On S_{R}(H)-blocks II^{1}

Yoshimasa Hieda

1. Introduction

Let G be a finite group, p a prime divisor of the order of G and (K, R, k) a p-modular system, i.e., R is a complete discrete valuation ring with maximal ideal (π), K is the quotient field of R of characteristic 0 and k(:= R/(π)) is the residue class field of R of characteristic p. Moreover, we assume that K contains the |G|th roots of unity.

For a subset X of G, X̂ denotes the sum of all elements of X in the group algebra 𝔬G, where 𝔬 is R, K or k.

We consider the Hecke algebra S_{𝔬}(H) := End_{𝔬G}(Ĥ𝔬G) for a subgroup H of G.

As e_{H} := Ĥ/|H| is the idempotent of KG (which is the central primitive idempotent of KH correponding to the trivial character 1_{H} of H), S_{K}(H) = e_{H}KGe_{H}, where we identify f ∈ S_{K}(H) with f(e_{H}) ∈ e_{H}KGe_{H} as usual.

For χ ∈ Irr(G), let e_{χ} be the central primitive idempotent of KG corresponding to χ and put Φ_{H}^{G} := {χ ∈ Irr(G); (χ_{|_{H}}, 1_{H})_{H} ≠ 0}. Then we have that {e_{χ}e_{H}; χ ∈ Φ_{H}^{G}} is the set of all central primitive idempotents of S_{K}(H) in KG (see [C-R, (11.26) Corollary]).

As S_{R}(H) ⊂ K ⊗_{R} S_{R}(H) = S_{K}(H), for a central idempotent ε of S_{R}(H), there exists a non-empty subset β of Φ_{H}^{G} such that ε = ∑_{χ∈β} e_{χ}e_{H}. Here the element of this form is denoted by ε_{β} and if ε_{β} is a centrally primitive, β (or ε_{β}S_{R}(H)) is called an S_{R}(H)-block (cf. Definition 2.1).

Also, we have S_{R}(H)/πS_{R}(H) ≃ S_{k}(H) as ĤRG is a permutation module. Hence the set of S_{R}(H)-blocks corresponds bijectively to the set of S_{k}(H)-blocks.

On the other hand, the multiplication induces the K-algebra homomorphism ϕ from Z(KG) to Z(S_{K}(H)). Using the map ϕ, G.R.Robinson [Ro] has proved that Z(S_{R}(H)) ≃ A_{R}(H) as R-algebras, where A_{R}(H) denotes the endomorpism ring End_{R[G×G]}(RGĤRG). Then so each S_{R}(H)-block corresponds to a (central) primitive idempotent of A_{R}(H) and to a unique indecomposable direct summand M_{β} of RGĤRG as R[G × G]-modulue. Therefore we can define a defect group for each S_{R}(H)-block in G × G (see Definition 2.2 and Remark 2.2).

Now we recall that for any S_{R}(H)-block β there exists a unique p-block B such that β ⊂ Irr(B) (see Proposition 2.3). Also if e_{B} is a central primitive idempotent, i.e., a block idempotent, of RG with the condition ϕ(e_{B}) ≠ 0, then ϕ(e_{B}) = ∑_{β∈𝔅} ε_{β} as ϕ(e_{B}) ∈ Z(S_{R}(H)), where 𝔅 is the suitable non-empty subset of S_{R}(H)-blocks. Hence Irr(B) ∩ Φ_{H}^{G} = ∑_{β∈𝔅} β is a (disjoint) union of S_{R}(H)-blocks.

In this note, we shall exhibit some results on S_{R}(H)-blocks (which are mainly proved in [Ro] and [H-T]) and show the following example :

1The final and detailed version of this note will be submitted for publication elsewhere.

Example 8 Let p be an odd prime, G := 𝔄_{p} and H := 𝔄_{t} for \frac{p + 1}{2} < t ≤ p, where 𝔄_{n} denotes the alternating group of degree n. Then

(1) Irr(B_{0}) ∩ Φ_{H}^{G} = {[p − i, 1^{i}]_{|_{G}}; 0 ≤ i ≤ p − t}, where [p − i, 1^{i}] is an irreducible character of 𝔖_{p}.

(2) β_{0} = Irr(B_{0}) ∩ Φ_{H}^{G} and δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

The notation is almost standard. Concerning some basic facts and terminologies used here, we refer to [C-R] and [N-T] for example.

2. Definitions and G.R.Robinson’s results on S_{R}(H)-blocks

At first we recall the definition of S_{R}(H)-blocks by G.R.Robinson [Ro] (he call them A_{R}(H)-blocks), which is equivalent to our definition (see section 1).

Definition 2.1. ([Ro]) For any central primitive idempotent ε of S_{R}(H) (or A_{R}(H)), there exists a minimal non-empty subset β of Φ_{H}^{G} which satisfies the following two conditions :

(a) ε = ϕ(e_{β}), where e_{β} := ∑_{χ∈β} e_{χ}.

(b) \frac{1}{|G|} ∑_{χ∈β} χ(Ĥg)χ(1) ∈ R for any g ∈ G.

We call such sets S_{R}(H)-blocks (A_{R}(H)-blocks) of irreducible characters of G.

Remark 2.1. (1) The above condition (b) is equivalent to the next condition :

(b)′ \frac{1}{|G|} ∑_{χ∈β} χ(Ĥx)χ(1) ∈ R for any x ∈ H \G/H, where H \G/H is a complete

set of representatives for (H, H)-double cosets of G.

(2) There exists a unique S_{R}(H)-block for any χ ∈ Φ_{H}^{G}. In particular, the trivial character 1_{G} of G is always in Φ_{H}^{G} for any subgroup H of G. Then there exists the S_{R}(H)-block which has 1_{G}. So such S_{R}(H)-block is called the principal S_{R}(H)- block and is denoted by β_{0}.

According to [Ro], we define defect groups for S_{R}(H)-blocks.

Definition 2.2. ([Ro]) For any S_{R}(H)-block β, there exists a minimal subgroup D of G × G and λ ∈ End_{RD}(RGĤRG) with the condition : Tr_{D}^{G×G}(λ) = ε_{β}.

Then D is called a defect group for β and denoted by δ_{H}(β).

Remark 2.2. ([Ro, the above remark of Lemma 2.1]) By the theory of G-algebras, we have

(1) δ_{H}(β) is a p-subgroup of G × G and uniquely determined up to G × G-conjugacy.

(2) δ_{H}(β) is a vertex of M_{β}.

The next proposition tells us some relations between p-blocks and S_{R}(H)-blocks.

Proposition 2.1. ([Ro, Remark of Lemma 2.1]) If H = {1}, then Irr(B) is an S_{R}({1})- block for any p-block B of G. Moreover, a defect group of an S_{R}({1})-block Irr(B) is the diagonal subgroup δ(B)^{Δ} := {(x, x) ∈ G × G; x ∈ δ(B)}, where δ(B) is a (usual) defect group of B.

The following property is one of the characterization of p-blocks.

Corollary 2.2. For any p-block B of G, Irr(B) is a minimal subset of Irr(G) which

satisfies the following condition : \frac{1}{|G|} ∑_{χ∈Irr(B)} χ(x)χ(1) ∈ R for any x ∈ G.

Proof. We take H = {1}. Then the statement is clear from Definition 2.1 and the above proposition. □

Now we exhibit some important and useful results by G.R.Robinson in [Ro].

Proposition 2.3. ([Ro, Lemma 2.1,Lemma 2.3(i),(iii),Corollary 2.4])

(1) For any S_{R}(H)-block β and x, y ∈ G, \frac{|δ_{H}(β)|}{|C_{G}(x)||C_{G}(y)|} ∑_{χ∈β} χ(x)χ(y) ∈ R.

In particular, \frac{|δ_{H}(β)|}{|G × G|} ∑_{χ∈β} χ(1)^{2} ∈ R.

(2) β is contained in a single p-block B of G in the usual sence, and if B has a defect group D, then δ_{H}(β) is contained (up to conjugacy) in D × D.

(3) Let C := core_{G}(H). Then there is a bijection between the set of S_{R}(H)-blocks of G and the set of S_{R}(H/C)-block of G/C. In particular, if C = H, i.e., H is normal in G, then the S_{R}(H)-blocks of G are precisely the p-blocks of R[G/H].

(4) For the principal S_{R}(H)-block β_{0},β_{0} = {1_{G}} if and only if H contains a Sylow p-subgroup of G.

Corollary 2.4. (see [H1, Corollary 3]) The followings hold.

(1) If ∑_{χ∈β} χ(1)^{2} is prime to p for an S_{R}(H)-block β, then a defect group of β is a Sylow p-subgroup of G × G. In particular, if H contains a Sylow p-subgroup of G, then a defect group of β_{0} is a Sylow p-subgroup of G × G.

(2) If χ ∈ Φ_{H}^{G} is in p-block B of defect 0, then {χ} = Irr(B) is an S_{R}(H)-block and whose defect group is unit element {(1, 1)} of G × G.

3. Othogonality Relation

In this section we assume \underline{H is a p′-subgroup of G} and consider only those blocks such that ϕ(e_{B}) ≠ 0.

In this case e_{H} ∈ RG, i.e., ĤRG = e_{H}RG is a projective RG-module and kH is a semisimple k-algebra.

For any φ ∈ IBr(G), let S_{φ} (resp. P_{φ}) be an irreducible kG-module (resp. an indecomposable projective RG-module) corresponding to φ and Ψ_{H}^{G} := {φ ∈ IBr(G); k_{H}|S_{φ}_{↓H}}. Here we mention that Ψ_{H}^{G} = {φ ∈ IBr(G); P_{φ}|e_{H}RG} by Robinson’s reciprocity. We let furthermore β_{i}^{∗} := {φ ∈ IBr(B); P_{φ}|ε_{β_{i}}(e_{H}RG)} for 0 ≤ i ≤ t, where 𝔅 = {β_{i}}_{i=0}^{t} (see section 1).

Proposition 3.1. (see [H-T, Proposition 3]) The followings hold.

(1) IBr(B) ∩ Ψ_{H}^{G} = ⋃_{i=0}^{t} β_{i}^{∗}.

(2) The decomposition matrix D_{B} of B has the following form :

(3.1) D_{B} = (\begin{matrix}D_{β_{0}} & 0 & ⋯ & 0 & ∗ \\ 0 & D_{β_{1}} & ⋯ & 0 & ∗ \\ ⋮ & ⋮ & ... & ⋮ & ∗ \\ 0 & 0 & ⋯ & D_{β_{t}} & ∗ \\ 0 & 0 & ⋯ & 0 & ∗ \\ ⋮ & ⋮ & ⋯ & ⋮ & ∗ \\ 0 & 0 & ⋯ & 0 & ∗ \end{matrix}) = (D_{B}′|D_{B}′′) ,

where D_{B}′ denotes the set of the first |IBr(B) ∩ Ψ_{H}^{G}| columns of D_{B} and D_{B}′′ the rest.

Corollary 3.2. (see [H-T, Lemma 7 and Corollary 8]) If Φ_{H}^{G} = Irr(G), then Irr(B) is an S_{R}(H)-block for any p-block B. Moreover, if Irr(B) ⊂ Φ_{H}^{G}, then Irr(B) is an S_{R}(H)-block.

Remark 3.1. Let G := 𝔄_{5}:the alternating group of degree 5, C_{3} := ⟨(1 2 3)⟩ ∈ Syl_{3}(G) and p = 3. (So C_{3} is not a 3′-subgroup. ) Then Φ_{C_{3}}^{G} = Irr(G). But Irr(B_{0}) = β_{0} ∪ β_{1},where β_{0} = {1_{G}} and β_{1} = {χ_{2}, χ_{3}} with χ_{2}(1) = 4 and χ_{3}(1) = 5. Then the above statement is not true in general.

In the rest of this note 𝔖_{n} (resp. 𝔄_{n}) denotes the symmetric (resp. the alternating) group of degree n. For a partition of n the Young diagram associated with λ is denoted by [λ] and λ′ denotes the conjugate of λ. (So [λ′] is the transposed diagram of [λ].) Also, the same notation [λ] means an irreducible character of 𝔖_{n} corresponding to the Young diagram [λ]. (For example [n] := [(n)] means the trivial character 1_{𝔖_{n}}, too.) Also B_{0} denotes the principal p-block of G.

Using the above notations, we have the following examples.

Example 1. Let G = 𝔖_{4}, H := ⟨(1 2)⟩ and p = 3.

Then Φ_{H}^{G} = Irr(G) \ {[1^{4}]} = β_{0} ∪ β_{1} ∪ β_{2}, where β_{0} = {[4], [2^{2}]} = Irr(B_{0}) ∩ Φ_{H}^{G}, β_{1} = {[3, 1]} = Irr(B_{1}) and β_{2} = {[2, 1^{2}]} = Irr(B_{2}). Moreover, δ_{H}(β_{0}) ∈ Syl_{3}(G × G) and δ_{H}(β_{1}) = {(1, 1)} for i = 1, 2 by Corollary 2.4. Also the decomposition matrices D_{B_{0}} and D_{β_{0}} are followings :

D_{B_{0}} = [4] [2^{2}] [1^{4}] (\begin{matrix}1 & 0 \\ 1 & 1 \\ 0 & 1 \end{matrix}) , D_{β_{0}} = {[4] \atop [2^{2}]} (\begin{matrix}1 \\ 1 \end{matrix}) .

Example 2. (see [H-T, Remark 12] and [H1, Example 10])

Let G := 𝔖_{5}, H := ⟨(1 2)(3 4), (1 3)(2 4)⟩ and p = 5. Then Irr(B_{0}) ∩ Φ_{H}^{G} = β_{0} ∪ β_{1} with β_{0} = {[5], [4, 1]} and β_{1} = {[2, 1^{3}], [1^{5}]}. Moreover, δ_{H}(β_{i}) ∈ Syl_{5}(G × G) (i = 0, 1) by Corollary 2.4. Also the decomposition matrix of B_{0} is the following form :

D_{B_{0}} = \begin{matrix}[5] \\ [4, 1] \\ [3, 1^{2}] \\ [2, 1^{3}] \\ [1^{5}] \end{matrix} (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix}) = \begin{matrix}[5] \\ [4, 1] \\ [2, 1^{3}] \\ [1^{5}] \\ [3, 1^{2}] \end{matrix} (\begin{matrix}1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{matrix}) .

So the decomposition matrices of S_{R}(H)-blocks in B_{0} are followings:

D_{β_{0}} = \begin{matrix}[5] \\ [4, 1] \end{matrix} (\begin{matrix}1 \\ 1 \end{matrix}) , D_{β_{1}} = \begin{matrix}[2, 1^{3}] \\ [1^{5}] \end{matrix} (\begin{matrix}1 \\ 1 \end{matrix}) .

The next theorem is the orthogonality relation for the S_{R}(H)-block.

Theorem 3.3. ([H-T, Theorem 5]) Let β be an S_{R}(H)-block. Then we have ∑_{χ∈β} χ(xe_{H})χ(y) = 0 for any y ∈ G−G_{p′} and x ∈ G_{p′} such that ⟨x, H⟩ is a p′-subgroup.

Remark 3.2. (see [H-T, Remark 12] and [H1, Remark 6-1])

(1) If H = {1} in the above theorem, we get the second orthogonality relation for the (usual) p-block.

(2) We recall Example 2 and take x := (4 5) ∈ G_{5′} and y := (1 2 3 4 5) ∈ G − G_{5′} in the above theorem. Then Hx ⊂ HxH ⊂ G_{5′} but ⟨H, x⟩ = G ⊄ G_{5′}. This means that the assumption, ⟨x, H⟩ is a p′-subgroup, is not satisfied. Moreover, ∑_{χ∈β_{0}} χ(xe_{H})χ(y) ≠ 0. So the above theorem is not true for any p-regular element x ∈ G.

(3) \underline{Question} Does the following statement hold?

∑_{χ∈β} χ(xe_{H})χ(y) ≡ 0 (mod(π)) for any x ∈ G_{p′} and y ∈ G − G_{p′}.

4. Some examples of the principal S_{R}(H)-blocks

We use the same notations in section 3.

Example 3. ([H2, Example 1]) Let H := G and chark = p.

(1) Φ_{G}^{G} = {1_{G}}.

(2) β_{0} = Φ_{G}^{G} and δ_{G}(β_{0}) ∈ Syl_{p}(G × G).

Example 4. ([H1, Example 12]) Let G := 𝔖_{n} and H := 𝔖_{n−1}, (n ≥ 2).

(1) Φ_{H}^{G} = {[n], [n − 1, 1]}.

(2) (a) If p does not devide n, then β_{0} = {[n]} and δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

(b) If p devides n, then β_{0} = {[n], [n − 1, 1]}(= Φ_{H}^{G}). In particular, if p is odd prime, then δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

For the alternating groups the same statements hold. (We notice that we correct the range of n in [H2, Example 5].)

Example 5. (cf. [H2, Example 5]) Let G := 𝔄_{n} and H := 𝔄_{n−1}, (n ≥ 4).

(1) Φ_{H}^{G} = {1_{G}, χ}, where χ(1) = n − 1.

(2) (a) If p does not divide n, then β_{0} = {1_{G}} and δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

(b) If p divides n, then β_{0} = {1_{G}, χ}(= Φ_{H}^{G}). In particular, if p is odd prime, then δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

If G is the symmetric group of degree p, then we have the following two examples.

Example 6. ([H1, Example 11]) Let G := 𝔖_{p}, H := 𝔖_{t} (1 ≤ t ≤ p) and chark = p. Then

(1) β_{0} = Irr(B_{0}) ∩ Φ_{H}^{G} = {[p − i, 1^{i}]; 0 ≤ i ≤ p − t}.

(2) δ_{H}(β_{0}) =_{G×G} \begin{cases}P^{Δ} & t = 1 \\ P × P & 2 ≤ t ≤ p \end{cases}, where P is a Sylow p-subgroup of G.

Example 7. ([H2, Example 7]) Let p be an odd prime, G := 𝔖_{p} and H := 𝔄_{t} for

\frac{p + 1}{2} < t ≤ p.

(1) Irr(B_{0}) ∩ Φ_{H}^{G} = {[p − i, 1^{i}]; 0 ≤ i ≤ p − t} ∪ {[j + 1, 1^{p−j−1}]; 0 ≤ j ≤ p − t}.

(2) β_{0} = {[p − i, 1^{i}]; 0 ≤ i ≤ p − t} and δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

One of the purposes of this note is to show the next example.

Example 8. Let p be an odd prime, G := 𝔄_{p} and H := 𝔄_{t} for \frac{p + 1}{2} < t ≤ p. Then

(1) Irr(B_{0}) ∩ Φ_{H}^{G} = {[p − i, 1^{i}]_{|_{G}}; 0 ≤ i ≤ p − t}, where [p − i, 1^{i}] is an irreducible character of 𝔖_{p}.

(2) β_{0} = Irr(B_{0}) ∩ Φ_{H}^{G} and δ_{H}(β_{0}) ∈ Syl_{p}(G × G).

Proof. We may assume that t ≠ p by Example 3. Put G̃ := 𝔖_{p} and B̃_{0} (resp. β̃_{0}) denotes the principal p-block (resp. the principal S_{R}(H)-block) of G̃. At first we mention

that [p − i, 1^{i}]_{|_{G}} = [i + 1, 1^{p−i−1}]_{|_{G}} is irreducible for any 0 ≤ i ≤ p − t as \frac{p + 1}{2} < t ≤ p

and [i + 1, 1^{p−i−1}] is the conjugate of [p − i, 1^{i}].

(1) As Irr(B_{0}) ∩ Φ_{H}^{G} = {χ̃_{|_{G}}; χ̃ ∈ Irr(B̃_{0}) ∩ ΦG̃ ; H} and Example 7(1), the assertion holds.

(2) We may only prove the first half because the later half follows from Example 7(2). Now put I := {[p − i, 1^{i}]; [p − i, 1^{i}]_{|_{G}} ∈ β_{0}} ⊂ β̃_{0}. So we may show that I = β̃_{0} by (1). As p > 2, for any x ∈ H \ G/H,

\frac{1}{|G̃|} ∑_{χ̃∈I} χ̃(Ĥx)χ̃(1) = \frac{1}{2|G|} ∑_{χ̃∈I} χ̃_{|_{G}}(Ĥx)χ̃_{|_{G}}(1) = \frac{1}{2|G|} ∑_{χ∈β_{0}} χ(Ĥx)χ(1) ∈ R,

where χ := χ̃_{|_{G}}. Also, \frac{1}{|G̃|} ∑_{χ̃∈I′} χ̃(Ĥx)χ̃(1) ∈ R for I′ := \{[i + 1, 1^{p−i−1}]; [p − i, 1^{i}] ∈ I\}.

Therefore \frac{1}{|G̃|} ∑_{χ̃∈I∪I′} χ̃(Ĥx)χ̃(1) ∈ R as I ∩ I′ = ∅.

On the other hand, for τ := (1 2) ∈ G̃/G, \frac{1}{|G̃|} ∑_{χ̃∈I} χ̃({ \atop Hxτ})χ̃(1) = −\frac{1}{|G̃|} ∑_{χ′∈I′} χ̃′({ \atop Hxτ})χ̃′(1).

Then \frac{1}{|G̃|} ∑_{χ̃∈I∪I′} χ̃({ \atop Hxτ})χ̃(1) = 0. Hence I ∪ I′ is a union of S_{R}(H)-blocks of G̃ as

H \ G̃/H = {x, xτ; x ∈ H \ G/H}. Thus β̃_{0} ⊂ I ∪ I′. Moreover, β̃_{0} ⊂ I since β̃_{0} ∩ I′ = ∅. This means I = β̃_{0} and the assertion holds. □

References

[C-R] C. W. Curtis and I. Reiner : Methods of Representation Theory with Application to Finite Groups

and Orders-Volume I, John Wiley and Sons, New York, 1981.

[H1] Y. Hieda : Some examples of S_{R}(H)-blocks, Bul. of Osaka Pref. of Tech., vol.34,73-76, 2000.

[H2] Hieda, Y. : Some examples of S_{R}(H)-blocks, Proc. of the 33th Symposium On Ring Theory and Representation Theory, 29-35, 2001.

[H-T] Y. Hieda and Y. Tsushima : On S_{R}(H)-Blocks for Finite Groups, J. of Algebra, vol.202, 583-588,

1998.

[Ja] G. D. James : The Representation Theory of Symmetric Groups, Lecture notes in Mathematics 682, 1978.

[N-T] H. Nagao and Y. Tsushima : Representations of Finite Groups, Academic Press, 1989.

[Ro] G. R. Robinson : Some Remarks on Hecke Algebras, J. of Algebra, vol.163, 806-812, 1994.

Department of Liberal Arts

Osaka Prefectural College of Technology Neyagawa, Osaka 572-8572 JAPAN Email: hieda@las.osaka-pct.ac.jp