m, r], let i (resp. j ) be the minimal index such that a_{i} ≠ 0 (resp. a_{j} = r) and define

(a) a′ = [0, r − a_{m−1}, ⋯, r − a_{1}, r];

(b) a_{⊢} = [0, ⋯, 0, a_{i} − 1, ⋯, a_{m} − 1];

(c) a_{⊣} = [a_{0}, a_{1}, ⋯, a_{j−1}, r − 1, ⋯, r − 1]. The notations we choose here are symmetric: if λ = Θ(a), then i (resp. j ) is the minimal (resp. maximal) index with λ_{i} ≠ 0 (resp. λ_{j} ≠ 0), and we have a′ = [(λ_{m}, ⋯, λ_{1})] = [λ^{∘}], a_{⊢} = [(0, ⋯, 0, λ_{i} − 1, λ_{i+1}, ⋯, λ_{m})] = [λ_{⊢}], and a_{⊣} = [(λ_{1}, ⋯, λ_{j−1}, λ_{j} − 1, 0, ⋯, 0)] = [λ_{⊣}]. Since (λ_{⊢})^{∘} = (λ^{∘})_{⊣} and (λ_{⊣})^{∘} = (λ^{∘})_{⊢}, it follows that

(d) (a_{⊢})′ = (a′)_{⊣} and (a_{⊣})′ = (a′)_{⊢}.

Let 𝔖_{r} = 𝔖_{{1,··· ,r}} be the symmetric group on r letters as in the introduction. Each element w ∈ 𝔖_{r} can be expressed as a product of s_{i_{1}} ⋯ s_{i_{k}}, where s_{i} = (i, i + 1) are basic transpositions. If k is minimal, such an expression is called a reduced expression of w. The number k is defined to be the length l(w) of w. It is independent of the reduced expression of w and l(w) = #{(i, j) | i < j, (i)w > (j)w}. For a = [a_{i}] ∈ Λ[m, r] with λ = Θ(a), let

𝔖_{a} = 𝔖_{λ} = 𝔖_{{1,··· ,a_{1}}} × 𝔖_{{a_{1}+1,··· ,a_{2}}} × ⋯ × 𝔖_{{a_{m−1}+1,··· ,a_{m}}}.

be the Young subgroup of 𝔖_{r} corresponding to a, and D_{a} = D_{λ} the set of distinguished representatives of right 𝔖_{λ}-cosets. Similarly, for a, b ∈ Λ[m, r], D_{a,b} = D_{a} ∩ D_{b}^{−1} denotes the set of distinguished representatives of 𝔖_{a}-𝔖_{b} double cosets.

For positive integers i, j, let s_{i,i} = 1 and

s_{i,j} = \{ \begin{matrix}s_{i−1} ⋯ s_{j}, & if i > j, \\ s_{i} ⋯ s_{j−1}, & if i < j. \end{matrix}

(1.3) Lemma. (a) We have 𝔖_{r} = ∪_{i=1}^{r}s_{i,r}𝔖_{r−1}, where s_{i,r}, 1 ⩽ i ⩽ r, are distinguished 𝔖_{r−1}-coset representatives in 𝔖_{r}.

(b) Let a ∈ Λ[m, r] and d ∈ D_{a,a′}. Then d = s_{a_{j},r}d_{1} for some j with a_{j−1} < a_{j} and d_{1} ∈ D_{b,(a_{⊢})′}, where b = [a_{0}, ⋯, a_{j−1}, a_{j} − 1, a_{j+1} − 1, ⋯, a_{m} − 1].

Proof. The statement (a) is well-known. Since d ∈ D_{a′}^{−1}, we have d_{1} ∈ D_{a′}^{−1} ∩ H(𝔖_{r−1}). Therefore, d_{1} ∈ D_{(a′)_{⊣}}^{−1} = D_{(a_{⊢})′}^{−1} by (1.2)(d). On the other hand, d ∈ D_{a} implies that i = a_{j} for some 1 ⩽ j ⩽ m. Take the minimal j with a_{j} = i. Then a_{j−1} ≠ a_{j}, and b ∈ Λ[m, r − 1]. If d_{1} ∉ D_{b}, then there is a s_{k} ∈ 𝔖_{b} such that d_{1} = s_{k}d_{2} with l(d_{1}) = l(d_{2})+1, and d = s_{a_{j},r}s_{k}d_{2}. If k ⩾ a_{j}, then d = s_{k+1}s_{a_{j},r}d_{2}, contrary to d ∈ D_{a}. If k < a_{j}, then k ⩽ a_{j} − 2 since s_{k} ∈ 𝔖_{b}. Thus, d = s_{k}s_{a_{j},r}d_{2}, a contradiction again. Therefore, d_{1} ∈ D_{b}, and d_{1} ∈ D_{b,(a_{⊢})′}. □

Let w_{i,j} = s_{i+1,1}s_{i+2,2} ⋯ s_{i+j,j}. Then w_{i,j} is the following permutation

(1.4)  w_{i,j} = ( \begin{matrix}1 & ··· & i & i+1 & ··· & i+j \\ j+1 & ··· & j+i & 1 & ··· & j \end{matrix} ) .

Let k be a non-negative integer. Define the k-shifted elements s_{i,j}^{(k)}, w_{i,j}^{(k)} by setting s_{i,j}^{(k)} = s_{i+k,j+k} and w_{i,j}^{(k)} = s_{i+1,1}^{(k)}s_{i+2,2}^{(k)} ⋯ s_{i+j,j}^{(k)}, and w_{i,j} = w_{i,j}^{(k)} = 1, if i = 0 or j = 0. Note that w_{i,j}^{(k)} is a permutation on {k+1, ⋯, k+i+j}, and explicitly,

(1.5)  w_{i,j}^{(k)} = ( \begin{matrix}k+1 & ··· & k+i & k+i+1 & ··· & k+i+j \\ k+j+1 & ··· & k+j+i & k+1 & ··· & k+j \end{matrix} ) .

Obviously, we have (s_{i,j}^{(k)})^{−1} = s_{j,i}^{(k)} and (w_{i,j}^{(k)})^{−1} = w_{j,i}^{(k)} For a = [a_{i}] ∈ Λ[m, r] with λ = Θ(a), let w_{a} ∈ 𝔖_{r} be defined by

(1.6)  (a_{i−1} + l)w_{a} = r − a_{i} + l for all i with a_{i−1} < a_{i}, 1 ⩽ l ⩽ a_{i} − a_{i−1}.

In particular, we have (a_{i})w_{a} = r−a_{i−1} if a_{i−1} < a_{i}. For example, for a = [0, i, i+j], w_{a} = w_{i,j}, and for b = [0, 2, 5, 9],

w_{b} = ( \begin{matrix}1 2 & 3 4 5 & 6 7 8 9 \\ \underline{8 9} & \underline{5 6 7} & \underline{1 2 3 4} \end{matrix} ) .

(1.7) Lemma. For a = [a_{i}] ∈ Λ[m, r] with λ = Θ(a),

\begin{aligned}w_{a} & = w_{a_{m−1},λ_{m}}^{(0)}w_{a_{m−2},λ_{m−1}}^{(λ_{m})}w_{a_{m−3},λ_{m−2}}^{(λ_{m}+λ_{m−1})} ⋯ w_{a_{1},λ_{2}}^{(λ_{m}+···+λ_{3})} \\ & = w_{a_{m−1},a_{m}−a_{m−1}}^{(0)}w_{a_{m−2},a_{m−1}−a_{m−2}}^{(a_{m}−a_{m−1})} ⋯ w_{a_{1},a_{2}−a_{1}}^{(a_{m}−a_{2})}. \end{aligned}

Proof. The result follows immediately from (1.4) and (1.5). □

We list some properties for the elements w_{a}. First, w_{a} is distinguished and turns 𝔖_{a} into 𝔖_{a′}.

(1.8) Lemma. For any a = [a_{i}] ∈ Λ[m, r], we have (a) w_{a}^{−1} = w_{a′} and w_{a} ∈ D_{a}. Hence w_{a} ∈ D_{a,a′}.

(b) w_{a}^{−1}𝔖_{a}w_{a} = 𝔖_{a′}. In particular, w_{a}^{−1}s_{j}w_{a} = s_{(j)w_{a}} for all j ⩾ 1 with j ≠ a_{i}.

Proof. The first assertion in (a) follows from definition. Consider the root system Φ of type A_{r−1} and its subsystem Φ_{a} whose Coxeter graph is obtained by removing all a_{i}-th vertices from that of Φ. For any s_{i} ∈ 𝔖_{r}, there is a simple root α_{i} = e_{i} − e_{i+1} with s_{i} = s_{α_{i}} and w(α_{i}) = e_{(i)w^{−1}} − e_{(i+1)w^{−1}} (see, e.g., [Hum]). From (1.6), we have (a_{j})w_{a} = r − a_{j−1} for a_{j−1} < a_{j}, and (i + 1)w_{a} = (i)w_{a} + 1 if i ≠ a_{j} for all 1 ⩽ j ⩽ m. Thus, w_{a}^{−1}(α_{i}) is a simple root. Therefore, w_{a}^{−1} stabilizes the positive root system of Φ_{a}, and consequently, w_{a} ∈ D_{a} and hence w_{a} ∈ D_{a,a′}, proving (a). Because w_{a}^{−1}s_{i}w_{a} = s_{(i)w_{a}} ∈ 𝔖_{a′} is a basic transposition, w_{a}^{−1}𝔖_{a}w_{a} ⊆ 𝔖_{a′}. Therefore, by (a), w_{a}^{−1}𝔖_{a}w_{a} = 𝔖_{a′}, proving (b). □

We now look at the relation between w_{a} and w_{a_{⊣}}. For a = [a_{i}] ∈ Λ[m, r] with minimal index k such that a_{k} = r, we have (see (1.2(c)) a_{⊣} = [0, a_{1}, ⋯, a_{k−1}, r − 1, ⋯, r − 1] ∈ Λ[m, r − 1]. Let a_{i} ∈ Λ[m, r], i = 1, ⋯, m, be defined by

(1.9)  a_{i} = a_{⊣} + 1_{i}, where 1_{i} = [0, \underbrace{0, ⋯, 0}_{i−1}, 1, ⋯, 1] ∈ Λ[m, 1].

Then a_{i} ∈ Λ[m, r] with a_{1} ≻ a_{2} ≻ ⋯ ≻ a_{m} and a_{k} = a.

(1.10) Lemma. Write a_{⊣} = [b_{0}, b_{1}, ⋯, b_{m}] ∈ Λ[m, r − 1]. Then, for any 1 ⩽ i ⩽ m, w_{a_{i}} = s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} with l(w_{a_{i}}) = l(s_{b_{i}+1,r})+l(w_{a_{⊣}})+l(s_{r,r−b_{i−1}}).

Proof. We prove (l)w_{a_{i}} = (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} for every l with 1 ⩽ l ⩽ r.

Assume that l ⩽ b_{i}. Then there is a j ⩽ i with b_{j−1} < l ⩽ b_{j}. Write l = b_{j−1} + l′ ⩽ b_{j}. Then (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} = (l)w_{a_{⊣}}s_{r,r−b_{i−1}} = ((r − 1) − b_{j} + l′)s_{r,r−b_{i−1}} using (1.6). Because r − 1 − b_{j} + l′ ⩾ r − b_{i−1} for j < i and r − 1 + l′ − b_{i} < r − b_{i−1}, we have

(l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} = \{ \begin{matrix}r − b_{j} + l′ & if j < i \\ (r − 1) − b_{j} + l′ & if j = i. \end{matrix}

Thus, (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} = (l)w_{a_{i}}. If l = b_{i} + 1, then (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} = r − b_{i−1} = (l)w_{a_{i}}. Assume l > b_{i} + 1. Then there is a j with j ⩾ i and b_{j} < l − 1 ⩽ b_{j+1}. Write l − 1 = b_{j} + l′. Because r − 1 − b_{j+1} + l′ < r − b_{i−1}, we have (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}} = (l − 1)w_{a_{⊣}}s_{r,r−b_{i−1}} = r − 1 − b_{j+1} + l′. In this case, l = (b_{j} + 1) + l′ and (l)w_{a_{i}} = r − (b_{j+1} + 1) + l′ = (l)s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}}. So w_{a_{i}} = s_{b_{i}+1,r}w_{a_{⊣}}s_{r,r−b_{i−1}}. The length formula is obviously. □

The elements w_{a} ∈ 𝔖_{r} and w_{a_{⊢}} ∈ 𝔖_{r−1} are related as follows.

(1.11) Corollary. Let a = [a_{i}] ∈ Λ[m, r] and suppose that k is the minimal index with a_{k} ≠ 0. Then,

(a) w_{a} = s_{a_{k},r}w_{a_{⊢}} with l(w_{a}) = l(w_{a_{⊢}}) + l(s_{a_{k},r}).

(b) w_{a′} = w_{(a_{⊢})′}s_{r,a_{k}} with l(w_{a′}) = l(w_{(a_{⊢})′}) + l(s_{r,a_{k}}).

Proof. It follows immediately from (1.2)(d) and (1.10), or from (1.8) and (1.3). □

2. Some zero divisors. Let W = W_{m}^{r} be the group defined in the introduction. Then W is the group with r generators {s_{0}, s_{1}, s_{2}, ⋯, s_{r−1}} = S and relations:

s_{0}^{m} = s_{i}^{2} = 1, for all 1 ⩽ i ⩽ r − 1,
s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}, for 1 ⩽ i ⩽ r − 2,
s_{0}s_{1}s_{0}s_{1} = s_{1}s_{0}s_{1}s_{0}
s_{i}s_{j} = s_{j}s_{i}, if 0 ⩽ i ⩽ j − 2 ⩽ r − 3.

Let t_{1} = s_{0} and t_{i} = s_{i−1}t_{i−1}s_{i−1}, 2 ⩽ i ⩽ r. Then t_{i}t_{j} = t_{j}t_{i}, t_{i}^{m} = 1 for 1 ⩽ i, j ⩽ r and {t_{i} | 1 ⩽ i ⩽ r} generates the bottom C. We will identify the subgroup of W generated by {s_{i} | 1 ⩽ i ⩽ r − 1} with 𝔖_{r}. Note that the group W_{m}^{r} is the symmetry group (resp. hyperoctahedral group) if m = 1 (resp. m = 2).

A deformation of the group algebra of W has been given recently by Ariki and Koike [AK]. Let R be a commutative ring with 1 and q, q^{−1}, u_{1}, …, u_{m} ∈ R. The Ariki-Koike algebra H = H_{R} = H_{m}^{r} associated to the group W is an associative algebra over R with generators T_{i} := T_{s_{i}}, 0 ⩽ i ⩽ r − 1, subject to the relations:

(2.1)  \{ \begin{matrix}(1) & T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0}, \\ (2) & T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}, for 1 ⩽ i ⩽ r − 2 \\ (3) & T_{i}T_{j} = T_{j}T_{i}, if |i − j| ⩾ 2 \\ (4) & (T_{i} − q)(T_{i} + 1) = 0, if i ≠ 0 \\ (5) & (T_{0} − u_{1}) ⋯ (T_{0} − u_{m}) = 0. \end{matrix}

Let L_{1} = T_{0} and L_{i} = q^{−1}T_{i−1}L_{i−1}T_{i−1}, 2 ⩽ i ⩽ r. Then L_{i}L_{j} = L_{j}L_{i}, 1 ⩽ i, j ⩽ r. The elements L_{i}, 1 ⩽ i ⩽ r, generate an abelian subalgebra of H. Since L_{i}^{m} ≠ 1 in general, this subalgebra contains a proper submodule of rank |C|. So it cannot be served as a “bottom” of H. However, H is R-free of rank |W | with basis [AK]:

(2.2)  {L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} | w ∈ 𝔖_{r}, and 0 ⩽ c_{i} ⩽ m − 1, ∀i}.

Let H be the subalgebra generated by T_{i} for all i ⩾ 1. For a Young subgroup 𝔖_{a}, let H(𝔖_{a}) be the corresponding subalgebra. By (2.2), we have H = ⊕_{c}L^{c}H, where c = (c_{1}, ⋯, c_{r}) and L^{c} = L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}. Thus, for every such c, we have a projection map

(2.3)  pr_{c} : H → L^{c}H.

The isomorphism from H to the algebra H^{op} opposite to H induces an antiautomorphism

(2.4)  ι : H → H such that ι(T_{i}) = T_{i}.

Clearly, ι(L_{i}) = L_{i}. We need the following commutator relations.

(2.5) Proposition. Let H be the Ariki-Koike algebra over a commutative ring R. Then

(a) T_{i} commutes with L_{j} if j ≠ i, i + 1.

(b) T_{i} commutes with L_{i}L_{i+1} and L_{i} + L_{i+1}.

(c) T_{i} commutes with ∏_{j=1}^{k}(L_{j} − a) for all a ∈ R and i ≠ k.

(d) L_{i}^{k} = q^{−1}T_{i−1}L_{i−1}^{k}T_{i−1} + (1 − q^{−1}) ∑_{c=1}^{k−1} L_{i}^{c}L_{i−1}^{k−c}T_{i−1} if 1 < i ⩽ r and k ⩾ 1.

Proof. See [AK, (3.3)] for statements (a-c), and [MM, (3.6)] for (d) . □

Following Graham and Lehrer [GL, (5.4) ] (or [DJM, (3.1)]), we define for a = [a_{i}] ∈ Λ[m, r],

(2.6)  \{ \begin{matrix}π_{a} = π_{a_{1}}(u_{2}) ⋯ π_{a_{m−1}}(u_{m}),  π̃_{a} = π_{a_{1}}(u_{m−1}) ⋯ π_{a_{m−1}}(u_{1}), \\ where π_{0}(x) = 1 and π_{a}(x) = ∏_{j=1}^{a}(L_{j} − x), ∀a > 0, x ∈ R. \end{matrix}

Note that π_{a} = π_{a_{⊣}} if a_{m−1} ≠ r and π̃_{a′} = π̃_{(a_{⊢})′} if a_{1} ≠ 0. Also, if, for x_{1}, ⋯, x_{m−1} ∈ R, we put π(a; x_{1}, ⋯, x_{m−1}) = π_{a_{1}}(x_{1}) ⋯ π_{a_{m−1}}(x_{m−1}), then π_{a} = π(a; u_{2}, ⋯, u_{m}) and π̃_{a} = π(a; u_{m−1}, ⋯, u_{1}).

(2.7) Corollary. (a) For a ∈ Λ[m, r], π_{a} and π̃_{a} commute with any element in H(𝔖_{a}). In particular, for any x ∈ R, π_{r}(x) is in the centre of H.

(b) Assume a_{j−1} < i ⩽ a_{j} for some 1 ⩽ j ⩽ m. Then

π_{a}T_{i,r} = T_{i,a_{j}}(L_{a_{j}} − u_{j+1})T_{a_{j},a_{j+1}} ⋯ (L_{a_{m−1}} − u_{m})T_{a_{m−1},a_{m}}π_{b},

where b = [0, ⋯, 0, a_{k}, ⋯, a_{j−1}, a_{j} − 1, ⋯, a_{m} − 1] ∈ Λ[m, r − 1] and T_{i,j} = T_{s_{i,j}}.

Proof. The statement (a) follows from (2.5)(c). Noting that T_{i,j} = T_{i} ⋯ T_{j−1} for i < j, we obtain (b) immediately from (2.5)(a)-(c).

We now prove the following useful result.

(2.8) Lemma. For a, b ∈ Λ[m, r], we have π_{a}Hπ̃_{b′} = 0 and π̃_{a}Hπ_{b′} = 0 unless a ≼ b.

Proof. Using anti-automorphism ι of H in (2.4) and the fact b′ ≼ a′ if and only if a ≼ b, we see that both assertions in (2.8) are equivalent. Therefore, we only need to prove π_{a}Hπ̃_{b′} = 0 unless a ≼ b.

We apply induction on r. Let r = 1. If a ⋠ b, then there is an i with a_{i} > b_{i}. Since b_{i}, a_{i} ⩽ 1 for all i, we have b_{i} = 0 and a_{i} = 1. By (2.6), ∏_{k=i}^{m−1}(L_{1} − u_{k+1}) = ∏_{k=i+1}^{m}(L_{1} − u_{k}) ( resp. ∏_{k=1}^{i}(L_{1} − u_{k}) ) is a factor of π_{a} (resp. π̃_{b′}). Therefore, π_{a}Hπ̃_{b′} = 0 by (2.1)(5), and (2.8) is true for r = 1. Assume π_{a}Hπ̃_{b′} = 0 for all a, b ∈ Λ[m, r − 1] with a ⋠ b.

Let i and j be the minimal indices with b_{i} ≠ 0 and a_{j} ≠ 0, respectively. Because b_{k} = 0 for all k < i, π_{b_{m}−b_{k}}(u_{k}) = π_{r}(u_{k}), which are in the centre of H (see (2.7)(a)). Therefore, for any w ∈ 𝔖_{r}, π_{a}T_{w}π̃_{b′} contains a factor ∏_{k=j+1}^{m}(L_{1} − u_{k}) ∏_{k=1}^{i−1}(L_{1} − u_{k}). By (2.1)(5), π_{a}Hπ̃_{b′} = 0 unless i ⩽ j. On the other hand, take w ∈ 𝔖_{r} with π_{a}T_{w}π̃_{b′} ≠ 0. Write w = dy with y ∈ 𝔖_{r−1} and d = s_{k,r} for some 1 ⩽ k ⩽ r (see (1.3)). By (2.7)(b), π_{a}T_{d} = hπ_{a_{⊢}} for some h ∈ H. By (2.6), π̃_{b′} = π̃_{(b_{⊢})′}(L_{r} − u_{i−1}) ⋯ (L_{r} − u_{1}). So, π_{a}T_{w}π̃_{b′} ≠ 0 implies π_{a_{⊢}}T_{y}π̃_{(b_{⊢})′} ≠ 0. Now, by induction, a_{⊢} ≼ b_{⊢}, which implies a ≼ b since i ⩽ j. □

3. Idempotents. In this section, idempotents e_{a} for all a ∈ Λ[m, r] will be constructed under certain conditions.

(3.1) Proposition. For any a ∈ Λ[m, r], let v_{a} = π_{a}T_{w_{a}}π̃_{a′}. Then we have:

(a) π_{a}Hπ̃_{a′} = v_{a}H(𝔖_{a′}) = H(𝔖_{a})v_{a}. Moreover, v_{a}T_{i} = T_{(i)w_{a}^{−1}}v_{a} for any s_{i} ∈ 𝔖_{a′}.

(b) v_{a}L_{i} ∈ v_{a}H(𝔖_{i}) ∩ v_{a}H(𝔖_{a′}) and L_{i}v_{a} ∈ H(𝔖_{i})v_{a} ∩ H(𝔖_{a})v_{a} for every i = 1, ⋯, r. In particular, v_{a}L_{i} = u_{j}v_{a} if i = r −a_{j} +1 for some j with a_{j−1} < a_{j}.

(c) π_{a}Hπ̃_{a′} = π_{a}Hπ̃_{a′}.

(d) v_{a}H = v_{a}H.

Proof. Let d ∈ D_{a,a′}. We prove π_{a}T_{d}π̃_{a′} = 0 unless d = w_{a}. Obviously, this result is true for r = 1, and assume r > 1 and the result is true for r − 1. By (1.3), d = s_{i,r}d_{1} with d_{1} ∈ 𝔖_{r−1} and i ≥ a_{k}, where k is the minimal index with a_{k} ≠ 0. If i > a_{k}, then there is an index j with a_{j−1} < i ⩽ a_{j}, j > k. By (2.7)(b),

π_{a}T_{d} = T_{i,a_{j}}(L_{a_{j}} − u_{j+1})T_{a_{j},a_{j+1}}(L_{a_{j+1}} − u_{j+2}) ⋯ (L_{a_{m−1}} − u_{m})T_{a_{m−1},a_{m}}π_{b}T_{d_{1}},

where b = [0, ⋯, 0, a_{k}, ⋯, a_{j−1}, a_{j} − 1, ⋯, a_{m} − 1] ∈ Λ[m, r − 1] and b ⋠ a_{⊢}. By (2.6), π̃_{a′} = π̃_{(a_{⊢})′}(L_{r} − u_{k−1}) ⋯ (L_{r} − u_{1}). We have π_{b}T_{d_{1}}π̃_{(a_{⊢})′} = 0 by (2.8). So π_{a}T_{d}π̃_{a′} = 0, a contradiction, proving i = a_{k}. From (1.3)(b), we have d_{1} ∈ D_{a_{⊢},(a_{⊢})′}. Because

\begin{matrix}π_{a}T_{d}π̃_{a′} = & (L_{a_{k}} − u_{j+1})T_{a_{k},a_{k+1}}(L_{a_{k+1}} − u_{k+2}) ⋯ (L_{a_{m−1}} − u_{m})T_{a_{m−1},a_{m}} \\ π_{a_{⊢}}T_{d_{1}}π̃_{(a_{⊢})′}(L_{r} − u_{k−1}) ⋯ (L_{r} − u_{1}), \end{matrix}

we have π_{a_{⊢}}T_{d_{1}}π̃_{(a_{⊢})′} ≠ 0. By induction, d_{1} = w_{a_{⊢}}, and therefore, d = w_{a} by (1.11). Now, the first assertion in (a) follows immediately from the 𝔖_{a}-𝔖_{a′} double coset decomposition of 𝔖_{r}.

Write h_{a} = T_{w_{a}}. By (1.8), we have h_{a}T_{i} = T_{(i)w_{a}^{−1}}h_{a} for every s_{i} ∈ 𝔖_{a′}. Therefore, v_{a}T_{i} = π_{a}h_{a}T_{i}π̃_{a′} = π_{a}T_{(i)w_{a}^{−1}}h_{a}π̃_{a′} = T_{(i)w_{a}^{−1}}v_{a}, proving the second assertion in (a).

To see (b), we first treat the case i = a_{m} − a_{j} + 1 with j minimal. Then a_{j} > a_{j−1}, and b = [a_{0}, a_{1}, ⋯, a_{j−1}, a_{j} − 1, a_{j+1}, ⋯, a_{m}] ∈ Λ[m, r]. Obviously, b ⋡ a. By (2.8), π_{a}h_{a}π̃_{b′} = 0. By (2.6), we have π̃_{a′}L_{i} = π̃_{b′} + u_{j}π̃_{a′}, where b′ = [0, a_{m} − a_{m−1}, ⋯, a_{m} − a_{j−1}, a_{m} − (a_{j} − 1), ⋯, a_{m} − a_{1}, a_{m}]. Thus, v_{a}L_{i} = π_{a}h_{a}π̃_{b′} + u_{j}v_{a} = u_{j}v_{a}. Now, assume i ≠ a_{m} − a_{j} + 1, 1 ⩽ j ⩽ m. Then there is a k with a_{m} − a_{k} + 1 < i ⩽ a_{m} − a_{k−1}. Write i = (a_{m} − a_{k} + 1) + l. Then L_{i} = q^{−l}T_{i,a_{m}−a_{k}+1}L_{a_{m}−a_{k}+1}T_{a_{m}−a_{k}+1,i}. By (a), v_{a}L_{i} = u_{k}q^{−l}T_{i,a_{m}−a_{k}+1}T_{a_{m}−a_{k}+1,i} ∈ v_{a}H(𝔖_{i}) ∩ v_{a}H(𝔖_{a′}). By the same arguments, one can prove L_{i}v_{a} ∈ H(𝔖_{i})v_{a} ∩ H(𝔖_{a})v_{a}.

By (b) and (a), L_{i}v_{a}H(𝔖_{a′}) ⊆ H(𝔖_{a})v_{a}H(𝔖_{a′}) = v_{a}H(𝔖_{a′}) for every i = 1, ⋯, r. Thus (c) follows immediately from (2.2).

For arbitrary v_{a}T_{w} ∈ v_{a}H with w ∈ 𝔖_{r}, write w = dy with y ∈ 𝔖_{(2,··· ,r)} and d ∈ 𝔖_{r}/𝔖_{(2,··· ,r)}. Then d = T_{i,1} with 1 ⩽ i ⩽ r. By (b), v_{a}T_{w}T_{0} = v_{a}T_{i,1}T_{0}T_{y} ∈ v_{a}H. Thus v_{a}H is stable under the right action of T_{0}, and v_{a}H = v_{a}H. □

Recall from (2.7) that T_{i,j} = T_{s_{i,j}} for i ≠ 0 ≠ j. For a ∈ Λ[m, 0], let v_{a} = 1.

(3.2) Lemma. Let x_{1}, ⋯, x_{m−1} be elements in the commutative ring R and a ∈ Λ[m, r] with a_{1} ≠ 0. Then there is an integer c depending on a such that

T_{a_{m},a_{m−1}}(L_{a_{m−1}} − x_{m−1}) ⋯ T_{a_{2},a_{1}}(L_{a_{1}} − x_{1}) − q^{c}L_{r}^{m−1}T_{a_{1},a_{m}}^{−1}

is in the free R-submodule spanned by {L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} | w ∈ 𝔖_{r}, c_{j} < m − 1, ∀j}.

Proof. Let U_{i} (1 ⩽ i ⩽ m − 1) be the free R-submodule spanned by

{L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} | w ∈ 𝔖_{{a_{i},a_{i}+1,··· ,r}}, c_{j} < m − i, ∀j}.

We claim that, for any i = 1, ⋯, m − 1,

(3.3)  T_{a_{m},a_{m−1}}(L_{a_{m−1}} − x_{m−1}) ⋯ T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i}) − q^{c}L_{r}^{m−i}T_{a_{i},a_{m}}^{−1} ∈ U_{i}.

Apply induction on i. The result for i = m − 1 is true since T_{a_{m},a_{m−1}}(L_{a_{m−1}} − x_{m−1}) = q^{a_{m}−a_{m−1}}L_{a_{m}}T_{a_{m−1},a_{m}}^{−1} − x_{m−1}T_{a_{m},a_{m−1}}. For i < m − 1, we have, by induction,

T_{a_{m},a_{m−1}}(L_{a_{m−1}} − x_{m−1}) ⋯ T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i})
=(q^{c}L_{r}^{m−i−1}T_{a_{i+1},a_{m}}^{−1} + h)T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i})
=q^{c}L_{r}^{m−i−1}T_{a_{i+1},a_{m}}^{−1}T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i}) + hT_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i})

for some h ∈ U_{i+1}. We may assume h = L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} with w ∈ 𝔖_{{a_{i+1},··· ,r}} and c_{j} < m − (i + 1), ∀j, without loss of generality. Write w = s_{k,a_{i+1}}y with y ∈ 𝔖_{{a_{i+1}+1,··· ,r}} (cf. (1.3)). Then T_{w}T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i+1}) = q^{k−a_{i}}L_{k}T_{a_{i},k}^{−1}T_{y} − x_{i+1}T_{k,a_{i}}T_{y}. Thus, we have hT_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i+1}) ∈ U_{i}. Noting qT_{i}^{−1} = T_{i} − (q − 1) and L_{r}^{m−i−1}T_{r,a_{i+1}}T_{a_{i+1},a_{i}}L_{a_{i}} = q^{r−a_{i}}L_{r}^{m−i}T_{a_{i},r}^{−1}, we may write

L_{r}^{m−i−1}T_{a_{i+1},a_{m}}^{−1}T_{a_{i+1},a_{i}}(L_{a_{i}} − x_{i+1}) = q^{c′}L_{r}^{m−i}T_{a_{i},r}^{−1} + h′

for some h′ ∈ U_{i} and c′ ∈ ℤ. Therefore, (3.3) holds for i, proving the claim. Now, the required result follows from the case i = 1. □

(3.4) Theorem. Let H be the Ariki-Koike algebra over R. For any a ∈ Λ[m, r], v_{a}H is a free R-module with basis {v_{a}T_{w} | w ∈ 𝔖_{r}}.

Proof. For h = (m − 1, ⋯, m − 1), let pr_{h} be the projection defined in (2.3). First, we prove

(3.5)  pr_{h}(v_{a}) = q^{c} ∏_{j=1}^{r} L_{j}^{m−1}h_{a′}^{−1}

for some integer c ∈ ℤ depending on a. We prove (3.5) by induction on r. Obviously, the result is true for r = 1. Assume now r > 1 and (3.5) holds for r − 1. For a ∈ Λ[m, r] with minimal index k, a_{k} = r, recall

a′ = [0, 0, ⋯, 0, r − a_{k−1}, ⋯, r − a_{1}, r],
a_{⊣} = [0, a_{1}, ⋯, a_{k−1}, r − 1, ⋯, r − 1],
(a_{⊣})′ = [0, ⋯, 0, r − 1 − a_{k−1}, ⋯, r − 1 − a_{1}, r − 1], and
π_{a} = π_{a_{⊣}}(L_{r} − u_{k+1}) ⋯ (L_{r} − u_{m}).

Then, noting from (1.10) w_{a} = w_{a_{k}} = w_{a_{⊣}}s_{r,r−a_{k−1}}, we have

\begin{matrix}v_{a} = & v_{a_{⊣}}(L_{r} − u_{k+1}) ⋯ (L_{r} − u_{m}) \\ T_{r,r−a_{1}}(L_{r−a_{1}} − u_{1}) ⋯ T_{r−a_{k−2},r−a_{k−1}}(L_{r−a_{k−1}} − u_{k−1}). \end{matrix}

Thus, applying (3.2) to [0, r − a_{k−1}, ⋯, r − a_{1}, r, ⋯, r], we have

v_{a} = v_{a_{⊣}}(q^{c_{2}}L_{r}^{m−1}T_{r−a_{k−1},r}^{−1} + h_{1})

where c_{2} ∈ ℤ, h_{1} ∈ U_{1} (see the proof of (3.2)). On the other hand, by induction, we have, for some integer c_{1}, v_{a_{⊣}} = q^{c_{1}} ∏_{j=1}^{r−1} L_{j}^{m−1}h_{(a_{⊣})′}^{−1} + h, where h

is a linear combinations of elements L_{1}^{d_{1}} ⋯ L_{r−1}^{d_{r−1}}T_{w} with w ∈ 𝔖_{r−1}. Since L_{r} commutes with any elements in H_{m}^{r−1}, we have pr_{h}(q^{c_{1}} ∏_{j=1}^{r−1} L_{j}^{m−1}h_{(a_{⊣})′}^{−1}h_{1}) = 0, pr_{h}(hq^{c_{2}}L_{r}^{m−1}T_{r,r−a_{k}}^{−1}) = 0 and pr_{h}(hh_{1}) = 0. Therefore, (3.5) holds (cf. (1.11)(a)).

Now we are ready to prove the set {v_{a}T_{w} | w ∈ 𝔖_{r}} is linearly independent. Suppose ∑_{w∈𝔖_{r}} c_{w}v_{a}T_{w} = 0. Then, by (3.5), there is a c ∈ ℤ depending on a such that

pr_{h}( ∑_{w∈𝔖_{r}} c_{w}v_{a}T_{w}) = ∑_{w∈𝔖_{r}} c_{w}q^{c}L_{1}^{m−1} ⋯ L_{r}^{m−1}h_{a′}^{−1}T_{w} = 0.

Because h_{a′}^{−1} and q are invertible, we have c_{w} = 0 for every w ∈ 𝔖_{r} by (2.2). Therefore, {v_{a}T_{w} | w ∈ 𝔖_{r}} is a linearly independent set. By (3.1)(d), v_{a}H = v_{a}H. Thus, v_{a}H is a free R-module with basis {v_{a}T_{w} | w ∈ 𝔖_{r}}. □

(3.6) Proposition. Let a ∈ Λ[m, r] and write v_{a}T_{w_{a′}} v_{a} = v_{a}z_{a′} = z_{a}v_{a}, where z_{a} ∈ H(𝔖_{a}) and z_{a′} ∈ H(𝔖_{a′}). Then z_{a} (resp. z_{a′}) is in the center of the algebra H(𝔖_{a}) (resp. H(𝔖_{a′}))

Proof. The existence of z_{a} and z_{a′} follows from (3.1a,c). We need only prove that z_{a′} is in the center of H(𝔖_{a′}). One may prove that z_{a} is in the center of H(𝔖_{a}), similarly.

Let T_{i} ∈ H(𝔖_{a′}). Then i ≠ a_{m} − a_{j} for 1 ⩽ j ⩽ m. By (3.1)(a), v_{a}h_{a′}v_{a}T_{i} = v_{a}h_{a′}T_{(i)w_{a}^{−1}}v_{a} = v_{a}T_{i}h_{a′}v_{a} = T_{(i)w_{a}^{−1}}v_{a}z_{a′} = v_{a}T_{i}z_{a′}. Therefore, v_{a}z_{a′}T_{i} = v_{a}T_{i}z_{a′}. By (3.4), z_{a′}T_{i} = T_{i}z_{a′}. So, z_{a′} is in the center of H(𝔖_{a′}). □

(3.7) Corollary. We have T_{w_{a}}z_{a′} = z_{a}T_{w_{a}}.

Proof. By (1.8), we may write T_{w_{a}}z_{a′} = zT_{w_{a}} for some z ∈ H. Then v_{a}z_{a′} = zv_{a}, and hence z = z_{a} by (3.4). □

(3.8) Corollary. For a ∈ Λ[m, r], v_{a}Hv_{a} = v_{a}z_{a′}H(𝔖_{a′}) = H(𝔖_{a})z_{a}v_{a}.

Proof. Let ι be the anti-automorphism of H defined in (2.4). By (3.1a), we have π̃_{a′}Hπ_{a} = ι(π_{a}Hπ̃_{a′}) = ι(H(𝔖_{a})v_{a}) = π̃_{a′}h_{a′}π_{a}H(𝔖_{a}). So,

v_{a}Hv_{a} = v_{a}Hv_{a} = π_{a}h_{a} (π̃_{a′}Hπ_{a}) h_{a}π̃_{a′}
= π_{a}h_{a}π̃_{a′}h_{a′}π_{a}H(𝔖_{a})h_{a}π̃_{a′}
= v_{a}h_{a′}v_{a}H(𝔖_{a′}) = v_{a}z_{a′}H(𝔖_{a′}). □

(3.9) Proposition. Let a ∈ Λ[m, r]. Then z_{a} (equivalently z_{a′}) is invertible if and only if v_{a}H is a projective right H-module.

Proof. Let J = v_{a}H. By (3.1)(d), we have J^{2} = v_{a}z_{a′}H. By (3.4), the map from H to v_{a}H sending h → v_{a}h for h ∈ H is injective. Therefore, J^{2} = J if and only if z_{a′}H = H, which is equivalent to say that z_{a′} is invertible.

If J is projective, then J = eH for some idempotent e ∈ H, and J = J^{2}. So, z_{a′}, and hence z_{a}, is invertible. Conversely, assume z_{a} is invertible. Let e_{a} = v_{a}h_{a′}z_{a}^{−1}, where h_{a′} = T_{w_{a′}}. Then

e_{a}^{2} = v_{a}h_{a′}z_{a}^{−1}v_{a}h_{a′}z_{a}^{−1} = v_{a}h_{a′}z_{a}^{−1}π_{a}h_{a}π̃_{a′}h_{a′}z_{a}^{−1}
= v_{a}h_{a′}π_{a}z_{a}^{−1}h_{a}π̃_{a′}h_{a′}z_{a}^{−1}
= v_{a}h_{a′}v_{a}z_{a′}^{−1}h_{a′}z_{a}^{−1} = v_{a}h_{a′}z_{a}^{−1} = e_{a}.

Thus e_{a} is an idempotent and J = v_{a}H = e_{a}H. Therefore, v_{a}H is projective. □

(3.10) Proposition. Let a, b ∈ Λ[m, r] and assume that z_{a}, z_{b} are invertible. Then e_{a}He_{b} = 0 unless a = b. Moreover, we have e_{a}He_{a} ≅ H(𝔖_{a}).

Proof. By (2.8) and (3.1)(c), v_{a}Hv_{b} ≠ 0 implies a ≼ b and a′ ≼ b′. So a = b. The second assertion follows from the following equality:

\begin{aligned}e_{a}He_{a} & = v_{a}Hv_{a}h_{a′}z_{a}^{−1} = v_{a}H(𝔖_{a′})h_{a′} by (3.8) \\ & = v_{a}h_{a′}H(𝔖_{a}) ≅ H(𝔖_{a}). □ \end{aligned}

(3.11) Corollary. Assume that z_{a} are invertible for all a ∈ Λ[m, r] and let ε = ∑_{a∈Λ[m,r]} e_{a}. Then εHε ≅ ⊕_{a∈Λ[m,r]}H(𝔖_{a}).

Proof. The result follows from (3.10). □

4. The Poincaré polynomial of W. To generalize the Morita equivalence (1) given in the introduction, we need two more ingredients: First, we want to know when the hypothesis in (3.11) holds. This leads to the introduction of the Poincaré polynomial of the bottom; second, we need to prove that the direct sum ⊕_{a∈Λ[m,r]}v_{a}H is a projective generator for the category H-mod. The latter requires that R is an integral domain.

(4.1) Lemma. For r_{i} = (0, \underbrace{0 ⋯ 0}_{i−1}, r ⋯, r) ∈ Λ[m, r], let z_{r′_{i}} ∈ 𝔖_{r′_{i}} defined in

(3.6). Then z_{r′_{i}} = ∏\begin{matrix}1⩽j⩽m \\ j≠i \end{matrix} ∏_{k=1}^{r}(u_{i}q^{1−k}T_{k,1}T_{1,k} − u_{j}).

Proof. Since w_{r_{i}} = 1, we have h_{r_{i}} = T_{w_{r_{i}}} = 1. It follows from (2.1)(5) and (2.7)(a) that

v_{r_{i}}^{2} = v_{r_{i}} ∏_{\begin{matrix}1⩽j⩽m \\ j≠i \end{matrix}} ∏_{k=1}^{r}(u_{i}q^{1−k}T_{k,1}T_{1,k} − u_{j}).

On the other hand, by (3.6), we have v_{r_{i}}^{2} = v_{r_{i}}z_{r′_{i}}. Now, the result follows from (3.4) immediately. □

(4.2) Definition. For positive integers m, r and i = 1, ⋯, m, let

\begin{matrix}f_{m,r,i} = f_{m,r,i}(u_{1}, ⋯ u_{m}, q) = ∏_{\begin{matrix}1⩽j⩽m \\ i≠j \end{matrix}} ∏_{k=1−r}^{r−1}(u_{i}q^{k} − u_{j}), \\ f_{m,r} = f_{m,r}(u_{1}, ⋯, u_{m}, q) = ∏_{i=1}^{m−1} ∏_{j=i+1}^{m} ∏_{k=1−r}^{r−1}(u_{i}q^{k} − u_{j}). \end{matrix}

We shall see below that the polynomial f_{m,r} can be viewed as the Poincaré polynomial of the bottom C. Let d_{𝔖_{r}} be the Poincaré polynomial of 𝔖_{r}, i.e., d_{𝔖_{r}} = ∑_{w∈𝔖_{r}} q^{l(w)}, then the polynomial d_{W} = f_{m,r}d_{𝔖_{r}} is called the Poincaré polynomial of the complex reflection group W.

(4.3) Proposition. Maintain the notation introduced above. The element z_{r′_{i}} is invertible if and only if f_{m,r,i} is invertible in R.

Proof. By (4.1), we see that the invertiblity of z_{r′_{i}} is equivalent to the invertiblity of h_{ij} = ∏_{k=1}^{r}(u_{i}q^{1−k}T_{k,1}T_{1,k} − u_{j}) for all j ≠ i. By [DJ2, (4.3)], we see that, if R is a field, then h_{ij} in invertible if and only if f_{m,r,i} is invertible. The general case follows by an argument similar to the one for [DJ2, (4.5)]. □

We need some preparation in order to get the main result of this paper. As before, we assume that R is a commutative ring (with 1). For any a ∈ Λ[m, r] and x_{1}, ⋯, x_{m−1} ∈ R, recall the element defined after (2.6).

π(a) = π(a; x_{1}, ⋯, x_{m−1}) = π_{a_{1}}(x_{1}) ⋯ π_{a_{m−1}}(x_{m−1}).

Write π(a) as a polynomial in L_{i}. The degree of this polynomial is denoted deg_{i}(π(a)).

(4.4) Lemma. Maintain the notation above and assume x_{1}, ⋯, x_{m} are a permutation of u_{1}, ⋯, u_{m}. For a ∈ Λ[m, r], the right ideal π(a)H is spanned by B_{a} = {π(a)L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} | deg_{j} π(a) + c_{j} ⩽ m − 1, ∀j ⩽ r, w ∈ 𝔖_{r}}.

Proof. Let M_{a} be the R-submodule spanned by B_{a}. Then M_{a} ⊆ π(a)H. Since π(a) ∈ M_{a}, it suffices to prove that M_{a} is a right H-module, or equivalently M_{a}T_{0} ⊆ M_{a}.

For π(a)L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{w} ∈ B_{a}. We write w = s_{i,1}y with y ∈ 𝔖_{{2,··· ,r}} (cf. (1.3)). Since M_{a} is a right H-module and T_{0}T_{y} = T_{y}T_{0}, we only need to prove π(a)L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}T_{i,1}T_{0} ∈ M_{a}, which is equivalent to

(4.5)  π(a)L_{1}^{c_{1}} ⋯ L_{r}^{c_{r}}L_{i} ∈ M_{a} for every 1 ⩽ i ⩽ r.

If a_{m−1} = 0, then π(a) = 1 and π(a)H = H. By (2.2), (4.5) holds in this case. We now assume a_{m−1} ≠ 0. Apply induction on i. The last relation in (2.1) implies (4.5) for i = 1. (If deg_{1}(π(a)) + c_{1} + 1 = m, then π(a) contains part of the product in (2.1)(5) involving parameters x_{i}, ⋯, x_{m−1} Write L_{1}^{c_{1}+1} = ∏_{k}(L_{1} − u_{i_{k}}) + ∑_{j⩽c_{1}} α_{j}L_{1}^{j}. where {u_{i_{j}}} = {x_{1}, ⋯, x_{i−1}, x_{m}}. Now, (2.1)(5) implies (4.5) in this case.)

We assume now that i > 1 and (4.5) holds for all L_{j} with j < i. The case for deg_{i} π(a) + c_{i} < m − 1 is trivial. Suppose deg_{i} π(a) + c_{i} = m − 1. Let k be the integer with a_{k−1} < i ⩽ a_{k}. Then, deg_{i} π(a) = m − k and π_{a_{j}}(x_{j})T_{i,1} = T_{i,1}π_{a_{j}}(x_{j}) for all j ⩾ k. On the other hand, by (2.5)(d) and induction on i, we have for some c

(4.6)  L_{i}^{l} − q^{c}T_{i,1}L_{1}^{l}T_{1,i} ∈ U_{i,l}

where U_{i,l} is the free R-submodule spanned by {L_{1}^{d_{1}} ⋯ L_{i}^{d_{i}}T_{w} | 0 ⩽ d_{j} < l, ∀j ≠ i, w ∈ 𝔖_{i}}. In particular, L_{i}^{c_{i}+1} = q^{c}T_{i,1}L_{1}^{c_{i}+1}T_{1,i} + h for some c and h ∈ U_{i,c_{i}+1}. So we have

π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}L_{i}^{c_{i}}L_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}}L_{i}
=π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}L_{i}^{c_{i}+1}L_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}}
=π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}(q^{c}T_{i,1}L_{1}^{c_{i}+1}T_{1,i} + h)L_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}}
=q^{c}π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}T_{i,1}L_{1}^{c_{i}+1}T_{1,i}L_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}}
+ π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}hL_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}}
=X_{1} + X_{2}

We now prove that each of the last two terms X_{i} above is in M_{a}. For h = L_{1}^{d_{1}} ⋯ L_{i}^{d_{i}}T_{w} ∈ U_{i,c_{i}+1}, we have T_{w}L_{j} = L_{j}T_{w} for all j ⩾ i + 1 and d_{i} < c_{i} + 1, and so, deg_{i} π(a) + d_{i} < deg_{i} π(a) + c_{i} + 1 = m. On the other hand, we have, by definition, deg_{j} π(a) + c_{j} ⩽ m − 1 for all 1 ⩽ j ⩽ m. Therefore, by inductive hypothesis,

π(a)L_{1}^{c_{1}} ⋯ L_{i−1}^{c_{i−1}}(L_{1}^{d_{1}} ⋯ L_{i}^{d_{i}}T_{w})L_{i+1}^{c_{i+1}} ⋯ L_{r}^{c_{r}} = π(a) ∏_{j<i} L_{j}^{c_{j}+d_{j}}L_{i}^{d_{i}} ∏_{j>i} L_{j}^{c_{j}}T_{w} ∈ M_{a}.

This proves X_{2} ∈ M_{a}.

Since i ⩽ a_{k}, we have

\begin{aligned}X_{1} & = ∏_{j=1}^{i−1} L_{j}^{c_{j}} ∏_{j=1}^{k−1} π_{a_{j}}(x_{j}) (T_{i,1} ∏_{j=k}^{m−1} π_{a_{j}}(x_{j})L_{1}^{c_{i}+1}T_{i,1}) ∏_{j=i+1}^{r} L_{j}^{c_{j}} \\ & = X_{11}(X_{12})X_{13}. \end{aligned}

where X_{11} (resp. X_{13}) denotes the first two products (resp. last product) and X_{12} denotes the element in the parenthesis. As in the argument for i = 1, L_{1}^{c_{i}+1} ∏_{j=k}^{m−1} π_{a_{j}}(x_{j}) is an R-linear combinations of L_{1}^{l} ∏_{j=k}^{m−1} π_{a_{j}}(x_{j}) with l < c_{i}+1, while, by (4.6), L_{i}^{l} − T_{i,1}L_{1}^{l}T_{1,i} = h ∈ U_{i,l}. Thus, T_{i,1}L_{1}^{l} ∏_{j=k}^{m−1} π_{a_{j}}(x_{j})T_{i,1} = ∏_{j=k}^{m−1} π_{a_{j}}(x_{j})L_{i}^{l} + ∏_{j=k}^{m−1} π_{a_{j}}(x_{j})h. Hence X_{1} can be expressed as a linear combination of X_{11}(∏_{j=k}^{m−1} π_{a_{j}}(x_{j})L_{i}^{l})X_{13} and X_{11}(∏_{j=k}^{m−1} π_{a_{j}}(x_{j})h)X_{13}, where l ⩽ c_{i} and h ∈ U_{i,l}. Since l ⩽ c_{i}, we have clearly X_{11}(∏_{j=k}^{m−1} π_{a_{j}}(x_{j})L_{i}^{l})X_{13} ∈ M_{a}. By inductive hypothesis, we have X_{11}(π_{a_{j}}(x_{j})h)X_{13} ∈ M_{a}, too. Therefore, X_{1} ∈ M_{a}, proving (4.5). □

For a = [a_{i}] ∈ Λ[m, r], let a_{i} be defined as in (1.9). Recall h_{a} = T_{w_{a}}.

(4.7) Proposition. For a ∈ Λ[m, r], write a_{⊣} = [b_{0}, ⋯, b_{m}] as in (1.10). Let V_{i} = π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}H.

Then

(a) V_{1} = v_{a_{⊣}}H and V_{m} = v_{a_{m}}H.

(b) The set B_{i} = {π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{r−b_{i−1}}^{c}T_{w} | c ⩽ m − i, w ∈ 𝔖_{r}} is a basis of V_{i}. In particular, the rank of V_{i} is (m − i + 1)r!.

(c) V_{i+1} is a pure R-submodule of V_{i}.

Proof. We first note that b_{i} = a_{i}, 1 ⩽ i ⩽ k − 1, b_{i} = r − 1, k ⩽ i ⩽ m, where k is the minimal index with a_{k} = r. Note also from (1.9) and (1.2) that

\begin{aligned}a′_{1} & = [0, ⋯, 0, r − a_{k−1} − 1, ⋯, r − a_{1} − 1, r] \\ (a_{⊣})′ & = [0, ⋯, 0, r − a_{k−1} − 1, ⋯, r − a_{1} − 1, r − 1]. \end{aligned}

Thus, π̃_{a′_{1}} = π̃_{(a_{⊣})′} by definition (2.6), and V_{1} = v_{a_{⊣}}H. Because h_{a_{⊣}}T_{r,r−b_{m−1}} = h_{a_{m}} (see (1.10)) and π_{a_{m}} = π_{a_{⊣}}, we have V_{m} = v_{a_{m}}H, proving (a).

For (b), we first prove that B_{i} spans V_{i}. Let V′_{i} be the submodule spanned by B_{i}. Applying (4.4) to π̃_{a′_{i}}H, it suffices to prove

(4.8)  π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{1}^{c_{1}} ⋯ L_{r−b_{i−1}}^{c_{r−b_{i−1}}} ⋯ L_{r}^{c_{r}} ∈ V′_{i},

where c_{j} ⩽ m−1−deg_{j} π̃_{a′_{i}}, ∀j. Since deg_{r−b_{i−1}} π̃_{a′_{i}} = i−1, we have, in particular, c_{r−b_{i−1}} ⩽ m − i.

We first look at the elements π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{j} with j ≠ r − b_{i−1}. Suppose j < r − b_{i−1} and write T_{r,r−b_{i−1}}π̃_{a′_{i}} = T_{r,r−b_{i−1}}π̃_{(a_{⊣})′}(L_{r−b_{1}} − u_{1}) ⋯ (L_{r−b_{i−1}} − u_{i−1}) = π̃_{(a_{⊣})′}h_{i}, where h_{i} = T_{r,r−b_{1}}(L_{r−b_{1}} − u_{1}) ⋯ T_{r−b_{i−2},r−b_{i−1}}(L_{r−b_{i−1}} − u_{i−1}). Then L_{j}h_{i} = h_{i}L_{j} and h_{i}T_{l} = T_{l}h_{i} for all s_{l} ∈ 𝔖_{j}. Thus, by (3.1)(b),

(4.9)  π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{j} = v_{a_{⊣}}L_{j}h_{i} ∈ π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}H(𝔖_{j}).

Suppose now j ⩾ r − b_{i−1} + 1. Choose l such that r − b_{l} + 1 ⩽ j ⩽ r − b_{l−1}. Then l ⩽ i−1 and l is the minimal index having property b_{l} = b_{i−1}. Using definition (2.6), we have π̃_{a′_{i}}(L_{r−b_{l}+1} − u_{l}) = π̃_{c′}, where c = [0, b_{1}, ⋯, b_{l−1}, b_{l} − 1, b_{l+1}, ⋯, b_{m}] + 1_{i} ∈ Λ[m, r] (see (1.9)). Since π_{a_{⊣}} = π_{a_{m}} and a_{m} ⋠ c (as l < i), we have, by (2.8) that π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}(L_{r−b_{l}+1} − u_{l}) = 0. Therefore, for some c,

(4.10)  π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{j} = u_{l}q^{c}π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}T_{j,r−b_{l}+1}T_{r−b_{l}+1,j}.

Now, noting (2.5)(a), we see that (4.8) follows from (4.9) and (4.10), since the elements in H(𝔖_{j}) occurring in (4.8) and the element T_{j,r−b_{l}+1}T_{r−b_{l}+1,j} occurring in (4.9) commute with L_{j+1} and L_{r−b_{i−1}}.

To see the linear independence, We apply induction on i. For i = 1, suppose

∑_{\begin{matrix}0⩽c⩽m−1 \\ w∈𝔖_{r} \end{matrix}} f_{c,w}v_{a_{⊣}}L_{r}^{c}T_{w} = 0.

Since pr_{h}(v_{a_{⊣}}) = q^{d} ∏_{j=1}^{r−1} L_{j}^{m−1}h_{(a_{⊣})′}^{−1} for some d ∈ ℤ depending on a_{⊣} (see (3.5)), we have

pr_{c}( ∑_{\begin{matrix}0⩽c⩽m−1 \\ w∈𝔖_{r} \end{matrix}} f_{c,w}v_{a_{⊣}}L_{r}^{c}T_{w}) = q^{d} ∑_{w∈𝔖_{r}} f_{c,w}L_{1}^{m−1} ⋯ L_{r−1}^{m−1}L_{r}^{c}h_{(a_{⊣})′}^{−1}T_{w} = 0,

where c = (m − 1, ⋯, m − 1, c). Thus, f_{c,w} = 0 for all c and w, since both q and h_{(a_{⊣})′}^{−1} are invertible. Therefore, B_{1} is a basis of V_{1}.

We assume now that B_{i} is a linearly independent set. We hope to prove that the set B_{i+1} is linearly independent, too. Using induction on l, we can easily prove

T_{a,b}L_{b}^{l} = q^{a−b}L_{a}^{l}T_{b,a}^{−1} + terms involving L_{a}^{c_{a}} ⋯ L_{b}^{c_{b}}T_{w} for a ⩾ b

where w ∈ 𝔖_{{b,b+1,··· ,a}} and all c_{i} < l. Using this and noting π̃_{a′_{i+1}} = π̃′_{a_{i}}(L_{r−b_{i}} − u_{i}), we have

(4.11)

π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i}}π̃_{a′_{i+1}}L_{r−b_{i}}^{c}T_{w}
=q^{a_{i}−a_{i−1}}π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{r−b_{i−1}}^{c+1}T_{r−b_{i},r−b_{i−1}}^{−1}T_{w}+
+ terms involving π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{r−b_{i−1}}^{l}T_{z},

where l < c+1 and z ∈ 𝔖_{r}. By induction, B_{i} is a basis of V_{i}. Because T_{r−b_{i},r−b_{i−1}} is invertible, the set {π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}L_{r−b_{i−1}}^{c}T_{r−b_{i},r−b_{i−1}}^{−1}T_{w} | c ⩽ m − i, w ∈ 𝔖_{r}} is linearly independent in V_{i}. Therefore, the condition l < c + 1 in (4.11) implies that B_{i+1} is linear independent. This completes the proof of (4.7)(b).

We now prove the purity in (c). Since

π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i}}π̃_{a′_{i+1}} = π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}}T_{r−b_{i−1},r−b_{i}}(L_{r−b_{i}} − u_{i}),

V_{i+1} is a submodule of V_{i}. Write h = ∑ f_{c,w}π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i}}π̃_{a′_{i+1}}L_{r−b_{i}}^{c}T_{w} for h ∈ V_{i+1}. To show that V_{i+1} is a pure R-submodule, we need to show that x ∈ R divides any of f_{c,w} if h ∈ xV_{i}. We prove it by induction on c_{0}, the highest degree of L_{r−b_{i}} in the expression of h. Since B_{i} is a basis of V_{i} and q is invertible, x divides f_{c_{0},w} by (4.11). Thus, h − ∑ f_{w,c_{0}}π_{b}h_{b}T_{r,r−b_{i}}π̃_{a′_{i+1}}L_{r−b_{i}}^{c}T_{w} ∈ xV_{i}. Now, the result follows from induction. □

From here onwards, we assume that R is an integral domain.

(4.12) Lemma. Keep the notation above. We have short exact sequence

0 → V_{i+1} → V_{i} → v_{a_{i}}H → 0.

Proof. Let

h_{i} = (L_{b_{i}+1} − u_{i+1})T_{b_{i}+1,b_{i+1}+1} ⋯ (L_{b_{k−1}+1} − u_{k})T_{b_{k−1}+1,r} ∏_{l=k+1}^{m}(L_{r} − u_{l}),

where, as usual, k is the minimal index such that a_{k} = r. Then, by (2.5)(a)-(c), we have h_{i}π_{a_{⊣}} = π_{a_{i}}T_{b_{i}+1,r}. Define ψ_{i} : V_{i} → v_{a_{i}}H by setting ψ_{i}(h) = h_{i}h, h ∈ V_{i}. Note that ψ_{i} is well-defined by (1.10), and clearly, ψ_{i} is surjective. Because a_{i} ≻ a_{i+1}, we have V_{i+1} ⊆ ker ψ_{i} by (2.8) and (3.1)(c). By (3.4) and (4.7)(b), we have dim V_{i} = dim V_{i+1} + dim v_{a_{i}}H, forcing V_{i+1} = ker ψ_{i} over the quotient field F of R. However, by (4.7)(c), V_{i+1} is a pure R-submodule of V_{i}. Thus V_{i+1} = ker ψ_{i}, proving the required short exact sequence. □

(4.13) Lemma. Let a = [a_{i}] ∈ Λ[m, r] and write a_{⊣} = [b_{i}]. Assume k is the minimal index with a_{k} = r. Define

Ṽ_{i} = π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}} ∏_{j=i+1}^{m} π_{r−b_{i−1}}(u_{j})H, for i = k, ⋯, m − 1.

If f_{m,r} is a unit in the integral domain R, then we have V_{i} ≅ V_{i+1}⊕Ṽ_{i} and Ṽ_{i} ≅ v_{a_{i}}H for all i = k, ⋯, m − 1.

Proof. Let ψ_{i} : V_{i} → v_{a_{i}}H, i ⩾ k, be the H-module homomorphism defined in the proof of (4.12). Then, ψ_{i}(Ṽ_{i}) = v_{a_{i}} ∏_{j=i+1}^{m} π_{r−b_{i−1}}(u_{j})H. By (3.1)(b), we have v_{a_{i}}L_{1} = u_{i}v_{a_{i}} for all i ⩾ k. Also, for j < r − a_{k−1}, s_{j} ∈ 𝔖_{a′_{k}}. So v_{a_{i}}T_{j} = T_{j′}v_{a_{i}} (see (3.1)(a)), and hence, v_{a_{k}}L_{l} = u_{k}q^{1−l}v_{a_{k}}T_{l,1}T_{1,l} for all l = 1, ⋯, r − b_{k−1}, noting a_{k−1} = b_{k−1}. Therefore, we have

ψ_{i}(Ṽ_{i}) = v_{a_{i}} ∏_{j=i+1}^{m} π_{r−b_{i−1}}(u_{j})H = v_{a_{i}} ∏_{j=i+1}^{m} ∏_{l=1}^{r−b_{i−1}}(q^{1−l}u_{i}T_{l,1}T_{1,l} − u_{j})H.

Since ∏_{j=i+1}^{m} ∏_{l=1}^{r−b_{i−1}}(q^{1−l}u_{i}T_{l,1}T_{1,l}−u_{j}) is a factor of z_{r′_{i}} (see (4.1)), and f_{m,r} ∈ R is a unit, ∏_{l=1}^{r−b_{i−1}}(q^{1−l}u_{i}T_{l,1}T_{1,l} − u_{j}) is invertible by (4.3). Therefore, ψ_{i}(Ṽ_{i}) = v_{a_{i}}H. Thus, by (4.12), the rows of the following commutative diagram

0 → V_{i+1} → Ṽ_{i} + V_{i+1} →^{ψ_{i}} v_{a_{i}}H → 0
∥  ↓  ∥
0 → V_{i+1} →  V_{i}  →^{ψ_{i}} v_{a_{i}}H → 0.

are exact. So short-five lemma implies Ṽ_{i} + V_{i+1} = V_{i}. On the other hand, consider the element x = π̃_{a′_{i}} ∏_{j=i+1}^{m} π_{r−b_{i−1}}(u_{j}) and the multiplication xL_{r−b_{i−1}}. Clearly, x has a factor (L_{1}−u_{1}) ⋯ (L_{1}−u_{i−1})(L_{1}−u_{i+1}) ⋯ (L_{1}−u_{m}). So, if i > k, then r− b_{i−1} = 1, and hence, xL_{1} = u_{i}x, while, for i = k, we have xT_{r−b_{k−1},1} = T_{r−b_{k−1},1}x and consequently, xL_{r−b_{k−1}} = q^{c}u_{i}xT_{r−b_{k−1},1}T_{1,r−b_{k−1}}. Therefore, by (4.7)(b), Ṽ_{i} is spanned by {π_{a_{⊣}}h_{a_{⊣}}T_{r,r−b_{i−1}}π̃_{a′_{i}} ∏_{j=i+1}^{m} π_{r−b_{i−1}}(u_{j})T_{w} | w ∈ 𝔖_{r}}, which is linearly independent, since its image under the homomorphism ψ_{i} is linearly independent. Hence, Ṽ_{i} is R-free of rank r!, forcing the sum Ṽ_{i} + V_{i+1} is a direct sum, i.e., V_{i} = Ṽ_{i} ⊕ V_{i+1}, and consequently, Ṽ_{i} ≅ v_{a_{i}}H. □

We now prove the main result of the paper. The case when H is the Hecke algebra of type B was first obtained in [DJ2].

(4.14) Theorem. Let H = H_{m}^{r} be the Ariki-Koike algebra over a integral domain R. Assume f_{m,r} ∈ R is a unit.

(a) For every a ∈ Λ[m, r_{1}] with r_{1} ⩽ r, the right ideal v_{a}H is projective.

(b) For any given r_{1} ⩽ r, each projective indecomposable H-module is isomorphic to a direct summand of v_{a}H for some a ∈ Λ[m, r_{1}].

(c) The categories of H-modules and ⊕_{λ∈Λ(m,r)}H(𝔖_{λ})-modules are Morita equivalent.

Proof. For r_{1} ⩽ r, H is free over H_{m}^{r_{1}}. So the functor −⊗_{H_{m}^{r_{1}}} H is exact. Therefore, it suffices to prove (a) for r_{1} = r.

We apply induction on r. For r = 1, we have a = r_{i} for some i. By the invertibility of f_{m,r}, (4.3) and (3.8), we see that v_{a}H is projective. Assume now that r > 1 and the result holds for r − 1. We now prove the projectivity of v_{a}H by downward induction on the partial ordering ≽. If a = r_{1} = [0, r, ⋯, r], the maximal element of Λ[m, r], then v_{a}H is projective by (4.3) and (3.8). Suppose now that r_{1} ≻ a and that v_{b}H is projective for all b ≻ a. Let a_{⊣} and a_{i}, 1 ⩽ i ⩽ m be defined as in (1.2) and (1.9), and k the minimal index with a_{k} = r. Then k ⩾ 2 and a_{k} = a. By induction, v_{a_{j}}H, 1 ⩽ j ⩽ k − 1, are projective, and v_{a_{⊣}}H is projective. Using the short exact sequence in (4.12), we have v_{a_{⊣}}H = ⊕_{j=1}^{k−1}v_{a_{j}}H ⊕ V_{k} and hence, V_{k} is projective. By (4.13), we have V_{k} ≅ V_{k+1} ⊕ v_{a_{k}}H, and consequently, v_{a}H = v_{a_{k}}H is projective, proving (a). Note from (4.7) and (4.12-3) that we have for any r > 0 and a ∈ Λ[m, r]

(4.15)  v_{a_{⊣}}H = ⊕_{i=1}^{m}v_{a_{i}}H.

To see (b), we apply the tensor functor − ⊗_{H_{m}^{r_{1}}} H to (4.15) and obtain, for 1 ⩽ r_{1} ⩽ r and a ∈ Λ[m, r_{1}]

v_{a_{⊣}}H = ⊕_{j=1}^{m}v_{a_{j}}H,

from which (b) follows.

Putting r_{1} = r in both (a) and (b), we see that the H-module ⊕_{a∈Λ[m,r]}v_{a}H is a projective generator. By standard results (see, e.g., [CR, (3.54)]) the categories of H-modules and εHε-modules are Morita equivalent (cf. (3.11)). □

5. Some applications. We first recall the notion of multi-compositions and multipartitions of r. By definition, λ = (λ^{(1)}, ⋯, λ^{(m)}) is called an m-composition (resp. m-partition) of r if λ^{(i)} is a composition (resp. partition) for every i with 1 ⩽ i ⩽ m and |λ| = ∑_{i=1}^{m} |λ^{(i)}| = r. Recall that Λ(n, r) (resp. Λ(n, r)^{+}) is the set of all compositions (resp. partitions) of r with n parts, and let, for m > 0 and μ = (μ_{1}, ⋯, μ_{m}) ∈ Λ(m, r),

\begin{matrix}Λ(n, μ) = Λ(n_{r}, μ_{1}) × ... × Λ(n_{r}, μ_{m−1}) × Λ(n, μ_{m}) \\ Λ_{m}(n, r) = ⋃_{μ∈Λ(m,r)} Λ(n, μ) \end{matrix}

where n_{r} is the maximum of n and r. Define Λ_{m}(n, r)^{+} similarly. The set Λ_{m}(n, r) can be identified with Λ(N, r) by concatenation, where N = (m − 1)n_{r} + n. To distinguish them, for λ ∈ Λ_{m}(n, r), let λ̄ denote the corresponding element in Λ(N, r).

For λ ∈ Λ_{m}(n, μ), let a = [a_{0}, ⋯, a_{m}] be the cumulative norm sequence of λ, denoted cns(λ), where a_{0} = 0 and a_{i} = μ_{1} + ⋯ + μ_{i} for 1 ⩽ i ⩽ m. Let e be the minimal integer l such that

1 + q + q^{2} + ⋯ + q^{l−1} = 0.

If such an integer l does not exist, then set e = ∞. A partition λ is called eregular if λ has no non-zero part occurring e or more times. An m-partition λ = (λ^{(1)}, ⋯, λ^{(m)}) is e-regular if each λ^{(i)}, 1 ⩽ i ⩽ m, is e-regular.

Let H_{F} be the Hecke algebra of type A_{r−1} over a field F, in which q is a primitive e-th root of 1. Let S^{λ} be the Specht module with respect to λ. Then, by [DJ1], S^{λ} has simple head D^{λ}, if λ is e-regular. (Note, for e > r, D^{λ} = S^{λ}.) For a multipartition λ = (λ^{(1)}, ⋯, λ^{(m)}) of r with cns(λ) = a, let S^{λ^{(1)}} ⋯ S^{λ^{(m)}} (≅ ⊗_{i=1}^{m}S^{λ^{(i)}}) be the corresponding Specht module for H_{F} (𝔖_{a}), and let S^{λ} be the right ideal e_{a}S^{λ^{(1)}} ⋯ S^{λ^{(m)}}H_{F} of H_{F}. By (4.14)(c), we know that the H-module S^{λ} and εHε-module S^{λ}ε have isomorphic submodule lattices. Since S^{λ}ε ≅ S^{λ^{(1)}} ⋯ S^{λ^{(m)}} and the latter has simple head if λ is e-regular, so is S^{λ}. Let D^{λ} denote the simple head of S^{λ}. The following is an immediate consequence of (4.14)(c) (see [DJ2, (5.3)] for the case m = 2).

(5.1) Theorem. Let F be a field and f_{m,r} ≠ 0 in F. Then the set

{D^{λ} | λ ∈ Λ_{m}(r, r)^{+} e-regular}

is a complete set of simple H_{F} -modules.

The following result has been proved in [Ari]. Recall the Poincaré polynomial d_{W} introduced in (4.2)

(5.2) Theorem. Let H_{F} be the Ariki-Koike algebra over a field F. Then H_{F} is semisimple if and only if d_{W} ≠ 0.

Proof. Note that d_{W} ≠ 0 is equivalent to f_{m,r} ≠ 0 and e > r. By (4.3), we have f_{m,r} ≠ 0 if and only if f_{m,r,i} ≠ 0 for all i. If f_{m,r} ≠ 0, then the categories of H-modules and εHε-modules are Morita equivalent. By (3.10) and [DJ1, 4.3], εHε is semisimple if and only if e > r, proving the “if” part. Conversely, it suffices to look at the case f_{m,r} = 0. Then f_{m,i,r} = 0 for some i, and z_{r′_{i}} is not invertible. Therefore, v_{r′_{i}}H is not an idempotent ideal by (3.9). Thus, H is not semisimple. □

We finally look at a Morita theorem between q-Schur^{m} algebras and q-Schur algebras (i.e., the q-Schur^{1} algebras). For any λ ∈ Λ_{m}(n, r) with cns(λ) = a, let x_{λ} = π_{a}xλ̄.

(5.3) Lemma. Let R be an integral domain and assume that f_{m,r} ∈ R is a unit. For any λ ∈ Λ_{m}(n, r) with cns(λ) = a, we have e_{a}x_{λ}H = e_{a}xλ̄H.

Proof. Since π_{a}H ⊆ H, we have e_{a}x_{λ}H ⊆ e_{a}xλ̄H. By (1.8), (3.1)(a) and (3.6)- (3.7), we have e_{a}xλ̄ = v_{a}h_{a′}z_{a}^{−1}xλ̄ = v_{a}h_{a′}xλ̄z_{a}^{−1} = v_{a}hh_{a′}z_{a}^{−1} = xλ̄v_{a}h_{a′}z_{a}^{−1} = xλ̄e_{a}, where h ∈ H(𝔖_{a′}) given by hh_{a′} = h_{a′}xλ̄. Thus, e_{a}x_{λ}H = xλ̄e_{a}π_{a}H = xλ̄v_{a}h_{a′}π_{a}H ⊇ xλ̄v_{a}h_{a′}v_{a}H = xλ̄v_{a}z_{a′}H = xλ̄e_{a}H, proving (5.3). □

Some special case for m = 2 of the second part of the following result has been discussed by Gruber and Hiss [GH] in the context of representations finite groups of Lie type.

(5.4) Corollary. Let H be the Ariki-Koike algebra over an integral domain R, in which f_{m,r} is a unit. Let a = cns(λ) for λ ∈ Λ_{m}(n, r). Then

End_{H} (⊕_{λ∈Λ_{m}(n,r)}e_{a}x_{λ}H) = End_{H} (⊕_{λ∈Λ_{m}(n,r)}e_{a}xλ̄H)
≅ ⊕_{μ∈Λ(m,r)}End_{H(𝔖_{μ})} (⊕_{λ∈Λ(n,μ)}xλ̄H(𝔖_{μ})) .

Proof. The first equality follows from (5.3). Using standard results (see [AF, (21.2)] or [DPS, (0.1)]), we know that

End_{H} (⊕_{λ∈Λ_{m}(n,r)}e_{a}xλ̄H) ≅ End_{εHε} (⊕_{λ∈Λ_{m}(n,r)}e_{a}xλ̄Hε) .

For λ = (λ^{(1)}, ⋯, λ^{(m)}) ∈ Λ_{m}(n, r), write cns(λ) = a = [a_{i}]. Then we have xλ̄H(𝔖_{a}) ≅ x_{λ^{(1)}}H(𝔖_{a_{1}}) ⊗_{R} ⋯ ⊗_{R} x_{λ^{(m)}}H(𝔖_{a_{m}−a_{m−1}}), and by the proof of (5.3), e_{a}xλ̄ = xλ̄e_{a}. Therefore, e_{a}xλ̄Hε = xλ̄e_{a}H(𝔖_{a}) ≅ xλ̄H(𝔖_{a}). By (3.10), we have

End_{εHε} (⊕_{λ∈Λ_{m}(n,r)}e_{a}xλ̄Hε) ≅ ⊕_{a∈Λ[m,r]}End_{e_{a}He_{a}} (⊕_{λ∈Λ(n,Θ(a))}xλ̄e_{a}He_{a})
≅ ⊕_{μ∈Λ(m,r)}End_{H(𝔖_{μ})} (⊕_{λ∈Λ(n,μ)}xλ̄H(𝔖_{μ})) .

Here Θ is defined in (1.1). □

The endomorphism algebra S_{R}^{m}(n, r) = End_{H} (⊕_{λ∈Λ_{m}(n,r)}x_{λ}H) is called a q-Schur^{m} algebra in [DR2], which is a cyclotomic q-Schur algebra in the sense of [DJM]. It is proved in [DR2] that a Borel type subalgebra of a q-Schur^{m} algebra is isomorphic to a Borel subalgebra of a q-Schur algebra. Now, with the invertibility of the polynomial f_{m,r}, we establish below a Morita equivalence between the module categories of a q-Schur^{m} algebra and a direct sum of tensor products of certain q-Schur algebras.

(5.5) Theorem. Let H = H_{O} be the Ariki-Koike algebra over a discrete valuation ring O. Then the q-Schur^{m} algebra S_{O}^{m}(n, r) is Morita equivalent to the algebra

⨁_{μ∈Λ(m,r)} S_{O}^{1}(n_{r}, μ_{1}) ⊗ ... ⊗ S_{O}^{1}(n_{r}, μ_{m−1}) ⊗ S_{O}^{1}(n, μ_{m}).

Proof. We first note the isomorphism

(5.6) ⨁_{μ} End_{H_{O}(𝔖_{μ})} (⊕_{λ∈Λ(n,μ)}xλ̄H_{O}(𝔖_{μ})) ≅ ⨁_{μ} ⊗_{i=1}^{m−1}S_{O}^{1}(n_{r}, μ_{i}) ⊗ S_{O}^{1}(n, μ_{m}),

where μ ∈ Λ(m, r). The q-Schur algebra of bidegree (n, r) is a (integral) quasihereditary algebra, whose simple modules are parametrized by Λ(n, r)^{+} (see [DS, §2]). Therefore, by [Wi, (1.3)] (or a cellular basis argument [GL],[DR1]), the algebra

⨁_{μ∈Λ(m,r)} ⊗_{i=1}^{m−1}S_{O}^{1}(n_{r}, μ_{i}) ⊗ S_{O}^{1}(n, μ_{m})

is a quasi-hereditary algebra, whose simple modules are indexed by Λ_{m}(n, r)^{+}. Thus, PIMs are indexed by Λ_{m}(n, r)^{+}, too. Using Fitting Lemma, the nonisomorphic indecomposable direct summands (The existence of these modules follows from Heller’s result, [CR, (30.18iii)].) of ⊕_{λ∈Λ_{m}(n,r)}xλ̄e_{a}H_{O}e_{a} are indexed by Λ_{m}(n, r)^{+}. Therefore, the non-isomorphic indecomposable direct summands of ⊕_{λ∈Λ_{m}(n,r)}e_{a}x_{λ}H_{O} are indexed by Λ_{m}(n, r)^{+} by (5.4). Since e_{a}xλ̄ = xλ̄e_{a} and e_{a}^{2} = e_{a}, we have e_{a}x_{λ}H_{O} ⊕ (1 − e_{a})x_{λ}H_{O} = x_{λ}H_{O}. It follows that every direct summand of e_{a}x_{λ}H_{O} is a direct summand of x_{λ}H_{O}. Now, by the quasi-heredity of the q-Schur^{m} algebra (see [DR2, (5.10)]), the non-isomorphic indecomposable direct summands of ⊕_{λ∈Λ_{m}(n,r)}x_{λ}H_{O} are indexed by Λ_{m}(n, r)^{+}. Therefore, both ⊕_{λ∈Λ_{m}(n,r)}x_{λ}H_{O} and ⊕_{λ∈Λ_{m}(n,r)}e_{a}x_{λ}H_{O} have the same non-isomorphic indecomposable direct summands. Consequently, the q-Schur^{m} algebra S_{O}^{m}(n, r) is Morita equivalent to End_{H_{O}} (⊕_{λ∈Λ_{m}(n,r)}e_{a}x_{λ}H_{O}). Now, the required Morita equivalence follows from (5.4) and (5.6). □

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