MORITA EQUIVALENCES OF CYCLOTOMIC HECKE ALGEBRAS OF TYPE G(r, p, n)

JUN HU AND ANDREW MATHAS

Abstract. We prove a Morita reduction theorem for the cyclotomic Hecke algebras ℋ_{r,p,n}(q, Q) of type G(r, p, n). As a consequence, we show that computing the decomposition numbers of ℋ_{r,p,n}(Q) reduces to computing the p′- splittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras ℋ_{r′,p′,n′}(Q′), where 1 ≤ r′ ≤ r, 1 ≤ n′ ≤ n, p′ | p and where the parameters Q′ are contained in a single (ϵ, q)-orbit and ϵ is a primitive p′th root of unity.

1. Introduction

Motivated by “generic features” of the representation theory of finite reductive groups Broué and Malle [3] attached a cyclotomic Hecke algebra to each complex reflection group. These algebras have many good properties and, conjecturally, they arise as the endomorphism algebras of Deligne–Lusztig representations.

This paper is concerned with the cyclotomic Hecke algebras of type G(r, p, n) with p > 1. These algebras were first considered by Broué and Malle [3] and by Ariki [1] in the semisimple case. These algebras have been studied extensively in the non–semisimple case, notably by the first author [9–12] and by Genet and Jacon [19]. In particular, the simple modules of these algebras have been classified over any field of characteristic coprime to p.

In the case p = 1 the cyclotomic Hecke algebras of type G(r, 1, n) are known as the Ariki–Koike algebras. These algebras are much better understood; see [17] and the references therein. The highlight of this theory is Ariki’s celebrated theorem which says that the decomposition numbers of these algebras in characteristic zero can be computed using the canonical bases of the higher level Fock spaces of the quantized affine special linear groups. Another fundamental result for the Ariki–Koike algebras is the Morita equivalence theorem of Dipper and the second author [6] which says that, up to Morita equivalence, these algebras are determined by the q–orbits of their parameters.

The first main result in this paper gives an analogue of the Dipper–Mathas Morita equivalence theorem for the Hecke algebras of type G(r, p, n). To state this result explicitly, fix positive integers r, p and n with r = pt, for some integer t, and let K be an algebraically closed field of characteristic coprime to p. Fix parameters q, Q_{1}, …, Q_{t} ∈ K^{×} and let Q = (Q_{1}, …, Q_{t}). Let ℋ_{r,n}(Q) be the Ariki–Koike algebra and let ℋ_{r,p,n}(Q) be the Hecke algebra of type G(r, p, n) with parameters q and Q. The algebra ℋ_{r,n}(Q) is equipped with an automorphism σ of order p and ℋ_{r,p,n}(Q) is the fixed point subalgebra of ℋ_{r,n}(Q) under σ. There is a second automorphism τ on ℋ_{r,n} which fixes ℋ_{r,p,n} setwise. For the precise definitions see section 2.

Fix a primitive pth root of unity ε in K and say that Q_{i} and Q_{j} are in the same (ε, q)–orbit if Q_{i} = ε^{a}q^{b}Q_{j}, for some integers a, b ∈ ℤ. Given two ordered tuples X = (X_{1}, …, X_{s}) and Y = (Y_{1}, …, Y_{l}) of elements of K^{×} set X ∨ Y = (X_{1}, …, X_{s}, Y_{1}, …, Y_{l}).

2000 Mathematics Subject Classification. 20C08, 20C30.

Key words and phrases. Cyclotomic Hecke algebras, Morita equivalence.

Suppose that A is an algebra and that ℤ_{p} is a group which acts on A as a group of algebra automorphisms. Let A ⋊ ℤ_{p} be the algebra with elements { ag | a ∈ A and g ∈ ℤ_{p} } and with multiplication

ag · bh = ab^{g} · gh, for a, b ∈ A and g, h ∈ ℤ_{p}.

The first main result of this paper is the following.

Theorem A. Suppose that Q = Q_{1} ∨ ⋯ ∨ Q_{κ}, where Q_{i} ∈ Q_{α} and Q_{j} ∈ Q_{β} are in the same (ε, q)–orbit only if α = β. Let t_{α} = |Q_{α}|, for 1 ≤ α ≤ κ. Then the following two algebras are Morita equivalent:

a) ℋ_{r,p,n}(Q),

b) ⨁_{\begin{matrix}b_{1},··· ,b_{κ}≥0 \\ b_{1}+···+b_{κ}=n \end{matrix}} (ℋ_{pt_{1},b_{1}}(Q_{1}) ⊗ ⋯ ⊗ ℋ_{pt_{κ},b_{κ}}(Q_{κ})) ⋊ ℤ_{p},

In part (b), each of the algebras ℋ_{pt_{α},b_{α}}(Q_{α}) has an automorphism σ_{α} of order p and, in the direct sum, the automorphism σ_{1}⊗⋯⊗σ_{κ} acts diagonally on the algebra ℋ_{pt_{1},b_{1}}(Q_{1}) ⊗ ⋯ ⊗ ℋ_{pt_{κ},b_{κ}}(Q_{κ}). Observe that ⟨σ_{1} ⊗ ⋯ ⊗ σ_{κ}⟩ ≅ ℤ_{p}.

The second result of this paper uses Theorem A to prove a reduction theorem for computing the decomposition numbers of ℋ_{r,p,n}. In order to state this result fix a modular system (F, O, K) “with parameters”. That is, we fix an algebraically closed field F of characteristic zero, a discrete valuation ring O with maximal ideal π and residue field K ≅ O/π, together with parameters q̂, Q̂_{1}, …, Q̂_{t} ∈ O^{×} such that q = q̂ + π and Q_{i} = Q̂_{i} + π for each i. Let ℋ_{r,p,n}^{F} = ℋ_{r,p,n}^{F}(q̂, Q̂) be the Hecke algebra of type G(r, p, n) over F with parameters q̂ and Q̂ = (Q̂_{1}, …, Q̂_{t}) and similarly let ℋ_{r,p,n}^{O} = ℋ_{r,p,n}(q̂, Q̂) and write ℋ_{r,p,n}^{K} = ℋ_{r,p,n}. We assume that ℋ_{r,p,n}^{F} is semisimple. By freeness we have that ℋ_{r,p,n}^{F} ≅ ℋ_{r,p,n}^{O} ⊗_{O} F and ℋ_{r,p,n}^{K} ≅ ℋ_{r,p,n}^{O} ⊗_{O} K. Thus, by choosing O–lattices we can talk of modular reduction from ℋ_{r,p,n}^{F}–Mod to ℋ_{r,p,n}^{K}–Mod.

Using the definitions below, it is straightforward to check that the automorphisms σ and τ commute with modular reduction. Thus, we have compatible automorphisms σ and τ on ℋ_{r,n}^{F} and on ℋ_{r,n}^{K}.

Let R ∈ {F, K} and let M be an ℋ_{r,p,n}^{R}–module. Then we define a new ℋ_{r,p,n}^{R}– module M^{τ} by “twisting” the action of ℋ_{r,p,n}^{R}using the automorphism τ. Explicitly, M^{τ} = M as a vector space and the ℋ_{r,p,n}^{R}–action on M^{τ} is defined by

m · h = mτ(h), for all m ∈ M and h ∈ ℋ_{r,p,n}^{R}.

Since τ^{p} is an inner automorphism of the algebra ℋ_{r,p,n}^{R}, it follows that M ≅ M^{τ^{p}} for any ℋ_{r,p,n}^{R}–module M. Therefore, there is a natural action of the cyclic group ℤ_{p} on the set of isomorphism classes of ℋ_{r,p,n}^{R}–modules. We define the inertia group of M to be G_{M} = { k | 1 ≤ k ≤ p, M ≅ M^{τ^{k}} } ≤ ℤ_{p}.

If A is any algebra let Irr(A) be the complete set of isomorphism classes of irreducible A–modules. We are interested in the inertia group G_{S} and G_{D}, for S ∈ Irr(ℋ_{r,p,n}^{F}) and D ∈ Irr(ℋ_{r,p,n}^{K}).

1.1. Definition. Suppose that S ∈ Irr(ℋ_{r,p,n}^{F}) and D ∈ Irr(ℋ_{r,p,n}^{K}). The decomposition number [S : D] is a p-splittable decomposition number of ℋ_{r,p,n} if G_{S} = {0} = G_{D}.

The second main result of this paper is the following:

Theorem B. Suppose that q ≠ 1. Then the decomposition numbers of the cyclotomic Hecke algebras of type G(r, p, n) are completely determined by the p′-splittable decomposition numbers of certain cyclotomic Hecke algebras ℋ_{r′,p′,n′}(Q′), where p′ divides p, 1 ≤ r′ ≤ r, 1 ≤ n′ ≤ n and where the parameters Q′ are contained in a single (ε, q)–orbit.

The proof of Theorem B explicitly describes the algebras ℋ_{r′,p′,n′}(Q′) and the parameters Q′ which appear in this reduction. Thus, once the p′-splittable decomposition numbers are known this result gives an algorithm for computing the decomposition matrices of the cyclotomic Hecke algebras of type G(r, p, n).

This paper is organized as follows. In the next section we define the cyclotomic Hecke algebras of type G(r, p, n) and prove the Morita equivalence result for the Hecke algebras of type G(r, 1, n) which underpins all of the results in this paper. In the third section we apply the results for the algebras of type G(r, 1, n) to prove Theorem A. The fourth section of the paper uses Clifford theory to show that if an algebra can be written as a semidirect product then its decomposition numbers are determined by a suitable family of p′-splittable decomposition numbers. This result is then applied in section 5 to prove Theorem B.

2. Morita equivalence theorems for Hecke algebras of type G(r, 1, n)

In this section we define the cyclotomic Hecke algebras and set our notation. We then recall and generalize the Morita equivalence results that we need for the cyclotomic algebras of type G(r, 1, n).

Throughout this paper we fix positive integers r, p and n such that r = pt for some integer t. Let K be an algebraically closed field which contains a primitive pth root of unity ε. In particular, the characteristic of K is coprime to p. Fix parameters q, Q_{1}, ⋯, Q_{t} ∈ K^{×} and, as in the introduction, let Q := (Q_{1}, ⋯, Q_{t}) and write |Q| = t.

Let ℋ_{r,n}(Q) be the cyclotomic Hecke algebra of type G(r, 1, n). As a K-algebra ℋ_{r,n}(Q) is generated by T_{0}, T_{1}, ⋯, T_{n−1} subject to the relations:

(T_{0}^{p} − Q_{1}^{p})(T_{0}^{p} − Q_{2}^{p}) ⋯ (T_{0}^{p} − Q_{t}^{p}) = 0,
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
(T_{i} + 1)(T_{i} − q) = 0, 1 ≤ i ≤ n − 1,
T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}, 1 ≤ i ≤ n − 2,
T_{i}T_{j} = T_{j}T_{i}, 0 ≤ i < j − 1 ≤ n − 2.

That is, ℋ_{r,n}(Q) is the cyclotomic Hecke algebra of type G(r, 1, n) with parameters {Q′_{1}, …, Q′_{r}}, where Q′_{i+pj} = ε^{i}Q_{j+1} for 0 ≤ i < p and 0 ≤ j < t.

2.1. Definition. The cyclotomic Hecke algebra of type G(r, p, n) is the subalgebra ℋ_{r,p,n}(Q) of ℋ_{r,n}(Q) which is generated by the elements T_{0}^{p}, T_{u} = T_{0}^{−1}T_{1}T_{0} and T_{1}, T_{2}, ⋯, T_{n−1}.

When the choice of parameters q, Q is clear we write ℋ_{r,p,n} = ℋ_{r,p,n}(Q) and ℋ_{r,n} = ℋ_{r,n}(Q). When we want to emphasize the coefficient ring we write ℋ_{r,p,n}^{K} = ℋ_{r,p,n}^{K}(Q) and ℋ_{r,n}^{K} = ℋ_{r,n}^{K}(Q), respectively.

Let 𝔖_{n} be the symmetric group of degree n. As the type A braid relations hold in ℋ_{r,n} for each w ∈ 𝔖_{n} there is a well–defined element T_{w} ∈ ℋ_{r,n}, where T_{w} = T_{i_{1}} … T_{i_{k}} whenever k is minimal such that w = (i_{1}, i_{1} + 1) … (i_{k}, i_{k} + 1). Set L_{1} = T_{0} and L_{k+1} = q^{−1}T_{k}L_{k}T_{k}, for k = 1, …, n − 1. Then Ariki and Koike [2, Theorem 3.10] showed that

{L_{1}^{c_{1}} … L_{n}^{c_{n}}T_{w} | w ∈ 𝔖_{n} and 0 ≤ c_{i} < r}

is a basis of ℋ_{r,n}.

All of the Morita equivalences in Theorem A are a consequence of the following result for the cyclotomic Hecke algebras ℋ_{r,n}.

2.2. Theorem (Dipper–Mathas [6, Theorem 1.1]). Suppose that Q = Q_{1}∨⋯∨Q_{κ}, such that α = β whenever Q_{i} ∈ Q_{α}, Q_{j} ∈ Q_{β} and Q_{i} = q^{a}Q_{j} for some a ∈ ℤ. Let t_{α} = |Q_{α}|. Then ℋ_{r,n}(Q) is Morita equivalent to the algebra

⨁_{\begin{matrix}b_{1},...,b_{κ}≥0 \\ b_{1}+···+b_{κ}=n \end{matrix}} ℋ_{pt_{1},b_{1}}(Q_{1}) ⊗ ⋯ ⊗ ℋ_{pt_{κ},b_{κ}}(Q_{κ}).

As noted in [6] the proof of Theorem 2.2 quickly reduces to the case κ = 2, so only this case is considered in [6]. Unfortunately, to prove Theorem A we need detailed information about the bimodule which induces the Morita equivalence of Theorem 2.2 for arbitrary κ ≥ 1. Consequently, we need to generalize the results of [6] and construct the bimodule which induces the Morita equivalence of Theorem 2.2 (in the special case when Q is partitioned into a disjoint union of (ε, q)– orbits). In constructing this bimodule we refer the reader back to [6] whenever the details are not substantially different from the case κ = 2.

First, fix non-negative integers a and b with a + b ≤ n and an integer s with 1 ≤ s ≤ t. Define

v_{a,b}(s) = ∏_{k=1}^{s}(L_{1}^{p} − Q_{k}^{p}) … (L_{a}^{p} − Q_{k}^{p}) · T_{w_{a,b}} · ∏_{k=s+1}^{t}(L_{1}^{p} − Q_{k}^{p}) … (L_{b}^{p} − Q_{k}^{p}),

where w_{a,b} = (s_{a+b−1} … s_{1})^{b}. (So v_{n−b,b}(s) is the element v_{b} of [6, Definition 3.3].) It may help the reader to observe that if we write w_{a,b} ∈ 𝔖_{a+b} as a permutation in two-line notation then

w_{a,b} = ( \begin{matrix}1 & ⋯ & a & a + 1 & ⋯ & a + b \\ b + 1 & ⋯ & a + b & 1 & ⋯ & b \end{matrix} ).

We will use the following notation extensively.

Notation. Given any sequence a = (a_{1}, …, a_{k}) and integers 1 ≤ i ≤ j ≤ k we set a_{i,j} = a_{i} + ⋯ + a_{j}. If i < j then let a_{j,i} = 0.

Until further notice we fix a partition Q = Q_{1} ∨⋯∨Q_{κ} of Q such that Q_{i} ∈ Q_{α} and Q_{j} ∈ Q_{β} are in the same (ε, q)-orbit only if α = β. Set t = (t_{1}, …, t_{κ}) where t_{α} = |Q_{α}|, for 1 ≤ α ≤ κ. Without loss of generality we assume that Q_{α} = (Q_{t_{1,α−1}+1}, …, Q_{t_{1,α}}), for α = 1, …, κ (set t_{0} = 0).

Let Λ(n, κ) = { b = (b_{1}, …, b_{κ}) | b_{1,κ} = n and b_{α} ≥ 0 for 1 ≤ α ≤ κ } be the set of compositions of n into κ parts. If b ∈ Λ(n, κ) then, for convenience, we set b_{κ+1} = 0.

2.3. Definition. Suppose that b ∈ Λ(n, κ). Define

v_{b} = v_{b_{κ},b_{1,κ−1}}(t_{1,κ−1})v_{b_{κ−1},b_{1,κ−2}}(t_{1,κ−2}) … v_{b_{2},b_{1,1}}(t_{1,1})

and let w_{b} = w_{b_{κ},b_{1,κ−1}}w_{b_{κ−1},b_{1,κ−2}} … w_{b_{2},b_{1,1}}. Define V^{b} = v_{b}ℋ_{r,n}.

Note that v_{b} depends crucially on our fixed partition Q = Q_{1} ∨ ⋯ ∨ Q_{κ} of Q, so we should really write v_{b} = v_{b}(Q_{1}, …, Q_{κ}). Our first goal is to understand V^{b}.

The key property of the elements v_{b} is the following.

2.4. Proposition. Suppose that b ∈ Λ(n, κ). Then a) T_{i}v_{b} = v_{b}T_{w_{b}(i)}, whenever 1 ≤ i < n and i ≠ b_{α,κ} for α = 1, …, κ.

b) L_{k}v_{b} = v_{b}L_{w_{b}(k)}, whenever 1 ≤ k ≤ n.

Proof. After translating notation, [6, Prop. 3.4] says that if 1 ≤ s ≤ t and a and b are non–negative integers with a + b ≤ n then

(2.5)  T_{i}v_{a,b}(s) = v_{a,b}(s)T_{w_{a,b}(i)} and L_{k}v_{a,b}(s) = v_{a,b}(s)L_{w_{a,b}(k)}

whenever 1 ≤ i < a + b, i ≠ a, and 1 ≤ k ≤ a + b. This is precisely the special case of the Proposition when κ = 2. The general case follows from this result together with the observation that T_{i}v_{a,b}(s) = v_{a,b}(s)T_{i} and L_{k}v_{a,b}(s) = v_{a,b}(s)L_{k} whenever a + b < i < n and a + b < k ≤ n for non–negative integers a and b. □

Observe that v_{b}T_{j} = T_{w_{b}^{−1}(j)}v_{b} and v_{b}L_{m} = L_{w_{b}^{−1}(m)}v_{b} by Proposition 2.4, for 1 ≤ j < n, 1 ≤ m ≤ n with j ≠ b_{1,α} for α = 1, …, κ.

2.6. Lemma. Suppose that b ∈ Λ(n, κ), 1 ≤ α ≤ κ and b_{α} ≠ 0. Then

∏_{Q_{i}∈Q_{α}}(L_{1+b_{α+1,κ}}^{p} − Q_{i}^{p}) · v_{b} = 0 = v_{b} · ∏_{Q_{i}∈Q_{α}}(L_{b_{1,α−1}+1}^{p} − Q_{i}^{p})

Proof. Recall that ∏_{i=1}^{t}(L_{1}^{p} − Q_{i}^{p}) = 0 since L_{1} = T_{0}. Therefore, it follows from the definitions that if b_{κ} ≠ 0 then

∏_{Q_{i}∈Q_{κ}}(L_{1}^{p} − Q_{i}^{p}) · v_{b_{κ},b_{1,κ−1}}(t_{1,κ−1}) = 0.

Hence, ∏_{Q_{i}∈Q_{κ}}(L_{1}^{p} − Q_{i}^{p}) · v_{b} = 0. Similarly, if b_{1} ≠ 0, then

v_{b} · ∏_{Q_{j}∈Q_{1}}(L_{1}^{p} − Q_{j}^{p}) = 0.

Now suppose that 1 ≤ α < κ, b_{α} ≠ 0, and set L(α) = ∏_{Q_{i}∈Q_{α}}(L_{1+b_{α+1,κ}}^{p} − Q_{i}^{p}). Then, using (2.5),

L(α)v_{b} = ∏_{β=α}^{κ−1} v_{b_{β+1},b_{1,β}}(t_{1,β}) · ∏_{Q_{i}∈Q_{α}}(L_{1}^{p} − Q_{i}^{p}) · ∏_{β=1}^{α−1} v_{b_{β+1},b_{1,β}}(t_{1,β}),

where the terms in the two outside products appear in order with β decreasing from left to right. Using the definitions, ∏_{k=t_{1,α}+1}^{t}(L_{1}^{p} − Q_{k}^{p}) is a right divisor

of v_{b_{α+1},b_{1,α}}(t_{1,α}), whereas ∏_{k=1}^{t_{1,α−1}}(L_{1}^{p} − Q_{k}^{p}) is a left divisor of v_{b_{α},b_{1,α−1}}(t_{1,α−1}). Hence, L(α)v_{b} = 0 by combining the relation ∏_{i=1}^{t}(L_{1}^{p} − Q_{i}^{p}) = 0 with the last displayed equation. The second statement

v_{b} · ∏_{Q_{i}∈Q_{α}}(L_{b_{1,α−1}+1}^{p} − Q_{i}^{p}) = 0,

for α = 2, …, κ, is equivalent to what we have just proved because v_{b}L_{b_{1,α−1}+1} = L_{w_{b}^{−1}(b_{1,α−1}+1)}v_{b} = L_{1+b_{α+1,κ}}v_{b} by Proposition 2.4. □

To proceed we recall the cellular basis of the algebras ℋ_{r,n}, and the associated combinatorics, introduced in [5]. A multipartition of n is an ordered r–tuple of partitions λ = (λ^{(1)}, …, λ^{(r)}) such that |λ^{(1)}| + ⋯ + |λ^{(r)}| = n. Let Λ_{n}^{+} be the set of multipartitions of n. Then Λ_{n}^{+} is a poset under the dominance order, where λ ⊵ μ if

∑_{a=1}^{s−1} |λ^{(a)}| + ∑_{j=1}^{i} λ_{j}^{(s)} ≥ ∑_{a=1}^{s−1} |μ^{(a)}| + ∑_{j=1}^{i} μ_{j}^{(s)},

for all 1 ≤ s ≤ r and all i ≥ 1.

The diagram of λ is the set [λ] = { (i, j, s) | 1 ≤ j ≤ λ_{i}^{(s)} for 1 ≤ s ≤ r }. A λ-tableau is a bijection 𝔱: [λ] ⟶ {1, 2, …, n}. The λ–tableau 𝔱 is standard if 𝔱(i, j, s) < 𝔱(i′, j′, s) whenever i ≤ i′, j ≤ j′ and (i, j, s) and (i′, j′, s) are distinct elements of [λ]. Let Std(λ) be the set of standard λ–tableaux. Observe that 𝔖_{n} acts from the right on the set of λ–tableaux. In particular, if w ∈ 𝔖_{n} and 𝔱 ∈ Std(λ) then 𝔱w is a λ–tableau, however, it is not necessarily standard.

If λ ∈ Λ_{n}^{+} let 𝔖_{λ} = 𝔖_{λ^{(1)}} ×⋯×𝔖_{λ^{(r)}} be the corresponding Young (or parabolic) subgroup of 𝔖_{n}. We set

x_{λ} = ∑_{w∈𝔖_{λ}} T_{w} and u_{λ}^{+} = ∏_{s=2}^{r} ∏_{k=1}^{|λ^{(1)}|+···+|λ^{(s−1)}|}(L_{k} − Q′_{s}).

Then x_{λ} and u_{λ}^{+} are commuting elements of ℋ_{r,n}. Next, if 𝔰 is a standard λ-tableau let d(𝔰) be the corresponding distinguished right coset representative of 𝔖_{λ} in 𝔖_{n}. Finally, given a pair (𝔰, 𝔱) of standard λ–tableaux define m_{𝔰𝔱} = T_{d(𝔰)}^{∗}x_{λ}u_{λ}^{+}T_{d(𝔱)}, where ∗ is the unique anti–isomorphism of ℋ_{r,n} which fixes T_{0}, …, T_{n−1}. Then

{ m_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ) for some λ ∈ Λ_{n}^{+} }

is a cellular basis of ℋ_{r,n} by [5, Theorem 3.26].

We can relate V^{b} to the combinatorics of the cellular basis {m_{𝔰𝔱}} by defining ω_{b} = (ω_{b}^{(1)}, …, ω_{b}^{(r)}) to be the multipartition with

ω_{b}^{(s)} = \begin{cases}(1^{b_{α}}), & if s = pt_{1,α} for some α, \\ (0), & otherwise. \end{cases}

From the definitions, u_{ω_{b}}^{+} = ∏_{α=1}^{κ−1} ∏_{Q_{i}∈Q_{α+1}∨···∨Q_{κ}}(L_{1}^{p} − Q_{i}^{p}) … (L_{b_{1,α}}^{p} − Q_{i}^{p}). Hence, using (2.5) we obtain the following.

2.7. Lemma. Suppose that b ∈ Λ(n, κ). Then v_{b} = v_{b}^{−}u_{ω_{b}}^{+}, where

v_{b}^{−} = ∏_{α=2}^{κ} ∏_{i=1}^{t_{1,α−1}}(L_{1}^{p} − Q_{i}^{p}) … (L_{b_{α}}^{p} − Q_{i}^{p}) · T_{w_{bα,b_{1,α−1}}}

where in the product α decreases in order from left to right.

Following [5] define M^{ω_{b}} = u_{ω_{b}}^{+}ℋ_{r,n}. By the Lemma, there is a surjective ℋ_{r,n}– module homomorphism θ_{b} : M^{ω_{b}} ⟶V^{b} given by θ_{b}(h) = v_{b}^{−}h, for all h ∈ M^{ω_{b}}.

Suppose that λ is a multipartition and that 𝔱 is a standard λ-tableau. For each integer k, with 1 ≤ k ≤ n, define comp_{𝔱}(k) = s if (i, j, s) is the unique node in [λ] such that 𝔱(i, j, s) = k.

2.8. Definition. Suppose that λ is a multipartition of n. Define

Std_{b}(λ) = { 𝔱 ∈ Std(λ) | comp_{𝔱}(k) ≤ pt_{1,α} if 1 ≤ k ≤ b_{1,α} }

and

Std_{b}^{+}(λ) = { 𝔱 ∈ Std(λ) | pt_{1,α−1} < comp_{𝔱}(k) ≤ pt_{1,α} if b_{1,α−1} < k ≤ b_{1,α} }

Then Std_{b}(λ) ≠ ∅ if only if ∑_{s=1}^{pt_{1,α}} |λ^{(s)}| ≥ b_{1,α} for 1 ≤ α ≤ κ and Std_{b}^{+}(λ) ≠ 0 if and only if ∑_{s=pt_{1,α−1}+1}^{pt_{1,α}} |λ^{(s)}| = b_{α}, for 1 ≤ α ≤ κ. Note that Std_{b}^{+}(λ) ⊆ Std_{b}(λ).

2.9. Lemma. Suppose that b ∈ Λ(n, κ).

a) M^{ω_{b}} has basis { m_{𝔰𝔱} | 𝔰 ∈ Std_{b}(λ), 𝔱 ∈ Std(λ) for some λ ∈ Λ_{n}^{+} }.

b) Suppose that 𝔰 ∈ Std_{b}(λ) \ Std_{b}^{+}(λ) and 𝔱 ∈ Std(λ). Then θ_{b}(m_{𝔰𝔱}) = 0.

Proof. Part (a) is a translation of [5, Theorem 4.14] into the current notation. See the proof of [6, Lemma 3.9] for more details.

For part (b) we follow the proof of [6, Lemma 3.10]. Let c = (c_{1}, …, c_{κ}), where c_{α} = |λ^{(pt_{1,α−1}+1)}| + ⋯ + |λ^{(pt_{1,α})}|, for 1 ≤ α ≤ κ. Then c_{1,α} ≥ b_{1,α}, for 1 ≤ α ≤ κ, since 𝔰 ∈ Std_{b}(λ) and c ≠ b since 𝔰 ∉ Std_{b}^{+}(λ). Choose β to be minimal such that c_{β} > b_{β}. Then 1 ≤ β < κ and if 1 ≤ α ≤ β then pt_{1,α−1} < comp_{𝔰}(k) ≤ pt_{1,α} if b_{1,α−1} < k ≤ b_{1,α} since c_{1} = b_{1}, …, c_{β−1} = b_{β−1} and 𝔰 ∈ Std_{b}(λ). Choose γ ≥ β to be minimal such that b_{γ} ≠ 0. Let w be a permutation of {b_{1,γ} + 1, b_{1,γ} + 2, …, n} of minimal length such that 𝔰′ = 𝔰w is a standard λ–tableau with pt_{1,α−1} < comp_{𝔰}(k) ≤ pt_{1,α} if c_{1,α−1} < k ≤ c_{1,α} for γ+1 ≤ α ≤ κ. (Such a permutation exists because we can first swap integers k, with c_{1,κ−1} < k ≤ n and comp_{𝔰}(k) ≤ pt_{1,κ−1}, with the integers l, where b_{1,κ−1} < l ≤ n and pt_{1,κ−1} < comp_{𝔰}(l) ≤ pt_{1,κ}; let 𝔰_{1} be the resulting λ–tableau. Then we swap integers k, with c_{1,κ−2} < k ≤ c_{1,κ−1} and comp_{𝔰_{1}}(k) ≤ pt_{1,κ−2}, with the integers l, where b_{1,κ−2} < l ≤ n and pt_{1,κ−2} < comp_{𝔰_{1}}(l) ≤ pt_{1,κ−1}; ⋯, and so on; compare [6, Lemma 3.10].) As a result, comp_{𝔰}(k) ≤ pt_{1,γ} if k ≤ c_{1,γ}. Then d(𝔰) = d(𝔰′)w, with the lengths adding, so that m_{𝔰𝔱} = T_{w}^{∗}m_{𝔰′𝔱}. Furthermore, by construction, there is a composition c′ ∈ Λ(n, κ) such that 𝔰′ ∈ Std_{c′}(λ), c′_{α} = c_{α}, for 1 ≤ α < β or γ ≤ α ≤ κ, and c′_{1,α} ≥ b_{1,α}, for 1 ≤ α ≤ κ. Hence, m_{𝔰′𝔱} ∈ M^{ω_{c′}} by part (a), so that m_{𝔰′𝔱} = u_{ω_{c′}}^{+} h for some h ∈ ℋ_{r,n}. Therefore, θ_{b}(m_{𝔰𝔱}) = v_{b}^{−}T_{w}^{∗}m_{𝔰′𝔱} = v_{b}^{−}T_{w}^{∗}v_{ω_{c′}}^{+} h.

To simplify the notation, for the remainder of the proof set w(α) = w_{b_{α},b_{1,α−1}} and u^{−}(m, s) = ∏_{i=1}^{s}(L_{1}^{p} − Q_{i}^{p}) … (L_{m}^{p} − Q_{i}^{p}), for 1 ≤ α ≤ κ, 1 ≤ m ≤ n and 1 ≤ s ≤ t. Similarly, set u^{+}(m, s) = ∏_{i=s}^{t}(L_{1}^{p} − Q_{i}^{p}) … (L_{m}^{p} − Q_{i}^{p}). Then

θ_{b}(m_{𝔰𝔱}) = v_{b}^{−}T_{w}^{∗}u_{ω_{c′}}^{+} h = ∏_{α=2}^{κ} u^{−}(b_{α}, t_{1,α−1})T_{w(α)} · T_{w}^{∗}u_{ω_{c′}}^{+} h,

where the product is taken in order with α decreasing from left to right. Now, u^{±}(m, s) commutes with T_{i} if i ≠ m. Therefore, since w is a permutation of {b_{1,γ} + 1, b_{1,γ} + 2, …, n}, we have

θ_{b}(m_{𝔰𝔱}) = ∏_{α=γ+1}^{κ} u^{−}(b_{α}, t_{1,α−1})T_{w(α)} · T_{w}^{∗} · ∏_{α=2}^{γ} u^{−}(b_{α}, t_{1,α−1})T_{w(α)} · u_{ω_{c′}}^{+} h,

for some w′ ∈ 𝔖_{n}, where again both products are ordered with α decreasing from left to right. By definition, u_{ω_{c′}}^{+} = ∏_{α=1}^{κ−1} u^{+}(c′_{1,α}, t_{1,α} + 1), where this product can be taken in any order. So,

∏_{α=2}^{γ} u^{−}(b_{α}, t_{1,α−1})T_{w(α)} · u_{ω_{c′}}^{+} = ∏_{α=2}^{γ} u^{−}(b_{α}, t_{1,α−1})T_{w(α)} · ∏_{α=1}^{κ−1} u^{+}(c′_{1,α}, t_{1,α} + 1).

Now, w(α) = w_{b_{α},b_{1,α−1}} ∈ 𝔖_{b_{1,α}}. So if 1 ≤ α < γ then T_{w(α)} commutes with u^{+}(c′_{1,γ}, t_{1,γ} + 1) since c′_{1,γ} = c_{1,γ} > b_{1,α}. Consequently, the last displayed equation contains u^{−}(b_{γ}, t_{1,γ−1})T_{w(γ)}u^{+}(c_{1,γ}, t_{1,γ} + 1) as a factor. Note that c_{1,γ} > b_{1,γ−1}. Looking at the definitions, this element is equal to

v_{b_{γ},b_{1,γ−1}}(t_{1,γ−1}) ∏_{s=t_{1,γ}+1}^{t}(L_{b_{1,γ−1}+1}^{p} − Q_{s}^{p}) … (L_{c_{1,γ}}^{p} − Q_{s}^{p}) = 0,

where the last equality comes from applying the right hand equation of Lemma 2.6 in the special case when κ = 2. Putting all of these equations together, we have shown that θ_{b}(m_{𝔰𝔱}) = 0, as required. □

Suppose that 𝔱 is a standard λ–tableau and that 1 ≤ k ≤ n. Let Shape_{k}(𝔱) be the multipartition with diagram 𝔱^{−1}({1, …, k}); that is, Shape_{k}(𝔱) is the multipartition given by the positions of {1, …, k} in 𝔱. If 𝔱 ∈ Std(λ) and 𝔳 ∈ Std(μ) then we write 𝔱 ⊵ 𝔳 if λ ▷ μ or if λ = μ and Shape_{k}(𝔱) ⊵ Shape_{k}(𝔳) for 1 ≤ k ≤ n. We extend this partial order to pairs of standard tableaux in the obvious way.

2.10. Lemma. Suppose that λ is a multipartition of n and that 𝔰 ∈ Std_{b}^{+}(λ) and 𝔱 ∈ Std(λ). Let 𝔰′ = 𝔰w_{b}^{−1}. Then there exists an invertible element u ∈ R such that

θ_{b}(m_{𝔰𝔱}) = um_{𝔰′𝔱} + ∑_{(𝔲,𝔳)▷(𝔰′,𝔱)} r_{𝔲𝔳}m_{𝔲𝔳},

for some r_{𝔲𝔳} ∈ R.

Proof. By [13, Prop. 3.7], if 1 ≤ k ≤ n and p(a − 1) < comp_{𝔱}(k) ≤ pa then

m_{𝔰𝔱}L_{k}^{p} = q^{p(j−i)}Q_{a}^{p}m_{𝔰𝔱} + ∑_{(𝔲,𝔳)▷(𝔰,𝔱)} r_{𝔲𝔳}m_{𝔲𝔳},

for some r_{𝔲𝔳} ∈ R. Using this formula we can compute θ(m_{𝔰𝔱}) = v_{b}^{−}m_{𝔰𝔱} directly, which shows that m_{𝔲𝔳} appears with non-zero coefficient in θ_{b}(m_{𝔰𝔱}) only if (𝔲, 𝔳) ⊵ (𝔰′, 𝔱). Finally, m_{𝔰′𝔱} appears with non-zero coefficient in θ_{b}(m_{𝔰𝔱}) because Q = Q_{1} ∨ ⋯ ∨ Q_{κ} is a partition of Q into (ε, q)-orbits. (Compare with the proof of [6, Lemma 3.11].) □

Suppose that λ is a multipartition of n. Let ℋ_{r,n}^{λ} be the module with basis {m_{𝔲𝔳}} where 𝔲 and 𝔳 range over the standard μ–tableaux with μ ▷ λ. It follows from the general theory of cellular algebras [16, Lemma 2.3] that ℋ_{r,n}^{λ} is a two–sided ideal of ℋ_{r,n}.

As a vector space, the Specht module (or cell module) S(λ) is the module with basis { m_{𝔰𝔱} + ℋ_{r,n}^{λ} | 𝔱 ∈ Std(λ) }. The theory of cellular algebras [16, 2.4] shows that S(λ) is an ℋ_{r,n}-module and that, up to isomorphism, S(λ) does not depend on the choice of 𝔰.

Finally, we need the classification of the blocks for ℋ_{r,n}. For each λ ∈ Λ_{n}^{+} define a “content function” c_{λ}: R⟶ℕ by

c_{λ}(x) = # { (i, j, a + pb) ∈ [λ] | 0 ≤ a < p and x = q^{j−i}ε^{a}Q_{b} } ,

for x ∈ R. Then the Specht modules S(λ) and S(μ) are in the same block only if c_{λ}(x) = c_{μ}(x), for all x ∈ R, by [8, Prop. 5.9(ii)]. (Although we will not need this we note that the converse is also true by [14, Theorem A].)

With the results that we have now proved we can complete the proof of Theorem 2.2 with only minor modifications of the arguments of [6]. Consequently, we sketch the rest of the proof and give references to [6] for those readers who require more detail.

2.11. Definition. Suppose that b ∈ Λ(n, κ) and that 𝔰 ∈ Std_{b}^{+}(λ), 𝔱 ∈ Std(λ) for some λ ∈ Λ_{n}^{+}.

a) Set v_{𝔰𝔱} = θ_{b}(m_{𝔰𝔱}) ∈ V^{b}.

b) If further 𝔱 ∈ Std_{b}^{+}(λ), then let θ_{𝔰𝔱} ∈ End_{R}(V^{b}) be the endomorphism θ_{𝔰𝔱}(v_{b}h) = v_{𝔰𝔱}h, for all h ∈ ℋ_{r,n}.

We remark that it is not clear from the definition that the maps θ_{𝔰𝔱} are well– defined.

2.12. Proposition. Suppose that b ∈ Λ(n, κ). Then:

a) V^{b} has basis

{ v_{𝔰𝔱} | 𝔰, ∈ Std_{b}^{+}(λ), 𝔱 ∈ Std(λ) for some λ ∈ Λ_{n}^{+} } .

b) If b ≠ c ∈ Λ(n, κ) then Hom_{ℋ_{r,n}} (V^{b}, V^{c}) = 0.

c) End_{ℋ_{r,n}} (V^{b}) is a vector space with basis

{ θ_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b}^{+}(λ) for some λ ∈ Λ_{n}^{+} } .

Proof. (a) This follows directly from Lemma 2.9(b) and Lemma 2.10.

(b) As in [6, Theorem 3.16] it follows from part (a) and the construction of the Specht modules that V^{b} has a filtration V^{b} = V_{1} ⊃ V_{2} ⋯ ⊃ V_{k} = 0 such that (1) V_{i}/V_{i+1} ≅ S(λ_{i}), for some λ_{i} ∈ Λ_{n}^{+}, and (2) if μ ∈ Λ_{n}^{+} then

# Std_{b}^{+}(μ) = # { 1 ≤ i < k | V_{i}/V_{i+1} ≅ S(μ) } .

Now, if b ≠ c and λ and μ are two multipartitions such that Std_{b}^{+}(λ) ≠ ∅ and Std_{c}^{+}(μ) ≠ ∅ then it is easy to see (cf. the proof of [6, Cor. 3.17]) that c_{λ} ≠ c_{μ}. Consequently, by the remarks before the Theorem, the Specht modules S(λ) and S(μ) are in different blocks. Therefore, all of the composition factors of V^{b} and V^{c} belong to different blocks, so Hom_{ℋ_{r,n}} (V^{b}, V^{c}) = 0.

(c) The proof is identical to that of [6, Theorem 3.19]. In outline, the argument is as follows. By [5, Theorem 6.16] and Lemma 2.9(a), End_{ℋ_{r,n}} (M^{ω_{b}}) has basis { φ_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b}^{+}(λ) for some λ ∈ Λ_{n}^{+} }, where φ_{𝔰𝔱}(u_{ω_{b}}^{+}h) = m_{𝔰𝔱}h for all h ∈ ℋ_{r,n}. As in the proof of part (b), the filtration 0 ⊆ ker θ_{b} ⊂ M^{ω_{b}} can be extended to a Specht filtration which is compatible with the Specht filtration of V^{b} ≅ M^{ω_{b}}/ ker θ_{b}. Using this Specht filtration and the classification of the blocks of ℋ_{r,n} given above it follows that all of the irreducible constituents of V_{b} and ker θ_{b} belong to different blocks. Therefore, the map θ_{b} : M^{ω_{b}} ⟶ V^{b} splits. Let θ_{b}^{−1} be a right inverse to θ_{b}. Then a straightforward calculation shows that

θ_{b}φ_{𝔰𝔱}θ_{b}^{−1} = \begin{cases}θ_{𝔰𝔱}, & if 𝔰, 𝔱 ∈ Std_{b}^{+}(λ), \\ 0, & otherwise. \end{cases}

As the maps { θ_{b}φ_{𝔰𝔱}θ_{b}^{−1} | 𝔰, 𝔱 ∈ Std_{b}^{+}(λ) } span End_{ℋ_{r,n}} (V^{b}), this proves part (c). □

Let ℋ_{b} = ℋ_{pt_{1},b_{1}}(Q_{1}) ⊗ ⋯ ⊗ ℋ_{pt_{κ},b_{κ}}(Q_{κ}). Let T_{i}^{(α)} = 1 ⊗ ⋯ ⊗ T_{i} ⊗ ⋯ ⊗ 1 be a generator of ℋ_{b}, where the T_{i} occurs in the α^{th} tensor factor, for 1 ≤ α ≤ κ. Then, as an algebra, ℋ_{b} is generated by the elements { T_{i}^{(α)} | 1 ≤ α ≤ κ and 0 ≤ i < b_{α} }.

We need some combinatorial machinery to describe the ℋ_{b}–modules. Let

Λ_{b}^{+} = { λ ∈ Λ_{n}^{+} | b_{α} = |λ^{(pt_{1,α−1}+1)}| + ⋯ + |λ^{(pt_{1,α})}|, for α = 1, …, κ } .

Note that Std_{b}^{+}(λ) ≠ ∅ if and only if λ ∈ Λ_{b}^{+}.

For λ ∈ Λ_{b}^{+} let λ_{b} = (λ_{b}^{(1)}, …, λ_{b}^{(κ)}), where λ_{b}^{(α)} = (λ^{(pt_{1,α−1}+1)}, …, λ^{(pt_{1,α})}). Then the Specht modules of ℋ_{b} are all of the form S(λ_{b}^{(1)}) ⊗ ⋯ ⊗ S(λ_{b}^{(κ)}), for λ ∈ Λ_{b}^{+}, and there is a natural bijection Std_{b}^{+}(λ) ≅ Std(λ_{b}^{(1)}) × ⋯ × Std(λ_{b}^{(κ)}).

Let 𝔖_{b} = 𝔖_{b_{1}} × ⋯ × 𝔖_{b_{κ}}, which we consider as a subgroup of 𝔖_{n} via the natural embedding. Let D_{b} be the set of distinguished (minimal length) right coset representatives for 𝔖_{b} in 𝔖_{n}. Observe that if λ ∈ Λ_{b}^{+} then

(2.13)  Std(λ) = ∐_{d∈D_{b}} Std_{b}^{+}(λ)d.

Recall that a progenerator, or projective generator, for an algebra A is a projective A–module V which contains every projective indecomposable A–module as a direct summand. The algebras A and End_{A}(V) are Morita equivalent and, moreover, every Morita equivalence arises in this way.

2.14. Proposition.

a) Let V = ⨁_{b∈Λ(n,κ)} V^{b}. Then V is a progenerator for ℋ_{r,n}.

b) Suppose that b ∈ Λ(n, κ). Then: i) V^{b} is a projective ℋ_{r,n}–module; ii) End_{ℋ_{r,n}} (V^{b}) ≅ ℋ_{b}; and iii) as left ℋ_{b}–modules, ℋ_{b} ≅ V_{0}^{b} and V^{b} = ⨁_{d∈𝔇_{b}} V_{0}^{b}T_{d}, where V_{0}^{b} is

the subspace of V^{b} with basis { v_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std_{b}^{+}(λ) for some λ ∈ Λ_{b}^{+} }.

Sketch of proof. Using Proposition 2.12 and (2.13) it is straightforward to show that ℋ_{r,n} ≅ ⨁_{b∈Λ(n,κ)} ⨁_{d∈D_{b}} T_{d}^{∗}V^{b} (see the proof of [6, Theorem 3.20]). Hence, V^{b} is a projective ℋ_{r,n}–module, proving b(i). Part (a) now follows because V^{b} ≅ T_{d}^{∗}V^{b}, for all d ∈ D_{b}.

Now consider the remaining parts of (b). By Proposition 2.4 and Lemma 2.6 there is an action of ℋ_{b} on V^{b} which is uniquely determined by letting the generator T_{i}^{(α)} of ℋ_{b} act as left multiplication by L_{1+b_{α+1,κ}} if i = 0 and by T_{i+1+b_{α+1,κ}} if 1 ≤ i < b_{α}. Thus, there is a map from ℋ_{b} into End_{ℋ_{r,n}} (V^{b}). The argument used to prove [6, Theorem 4.7] now shows that if λ ∈ Λ_{b}^{+} and 𝔰, 𝔱 ∈ Std_{b}^{+}(λ) then the map θ_{𝔰𝔱} ∈ End_{ℋ_{r,n}} (V^{b}) corresponds to left multiplication by the corresponding Murphy basis element of ℋ_{b}, where we use the bijection Std_{b}^{+}(λ) ≅ Std(λ_{b}^{(1)}) × ⋯×Std(λ_{b}^{(κ)}). That is if h ∈ ℋ_{r,p,n} then θ_{𝔰𝔱}(v_{b}h) = (m_{𝔰^{(1)}𝔱^{(1)}} ⊗⋯⊗m_{𝔰^{(κ)}𝔱^{(κ)}})v_{b}h, where 𝔲 ∈ Std_{b}^{+}(λ) maps to (𝔲^{(1)}, …, u^{(κ)}) under the bijection above; see the proof of [6, Lemma 4.6]. This shows that End_{ℋ_{r,n}} (V^{b}) ≅ ℋ_{b}. Finally, part b(iii) follows from b(ii) and the observation that V^{b} = ⨁_{d∈D_{b}} V_{0}^{b}T_{d}, as a left ℋ_{b}–module. □

Notice, in particular, that Theorem 2.2 is an immediate Corollary of the Proposition. We will also need the following result which follows directly from Proposition 2.14b(iii); compare [6, Remark 3.15].

2.15. Corollary. Suppose that b ∈ Λ(n, κ). Then

{ v_{b}L_{1}^{c_{1}} ⋯ L_{n}^{c_{n}}T_{w} | w ∈ 𝔖_{n} and 0 ≤ c_{i} < pt_{α} whenever b_{1,α−1} < i ≤ b_{1,α} }

is a basis of v_{b}ℋ_{r,n,}.

We now have the information that we need to start proving Theorem A from the introduction.

3. Morita equivalence theorems for algebras of type G(r, p, n)

In this section we prove Theorem A, the Morita reduction theorem for the Hecke algebras of type G(r, p, n), by analyzing the structure of V^{b} = v_{b}ℋ_{r,n} as an ℋ_{r,p,n}– module. We maintain our notation from the previous section. In particular, we fix a partitioning Q = Q_{1}∨⋯∨Q_{κ} of Q such that Q_{i} and Q_{j} are in different (ε, q)–orbits whenever Q_{i} ∈ Q_{α}, Q_{j} ∈ Q_{β} and α ≠ β.

Following Ariki [1], for each integer m with 1 ≤ m ≤ n, define:

S_{m} = \begin{cases}T_{0}^{p}, & if m = 1; \\ T_{0}^{−1}L_{m}, & if 2 ≤ m ≤ n. \end{cases}

The elements S_{1}, S_{2}, ⋯, S_{n} are the Murphy operators of ℋ_{r,p,n}. We need these elements to prove the following fundamental fact.

3.1. Lemma. The algebra ℋ_{r,p,n} has basis

{ L_{1}^{c_{1}} … L_{n}^{c_{n}}T_{w} | w ∈ 𝔖_{n}, 0 ≤ c_{i} < r and c_{1} + ⋯ + c_{n} ≡ 0 mod p } .

Proof. Ariki [1, Prop. 1.6] showed that ℋ_{r,p,n} is the subspace of ℋ_{r,n} with basis

\{S_{1}^{c_{1}} ⋯ S_{n}^{c_{n}}T_{w} | \begin{matrix}w ∈ 𝔖_{n}, 0 ≤ c_{i} < r for 2 ≤ i ≤ n, \\ and 0 ≤ pc_{1} − c_{2} − ⋯ − c_{n} < r \end{matrix}\}.

Applying the definitions S_{1}^{c_{1}} ⋯ S_{n}^{c_{n}}T_{w} = L_{1}^{pc_{1}−c_{2}−···−c_{n}}L_{2}^{c_{2}} … L_{n}^{c_{n}}T_{w}. Hence, the Lemma is just a reformulation of Ariki’s result. □

Recall from the introduction that there are two algebra automorphisms σ and τ of ℋ_{r,n}. The automorphism τ is the K-algebra automorphism of ℋ_{r,n} which is defined on generators of ℋ_{r,n} by

τ(T_{1}) = T_{0}^{−1}T_{1}T_{0} and τ(T_{i}) = T_{i}, for i ≠ 1.

The map σ is the K-algebra automorphism of ℋ_{r,n} which is determined by

σ(T_{0}) = εT_{0} and σ(T_{i}) = T_{i}, for i = 1, …, n − 1.

The reader can check that ℋ_{r,p,n} is the fixed point subalgebra of ℋ_{r,n} under σ.

Suppose that A is an algebra with an automorphism θ of order p. Define A⋊_{θ} ℤ_{p} to be the K-algebra with elements

{ aθ^{k} | a ∈ A and 0 ≤ k < p }

and with multiplication aθ^{k} · bθ^{l} = aθ^{k}(b)θ^{k+l}, for a, b ∈ A and 0 ≤ k, l < p. As in the introduction, if M is an A–module then we can define a new A–module M^{θ} which is isomorphic to M as a vector space but with the A–action twisted by θ. Formally, it is convenient to think of M^{θ} as the set of elements { mθ | m ∈ M } with A-action mθ · a = (mθ(a))θ, for m ∈ M and a ∈ A.

3.2. Lemma. Suppose that 0 ≤ b ≤ n. Then σ(v_{b}) = v_{b} and τ(v_{b}) ∈ v_{b}ℋ_{r,p,n}. Consequently, (v_{b}ℋ_{r,n})^{σ} = v_{b}ℋ_{r,n} and (v_{b}ℋ_{r,p,n})^{τ} = v_{b}ℋ_{r,p,n} as ℋ_{r,p,n}-modules.

Proof. Since σ(T_{i}) = T_{i}, for 1 ≤ i < n, and σ(L_{k}^{p}) = L_{k}^{p}, for 1 ≤ k ≤ n, we see that σ(v_{b}) = v_{b}. Furthermore, T_{0}v_{b} = v_{b}L_{b_{1,κ−1}+1} and v_{b}T_{0} = L_{b_{2,κ}+1}v_{b}, by Proposition 2.4, so τ(v_{b}) = T_{0}^{−1}v_{b}T_{0} = v_{b}L_{b_{1,κ−1}+1}^{−1}L_{1} = v_{b}S_{b_{1,κ−1}+1}^{−1} ∈ v_{b}ℋ_{r,p,n}. ¿From what we have proved, σ(v_{b}ℋ_{r,n}) = v_{b}ℋ_{r,n}. Consequently, the map v_{b}h ↦ σ(v_{b}h) = σ(v_{b})σ(h) defines a module isomorphism v_{b}ℋ_{r,n} ≅ (v_{b}ℋ_{r,n})^{σ}, for h ∈ ℋ_{r,n}. Similarly, (v_{b}ℋ_{r,p,n})^{τ} ≅ v_{b}ℋ_{r,p,n} as ℋ_{r,p,n}-modules. □

3.3. Proposition. Suppose that 0 ≤ b ≤ n. Then

\{v_{b}L_{1}^{c_{1}} ⋯ L_{n}^{c_{n}}T_{w} | \begin{matrix}w ∈ 𝔖_{n} and 0 ≤ c_{i} < pt_{α} whenever b_{1,α−1} < i ≤ b_{1,α} \\ and c_{1} + ⋯ + c_{n} ≡ 0 (mod p) \end{matrix}\}

is a basis of v_{b}ℋ_{r,p,n}. In particular, dim v_{b}ℋ_{r,p,n} = \frac{1}{p} dim v_{b}ℋ_{r,n}.

Proof. First, observe that dim v_{b}ℋ_{r,p,n} ≥ \frac{1}{p} dim v_{b}ℋ_{r,n} since ℋ_{r,n} is a free ℋ_{r,p,n}– module of rank p. By Corollary 2.15 the number of the elements given in the statement of the Proposition is exactly \frac{1}{p} dim v_{b}ℋ_{r,n}. Therefore, it suffices to show that the elements in the statement in the Proposition span v_{b}ℋ_{r,p,n}.

By Lemma 3.1 the module v_{b}ℋ_{r,p,n} is spanned by the elements

\{v_{b}L_{1}^{c_{1}} ⋯ L_{n}^{c_{n}}T_{w} | \begin{matrix}w ∈ 𝔖_{n}, 0 ≤ c_{i} < r for 1 ≤ i ≤ n \\ c_{1} + ⋯ + c_{n} ≡ 0 (mod p) \end{matrix}\}.

The elements L_{1}, …, L_{n} commute by [2, Lemma 3.3]. Therefore, to prove the Proposition it is enough to show for α = 1, …, κ that if b_{1,α−1} < i ≤ b_{1,α} then v_{b}L_{i−1}^{c} is a linear combination of terms of the form v_{b}L_{b_{1,α−1}+1}^{a_{b_{1,α−1}+1}} … L_{i−1}^{a_{i−1}}T_{w}, where 0 ≤ a_{j} < pt_{α} for all j and w is an element of the symmetric group on the letters {b_{1,α−1} + 1, …, i − 1}. We prove this by induction on i.

Fix α such that b_{α} ≠ 0 and 1 ≤ α ≤ κ. Suppose first that i = i_{0} ≤ b_{1,α}, where i_{0} = b_{1,α−1} + 1. Recall from Lemma 2.6 that

v_{b} · ∏_{Q_{i}∈Q_{α}}(L_{i_{0}}^{p} − Q_{i}^{p}) = 0.

Therefore, v_{b}L_{i_{0}}^{pt_{α}} can be written as a linear combination of the elements v_{b}L_{i_{0}}^{pk}, for 0 ≤ k < t_{α}. Note that, modulo p, we have not changed the exponent of L_{i_{0}}. Hence, we may assume that 0 ≤ c_{i} < pt_{α} when i = i_{0}. Now suppose that i_{0} < i ≤ b_{1,α}. Arguing by induction (see [2, Lemma 3.3]), it follows easily that

(3.4)  L_{i}^{c} = q^{−1}T_{i−1}L_{i−1}^{c}T_{i−1} + (1 − q^{−1}) ∑_{d=1}^{c−1} L_{i−1}^{c−d}L_{i}^{d}T_{i−1}.

Therefore, using Proposition 2.5,

v_{b}L_{i}^{c} = q^{−1}v_{b}T_{i−1}L_{i−1}^{c}T_{i−1} + (1 − q^{−1}) ∑_{d=1}^{c−1} v_{b}L_{i−1}^{c−d}L_{i}^{d}T_{i−1}

= q^{−1}T_{w_{b}^{−1}(i−1)}v_{b}L_{i−1}^{c}T_{i−1} + (1 − q^{−1}) ∑_{d=1}^{c−1} v_{b}L_{i−1}^{c−d}L_{i}^{d}T_{i−1}.

If c ≥ pt_{α} then, by induction on i, we can rewrite v_{b}L_{i−1}^{c} as a linear combination of terms of the form v_{b}L_{i_{0}}^{a_{i_{0}}} … L_{i−1}^{a_{i−1}}T_{w}, where 0 ≤ a_{j} < pt_{α} for all j and w is an element of the symmetric group on the letters {i_{0}, …, i − 1}. Now, L_{1}, …, L_{n} commute with each other, and T_{i−1} commutes with L_{j} if j ≠ i − 1, i, so

T_{w_{b}^{−1}(i−1)}v_{b}L_{i_{0}}^{a_{i_{0}}} … L_{i−1}^{a_{i−1}}T_{w}T_{i−1} = v_{b}T_{i−1}L_{i_{0}}^{a_{i_{0}}} … L_{i−1}^{a_{i−1}}T_{w}T_{i−1}
= v_{b}L_{i_{0}}^{a_{i_{0}}} … L_{i−2}^{a_{i−2}}T_{i−1}L_{i−1}^{a_{i−1}}T_{w}T_{i−1}

Hence, using (3.4) once again, we can rewrite v_{b}L_{i}^{c} as a linear combination of terms of the form v_{b}L_{i_{0}}^{a_{i_{0}}} … L_{i}^{a_{i}}T_{w}, where 0 ≤ a_{j} < pt_{α} for all j and w is an element of the symmetric group on the letters {i_{0}, …, i}. This proves our claim. Moreover, this completes the proof of the Proposition because, modulo p, the sums of the exponents of L_{1}, …, L_{n} are unchanged in all of the formulae above. □

If M is an ℋ_{r,p,n}–module let M↑_{ℋ_{r,p,n}}^{ℋ_{r,n}} = M ⊗_{ℋ_{r,p,n}} ℋ_{r,n} be the corresponding

induced ℋ_{r,n}–module. Similarly, if N is an ℋ_{r,n}–module let N↓_{ℋ_{r,p,n}}^{ℋ_{r,n}} be the restriction of N to ℋ_{r,p,n}. Since ℋ_{r,n} is free as an ℋ_{r,p,n}–module both induction and restriction are exact functors.

3.5. Corollary. a) v_{b}ℋ_{r,n} ≅ (v_{b}ℋ_{r,p,n})↑_{ℋ_{r,p,n}}^{ℋ_{r,n}} ,

b) (v_{b}ℋ_{r,n})↓_{ℋ_{r,p,n}}^{ℋ_{r,n}} ≅ (v_{b}ℋ_{r,p,n})^{⊕p},

c) (v_{b}ℋ_{r,n})↓_{ℋ_{r,p,n}}^{ℋ_{r,n}} ↑_{ℋ_{r,p,n}}^{ℋ_{r,n}} ≅ (v_{b}ℋ_{r,n})^{⊕p}.

Proof. Since ℋ_{r,n} = ⨁_{k=0}^{p−1} T_{0}^{k}ℋ_{r,p,n}, there is a surjective homomorphism from (v_{b}ℋ_{r,p,n})↑_{ℋ_{r,p,n}}^{ℋ_{r,n}} onto v_{b}ℋ_{r,n}. By Corollary 2.15 and Proposition 3.3 both modules have the same dimension so this map must be an isomorphism, proving (a). Part (b) now follows from Proposition 2.14b(iii); alternatively, use part (a) and Lemma 3.2. Part (c) follows from parts (a) and (b). □

3.6. Corollary. Suppose that 0 ≤ b ≤ n as above. Then

\frac{1}{p} dim End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}) = dim End_{ℋ_{r,n}} (v_{b}ℋ_{r,n}) = p dim End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,p,n}).

Proof. The left and right hand equalities follow using Corollary 3.5 and Frobenius reciprocity. □

We can now prove Theorem A from the introduction.

3.7. Theorem. Suppose that Q = Q_{1} ∨ ⋯ ∨ Q_{κ}, where Q_{i} ∈ Q_{α} and Q_{j} ∈ Q_{β} are in the same (ε, q)–orbit only if α = β. Then ℋ_{r,p,n} is Morita equivalent to the algebra

⨁_{b∈Λ(n,κ)} ℋ_{b} ⋊ ℤ_{p}

Proof. By Proposition 2.14, ⨁_{b∈Λ(n,κ)} V^{b} is a progenerator for ℋ_{r,n}. Hence, by restriction, it is also a progenerator for ℋ_{r,p,n}. By Proposition 2.12(b), Frobenius reciprocity and Corollary 3.5(c) if b ≠ c then Hom_{ℋ_{r,p,n}} (V^{b}, V^{c}) = 0. Therefore, ℋ_{r,p,n} is Morita equivalent to

End_{ℋ_{r,p,n}} ( ⨁_{b∈Λ(n,κ)} V^{b}) = ⨁_{b∈Λ(n,κ)} End_{ℋ_{r,p,n}} (V^{b}).

Hence, to prove the Proposition it suffices to show that

End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}) ≅ ℋ_{b} ⋊ ℤ_{p}.

for all b ∈ Λ(n, κ).

Recall that each of the algebras ℋ_{pt_{α},b_{α}}(Q_{α}), for 1 ≤ α ≤ κ, has an automorphism σ_{α} of order p. The automorphism σ_{1} ⊗ ⋯ ⊗ σ_{κ} acts diagonally on the algebra ℋ_{b}. Note that ⟨σ_{1} ⊗ ⋯ ⊗ σ_{κ}⟩ ≅ ℤ_{p}. By Proposition 2.14, we have that

ℋ_{b} ≅ End_{ℋ_{r,n}} (v_{b}ℋ_{r,n}) ↪ End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}).

On the other hand, σ(v_{b}) = v_{b} by Lemma 3.2. So, σ induces an automorphism of v_{b}ℋ_{r,p,n} which is given by v_{b}h ↦ σ(v_{b}h) = v_{b}σ(h), for all h ∈ ℋ_{r,p,n}. Hence, we have an injective map ℤ_{p} ↪ End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}). Since σ is an outer automorphism of ℋ_{r,p,n} it follows that we have an embedding

ℋ_{b} ⋊ ℤ_{p} ↪ End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}).

Using Corollary 3.6 to compare the dimensions on both sides of this equation, we conclude that End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,n}) ≅ ℋ_{b} ⋊ ℤ_{p}. □

If instead of V^{b} = v_{b}ℋ_{r,n} we consider the ℋ_{r,p,n}–module v_{b}ℋ_{r,p,n} then we obtain a second Morita reduction theorem for ℋ_{r,p,n}. To state this result, if b ∈ Λ(n, κ) set ℋ_{p,b} = ℋ_{pt_{1},p,b_{1}}(Q_{1}) ⊗ ⋯ ⊗ ℋ_{pt_{κ},p,b_{κ}}(Q_{κ}). Observe that ℋ_{p,b} is a subalgebra of ℋ_{b} and that dim ℋ_{b} = p^{κ} dim ℋ_{p,b}. Next, let ℋ′_{p,b} be the sub-

algebra of ℋ_{b} generated by ℋ_{p,b} and the elements { T_{0}^{(1)}(T_{0}^{(α)})^{−1} | 1 < α ≤ κ }.

3.8. Proposition. Suppose that Q = Q_{1}∨⋯∨Q_{κ}, where Q_{i} and Q_{j} are in different (ε, q)–orbits whenever Q_{i} ∈ Q_{α} and Q_{j} ∈ Q_{β} for some α ≠ β. Let b ∈ Λ(n, κ). Then ℋ_{r,p,n} is Morita equivalent to the algebra

⨁_{b∈Λ(n,κ)} ℋ′_{p,b}.

Proof. By Proposition 2.14(a), ⨁_{b} v_{b}ℋ_{r,n} is a progenerator for ℋ_{r,n}. Therefore, ⨁_{b} v_{b}ℋ_{r,p,n} is a progenerator for ℋ_{r,p,n} by Corollary 3.5(b). Furthermore, if b ≠ c ∈ Λ(n, κ) then Hom_{ℋ_{r,n}} (v_{b}ℋ_{r,n}, v_{c}ℋ_{r,n}) = 0 by Proposition 2.12(b). By Corollary 3.5 and Frobenius reciprocity, Hom_{ℋ_{r,p,n}} (v_{b}ℋ_{r,p,n}, v_{c}ℋ_{r,p,n}) = 0. Combining these results we see that ℋ_{r,p,n} and

End_{ℋ_{r,p,n}} ( ⨁_{b∈Λ(n,κ)} v_{b}ℋ_{r,p,n}) = ⨁_{b∈Λ(n,κ)} End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,p,n})

are Morita equivalent.

Fix b ∈ Λ(n, κ) and let E_{b} = End_{ℋ_{r,p,n}} (v_{b}ℋ_{r,p,n}). To complete the proof it is enough to show that E_{b} ≅ ℋ′_{p,b}. As a left ℋ_{b}–module, v_{b}ℋ_{r,n} is isomorphic to a direct sum of [𝔖_{n}:𝔖_{b}] copies of the regular representation of ℋ_{b} by Corollary 2.15. By Proposition 2.14b(iii), ℋ_{p,b} acts faithfully on v_{b}ℋ_{r,p,n} by restriction and this action commutes with the action of ℋ_{r,p,n} from the right. Hence, we can identify ℋ_{p,b} with a subalgebra of E_{b}.

By Lemma 3.2 and Proposition 2.14b(iii), τ(v_{b}) = L_{1}^{−1}L_{b_{2,κ}+1}v_{b} ∈ v_{b}ℋ_{r,p,n} acts on v_{b}ℋ_{r,p,n} in the same way that T_{0}^{(1)}(T_{0}^{(κ)})^{−1} = T_{0} ⊗ 1 ⊗ ⋯ ⊗ 1 ⊗ T_{0}^{−1} ∈ ℋ_{b} acts on v_{b}ℋ_{r,p,n}. More generally, for α = 1, …, κ − 1 let ρ_{α} be the automorphism of v_{b}ℋ_{r,p,n} given by left multiplication by L_{b_{2,κ}+1}L_{b_{α+1,κ}+1}^{−1} ∈ ℋ_{b}. By Proposition 2.4,

L_{b_{2,κ}+1}L_{b_{α+1,κ}+1}^{−1}v_{b} = v_{b}L_{1}L_{b_{1,α−1}+1}^{−1} = v_{b}S_{b_{1,α−1}+1}^{−1} ∈ v_{b}ℋ_{r,p,n}.

Therefore, ρ_{α} ∈ E_{b}, for 2 ≤ α ≤ κ, since ρ_{α} commutes with the action of ℋ_{r,p,n}. Thus, ρ_{α} coincides with the action of T_{0}^{(1)}(T_{0}^{(α)})^{−1} ∈ ℋ_{b} on V^{b} in Proposition 2.14. Consequently, ρ_{α} ∉ ℋ_{p,b} and the automorphisms ρ_{2}, …, ρ_{κ} commute. By definition, ℋ′_{p,b} is isomorphic to the subalgebra of E_{b} generated by ℋ_{p,b} and ρ_{2}, …, ρ_{κ}. Hence, ℋ′_{p,b} is a subalgebra of E_{b}.

For each α, the map ρ_{α}^{p} acts as left multiplication by (T_{0}^{(1)})^{p}(T_{0}^{(α)})^{−p} ∈ ℋ_{b}, so ρ_{α}^{p} ∈ ℋ_{p,b}. Note that the extensions of the endomorphisms ρ_{α}, …, ρ_{α}^{p−1} to End_{ℋ_{r,n}} (v_{b}ℋ_{r,n}), for 2 ≤ α ≤ κ, are all linearly independent since End_{ℋ_{r,n}} (V^{b}) ≅ ℋ_{b} by Proposition 2.14. As these maps act on different components of ℋ_{p,b} it follows that dim ℋ′_{p,b} = p^{κ−1} dim ℋ_{p,b}. However, by Lemma 3.6 and Proposition 2.14b(ii),

dim E_{b} = \frac{1}{p} dim End_{ℋ_{r,n}} (V^{b}) = \frac{1}{p} dim ℋ_{b} = p^{κ−1} dim ℋ_{p,b}.

Therefore, dim ℋ′_{p,b} = p^{κ−1} dim ℋ_{p,b} = dim E_{b}. So E_{b} ≅ ℋ′_{p,b}, as required. □

4. Splittable decomposition numbers

In this section we use Clifford theory to show that if an algebra can be written be a semidirect product then its decomposition numbers are determined by the corresponding“p′-splittable” decomposition numbers of a related family of algebras. First, we recall some general results about the representation theory of semidirect product algebras. The basic references for this topic are [4, 15, 18].

Let A be a finite dimensional algebra over an algebraically closed field K and suppose that θ is an algebra automorphism of A of order p. We identify ℤ_{p} with the group generated by θ and consider the algebra A ⋊ ℤ_{p}. Then, as a set,

A ⋊ ℤ_{p} = { aθ^{k} | a ∈ A and 0 ≤ k ≤ p }

and the multiplication in A ⋊ ℤ_{p} is defined by

(aθ^{k}) · (bθ^{m}) = aθ^{k}(b) · θ^{k+m},

for a, b ∈ A and 0 ≤ k, m < p. If H is a subgroup of ℤ_{p} we identify A ⋊ H with a subalgebra of A ⋊ ℤ_{p} in the natural way. In particular, by taking H = 1 we can view A as a subalgebra of A ⋊ ℤ_{p}. Moreover, there are natural induction and restriction functors between the module categories of all of these algebras.

Suppose that L is an A–module. Then we can twist L by θ to get a new simple A-module L^{θ}. As a vector space we set L^{θ} = L, and we define the action of A on L^{θ} by

v · a := vθ(a), for all v ∈ L and a ∈ A.

It is straightforward to check that L is irreducible if and only if L^{θ} is irreducible. The inertia group of L is the group

G_{L} := { θ^{k} ∈ ℤ_{p} | L ≅ L^{θ^{k}} } .

Then G_{L} is a subgroup of ℤ_{p} and, in particular, it is cyclic. Let l = |G_{L}|. Then p = lk and G_{L} is generated by θ^{k}. Recall that we have fixed a primitive pth root of unity ε ∈ K. Since K is algebraically closed we can choose an A-module isomorphism ϕ: L ⟶ L^{θ^{k}} such that ϕ^{l} = 1_{L}, the identity map on L. For each integer i ∈ ℤ define the (A⋊G_{L})-module L_{l,i} as follows: as a vector space L_{l,i} := L and the action of (A ⋊ G_{L}) on L_{l,i} is given by:

v · (aθ^{mk}) := ε^{mki}ϕ^{m}(va), for all m ∈ ℤ, v ∈ L_{l,i} and a ∈ A.

It is easy to check that L_{l,i} is an (A ⋊ G_{L})-module and, by definition, L_{l,i+l} ≅ L_{l,i} for all i ∈ ℤ.

Recall that Irr(A) is the complete set of isomorphism classes of simple A-modules. Let L ∈ Irr(A). Then, since L_{l,i} ↓_{A}≅ L, it follows that L_{l,i} is a simple (A ⋊ G_{L})- module. In fact, it is shown in [4, 15, 18] that

{L_{l,1}↑_{A⋊G_{L}}^{A⋊ℤ_{p}}, ⋯, L_{l,l}↑_{A⋊G_{L}}^{A⋊ℤ_{p}} | L ∈ Irr(A), l = |G_{L}|}

is a complete set of pairwise non–isomorphic simple (A ⋊ ℤ_{p})-modules.

4.1. Lemma. Suppose that L is a simple A–module and that H is a subgroup of G_{L}. Let l = |G_{L}|, h = |H| and fix i with 1 ≤ i ≤ l. Then L_{l,i}↓_{A⋊H}^{A⋊G_{L}} is an irreducible (A ⋊ H)-module and

L_{l,i}↓_{A⋊H}^{A⋊G_{L}}↑_{A⋊H}^{A×G_{L}} ≅ ⨁_{j=1}^{[G_{L}:H]} L_{l,i+hj}.

Proof. As remarked above, we can view A and A ⋊ H as subalgebras of A ⋊ G_{L}. Since L_{l,i}↓_{A}^{A⋊G_{L}} ≅ L is irreducible we see that L_{l,i}↓_{A⋊H}^{A⋊G_{L}} is irreducible. Next, observe that θ^{p/h} is a generator of H. Hence, by the argument e if 1 ≤ j ≤ l = |G_{L}| then L_{l,i+hj}↓_{A⋊H}^{A⋊G_{L}} ≅ L_{l,i}↓_{A⋊H}^{A⋊G_{L}} is irreducible. Therefore, by Frobenius reciprocity and Schur’s Lemma,

Hom_{A⋊G_{L}}(L_{l,i+hj}, L_{l,i}↓_{A⋊H}^{A⋊G_{L}} ↑_{A⋊H}^{A⋊G_{L}} ) ≅ Hom_{A⋊H}(L_{l,i+hj} ↓_{A⋊H}^{A⋊G_{L}}, L_{l,i}↓_{A⋊H}^{A⋊G_{L}} )
≅ K.

The lemma now follows by comparing dimensions. □

Now suppose that we have a modular system (F, O, K) such that A = A_{K} has an O-lattice A_{O} which is an O-algebra, θ can be lifted to an automorphism of A_{O} of order p, F is an algebraically closed field of characteristic zero, and A_{F} := A_{O} ⊗_{O} F is a (split) semisimple F -algebra. We abuse notation and write θ for the corresponding automorphism of A_{F}. Note that if H is a subgroup of ℤ_{p} then A_{K} ⋊ H ≅ (A_{O} ⋊ H) ⊗_{O} K, so that we also have a modular reduction system for the algebras A_{F} ⋊ H and A_{K} ⋊ H.

By definition, Irr(A_{F}) is the complete set of isomorphism classes of simple A_{F} - modules — the “semisimple” A_{F} –modules. As the automorphism θ lifts to A_{F} for each simple A_{F} –module S ∈ Irr(A_{F}) we have an inertia group G_{S} ≤ ℤ_{p} and, as above, we can define (A_{F} ⋊ G_{S})-modules S_{s,j}, for j ∈ ℤ where s = |G_{S}|. Consequently,

{S_{s,1}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}, ⋯, S_{s,s}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}} | S ∈ Irr(A_{F}) and s = |G_{S}|}

is a complete set of pairwise non-isomorphic simple (A_{F} ⋊ ℤ_{p})-modules.

Suppose that S ∈ Irr(A_{F}) and that D ∈ Irr(A_{K}) and let s = |G_{S}| and d = |G_{D}|. Given i and j with 1 ≤ i ≤ s and 1 ≤ j ≤ d, we want to determine the decomposition numbers

[S_{s,i}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}} : D_{d,j}↑_{A⋊G_{D}}^{A⋊ℤ_{p}}],

which gives the multiplicity of D_{d,j}↑_{A⋊G_{D}}^{A⋊ℤ_{p}} as an irreducible composition factor of a modular reduction of S_{s,i}↑_{A⋊G_{S}}^{A⋊ℤ_{p}}.

4.2. Definition. Suppose that S ∈ Irr(A_{F}) and D ∈ Irr(A_{K}) and set s = |G_{S}| and d = |G_{D}|. Then the pair (S, D) has cyclic decomposition numbers if

[S_{s,i} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i+1} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+1} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}],

for all i, j ∈ ℤ.

In the next section we show that all pairs of irreducible ℋ_{r,n}-modules have cyclic decomposition numbers.

4.3. Proposition. Let S ∈ Irr(A_{F}) and D ∈ Irr(A_{K}) and suppose that (S, D) have cyclic decomposition numbers. Set s = |G_{S}|, d = |G_{D}| and let d_{0} = gcd(s, d).

Then

[S_{s,i+d_{0}l} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}l′} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}]

for all i, i′j, j′, l, l′ ∈ ℤ.

Proof. Since the groups G_{S}, G_{D} and G_{0} = G_{S} ∩ G_{D} are all cyclic subgroups of ℤ_{p}, we have that |G_{0}| = gcd(|G_{S}|, |G_{D}|) = gcd(s, d) = d_{0}. Therefore, there exist integers u and v such that d_{0} = us + vd. Consequently, if 1 ≤ i ≤ s, 1 ≤ j ≤ d and l ∈ ℤ then

[S_{s,i} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}l} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i+us} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}l+us} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}]
= [S_{s,i+us} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}l+d_{0}−vd} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}]
= [S_{s,i} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}(l+1)} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}],

where the first equality follows since,by assumption, (S, D) has cyclic decomposition numbers. Similarly, we have that

[S_{s,i+d_{0}l} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i+d_{0}(l+1)} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}],

whenever 1 ≤ i ≤ s, 1 ≤ j ≤ d and l ∈ ℤ. Together these two identities imply that if 1 ≤ i, i′ ≤ s, 1 ≤ j, j′ ≤ d and l, l′ ∈ ℤ then

[S_{s,i+d_{0}l} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j+d_{0}l′} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i} ↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}}: D_{d,j} ↑_{A⋊G_{D}}^{A⋊ℤ_{p}}].

This completes the proof. □

4.4. Definition. Suppose that S ∈ Irr(A_{F}) and D ∈ Irr(A_{K}) and let s = |G_{S}| and d = |G_{D}| as above. Then the decomposition number [S_{s,i}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}} : D_{d,j}↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] is p-splittable if G_{S} = ℤ_{p} = G_{D}.

We will see in Corollary 5.6 below that this definition of p–splittable decomposition number agrees with Definition 1.1 when applied to the algebras ℋ_{r,p,n}.

4.5. Corollary. Suppose that S ∈ Irr(A_{F}) and D ∈ Irr(A_{K}) have cyclic decomposition numbers and let G_{0} = G_{S} ∩ G_{D}, s = |G_{S}| and d = |G_{D}|. Then

[S_{s,i}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}} : D_{d,j}↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = [S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} : D_{d,j}↓_{A⋊G_{0}}^{A⋊G_{D}}]

for all 1 ≤ i ≤ s and all 1 ≤ j ≤ d. In particular, these decomposition numbers are p′-splittable in the sense of Definition 4.4, where p′ = |G_{0}|.

Proof. Observe that if L and L′ are non–isomorphic simple (A ⋊ G_{0})-modules then the induced modules L↑_{A⋊G_{0}}^{A⋊ℤ_{p}} and L′↑_{A⋊G_{0}}^{A⋊ℤ_{p}} have no common irreducible composition factors. Next observe that if 1 ≤ i ≤ s then S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} is an irreducible (A ⋊ G_{0})- module by Lemma 4.1. Similarly, if 1 ≤ j ≤ d then D_{d,j}↓_{A⋊G_{0}}^{A⋊G_{S}} is an irreducible (A ⋊ G_{0})-module. The first claim now follows by combining this observation with Lemma 4.1 and Proposition 4.3. Finally, these decomposition numbers are p′- splittable because the inertia groups of the modules S_{s,i} and D_{d,j} inside G_{0} is exactly G_{0}. □

Corollary 4.5 reduces the calculation of decomposition numbers of the algebra A ⋊ ℤ_{p} to determining the p′-splittable decomposition numbers of the subalgebras A ⋊ ℤ_{p′}, where p′ divides p.

We now apply Corollary 4.5 in the special case where A has a tensor product decomposition. That is, we suppose that A = A^{(1)} ⊗ ⋯ ⊗ A^{(κ)}, for some finite dimensional K–algebras A^{(1)}, …, A^{(κ)} which are equipped with automorphisms θ_{1}, …, θ_{κ}, respectively, of order p. Set θ = θ_{1} ⊗⋯⊗θ_{κ}. We assume that the tensor product decomposition of A is compatible with the modular system (F, O, K) so that A_{F} = A_{F}^{(1)} ⊗ ⋯ ⊗ A_{F}^{(κ)}.

Fix an integer f ∈ {1, …, κ} and let S^{(f)} ∈ Irr(A_{F}^{(f)}) and D^{(f)} ∈ Irr(A_{f}) be irreducible modules. Then S = S^{(1)} ⊗ ⋯ ⊗ S^{(κ)} ∈ Irr(A_{F}) and D = D^{(1)} ⊗ ⋯ ⊗ D^{(κ)} ∈ Irr(A_{K}). Further, G_{S} = G_{S^{(1)}} ∩ ⋯ ∩ G_{S^{(κ)}}, and G_{D} = G_{D^{(1)}} ∩ ⋯ ∩ G_{D^{(κ)}}. Set s = |G_{S}| and d = |G_{D}|. Then s = |G_{S}| = gcd(|G_{S^{(1)}}|, …, |G_{S^{(κ)}}|) and d = |G_{D}| = gcd(|G_{D^{(1)}}|, …, |G_{D^{(κ)}}|).

Suppose that (S, D) has cyclic decomposition numbers and let G_{0} = G_{S} ∩ G_{D}. Then, by Corollary 4.5, the decomposition numbers of A ⋊ ℤ_{p} are completely determined by the decomposition numbers of the form [S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} : D_{d,j}↓_{A⋊G_{0}}^{A_{K}⋊G_{D}}], for 1 ≤ i ≤ s and 1 ≤ j ≤ d.

4.6. Theorem. Suppose that S = S^{(1)} ⊗ ⋯ ⊗ S^{(κ)} ∈ Irr(A_{F}) and D = D^{(1)} ⊗ ⋯ ⊗ D^{(κ)} ∈ Irr(A_{K}). Let s = |G_{S}|, d = |G_{D}| and G_{0} = G_{S} ∩ G_{D} and set d_{0} = |G_{0}|. Then

[S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} : D_{d,j}↓_{A_{K}⋊G_{0}}^{A_{K}⋊G_{D}}] = ∑_{\begin{matrix}0≤j_{1},...,j_{κ}<d_{0} \\ j_{1}+···+j_{κ}≡(κ−1)i+j (mod d_{0}) \end{matrix}} ∏_{α=1}^{κ}[S_{d_{0},i}^{(α)} : D_{d_{0},j_{α}}^{(α)}]

for all i and j with 1 ≤ i ≤ s and 1 ≤ j ≤ d.

Proof. Suppose that R ∈ \{F, K\}. Let k = \frac{p}{d_{0}} = |ℤ_{p}/G_{0}|, so that G_{0} = ⟨θ^{k}⟩. Consider the algebra

Â_{R} = RG_{0} ⊗ (A_{R}^{(1)} ⋊ ⟨θ_{1}^{k}⟩) ⊗ ⋯ ⊗ (A_{R}^{(κ)} ⋊ ⟨θ_{κ}^{k}⟩).

Then it is straightforward to check that there is an embedding of algebras A_{R} ⋊ G_{0} ↪ Â_{R} given by

(a_{1} ⊗ ⋯ ⊗ a_{κ})θ^{mk} ↦ θ^{mk} ⊗ (a_{1}θ_{1}^{mk}) ⊗ ⋯ ⊗ (a_{κ}θ_{κ}^{mk}),

for a_{1}, ⋯, a_{κ} ∈ A_{R} and m ∈ ℤ.

Let L = L^{(1)} ⊗ ⋯ ⊗ L^{(κ)} be a simple A_{R}-module, where L = S if R = F, or L = D if R = K. Let l = |G_{L}| and suppose that 1 ≤ i ≤ l. Then L_{l,i} is a simple (A_{R} ⋊ G_{L})-module. Recall that the modules L_{l,i} are defined using a fixed isomorphism ϕ: L ⟶ L^{θ^{p/l}} satisfying ϕ^{l} = id. We may assume that ϕ is compatible with the tensor decomposition of L; that is, ϕ = ϕ̃_{1} ⊗⋯⊗ϕ̃_{κ}, where for f = 1, …, κ, the order of the inertia group G_{L^{(f)}} of L^{(f)} in ℤ_{p} is l_{f}, ϕ̃_{f} = (ϕ_{f})^{l_{f} /l},

ϕ_{f} : L^{(f)} ⟶(L^{(f)})^{θ_{f}^{p/l_{f}}} is an isomorphism defining the module L_{l_{f} ,i}^{(f)}. Note that

L_{l_{f} ,i}^{(f)} ↓_{A_{R}^{(f)}⋊⟨θ_{f}^{k}⟩}^{A_{R}^{(f)}⋊G_{L^{(f)}}} ≅ L_{d_{0},i}^{(f)}.

Applying the definitions, given any integers i, i_{0}, i_{1}, …, i_{κ} with i ≡ i_{1} + ⋯ + i_{κ} − i_{0} (mod d_{0}) there is a natural isomorphism of (A_{R} ⋊ G_{0})-modules

(†)  L_{l,i}↓_{A_{R}⋊G_{0}}^{A_{R}⋊G_{L}} ≅ (ε^{−i_{0}} ⊗ L_{d_{0},i_{1}}^{(1)} ⊗ ⋯ ⊗ L_{d_{0},i_{κ}}^{(κ)})↓Â_{R} ; A_{R}⋊G_{0},

where ε^{−i_{0}} is the one dimensional representation of G_{0} = ⟨θ^{k}⟩ upon which θ^{k} acts as multiplication by ε^{−i_{0}}.

By (†), if 1 ≤ i ≤ s then S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} ≅ (ε^{−(κ−1)i} ⊗ S_{d_{0},i}^{(1)} ⊗ ⋯ ⊗ S_{d_{0},i}^{(κ)})↓Â_{F} ; A_{F} ⋊G_{0}.

Therefore, we can find the (A_{F} ⋊ G_{0})-module composition factors of S_{s,i}↓_{A_{F} ⋊G_{0}}^{A_{F} ⋊G_{S}} by first finding the Â_{K}-module composition factors of ε^{−(κ−1)i} ⊗ S_{d_{0},i}^{(1)} ⊗ ⋯ ⊗ S_{d_{0},i}^{(κ)} and then restricting to A_{K} ⋊G_{0}. The Â_{K}-module composition factors of ε^{−(κ−1)i} ⊗ S_{d_{0},i}^{(1)} ⊗ ⋯ ⊗ S_{d_{0},i}^{(κ)} are all of the form ε^{−(κ−1)i} ⊗ D_{d_{0},j_{1}}^{(1)} ⊗ ⋯ ⊗ D_{d_{0},j_{κ}}^{(κ)}, where 0 ≤ j_{1}, …, j_{κ} < d_{0}. Further, by (†), if j ≡ j_{1} + ⋯ + j_{κ} − (κ − 1)i (mod d_{0}) then

D_{d,j}↓_{A_{K}⋊G_{0}}^{A_{K}⋊G_{D}} ≅ (ε^{−(κ−1)i} ⊗ D_{d_{0},j_{1}}^{(1)} ⊗ ⋯ ⊗ D_{d_{0},j_{κ}}^{(κ)})↓Â_{K} ; A_{K}⋊G_{0}

The theorem now follows. □

4.7. Corollary. With the same notation as in Theorem 4.6, and suppose that S ∈ Irr(A_{F}) and D ∈ Irr(A_{K}) have cyclic decomposition numbers, then we have

[S_{s,i}↑_{A_{F} ⋊G_{S}}^{A_{F} ⋊ℤ_{p}} : D_{d,j}↑_{A⋊G_{D}}^{A⋊ℤ_{p}}] = ∑_{\begin{matrix}0≤j_{1},...,j_{κ}<d_{0} \\ j_{1}+···+j_{κ}≡(κ−1)i+j (mod d_{0}) \end{matrix}} ∏_{α=1}^{κ}[S_{d_{0},i}^{(α)} : D_{d_{0},j_{α}}^{(α)}]

for all i and j with 1 ≤ i ≤ s and 1 ≤ j ≤ d.

Proof. This follows from Corollary 4.5 and Theorem 4.6. □

5. Reduction theorems for the decomposition numbers of ℋ_{r,p,n}

We now combine the results of last two sections to show the decomposition numbers of the cyclotomic Hecke algebras ℋ_{r,p,n} are completely determined by the p′-splittable decomposition numbers of an explicitly determined family of cyclotomic Hecke algebras. Throughout this section we assume that q ≠ 1. We first show that the simple ℋ_{r,n}-modules always have cyclic decomposition numbers.

5.1. Lemma. The algebras ℋ_{r,p,n} and ℋ_{r,n} ⋊ ℤ_{p} are Morita equivalent.

Proof. As a right ℋ_{r,p,n}–module ℋ_{r,n} = ⨁_{k=0}^{p−1} T_{0}^{k}ℋ_{r,p,n} by Lemma 3.1. Consequently, ℋ_{r,p} is a progenerator for ℋ_{r,p,n}, so ℋ_{r,p,n} is Morita equivalent to End_{ℋ_{r,p,n}} (ℋ_{r,p}). Observe that σ ∈ End_{ℋ_{r,p,n}} (ℋ_{r,n}) since σ is trivial on ℋ_{r,p,n}. Furthermore, as vector spaces,

End_{ℋ_{r,p,n}} (ℋ_{r,n}) ≅ Hom_{ℋ_{r,p,n}} (ℋ_{r,n}↓_{ℋ_{r,p,n}}^{ℋ_{r,n}} , ℋ_{r,p,n}^{⊕p})
≅ Hom_{ℋ_{r,p,n}} (ℋ_{r,n}↓_{ℋ_{r,p,n}}^{ℋ_{r,n}} , ℋ_{r,p,n})^{⊕p}
≅ Hom_{ℋ_{r,n}} (ℋ_{r,n}, ℋ_{r,p,n}↑_{ℋ_{r,p,n}}^{ℋ_{r,n}} )^{⊕p}
≅ Hom_{ℋ_{r,n}} (ℋ_{r,n}, ℋ_{r,n})^{⊕p} ≅ ℋ_{r,n}^{⊕p}.

where the third isomorphism comes from Frobenius reciprocity. Hence, by counting dimensions, End_{ℋ_{r,p,n}} (ℋ_{r,n}) ≅ ℋ_{r,n} ⋊ ℤ_{p}. (Alternatively, apply Theorem A with κ = 1.) □

The proof of this Lemma gives a Morita equivalence from the category of (finite dimensional right) ℋ_{r,n}^{R} ⋊ ℤ_{p}–modules to the category of ℋ_{r,p,n}^{R}–modules. This functor sends the finite dimensional (ℋ_{r,n} ⋊ ℤ_{p})–module M to the ℋ_{r,p,n}–module

F (M) = M ⊗_{ℋ_{r,n}^{R}⋊ℤ_{p}} ℋ_{r,n}^{R}.

5.2. Lemma. Suppose that L is a simple ℋ_{r,n}^{R}–module. Then

L↓_{ℋ_{r,p,n}^{R}}^{ℋ_{r,n}^{R}} ≅ F (L↑_{ℋ_{r,n}^{R}}^{ℋ_{r,n}^{R}⋊ℤ_{p}} ).

Proof. Applying the definitions and standard properties of tensor products,

F (L↑_{ℋ_{r,n}^{R}}^{ℋ_{r,n}^{R}⋊ℤ_{p}} ) = (L ⊗_{ℋ_{r,n}^{R}} ℋ_{r,n}^{R} ⋊ ℤ_{p}) ⊗_{ℋ_{r,n}^{R}⋊ℤ_{p}} ℋ_{r,n}^{R} ≅ L = L↓_{ℋ_{r,p,n}^{R}}^{ℋ_{r,n}^{R}} .

□

5.3. Proposition ( [7, (2.2)], [19, (2.2)] and [9, (5.4), (5.5), (5.6)]). Suppose that L is a simple ℋ_{r,n}^{R}-module and let k > 0 be minimal such that L ≅ L^{σ^{k}}. Then

1 ≤ k ≤ p and l := \frac{p}{k} is the smallest positive integer such that L′ ≅ L′^{τ^{l}} whenever L′ is a simple ℋ_{r,p,n}-submodule of L.

Now fix an isomorphism ϕ: L⟶L^{σ^{k}} such that ϕ^{l} = 1 and for i ∈ ℤ define

L_{i} := {v ∈ L | ϕ(v) = ε^{−ik}v}.

Then L↓_{ℋ_{r,p,n}^{R}}^{ℋ_{r,n}^{R}} = L_{0} ⊕ ⋯ ⊕ L_{l−1}. Moreover, L_{i} = L_{i+l} and L_{i+1} ≅ L_{i}^{τ}, for any

i ∈ ℤ. Consequently, L↓_{ℋ_{r,p,n}^{R}}^{ℋ_{r,n}^{R}} ≅ L_{0} ⊕ L_{0}^{τ} ⊕ ⋯ ⊕ L_{0}^{τ^{l−1}}.

For each simple ℋ_{r,n}^{R}-module L we henceforth fix an isomorphism ϕ: L ⟶ L^{σ^{k}} such that ϕ^{l} = 1, where k and l = \frac{p}{k} as in the Lemma. Observe that l = |G_{L}|, where G_{L} is the inertia group of L. For each integer i we have defined ℋ_{r,p,n}–modules L_{i} and L_{l,i}. The next result gives the connection between these two modules.

5.4. Lemma. Suppose that L is a simple ℋ_{r,n}^{R}-module with inertia group G_{L} and let l = |G_{L}|. Then, for each i ∈ ℤ, we have

L_{i} ≅ F (L_{l,i}↑_{ℋ_{r,n}^{R}⋊G_{L}}^{ℋ_{r,n}^{R}⋊ℤ_{p}} ).

Proof. As in the proof of Lemma 5.2, F (L_{l,i}↑_{ℋ_{r,n}^{R}⋊G_{L}}^{ℋ_{r,n}^{R}⋊ℤ_{p}} ) ≅ L_{l,i} ⊗_{ℋ_{r,n}^{R}⋊G_{L}} ℋ_{r,n}^{R}.

Therefore, there is a natural map ψ: F (L_{l,i}↑_{ℋ_{r,n}^{R}⋊G_{L}}^{ℋ_{r,n}^{R}⋊ℤ_{p}} )⟶L given by ψ(v ⊗_{ℋ_{r,n}⋊ℤ_{d}}

h) = vh, for v ∈ L_{l,i} and h ∈ ℋ_{r,n}^{R}. Clearly, ψ ≠ 0 so it suffices to show that the image of ψ is contained in L_{i}. Now, if v ∈ L_{l,i} and h ∈ ℋ_{r,n}^{R} as above then, using the definition of L_{l,i}, we see that

ε^{ik}ϕ(vh) = v · (hσ^{k}) = ψ(v · (hσ^{k}) ⊗_{ℋ_{r,n}⋊ℤ_{d}} 1) = ψ(v ⊗_{ℋ_{r,n}⋊ℤ_{d}} h) = vh.

Hence, ψ(vh) ∈ L_{i} as required. □

5.5. Corollary. Suppose that S ∈ Irr(ℋ_{r,n}) and D ∈ Irr(ℋ_{r,n}^{K}) and let s = |G_{S}| and d = |G_{D}|. Then

[S_{s,i} ↑_{ℋ_{r,n}⋊G_{S}}^{ℋ_{r,n}⋊ℤ_{p}} : D_{d,j} ↑_{ℋ_{r,n}⋊G_{D}}^{ℋ_{r,n}⋊ℤ_{p}} ] = [S_{s,i+1} ↑_{ℋ_{r,n}⋊G_{S}}^{ℋ_{r,n}⋊ℤ_{p}} : D_{d,j+1} ↑_{ℋ_{r,n}⋊G_{D}}^{ℋ_{r,n}⋊ℤ_{p}} ],

for all i, j ∈ ℤ. That is, (S, D) has cyclic decomposition numbers.

Proof. Using Lemma 5.4 and Proposition 5.3 we have

[S_{s,i+1} ↑_{ℋ_{r,n}⋊G_{S}}^{ℋ_{r,n}⋊ℤ_{p}} : D_{d,j+1} ↑_{ℋ_{r,n}⋊G_{D}}^{ℋ_{r,n}⋊ℤ_{p}} ] = [S_{i+1} : D_{j+1}] = [S_{i}^{τ} : D_{j}^{τ}] = [S_{i} : D_{j}]
= [S_{s,i} ↑_{ℋ_{r,n}⋊G_{S}}^{ℋ_{r,n}⋊ℤ_{p}} : D_{d,j} ↑_{ℋ_{r,n}⋊G_{D}}^{ℋ_{r,n}⋊ℤ_{p}} ],

as required. □

Note that Proposition 5.3 and Lemma 5.4 also imply that the two notions of p–splittable decomposition numbers for the algebras ℋ_{r,p,n} coincide.

5.6. Corollary. Suppose that S ∈ Irr(ℋ_{r,n}^{F}) and D ∈ Irr(ℋ_{r,n}^{K}) and let s = |G_{S}| and d = |G_{D}|. Then the decomposition number

[S_{s,i}↑_{ℋ_{r,n}^{F}⋊G_{S}}^{ℋ_{r,n}^{F}⋊ℤ_{p}} : D_{d,j}↑_{ℋ_{r,n}^{K}⋊G_{D}}^{ℋ_{r,n}^{K}⋊ℤ_{p}} ]

is p–splittable in the sense of Definition 4.4 if and only if the decomposition number

[F (S_{s,i}↑_{ℋ_{r,n}^{F}⋊G_{S}}^{ℋ_{r,n}^{F}⋊ℤ_{p}} ) : F (D_{d,j}↑_{ℋ_{r,n}^{K}⋊G_{D}}^{ℋ_{r,n}^{K}⋊ℤ_{p}} )]

is p–splittable in the sense of Definition 1.1.

5.7. Theorem. Suppose that q ≠ 1. Then every decomposition number of ℋ_{r,p,n}(Q) is equal to a p′–splittable decomposition number of some cyclotomic Hecke algebra ℋ_{r,p′,n}(q, Q′), where p = kp′ and

Q′ = (Q_{1}, εQ_{1}, …, ε^{k−1}Q_{1}, Q_{2}, …, ε^{k−1}Q_{2}, …, Q_{t}, …, ε^{k−1}Q_{t}).

Proof. By Proposition 5.3 every irreducible ℋ_{r,p,n}^{F}-module is equal to S_{i} for some S ∈ Irr(ℋ_{r,n}^{F}) and with 1 ≤ i ≤ s = |G_{S}|. Similarly, every irreducible ℋ_{r,p,n}^{K}- module is equal to D_{j} for some D ∈ Irr(ℋ_{r,n}^{K}) and with 1 ≤ j ≤ s = |G_{D}|. Therefore, by Lemma 5.4, Corollary 5.5 and Corollary 4.5,

[S_{i} : D_{j}] = [S_{s,i}↓_{ℋ_{r,n}^{F}⋊G_{0}}^{ℋ_{r,n}^{F}⋊G_{S}} : D_{d,j}↓_{ℋ_{r,n}^{K}⋊G_{0}}^{ℋ_{r,n}^{F}⋊G_{D}} ],

where G_{0} = G_{S} ∩ G_{D}. Suppose that G_{0} = ⟨σ^{k}⟩ and write p = kp′. Then we have shown that [S_{i} : D_{j}] is a p′–splittable decomposition number of ℋ_{r,n} ⋊ ℤ_{p′}.

As at the beginning of section 2, write r = pt. Then r = p′kt and in ℋ_{r,p,n}(Q) the ‘order relation’ for T_{0} is

0 = ∏_{b=1}^{t} (T_{0}^{p} − Q_{b}^{p}) = ∏_{b=1}^{t} ∏_{a=0}^{k−1} (T_{0}^{p′} − (ε^{a}Q_{b})^{p′}).

Observe that the right hand side is the ‘order relation’ for T_{0}^{p′} in ℋ_{r,p′,n}(Q′). It now follows using Lemma 5.1 that ℋ_{r,n}⋊ℤ_{p′} is Morita equivalent to ℋ_{r,p′,n}(q, Q′), where the parameters Q′ are as given in the statement of the theorem. This completes the proof of the theorem. □

Proof of Theorem B. This follows from an recursive application of Theorem A, Corollary 4.7, Corollary 5.5 and Theorem 5.7. □

Theorem B gives a recursive algorithm for computing all of the decomposition numbers of a cyclotomic Hecke algebra ℋ_{r,p,n}(Q) in terms of the p′-splittable decomposition numbers of a family of “smaller” cyclotomic Hecke algebras ℋ_{r′,p′,n′}(Q′), where 1 ≤ r′ ≤ r, 1 ≤ n′ ≤ n, 1 ≤ p′ | p, and the parameters Q′ are contained in a single (ε, q)-orbit of Q. Therefore, the p′-splittable decomposition numbers of the cyclotomic Hecke algebras of type G(r′, p′, n′) completely determine the decomposition numbers of all cyclotomic Hecke algebras ℋ_{r,p,n}(Q).

Acknowledgments

The first author was supported partly by the National Natural Science Foundation of China (Project 10401005) and the Program NCET. He wishes to thank the School of Mathematics and Statistics at University of Sydney for their hospitality during his visit in 2006. Both authors were supported by an Australian Research Council discovery grant.

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Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China

E-mail address: junhu303@yahoo.com.cn

School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

E-mail address: a.mathas@usyd.edu.au