BLOCKS OF FINITE TYPE IN REDUCED ENVELOPING ALGEBRAS FOR CLASSICAL LIE ALGEBRAS

DANIEL K. NAKANO AND R. DAVID POLLACK

Abstract. In this paper the authors determine when a block for the reduced enveloping algebra of a classical Lie algebra corresponding to a character in Levi form has finite representation type. These results refine earlier work of Premet.

1 Introduction

1.1 Let A be a finite-dimensional algebra over a field k and let mod(A) be the set of finitedimensional A-modules. The algebra A has finite representation type if mod(A) contains finitely many indecomposable modules up to isomorphism. Otherwise, A has infinite representation type. Even if A has infinite representation type, one might still be able to classify the indecomposable modules. This may occur when A has tame representation type. One of the fundamental questions in studying the representation theory of finite-dimensional algebras involves determining the representation type of a given algebra. That is, if the algebra has finite, tame, or wild representation type.

Let 𝔤 be a restricted Lie algebra with pth-power map [p] defined over an algebraically closed field of characteristic p > 0 and let U(𝔤) be the universal enveloping algebra of 𝔤. For any χ ∈ 𝔤^{∗}, the reduced enveloping algebra corresponding to χ is the finite-dimensional associative algebra u(𝔤, χ) := U(𝔤)/⟨x^{[p]} − x^{p} − χ(x)^{p}⟩. We remark that any irreducible representation of 𝔤 will be an irreducible representation of u(𝔤, χ) for a unique choice of χ. When χ = 0 u(𝔤, 0) := u(𝔤) is the restricted enveloping algebra of 𝔤. Modules for this algebra are the restricted representations for 𝔤.

Much work has been accomplished for non-restricted representations for the classical Lie algebras. In [KW1] [KW2], Kac and Weisfeiler reduced the study of non-restricted representations of a restricted Lie algebra to the representation theory of these reduced enveloping algebras. The representation theory of these algebras was further developed by work of Friedlander and Parshall [FP1] [FP2] ten years ago. Recently, Premet [Pr1] [Pr2] proved earlier conjectures made by Kac and Weisfeiler on the dimensions of irreducible representations for non-restricted representations by using support varieties. As a by product of his work Premet classifies reduced enveloping algebras having finite representation type. These results encompass earlier work done by the second author on representation of the restricted enveloping algebra [Po]. Specific computations have been made by Jantzen for 𝔰𝔩_{𝔫} and 𝔰𝔬_{2𝔫+1} in [Jan3] [Jan4]. Surveys of these and other results have been written by Humphreys [Hu2] and Jantzen [Jan5]. Work on non-restricted representations for non-classical simple Lie algebras and their support varieties can be found in [FN].

Date: December 1998.

Research of the first author partially supported by NSF grant DMS-9800960.

Research of the second author was supported by a Queen’s sabbatical grant.

This paper will appear in the Journal of the London Mathematical Society.

The main purpose of this paper is to determine those blocks of u(𝔤, χ) for 𝔤 a classical Lie algebra and χ of Levi form which are of finite representation type. The definition of Levi form will be provided in Section 2. Later, we provide an alternative proof of Premet’s earlier finite type results for u(𝔤, χ).

1.2 Acknowledgements: The second author would like to acknowledge the Department of Mathematics and Statistics at Utah State University for their hospitality during his 1997-1998 sabbatical year. The authors thank Alexander Premet for useful comments and suggestions on an earlier version of this manuscript.

2 Standard Levi Form

2.1 Let G be a simple algebraic group over an algebraically closed field k of characterstic p > 0 and 𝔤 be the Lie algebra of G. Throughout this paper we will make the following standard assumptions given in [Jan5, §6]:

(A1) The group G is simply connected;

(A2) The prime p is good for 𝔤.

(A3) There exists a G invariant non-degenerate bilinear form on 𝔤. The assumption (A2) is equivalent to stating that for certain primes p, the following algebras are omitted:

• p = 2 and 𝔤 of types B_{n}, C_{n} (n ≥ 2), D_{n} (n ≥ 4);

• p = 2, 3 and 𝔤 of types E_{6}, E_{7}, F_{4} and G_{2};

• p = 2, 3, 5 and 𝔤 of type E_{8}. Assumption (A3) occurs in our situation if and only if 𝔤 is of type:

• A_{n} (n ≥ 1) for p ∤ n + 1;

• B_{n}, C_{n} (n ≥ 2), D_{n} (n ≥ 4), E_{7}, F_{4} for p ≠ 2;

• E_{6}, G_{2} for p ≠ 3;

• E_{8} for all primes. Therefore, by assuming (A1), (A2), and (A3) we are adding A_{n} when p | n + 1 to the list proscribed above by the assumption (A2). We also remark that if 𝔤 satisfies (A1)-(A3), then so does every Levi factor [Jan5, 6.5]. Under these conditions listed above, 𝔤 is a classical simple Lie algebra.

2.2 Notation: Let G be a simple algebraic group over k having irreducible root system Φ with respect to maximal torus T. Let B^{+} ⊃ T be the Borel subgroup defined by Φ^{+} and let B^{−} be the opposite Borel. Let 𝔼 be the Euclidean space associated with Φ. The inner product on 𝔼 will be denoted by ( , ). The set of simple roots will be denoted by Π. The Weyl group W is the group generated by reflections associated to root system Φ. Let w_{0} denote the long element in W.

Let X(T) be the integral weight lattice obtained from Φ with α^{∨} = 2α/(α, α) the coroot corresponding to α ∈ Φ. There is a partial ordering on X(T) defined by λ > μ if and only if λ − μ is a sum of positive roots. The set of dominant integral weights is defined by:

X(T)_{+} = {λ ∈ X(T) : 0 ≤ (λ, α^{∨}) for all α ∈ Π},

and the set of restricted weights is defined by:

X_{1}(T) = {λ ∈ X(T) : 0 ≤ (λ, α^{∨}) < p for all α ∈ Π}.

The Weyl group W acts on X(T) via the “dot action” given by w · λ = w(λ + ρ) − ρ where w ∈ W, λ ∈ X(T), and ρ is the half sum of positive roots. Since G is simple and G^{(1)} is simply connected, each λ ∈ X(T) can be expressed uniquely as λ = λ_{0} + pλ_{1} where λ_{0} ∈ X_{1}(T) and λ_{1} ∈ X(T) [Jan2, II 9.10]. Therefore, X_{1}(T) ≅ X(T)/pX(T) and so X_{1}(T) inherits the W dot action from X(T). For any λ ∈ X(T) set

Φ_{λ} = {α ∈ Φ : (λ + ρ, α^{∨}) ∈ pℤ}.

The set Φ_{λ} is a subroot system of Φ if Φ_{λ} ≠ ∅. Furthermore, an easy computation shows that for λ ∈ X(T) we have w(Φ_{λ}) = Φ_{w·λ}. Let W_{Φ_{λ}} be the subgroup of W generated by reflections σ_{α} where α ∈ Φ_{λ}. Moreover, for λ ∈ X_{1}(T), let Stab_{W} λ = {w ∈ W : w.λ = λ}. By [Jan2, II 6.3], one can see that W_{Φ_{λ}} = Stab_{W} λ for λ ∈ X_{1}(T).

2.3 Given the assumptions (A1)-(A3), the classical Lie algebra 𝔤 can be identified with 𝔤^{∗} via a G-isomorphism. The Jordan-Chevalley decomposition in 𝔤 can be used to produce a decomposition of characters χ = χ_{s} + χ_{n} where χ_{s} corresponds with the semisimple component and χ_{n} corresponds with the nilpotent component in 𝔤. Friedlander and Parshall [FP1, Thm. 3.2] proved a Morita equivalence theorem reducing the representation theory of u(𝔤, χ) to u(𝔩, ρ) where 𝔩 is a reductive subalgebra of 𝔤 and ρ is a nilpotent character of 𝔩. Hence, for questions involving representation type it is enough to study nilpotent characters for the classical Lie algebras.

The Lie algebra 𝔤 has a triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔱 ⊕ 𝔫^{+} where Lie B^{±} = 𝔟^{±} = 𝔱 ⊕ 𝔫^{±} and Lie T = 𝔱. Moreover, 𝔫^{+} = ∑_{α∈Φ^{+}} ke_{α} and 𝔫^{−} = ∑_{α∈Φ^{+}} kf_{α}. Suppose χ is a nilpotent character for the classical Lie algebra 𝔤. Without a loss of generality we may assume that χ(𝔟^{+}) = 0. We can now state when a nilpotent character is in standard Levi form.

Definition . Let 𝔤 be a classical Lie algebra and χ a nilpotent character such that χ(𝔟^{+}) = 0. The character χ is in standard Levi form if there exists a subset I of Π such that χ(f_{α}) = 1 for α ∈ I and χ(f_{β}) = 0 for β ∈ Φ^{+} − I. If I = Π then the character χ is called regular nilpotent.

In this paper we restrict our attention to only those characters in standard Levi form. The reason for limiting our consideration is that the indexing for the blocks of u(𝔤, χ) is only known (at this time) for characters in this form.

2.4 We now summarize some of the relevent representation theoretic results that will be needed for studying nilpotent and Levi form characters [Rud] [FP1] [FP2] [Jan5]. Let χ be a nilpotent character with χ(𝔟^{+}) = 0. If λ ∈ X_{1}(T) then λ can be regarded as a one-dimensional u(𝔟^{+})-module by letting 𝔫^{+} act trivially. Set

Z_{χ}(λ) = ind_{u(𝔟^{+})}^{u(𝔤,χ)}λ := u(𝔤, χ) ⊗_{u(𝔟^{+})} λ.

The standard results for ordinary Verma modules also hold for the module Z_{χ}(λ).

(2.4.1) dim_{k} Z_{χ}(λ) = p^{n} where n = |Φ^{+}| (2.4.2) Each irreducible module for u(𝔤, χ) is the quotient of some Z_{χ}(λ). Assume now that χ is in standard Levi form with the notation given in Definition 2.3. (2.4.3) Each Z_{χ}(λ) has a unique maximal submodule whose irreducible quotient is denoted by L_{χ}(λ). (2.4.4) The modules Z_{χ}(λ) and Z_{χ}(μ) belong to the same block in u(𝔤, χ) if and only if μ ∈ W · λ. (2.4.5) The module Z_{χ}(λ) ≅ Z_{χ}(μ) if and only if μ ∈ W_{I} · λ, where W_{I} is the subgroup of W generated by the reflections {σ_{α} : α ∈ I}. Furthermore, L_{χ}(λ) ≅ L_{χ}(μ) if and only if μ ∈ W_{I} · λ. Let Λ_{χ} be a set of W_{I} orbit representatives in X_{1}(T). This set is in one-to-one correspondence with the irreducible u(𝔤, χ)-modules. (2.4.6) Consequently, the blocks of u(𝔤, χ) are in one-to-one correspondence with the W orbits in X_{1}(T) and if B_{λ} is the block containing L_{χ}(λ), then B_{λ} contains |W · λ ∩ Λ_{χ}| non-isomorphic simple modules. If Φ_{λ} = ∅ then this number is |W |/|W_{I}|. (2.4.7) If χ is regular nilpotent then Z_{χ}(λ) is simple for all λ ∈ Λ_{χ}. Moreover, each block contains one simple module and is Morita equivalent to a local algebra. Let P_{χ}(λ) be the projective cover for the simple module Z_{χ}(λ). The number of times the simple module Z_{χ}(λ) appears as a composition factor of P_{χ}(λ) is |W |/|Stab_{W} λ|.

3 Regular nilpotent characters:

3.1 Throughout this section, χ will be a regular nilpotent character for a classical Lie algebra 𝔤. For any restricted Lie algebra 𝔥 and u(𝔥, χ)-module M, let l_{𝔥}(M) denote the radical length of M as a u(𝔥, χ)-module.

For each λ ∈ Λ_{χ}, let B_{λ} be the block of u(𝔤, χ) with unique simple module Z_{χ}(λ) and projective cover P_{χ}(λ). Since Z_{χ}((p−1)ρ) is a projective u(𝔤, χ)-module, so is Z_{χ}((p−1)ρ)⊗ L(σ) for any simple u(𝔤)-module L(σ). If σ = λ + ρ in X_{1}(T), then Z_{χ}(λ) is a submodule of Z_{χ}((p − 1)ρ) ⊗ L(σ) and P_{χ}(λ) must be a direct summand of Z_{χ}((p − 1)ρ) ⊗ L(σ). One should also observe that

Z_{χ}((p − 1)ρ) ⊗ L(σ) ≅ ind_{u(𝔟^{+})}^{u(𝔤,χ)}[(p − 1)ρ ⊗ L(σ)].

Let L(σ) = ⊕_{μ∈X(T)}L(σ)_{μ} be the decomposition of L(σ) into its weight spaces as T - module. If L(σ)_{μ} ≠ 0, then σ ≥ μ ≥ w_{0}σ [Jan2, II 2.4]. If μ = w_{0}σ + ∑_{α∈Π} k_{α}α, then set c(μ) = ∑_{α∈Π} k_{α} ≥ 0. Let M_{i} = ∑_{c(μ)≥i} L(σ)_{μ}. Then one has a filtration of u(𝔟^{+})-modules

L(σ) = M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ {0},  (3.1.1)

whose successive quotients are semisimple as u(𝔟^{+})-module. Since the simple u(𝔤, χ)- modules are of the form Z_{χ}(μ), and ind_{u(𝔟^{+})}^{u(𝔤,χ)} is an exact functor, the preceding filtration induces a filtration of Z_{χ}((p − 1)ρ) ⊗ L(σ):

Z_{χ}((p − 1)ρ) ⊗ L(σ) = Z_{χ}((p − 1)ρ) ⊗ M_{0} ⊃ Z_{χ}(p − 1)ρ) ⊗ M_{1} ⊃ ⋯ ⊃ {0}, (3.1.2)

whose successive quotients are semisimple as u(𝔤, χ)-module. We observe that l_{𝔟^{+}}(L(σ)) ≤ c(σ) + 1.

In order to show that a given block B_{λ} is not of finite representation type, or equivalently not a uniserial algebra, it suffices to demonstate that the l_{𝔤}(P_{χ}(λ)) is not equal to the composition factor multiplicity of Z_{χ}(λ) in P_{χ}(λ). By the analysis given in the previous paragraph one has

l_{𝔤}(P_{χ}(λ)) ≤ l_{𝔤}(Z_{χ}((p − 1)ρ) ⊗ L(σ)) ≤ l_{𝔟^{+}}((p − 1)ρ ⊗ L(σ)) = l_{𝔟^{+}}(L(σ)).

Therefore, by (2.4.7), B_{λ} is of infinite type, if c(σ)+1 < |W |/|Stab_{W} λ| where σ = λ+ρ. On the other hand, the strategy to show that certain blocks B_{λ} indeed have finite representation type will require the computation of the support variety of Z_{χ}(λ).

3.2 In this section we will prove some results that allow us to reduce our analysis to specific λ ∈ Λ_{χ}. Let α_{1}, α_{2}, …, α_{n} denote the simple roots in Π and let ω_{1}, ω_{2}, …, ω_{n} be the corresponding fundamental weights. The labelling and conventions given here will be the same as in [Hu1]. Let

\overline{C}_{ℤ} = \{λ ∈ X(T) : 0 ≤ (λ + ρ, α^{∨}) ≤ p for all α ∈ Φ^{+}\}.

Let Δ = {1, 2, …, n}. If S ⊆ Δ, then let Φ_{S} be the subroot system of Φ with base {α_{i} : i ∈ S}. Moreover, let W_{S} be the subgroup of W generated by {σ_{α_{i}} : i ∈ S}. The following notation will also be useful. Suppose there exists w ∈ W with w({α_{i} : i ∈ S}) ⊆ Π for some S ⊆ Δ, then set w(S) = {i ∈ Δ : α_{i} ∈ w({α_{i} : i ∈ S})}.

Proposition (A). Let λ ∈ X(T)_{+} with 0 ≤ (λ + ρ, β^{∨}) < p for all β ∈ Φ^{+}. If S = {i ∈ Δ : (λ + ρ, α_{i}^{∨}) = 0} then Φ_{λ} = Φ_{S}.

Proof. Since 0 ≤ (λ + ρ, β^{∨}) < p for all β ∈ Φ^{+}, it follows that λ + ρ = ∑_{i∈Δ−S} m_{i}ω_{i} where 0 < m_{i} ≤ p − 1 for i ∈ Δ − S. Let β ∈ Φ_{S} then β^{∨} = ∑_{j∈S} l_{j}α_{j}^{∨}. Consequently,

(λ + ρ, β^{∨}) = ( ∑_{i∈Δ−S} m_{i}ω_{i}, ∑_{j∈S} l_{j}α_{j}^{∨}) = 0.

Therefore, Φ_{S} ⊆ Φ_{λ}.

On the other hand, let β ∈ Φ_{λ} and β ∈ Φ^{+}. Then (λ+ρ, β^{∨}) ∈ pℤ, so by our hypotheses, (λ + ρ, β^{∨}) = 0. Moreover, β^{∨} = ∑_{j∈Δ} n_{j}α_{j}^{∨} where n_{j} ≥ 0. Thus,

(λ + ρ, β^{∨}) = ( ∑_{i∈Δ−S} m_{i}ω_{i}, ∑_{j∈Δ} n_{j}α_{j}^{∨}) = ∑_{i∈Δ−S} m_{i}n_{i} = 0.

Therefore, n_{i} = 0 for all i ∈ Δ − S, and β ∈ Φ_{S}. _{□}

Proposition (B). Let λ ∈ \overline{C}_{ℤ}. There exists w ∈ W and λ′ ∈ \overline{C}_{ℤ} such that

(a) λ′ + ρ = ∑_{i∈Δ−S} ω_{i};

(b) 0 ≤ (λ′ + ρ, β^{∨}) < p for all β ∈ Φ^{+};

(c) w(Stab_{W} λ) = Stab_{W} λ′ = W_{S} for some S ⊆ Δ.

Proof. Let λ ∈ \overline{C}_{ℤ}. If 0 ≤ (λ + ρ, β^{∨}) < p for all β ∈ Φ^{+} and S = \{i : (λ + ρ, α_{i}^{∨}) = 0\}, then by Proposition 3.2(A), Φ_{λ} = Φ_{S}. Let λ′ + ρ = ∑_{i∈Δ−S} ω_{i}. Then

0 ≤ (λ′ + ρ, β^{∨}) ≤ (λ + ρ, β^{∨}) < p

for all β ∈ Φ^{+}. Therefore, one may apply Proposition 3.2(A) again to conclude that Φ_{λ′} = Φ_{S}. Hence, Stab_{W} λ = Stab_{W} λ′ = W_{S}.

Now assume that (λ + ρ, β^{∨}) = p for some β ∈ Φ^{+}. Since λ ∈ \overline{C}_{ℤ}, (λ + ρ, β^{∨}) = p for β the highest short root in Φ^{+}. Express λ + ρ = ∑_{i∈Δ} n_{i}ω_{i}, and set T = {i ∈ Δ : n_{i} = 0}, and T′ = Δ − T. Therefore, λ + ρ = ∑_{i∈T′} n_{i}ω_{i}. Observe that Φ_{T} ⊆ Φ_{λ} and (λ + ρ, β^{∨}) = p for β the highest short root, so |T | < rank Φ_{λ}. Since p is good, there exists w ∈ W such that Φ_{w·λ} = w(Φ_{λ}) = Φ_{S} where S ⊆ Δ [Jan1, 2.7]. Let S′ = Δ − S. Therefore,

|T | < rank Φ_{λ} = rank Φ_{S} = |S|,

thus, |S′| < |T′|. Let λ′ + ρ = ∑_{i∈S′} ω_{i}. If

0 ≤ (λ′ + ρ, β^{∨}) < p  (3.2.1)

for β the highest short root, then by Proposition 3.2(A), Φ_{λ′} = Φ_{S} = w(Φ_{λ}). Hence, w(Stab_{W} λ) = Stab_{W} λ′ = W_{S}, and the statement of this proposition follows. Notice that (3.2.1) holds if p ≥ h where h is the Coxeter number. Our goal is to show that there exists w ∈ W such that (3.2.1) holds in general. We divide the work into four distinguishable cases. Throughout the following discussion we will refer to the tables in Section 6.

CASE 1: Types A, B, C: For type A_{n}, B_{n} and C_{n}, we have

(λ′ + ρ, β^{∨}) ≤ \begin{cases}|S′| & type A_{n} \\ 2|S′| & type B_{n} and C_{n}. \end{cases}

On the other hand,

(λ + ρ, β^{∨}) ≥ \begin{cases}|T′| & type A_{n} \\ 2(|T′| − 1) + 1 & type B_{n} and C_{n}. \end{cases}

Since |S′| < |T′|, it follows that 0 ≤ (λ′ + ρ, β^{∨}) < (λ + ρ, β^{∨}) ≤ p.

CASE 2: Type D: For type D_{n} a similar analysis to the previous case with the highest short root shows that

(λ′ + ρ, β^{∨}) ≤ 2|S′|;
(λ + ρ, β^{∨}) ≥ 2(|T′| − 3) + 3 = 2(|T′| − 1) − 1.

Suppose that |S′| < |T′| − 1. Then 0 ≤ (λ′ + ρ, β^{∨}) < (λ + ρ, β^{∨}) ≤ p. Therefore, we may assume that |S′| = |T′| − 1.

Now suppose that j ∈ S for j = 1, n − 1, or n. Then

(λ′ + ρ, β^{∨}) ≤ 2(|S′| − 1) + 1
= 2|S′| − 1
< 2(|T′| − 1) + 1
≤ (λ + ρ, β^{∨})
≤ p.

Therefore, we may assume that {1, n − 1, n} ⊆ S. Observe that if i, i + 1 ∈ S′, then there exists w ∈ W with w(S) = Ŝ ⊆ Δ with 1 ∈ Ŝ. Now one takes λ′ := ∑_{i∈Δ−Ŝ} ω_{i} with respect to Ŝ and applies the previous argument for Ŝ′ to verify the proposition in this case.

Hence, we have reduced ourselves to the case when (1) |S′| = |T′|−1, (2) {1, n−1, n} ⊆ S, and (3) i, i + 1 ∉ S′ for any i. The root system Φ_{S} can take on three possible forms:

Φ_{S} ≅ \begin{cases}A_{n_{1}} × A_{n_{2}} × ⋯ × A_{n_{r}} × D_{m} \\ A_{n_{1}} × A_{n_{2}} × ⋯ × A_{n_{r}} × A_{1} × A_{1} \\ A_{n_{1}} × A_{n_{2}} × ⋯ × A_{n_{r}} × A_{3}. \end{cases}

The number of irreducible components in each case respectively is r + 1, r + 2 and r + 1. In all three cases n = |S|+r. Let s = |{j ∈ T′ : j ∈ {2, …, n−2}|. Therefore, n = |T |+3+s. By combining these two expressions for n, one obtains the r − 2 = s. The number of irreducible components of T is at most s + 1, or equivalently r − 1. But |S′| = |T′| − 1 and Φ_{T} ⊊ Φ_{λ} ≅ Φ_{S}, so this information tells us that the number of components of Φ_{S} is at most r. Therefore, we cannot have the case when (1), (2) and (3) all hold simultaneously.

CASE 3: Type E, F: For E_{6}, E_{7}, E_{8}, and F_{4}. We have seen that if p ≥ h the theorem holds. If p = h − 1 (i.e. h − 1 = 11 (E_{6}), 17 (E_{7}), 29 (E_{8}), and 11 (F_{4})), then S′ = Δ, so S = ∅, contrary to our assumptions. Therefore, we may focus our attention on the cases when p < h − 1. Let M be the possible values of (λ′ + ρ, β^{∨}) where β is the highest short root. The tables in Section 6 (Appendix) indicate the possible subsets S ⊆ Π that may arise when (λ′ + ρ, β^{∨}) ≥ p. The main idea is to locate a subset Ŝ ⊆ Π such that w(S) = Ŝ for some w ∈ W and (λ′′ + ρ, β^{∨}) < (λ′ + ρ, β^{∨}) where λ′′ + ρ = ∑_{Δ−Ŝ} ω_{i}. If this can be accomplished one can recursively apply this procedure to eventually find Ŝ ⊆ Π such that 0 ≤ (λ′′ + ρ, β^{∨}) < p which would prove the proposition. This procedure can be accomplished for all subsets S in the tables except for the ones listed by (∗). These cases must be handled separately.

One of these cases (∗) involves the following situation. Consider the case E_{6} with S = {1, 3, 5, 6}. Then Φ_{S} = A_{2} × A_{2}. Now |T′| = 3 and there are six possibilities for λ + ρ, namely ω_{1} +ω_{2} +ω_{3}, ω_{1} +ω_{2} +ω_{5}, ω_{1} +ω_{3} +ω_{5}, ω_{2} +ω_{3} +ω_{6}, ω_{3} +ω_{5} +ω_{6}, and ω_{2} +ω_{5} +ω_{6}. But, one can directly verify that Φ_{λ} in each of these cases is not conjugate to A_{2} × A_{2}. Therefore, this case cannot occur under our assumptions. A similar argument can be made for all other (∗) cases which includes E_{7} for p = 7 and p = 5, and in E_{8} for p = 11, by looking at the possibilities for λ + ρ and showing that Φ_{λ} is not conjugate to Φ_{S}.

CASE 4: Type G: For type G_{2}, we have |S′| < |T′| ≤ 2. Therefore, |S′| ≤ 1, and 0 ≤ (λ′ + ρ, β^{∨}) ≤ 3 < p.

□

Theorem . Let λ ∈ X(T). There exists S ⊆ Δ and λ′ ∈ \overline{C}_{ℤ} such that

(a) λ′ + ρ = ∑_{i∈S} ω_{i};

(b) 0 ≤ (λ′ + ρ, β^{∨}) < p for all β ∈ Φ^{+};

(c) Stab_{W} λ′ = W_{Δ−S};

(d) B_{λ} is Morita equivalent to B_{λ′}.

Proof. Let λ ∈ X(T). Without a loss of generality, we may assume that λ ∈ \overline{C}_{ℤ} because \overline{C}_{ℤ} is a fundamental domain for the action of W_{p} and B_{λ} = B_{w·λ} for all w ∈ W_{p}. According to Proposition 3.2(B), there exists λ′ satisfying conditions (a) and (b) with λ′ + ρ = ∑_{i∈S} ω_{i} and

Stab_{W} (w · λ) = w(Stab_{W} λ) = Stab_{W} λ′ = W_{Δ−S}

for some w ∈ W. Thus, part (c) of this theorem is satisfied.

Now by a recent result proved by Premet, Mirkovic-Ryuminin [MR], and Jantzen-Soergel, for μ ∈ X_{1}(T),

End_{u(𝔤,χ)}(P_{χ}(μ)) ≅ S^{•}(𝔱)^{Stab_{W} μ} ⊗_{S^{•}(𝔱)^{W}} k_{0}.

Note that B_{μ} is Morita equivalent to End_{u(𝔤,χ)}(P_{χ}(μ)). Therefore, since Stab_{W} (w · λ) = Stab_{W} λ′, it follows that B_{w·λ} = B_{λ} and B_{λ′} are Morita equivalent. _{□}

Observe that these reductions show that it suffices to look at B_{λ} where λ = ∑_{i∈S} −ω_{i} where S ⊆ {1, 2, …, n}. We have λ = ∑_{i∈S} −ω_{i} if and only if σ = ∑_{i∈Y} ω_{i} where Y = {1, 2, …, n} − S because σ = λ + ρ in X(T).

3.3 We will first calculate c(ω_{i}), i = 1, 2, …, n for the Lie algebras of types A_{n}, B_{n}, C_{n} and D_{n}. The weight w_{0}ω_{i} for i = 1, 2, …, n in general can obtained from [Sa, p.140-141].

For type A_{n}, w_{0}ω_{i} = −ω_{n+1−i} for i = 1, 2, …, n. Therefore, by using the information given in [Hu1, p. 69],

c(ω_{i}) = (n + 1)i − i^{2}, i = 1, 2, …, n.  (3.3.1)

For type B_{n}, w_{0}ω_{i} = −ω_{i} for i = 1, 2, …, n. Thus,

c(ω_{i}) = \{ \begin{matrix}(2n + 1)i − i^{2} & i = 1, 2, …, n − 1 \\ \frac{n(n+1)}{2} & i = n. \end{matrix}  (3.3.2)

Similarly, for type C_{n}, w_{0}ω_{i} = −ω_{i} for i = 1, 2, …, n and

c(ω_{i}) = (2n)i − i^{2}, i = 1, 2, …, n.  (3.3.3)

Finally for type D_{n}, w_{0}ω_{i} = −ω_{i} for i = 1, 2, …, n − 2, w_{0}ω_{n−1} = −ω_{n}, and w_{0}ω_{n} = −ω_{n−1}. Therefore,

c(ω_{i}) = \{ \begin{matrix}(2n − 1)i − i^{2} & i = 1, 2, …, n − 2 \\ \frac{n(n−1)}{2} & i = n − 1, n. \end{matrix}  (3.3.4)

For the exceptional cases of rank greater than three, E_{6}, E_{7}, E_{8}, and F_{4}, the invariant c(ω_{i}) is given by the following tables. The Lie algebras of rank less than or equal to two will be handled separately in Section 3.11. The dimensions of each representation are provided for the sake of clarity.

\begin{matrix}E_{6} & ω_{1} & ω_{2} & ω_{3} & ω_{4} & ω_{5} & ω_{6} \\ w_{0}ω_{i} & −ω_{6} & −ω_{2} & −ω_{5} & −ω_{4} & −ω_{3} & −ω_{1} \\ c(ω_{i}) & 23 & 22 & 30 & 42 & 30 & 23 \\ dim_{k} L(ω_{i}) & 27 & 78 & 351 & 2925 & 351 & 27 \end{matrix}

\begin{matrix}E_{7} & ω_{1} & ω_{2} & ω_{3} & ω_{4} & ω_{5} & ω_{6} & ω_{7} \\ w_{0}ω_{i} & −ω_{1} & −ω_{2} & −ω_{3} & −ω_{4} & −ω_{5} & −ω_{6} & −ω_{7} \\ c(ω_{i}) & 34 & 49 & 66 & 96 & 75 & 52 & 27 \\ dim_{k} L(ω_{i}) & 133 & 912 & 8645 & 365750 & 27664 & 1539 & 56 \end{matrix}

\begin{matrix}E_{8} & ω_{1} & ω_{2} & ω_{3} & ω_{4} & ω_{5} & ω_{6} & ω_{7} & ω_{8} \\ w_{0}ω_{i} & −ω_{1} & −ω_{2} & −ω_{3} & −ω_{4} & −ω_{5} & −ω_{6} & −ω_{7} & −ω_{8} \\ c(ω_{i}) & 92 & 136 & 182 & 270 & 220 & 168 & 114 & 58 \\ dim_{k} L(ω_{i}) & 3875 & 147250 & 6696000 & 6899079264 & 146325270 & 2450240 & 30380 & 248 \end{matrix}

\begin{matrix}F_{4} & ω_{1} & ω_{2} & ω_{3} & ω_{4} \\ w_{0}ω_{i} & −ω_{1} & −ω_{2} & −ω_{3} & −ω_{4} \\ c(ω_{i}) & 22 & 42 & 30 & 16 \\ dim_{k} L(ω_{i}) & 52 & 1274 & 273 & 26 \end{matrix}

3.4 General bounds: Let Y be a subset of {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i}. One can express σ − w_{0}σ = ∑_{α∈Π} r_{α}α. Then

l_{𝔟^{+}}(L(σ)) ≤ ∑_{α∈Π} r_{α}α + 1 ≤ ∑_{i∈T} c(ω_{i}) + 1 ≤ ∑_{i=1}^{n} c(ω_{i}) + 1.

This upper bound, ∑_{i=1}^{n} c(ω_{i}) + 1, is given for each type in following tables.

\begin{matrix}A_{n} & B_{n} & C_{n} & D_{n} \\ \frac{1}{6}(n + 2)(n + 1)n + 1 & \frac{1}{3}n(n + 1)(2n + 1) + 1 & \frac{1}{6}n(n + 1)(4n − 1) + 1 & \frac{2}{3}n(n − 1)(n + 1) + 1 \end{matrix}

\begin{matrix}E_{6} & E_{7} & E_{8} & F_{4} \\ 171 & 400 & 1241 & 111 \end{matrix}

3.5 Type A_{n}, n ≥ 3: The following result provides a sufficient criterion for blocks in algebra of type A_{n} n ≥ 3 to have infinite representation type.

Proposition . Let 𝔤 be of type A_{n} where n ≥ 3. If Φ_{λ} is not type A_{n−1} or A_{n} then B_{λ} has infinite representation type.

Proof. Let Y ⊆ {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i}. Consider the inequalities

l_{𝔟^{+}}(L(σ)) ≤ \frac{1}{6}(n + 2)(n + 1)(n) + 1 < |W |/|Stab_{W} λ|.

The first inequality is obtained from Section 3.4. The second inequality will hold if rank(Φ_{λ}) ≤ n − 3 (or equivalently if |Y | ≥ 3).

Now assume that rank(Φ_{λ}) = n − 2 with n ≥ 3. A sharper bound is necessary for l_{𝔟^{+}}(L(σ)). Let σ = ω_{r} + ω_{s} where 1 ≤ r, s ≤ n. Then by equation (3.3.1), we have

l_{𝔟^{+}}(L(σ)) ≤ (n + 1)(r + s) − (r^{2} + s^{2}) + 1
≤ (n + 1)((\frac{n + 1}{2} − 1) + \frac{n + 1}{2}) − ((\frac{n + 1}{2} − 1)^{2} + (\frac{n + 1}{2})^{2}) + 1
= \frac{(n + 1)^{2}}{2} + 1
< (n + 1)!/(n − 1)!
≤ |W |/|Stab_{W} λ|

Note that the inequality from the first line to the second line is obtained by looking at the critical points of the function f(r, s) = (n + 1)(r + s) − (r^{2} + s^{2}) + 1.

Finally consider the case when rank(Φ_{λ}) = n − 1 and σ = ω_{r} with 1 < r < n (i.e. Φ_{λ} is not irreducible). In this situation L(σ) ≅ Λ^{r}(V) where V is the natural representation of A_{n} and

l_{𝔟^{+}}(L(σ)) ≤ (n + 1)r − r^{2} + 1 < \binom{n + 1}{r} = |W |/|Stab_{W} λ|.

This completes the proof. _{□}

3.6 Type B_{n}, n ≥ 3: For type B_{n}, we have an analogous result to Proposition 3.5.

Proposition . Let 𝔤 be of type B_{n}, n ≥ 3. If Φ_{λ} is not of type B_{n} or B_{n−1} then B_{λ} has infinite representation type.

Proof. Let Y ⊆ {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i}. Once again we consider the following string of inequalities.

l_{𝔟^{+}}(L(σ)) ≤ \frac{1}{3}(n)(n + 1)(2n + 1) + 1 < |W |/|Stab_{W} λ| = 2^{n} · n!/|Stab_{W} λ|.

The first inequality is given in Section 3.4. The last inequality will be satisfied if rank(Φ_{λ}) ≤ n − 3 or |Y | ≥ 3. Now consider the case when rank(Φ_{λ}) = n − 2 and σ = ω_{r} + ω_{s} 1 ≤ r < s ≤ n. In this case one has

l_{𝔟^{+}}(L(σ)) ≤ \begin{cases}(2n + 1)(r + s) − (r^{2} + s^{2}) + 1 & 1 ≤ r < s < n \\ (2n + 1)(r) − r^{2} + \frac{n(n+1)}{2} + 1 & 1 ≤ r < n, s = n \end{cases}

One can use a similar argument to the one given in the previous section for type A to conclude that

l_{𝔟^{+}}(L(σ)) ≤ \frac{(2n + 1)^{2}}{2} + 1 < \frac{2^{n} · n!}{2^{n−2}(n − 2)!} ≤ |W |/|Stab_{W} λ|

for 1 ≤ r < s < n. On the other hand, if 1 ≤ r ≤ n − 1 and s = n (n ≥ 3) then

l_{𝔟^{+}}(L(σ)) ≤ \frac{(2n + 1)^{2}}{4} + \frac{n(n + 1)}{2} + 1 < \frac{2^{n} · n!}{(n − 1)!} ≤ |W |/|Stab_{W} λ|.

Next, let rank(Φ_{λ}) = n − 1 (n ≥ 3) with Φ_{λ} not of type B_{n−1}. By applying (3.3.2) we have

l_{𝔟^{+}}(L(ω_{r})) ≤ \begin{cases}(2n + 1)r − r^{2} + 1 & 1 < r < n \\ \frac{n(n+1)}{2} + 1 & r = n \end{cases}

≤ \begin{cases}(nr − r^{2} + 1) + (n + 1)r & 1 < r < n \\ \frac{n^{2}+n}{2} + 1 & r = n \end{cases}

< \begin{cases}\binom{n}{r} + (2^{r} − 1)\binom{n}{r} = 2^{r}\binom{n}{r} & 1 < r < n \\ 2^{n} & 1 ≤ r < n, r = n. \end{cases}

Observe that |W |/|Stab_{W} λ| equals 2^{r} · \binom{n}{r} when 1 ≤ r < n and 2^{n} when r = n so the proof of the proposition is now complete. _{□}

3.7 Type C_{n}; n ≥ 3: For type C, we have the following result on blocks of infinite representation type.

Proposition . Let 𝔤 be a classical Lie algebra of type C_{n}, n ≥ 3. If Φ_{λ} is not of type C_{n} or C_{n−1} then B_{λ} is of infinite representation type.

Proof. Let Y ⊆ {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i} and consider the inequalities:

l_{𝔟^{+}}(L(σ)) ≤ \frac{1}{6}(n)(n + 1)(4n − 1) + 1 < |W |/|Stab_{W} λ| = 2^{n} · n!/|Stab_{W} λ|.

The first inequality holds by results given in Section 3.4. The second inequality will be valid as long as rank(Φ_{λ}) ≤ n − 3. For the case when rank(Φ_{λ}) = n − 2, set σ = ω_{r} + ω_{s} where 1 ≤ r < s ≤ n. Then

l_{𝔟^{+}}(L(σ)) ≤ (2n)(r + s) − (r^{2} + s^{2}) + 1 ≤ \frac{(2n)^{2}}{2} + 1 = 2n^{2} + 1

On the other hand |W |/|Stab_{W} λ| ≥ \frac{2^{n}·n!}{2^{n−2}·(n−2)!} = 4n(n − 1) so l_{𝔟^{+}}(L(σ)) < |W |/|Stab_{W} λ|.

Now suppose that rank(Φ_{λ}) = n − 1 and σ = ω_{r}, 1 < r ≤ n. Then

l_{𝔟^{+}}(L(σ)) ≤ (2n)(r) − r^{2} + 1 = (nr) + (nr − r^{2} + 1)

Furthermore,

|W |/|Stab_{W} λ| = \begin{cases}2^{r} · \binom{n}{r} & 1 ≤ r < n \\ 2^{n} & r = n \end{cases}

For 1 < r < n, l_{𝔟^{+}}(L(σ)) < |W |/|Stab_{W} λ| because 2^{r} · \binom{n}{r} = \binom{n}{r} + (2^{r} − 1)\binom{n}{r} with n ≤ \binom{n}{r} and r < 2^{r} − 1. In the case when r = n, we have for n ≥ 5,

l_{𝔟^{+}}(L(σ)) ≤ 2n(n) − n^{2} + 1 = n^{2} + 1 < 2^{n} = |W |/|Stab_{W} λ|.

In order to finish the proof, we need to consider L(ω_{n}) for n = 3, 4. For C_{3} , we refer the reader to [FH, p. 258] for details. The results used here are the ones independent of the characteristic of the field. The irreducible C_{3}-module L(ω_{3}) embeds into Λ^{3}(V) where dim_{k} V = 6. Note that dim_{k} L(ω_{3}) = 14 and dim_{k} Λ^{3}(V) = 20. The irreducible C_{3}-module V is uniserial as a u(𝔟^{+})-module with radical series:

V ⊃< w_{5}, w_{4}, …, w_{1} >⊃< w_{4}, …, w_{1} >⊃< w_{3}, w_{2}, w_{1} >⊃< w_{2}, w_{1} >⊃< w_{1} >⊃ {0}.

where the standard basis for V is {v_{1}, v_{2}, v_{3}, …, v_{6}} and w_{1} = v_{3}, w_{2} = v_{2}, w_{3} = v_{1}, w_{4} = v_{6}, w_{5} = v_{5} and w_{6} = v_{4}. This filtration induces a filtration of length 10 on Λ^{3}(V) with semisimple quotients:

Λ^{3}(V) = M_{15} ⊇ M_{14} ⊇ ⋯ ⊇ M_{6} ⊇ {0}

where M_{r} =< w_{i} ∧ w_{j} ∧ w_{l} : i + j + l = r >.

Now P_{χ}(λ) is a direct summand of ind_{u(𝔟^{+})}^{u(𝔤,χ)}L(σ)⊗(p−1)ρ where λ = (p−1)ω_{1}+(p−1)ω_{2},

thus a direct summand of ind_{u(𝔟^{+},χ)}^{u(𝔤,χ)}Λ^{3}(V) ⊗ (p − 1)ρ. Since induction is exact we have a filtration of P_{χ}(λ):

P_{χ}(λ) = N_{15} ⊇ N_{14} ⊇ ⋯ ⊇ N_{6} ⊇ {0}.

Here, N_{r} = (ind_{u(𝔟^{+})}^{u(𝔤,χ)}M_{r} ⊗ (p − 1)ρ) ∩ P_{χ}(λ). The W -conjugate weights of ω_{3} are ±ω_{3}, ±(ω_{3} − 2ω_{2}), ±(2ω_{2} − 2ω_{1} + ω_{3}), and ±(ω_{3} − 2ω_{1}). An analysis on the weights shows that N_{13} = N_{12} and N_{8} = N_{7}. Furthermore, N_{11}/N_{9} is semisimple so it follows that l_{𝔤}(P_{χ}(λ)) ≤ 7 < |W |/|Stab_{W} λ| = 8. Hence, P_{χ}(λ) is not uniserial and B_{λ} has infinite representation type.

The case involving C_{4} is similar to one for C_{3}. Let V be the 8-dimensional natural C_{4} representation with standard basis {v_{1}, v_{2}, …, v_{8}}. Set w_{1} = v_{4}, w_{2} = v_{3}, w_{3} = v_{2}, w_{4} = v_{1}, w_{5} = v_{8}, w_{6} = v_{7}, w_{7} = v_{6} and w_{8} = v_{5}. The module V as u(𝔟^{+})-module is uniserial with radical series:

V = W_{8} ⊃ W_{7} ⊃ ⋯ ⊃ W_{1} ⊃ {0}

where W_{j} =< w_{i} : 1 ≤ i ≤ j >. This induces a u(𝔟^{+})-filtration on Λ^{4}(V) of length 17 with semisimple quotients:

Λ^{4}(V) = M_{26} ⊃ M_{25} ⊃ ⋯ ⊃ M_{10} ⊃ {0}.

Here, M_{r} =< w_{i} ∧ w_{j} ∧ w_{l} ∧ w_{m} : i + j + l + m = r >.

We have L(ω_{4}) ↪ Λ^{4}(V) which implies that P_{χ}(λ) is a direct summand of ind_{u(𝔟^{+})}^{u(𝔤,χ)}Λ^{4}(V)⊗

(p − 1)ρ (λ = (p − 1)ω_{1} + (p − 1)ω_{2} + (p − 1)ω_{3}). Set N_{r} = (ind_{u(𝔟^{+})}^{u(𝔤,χ)}M_{r} ⊗ (p − 1)ρ) ∩ P_{χ}(λ). Then P_{χ}(λ) has a u(𝔤, χ)-filtration with semisimple quotients:

P_{χ}(λ) = N_{26} ⊇ N_{25} ⊇ ⋯ ⊇ N_{10} ⊇ {0}.

The weights contributing to composition factors of P_{χ}(λ) from Λ^{4}(V) are of the form W ω_{4}. But, M_{25}/M_{24} =< w_{4} ∧ w_{6} ∧ w_{7} ∧ w_{8} > and M_{12}/M_{11} =< w_{1} ∧ w_{2} ∧ w_{3} ∧ w_{5} >. The vectors in these modules do not have weights of the form W ω_{4}. Thus, N_{25} = N_{24} and N_{12} = N_{11} and

l_{𝔤}(P_{χ}(λ)) ≤ 15 < |W |/|Stab_{W} λ| = 16.

Consequently, P_{χ}(λ) is not uniserial and B_{λ} has infinite representation type. _{□} 3.8 Type D_{n}; n ≥ 4: We will now give a sufficient criterion for blocks for Lie algebras of type D to have infinite representation type. Observe that this criterion differs from the ones given for types A, B and C.

Proposition . Let 𝔤 be a classical Lie algebra of type D_{n}, n ≥ 4. If Φ_{λ} is not type D_{n} then B_{λ} has infinite representation type.

Proof. Let Y ⊆ {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i}. Consider the following inequalities

l_{𝔟^{+}}(L(σ)) ≤ \frac{2}{3}(n)(n − 1)^{2} + 1 < |W |/|Stab_{W} λ| = 2^{n−1} · n!/|Stab_{W} λ|.

The first inequality holds generally by results in Section 3.4. The second inequality if valid if rank(Φ_{λ}) ≤ n − 3 (or |Y | ≥ 3). Now suppose that rank(Φ_{λ}) = n − 2. If σ = ω_{r} + ω_{s} 1 ≤ r < s ≤ n then

l_{𝔟^{+}}(L(σ)) ≤ \begin{cases}(2n − 1)(r + s) − (r^{2} + s^{2}) + 1 & 1 ≤ r < s < n − 2 \\ (2n − 1)r − r^{2} + (2n − 1)(n − 2) − (n − 2)^{2} + 1 & 1 ≤ r < n − 2, s = n − 2 \\ \frac{n+1}{2} + (2n − 1)r − r^{2} & 1 ≤ r ≤ n − 2, s = n − 1, n \\ (n − 1)(n) + 1 & r = n − 1, s = n \end{cases}

In each of these cases we have

|W |/|Stab_{W} λ| = \begin{cases}2^{s} · \binom{n}{s} · \binom{s}{r} & 1 ≤ r < s < n − 2 \\ 2^{n−3} · n(n − 1)\binom{n−2}{r} & 1 ≤ r < n − 2, s = n − 2 \\ 2^{n−1} · \binom{n}{r} & 1 ≤ r < n − 2, s = n − 1, n \\ 2^{n−1} · n(n − 1) & r = n − 1, s = n \end{cases}

One can verify directly that l_{𝔟^{+}}(L(σ)) < |W |/|Stab_{W} λ|. If rank(Φ_{λ}) = n − 1 and σ = ω_{r} 1 ≤ r ≤ n then

l_{𝔟^{+}}(L(σ)) ≤ \begin{cases}(2n − 1) & r = 1 \\ (2n − 1)r − r^{2} + 1 & 1 < r < n − 2 \\ (2n − 1)(n − 2) − (n − 2)^{2} + 1 & r = n − 2 \\ \frac{n(n−1)}{2} + 1 & r = n − 1, n \end{cases}

and

|W |/|Stab_{W} λ| = \begin{cases}2n & r = 1 \\ 2^{r} · \binom{n}{r} & 1 < r < n − 2 \\ 2^{n−3} · n & r = n − 2 \\ 2^{n−1} & r = n − 1, n \end{cases}

Again one can verify that in each case that l_{𝔟^{+}}(L(ω_{r})) < |W |/|Stab_{W} λ|. _{□}

3.9 Exceptional cases: E and F: The following result demonstrates that all blocks for the exceptional algebras of types E and F other than the block containing the “Steinberg module” have infinite representation type.

Theorem . Let 𝔤 be a classical algebra of type E_{6}, E_{7}, E_{8}, or F_{4}. If λ ≠ (p − 1)ρ, then B_{λ} is of infinite representation type.

Proof. The verification will be done in a case by case manner. Let Y ⊆ {1, 2, …, n} and σ = ∑_{i∈Y} ω_{i}. For the algebra E_{6}, if |Stab_{W} λ| < 304, then

l_{𝔟^{+}}(L(σ)) ≤ 171 < |W |/|Stab_{W} λ| = 51840/|Stab_{W} λ|.

This will hold if rank(Φ_{λ}) ≤ 4. If Φ_{λ} is of type D_{5}, then

l_{𝔟^{+}}(L(ω_{j})) ≤ 24 < |W |/|Stab_{W} λ| = 51840/1920 = 27  (3.9.1)

for j = 1, 6. On the other hand, if Φ_{λ} is of type A_{5}, then

l_{𝔟^{+}}(L(ω_{2})) ≤ 23 < |W |/|Stab_{W} λ| = 51840/61 = 72.  (3.9.2)

Next we look at the case for E_{7}. If |Stab_{W} λ| < 7258, then

l_{𝔟^{+}}(L(σ)) ≤ 400 < |W |/|Stab_{W} λ| = 2903040/|Stab_{W} λ|.

In this case, this will be valid if rank(Φ_{λ}) ≤ 5. For the rank six subalgebras D_{6}, A_{6}, and E_{6} one has

l_{𝔟^{+}}(L(ω_{1})) ≤ 35 < |W |/2^{5} · 6! = 126  (3.9.3)
l_{𝔟^{+}}(L(ω_{2})) ≤ 50 < |W |/7! = 576  (3.9.4)
l_{𝔟^{+}}(L(ω_{7})) ≤ 35 < |W |/2^{7} · 3^{4} · 5 = 56  (3.9.5)

The E_{8} is similar to the previous cases. If |Stab_{W} λ| < 561426 then

l_{𝔟^{+}}(L(σ)) ≤ 1241 < |W |/|Stab_{W} λ| = 2^{14} · 3^{5} · 5^{2} · 7/|Stab_{W} λ|,

and this holds if rank(Φ_{λ}) ≤ 6. For the rank seven algebras, D_{7}, A_{7} and E_{7}, one has

l_{𝔟^{+}}(L(ω_{1})) ≤ 93 < |W |/2^{6} · 7! = 2160  (3.9.6)
l_{𝔟^{+}}(L(ω_{2})) ≤ 157 < |W |/8! = 17280  (3.9.7)
l_{𝔟^{+}}(L(ω_{8})) ≤ 59 < |W |/2^{10} · 3^{4} · 5 · 7 = 240  (3.9.8)

The final case is F_{4}. If Φ_{λ} = ∅, A_{1}, A_{1} × A_{1}, A_{2} or B_{2}, then |Stab_{W} λ| < 11. Therefore, in these cases

l_{𝔟^{+}}(L(σ)) ≤ 111 < |W |/|Stab_{W} λ| = 2^{7} · 3^{2}/|Stab_{W} λ|.

On the other hand, if Φ_{λ} is of type C_{3} or B_{3},

l_{𝔟^{+}}(L(ω_{1})) ≤ 23 < |W |/2^{3} · 3! = 24  (3.9.9)
l_{𝔟^{+}}(L(ω_{4})) ≤ 17 < |W |/2^{3} · 3! = 24.  (3.9.10)

This completes the proof because in each case l_{𝔤}(P_{χ}(λ)) ≤ l_{𝔟^{+}}(L(σ)) < |W |/|Stab_{W} λ| for Y ≠ ∅. _{□}

3.10 Support varieties of simple modules: Let 𝔩 be a Levi subalgebra of a classical Lie algebra 𝔤, and let 𝔭^{+} = 𝔩 ⊕ 𝔲^{+} be the corresponding parabolic subalgebra containing 𝔟^{+}. For any λ ∈ Λ_{χ} we have

Z_{χ}(λ) = ind_{u(𝔟^{+})}^{u(𝔤,χ)}λ = ind_{u(𝔭^{+})}^{u(𝔤,χ)}Z_{χ}^{𝔩}(λ).

Here Z_{χ}^{𝔩}(λ) is the Verma module for u(𝔩, χ). This module becomes a u(𝔭^{+})-module by letting 𝔲^{+} act trivially.

The cohomological support varieties of modules are useful devices for determining the representation type of algebras. Such a determination can obtained at times by simply knowing the dimension of such varieties. Let M be a u(𝔤, χ)-module. The support variety of M is the affine homogeneous variety |𝔤|_{M} in 𝔤 defined by

|𝔤|_{M} = {x ∈ 𝔤 : x^{[p]} = 0, and M is not free as u(kx, χ)-module} ∪ {0}.

Note that x ∈ |𝔤|_{M} if and only if dim_{k}(x − χ(x))^{p−1}.M ≠ (dim_{k} M)/p. Let 𝔤^{χ} = {x ∈ 𝔤 : χ([x, 𝔤]) = 0} be the centralizer of the character χ. The next result provides an upper bound on the dimension of the support varieties for Z_{χ}(λ), λ ∈ Λ_{χ}.

Theorem . Let 𝔤 be a classical Lie algebra, χ be a regular nilpotent character. Moreover, let 𝔩 be a Levi subalgebra such that Z_{χ}^{𝔩}(λ) is projective u(𝔩, χ)-module. Then

|𝔤|_{Z_{χ}(λ)} ⊆ 𝔤^{χ} ∩ 𝔲^{+}.

Proof. Let 𝔩 be a Levi subalgebra such that Z^{𝔩}(λ) is projective as u(𝔩, χ)-module and let 𝔭^{+} = 𝔩 ⊕ 𝔲^{+}. Since Z_{χ}(λ) ≅ ind_{u(𝔭^{+},χ)}^{u(𝔤,χ)}Z_{χ}^{𝔩}(λ), we have |𝔤|_{Z_{χ}(λ)} ⊆ |𝔭^{+}|_{Z_{χ}^{𝔩}(λ)} [FN, Prop. 4.1a]. Furthermore, |𝔤|_{Z_{χ}(λ)} ⊆ 𝔤^{χ} by [Pr2]. It suffices to show that |𝔭^{+}|_{Z_{χ}^{𝔩}(λ)} ⊆ 𝔲^{+}.

Let x = l + u where x ∈ |𝔭^{+}|_{Z_{χ}^{𝔩}(λ)}, l ∈ 𝔩 and u ∈ 𝔲^{+}. Since 𝔲^{+} acts trivially on Z_{χ}^{𝔩}(λ) and χ(x) = χ(l), it follows that

dim_{k}(x − χ(x))^{p−1}.Z_{χ}^{𝔩}(λ) = dim_{k}(l − χ(l))^{p−1}.Z_{χ}^{𝔩}(λ).

The module Z_{χ}^{𝔩}(λ) is projective as a u(𝔩, χ)-module so by the preceding statement Z_{χ}^{𝔩}(λ) is projective as a u(kx, χ)-module unless l = 0. Hence, l = 0 and x ∈ 𝔲^{+}. _{□}

3.11 Rank one and two cases: A weight λ ∈ Λ_{χ} is regular if Φ_{λ} = ∅. The existence of a regular weight is equivalent to that statement that p ≥ h where h is the Coxeter number associated to W [Jan2, II 6.2]. We now examine the representation type of the blocks for rank two classical simple Lie algebras.

Theorem . Let 𝔤 be a classical simple Lie algebra of rank less than or equal to two and χ be regular nilpotent.

(a) If 𝔤 is of type A_{1} then B_{λ} has finite representation type and is not semisimple if and only if Φ_{λ} = ∅.

(b) if 𝔤 is of type A_{2}, B_{2} or G_{2} then B_{λ} has finite representation type and is not semisimple if and only if Φ_{λ} is of type A_{1}.

Proof. (a) Let 𝔤 be of type A_{1}. If Φ_{λ} is of type A_{1} then B_{λ} is semisimple. On the other hand, if Φ_{λ} = ∅ then B_{λ} is a uniserial block whose projective indecomposable module has two isomorphic simple composition factors, so in this case B_{λ} is of finite type.

(b) Now let 𝔤 be A_{2}, B_{2} or G_{2}. If Φ_{λ} has rank two then B_{λ} is semisimple because P_{χ}(λ) has only one composition factor. Now suppose that Φ_{λ} has rank one (i.e. of type A_{1}). Let 𝔩 be the Levi subalgebra corresponding to Φ_{λ}. The module Z_{χ}^{𝔩}(λ) is projective as a u(𝔩, χ)-module, thus by Theorem 3.10

|𝔤|_{Z_{χ}(λ)} ⊆ 𝔤^{χ} ∩ 𝔲^{+} = ke_{π}.

where π is the highest long root. Thus, dim |𝔤|_{Z_{χ}(λ)} ≤ 1. The block B_{λ} cannot be semisimple because there are |W |/|Stab_{W} λ| composition factors in P_{χ}(λ). It follows that dim |𝔤|_{Z_{χ}(λ)} = 1 and by [Fa1, p. 58], B_{λ} has finite representation type.

Finally, suppose that Φ_{λ} = ∅. Therefore, λ is a regular weight and (𝔤^{χ})^{[p]} = 0. Therefore, by [Pr3] we have

𝔤^{χ} = |𝔤|_{⊕_{μ∈Λχ}Z_{χ}(μ)} = ⋃_{μ∈Λ_{χ}} |𝔤|_{Z_{χ}(μ)}.

If λ_{1} and λ_{2} are regular then B_{λ_{1}} and B_{λ_{2}} are Morita equivalent [Jan5, Prop. 11.21], and dim |𝔤|_{Z_{χ}(λ_{1})} = dim |𝔤|_{Z_{χ}(λ_{2})}. Hence,

2 = dim 𝔤^{χ} = max_{μ∈Γ}{dim |𝔤|_{Z_{χ}(λ)}, dim |𝔤|_{Z_{χ}(μ)}}

where Γ = {μ ∈ Λ_{χ} : rank(Φ_{μ}) ≥ 1}. If rank(Φ_{μ}) = 2 then |𝔤|_{Z_{χ}(μ)} = {0}. On the other hand, if rank(Φ_{μ}) = 1, then we may assume without a loss of generality (by conjugating and using the fact that Z_{χ}(μ) ≅ Z_{χ}(w · μ) for all w ∈ W) that there exists a Levi factor 𝔩 of type A_{1} for which Z_{χ}^{𝔩}(μ) is projective as u(𝔩, χ)-module. Let 𝔭^{+} = 𝔩 ⊕ 𝔲^{+} be the associated parabolic subalgebra. Then |𝔤|_{Z_{χ}(μ)} ⊆ 𝔤^{χ} ⊕ 𝔲^{+} and dim_{k} |𝔤|_{Z_{χ}(μ)} ≤ 1 for all μ ∈ Γ. Since dim_{k} 𝔤^{χ} = 2, it follows that dim_{k} |𝔤|_{Z_{χ}(λ)} = 2, and B_{λ} has infinite representation type. _{□}

4 Representation type

4.1 The first step to determining the representation type of blocks will be to show that the blocks of finite representation type in u(𝔤, χ) where 𝔤 classical and χ of Levi form can only occur when χ is regular nilpotent. Information on the linkage classes of weights under the action of W is important for this purpose.

Proposition . Let 𝔤 be a classical Lie algebra satisfying (A1)-(A3). If I is proper subset of Π and λ ≠ (p − 1)ρ in X_{1}(T) then there exists μ ∈ W · λ such that |W · μ| > |W_{I} · μ|.

Proof. The assumption (A3) insures that the form ( , ) on X(T) induces a non-degenerate bilinear form on X(T)/pX(T) ≅ X_{1}(T) ≅ 𝔱^{∗}. Therefore, if (λ + ρ, α^{∨}) ∈ pℤ for all α ∈ Π, then λ + ρ ∈ pX(T) and λ = (p − 1)ρ in X_{1}(T). Since this is not the case, it follows that (λ + ρ, α^{∨}) ∉ pℤ for some α ∈ Π.

Let α ∈ Π and α ∉ Φ_{λ}. If α is a long (resp. short) root then (A3) enables us to show there exists w ∈ W such that wα = β, where β is the highest long (resp. short) root [Hu1, 10.4 Lemma C]. Set μ = w · λ in X_{1}(T). Consider σ_{β} · μ ∈ W · μ. Since α ∉ Φ_{λ}, it follows that β = wα ∉ Φ_{w·λ} = Φ_{μ}. Consequently, (μ + ρ, β^{∨}) ∉ pℤ and

σ_{β} · μ = μ − (μ + ρ, β^{∨})β ≠ μ

in X_{1}(T). By using induction on the length of y it follows that for any y ∈ W_{I} we have y · λ = λ − ∑_{α∈I} d_{α}α. On the other hand, condition (A2) implies that β = ∑_{α∈Π} c_{α}α with c_{α} ∉ pℤ [Hu1, p.66]. The condition that I ⊊ S and shows that σ_{β} · μ ∉ W_{I} · μ. _{□}

4.2 Finite representation type: We can now state which blocks of u(𝔤, χ) for 𝔤 classical have finite representation type. The semisimple blocks are also indicated.

Theorem . Let 𝔤 be a classical Lie algebra satisfying (A1)-(A3), χ a nilpotent character in Levi form and λ ∈ Λ_{χ}.

(a) The block B_{λ} is semisimple if and only if λ = (p − 1)ρ.

(b) The block B_{λ} has finite representation type and is not semisimple if and only if χ is a regular nilpotent character, and one of the following holds:

(i) 𝔤 is type A_{1} and Φ_{λ} = ∅; (ii) rank 𝔤 = 2 and Φ_{λ} is of type A_{1}; (iii) 𝔤 is of type A_{n}, n ≥ 3 and Φ_{λ} is of type A_{n−1}; (iv) 𝔤 is of type B_{n}, n ≥ 3 and Φ_{λ} is of type B_{n−1};

(v) 𝔤 is of type C_{n}, n ≥ 3 and Φ_{λ} is of type C_{n−1}.

Proof. If λ = (p − 1)ρ then B_{λ} is semisimple by [FP1, Thm 4.3c]. Now assume that λ ≠ (p − 1)ρ. Suppose that χ is in Levi form, but not regular nilpotent. Let I = {α ∈ Π : χ(f_{α}) = 1} (see Definition 2.3). Since χ is not regular nilpotent we have I ⊊ Π. The number of non-isomorphic simple modules in B_{λ}, |B_{λ}|, is |W ·λ ∩Λ_{χ}| by (2.4.6). Proposition 4.1 shows that |B_{λ}| > 1. Therefore, by [Fa1, Thm. 3.2], B_{λ} has infinite representation type.

Suppose that χ is regular nilpotent and λ ≠ (p − 1)ρ. The statements (b)(i) and (b)(ii) hold by Theorem 3.11. We may now assume that the rank(𝔤) = n ≥ 3. According to Propositions 3.8 and 3.9, the block B_{λ} has infinite representation type for 𝔤 of type D_{n}, E_{6}, E_{7}, E_{8}, and F_{4}. Furthermore, by Propositions 3.5, 3.6 and 3.7, it follows that B_{λ} has infinite representation type in A_{n} (resp. B_{n} and C_{n}) if Φ_{λ} is not of type A_{n−1} (resp. B_{n−1} and C_{n−1}).

Let 𝔤 be of type A_{n} (resp. B_{n} and C_{n}) and Φ_{λ} be of type A_{n−1} (resp. B_{n−1} and C_{n−1}). Moreover, let 𝔩 be the Levi subalgebra corresponding to Φ_{λ}. The module Z_{χ}^{𝔩}(λ) is projective as a u(𝔩, χ)-module. Consequently, by Theorem 3.10 one has

|𝔤|_{Z_{χ}(λ)} ⊆ 𝔤^{χ} ∩ 𝔲^{+}.

In these cases one can directly verify that 𝔤^{χ} ∩ 𝔲^{+} = ke_{π} where π denotes the highest long root. Therefore, dim |𝔤|_{Z_{χ}(λ)} ≤ 1. The number of times the simple module Z_{χ}(λ) appears in P_{χ}(λ) is n + 1, 2n + 1 and 2n respectively for types A_{n}, B_{n} and C_{n} so the block B_{λ} cannot be semisimple. Consequently, dim |𝔤|_{Z_{χ}(λ)} = 1 and B_{λ} has finite representation type by [Fa1, p. 58]. _{□}

Remark (A). The reader should be made aware that the results given for classical Lie algebras in [Fa2, Thm 4.3] are not correct. Let 𝔤 be of type A_{3} and χ a regular nilpotent character. If Φ_{λ} is of type A_{2} then B_{λ} has finite representation type and P_{χ}(λ) is uniserial of length four. Conversely, the only way that the P_{χ}(λ) in u(𝔤, χ) (𝔤 of type A_{3}) can have four composition factors is when Φ_{λ} is of type A_{2}.

Remark (B). Jantzen [Jan6] shows that for type A_{n}, if χ is a regular nilpotent character and t = n − rank(Φ_{λ}) then dim_{k} Ext_{u(𝔤,χ)}^{1}(Z_{χ}(λ), Z_{χ}(λ)) ≥ t. Equality does hold for A_{2}. However, for A_{3} and Φ_{λ} of type A_{1} × A_{1},

dim_{k} Ext_{u(𝔤,χ)}^{1}(Z_{χ}(λ), Z_{χ}(λ)) ≥ 2 > t

where t = 1, for otherwise the block B_{λ} would be of finite type which is contrary to Theorem 4.2.

4.3 Premet’s results We will now indicate how Premet’s determination of the enveloping algebras of classical Lie algebras of finite representation type in the case when χ is a Levi form character follows from our results.

Theorem . [Pr3, Prop 5.3] Let 𝔤 be a classical simple Lie algebra satisfying (A1)-(A3) and χ be a nilpotent character in Levi form. The algebra u(𝔤, χ) has finite representation type if and only if χ is a regular nilpotent character and one of the following holds:

(i) 𝔤 is of type A_{1}; (ii) 𝔤 is of type A_{2}, p = 2; (iii) 𝔤 is of type B_{2}, p = 3; (iv) 𝔤 is of type G_{2}, p = 5

Proof. If χ is not regular nilpotent then by Theorem 4.2 all blocks of u(𝔤, χ) except for B_{(p−1)ρ} have infinite representation type. Therefore, u(𝔤, χ) can have finite representation type only in the case when χ is regular nilpotent. Now suppose χ is regular nilpotent and rank(𝔤) ≥ 3. Let λ = (p − 1, 0, p − 1, …, p − 1). By Theorem 4.2(b), the block B_{λ} of u(𝔤, χ) has infinite representation type. Consequently, the only finite type algebra happen possibly when rank(𝔤) ≤ 2 and χ regular nilpotent.

If rank(𝔤) = 1 (i.e. 𝔤 is of type A_{1}) then u(𝔤, χ) has finite representation type by Theorem 4.2(b)(i). Now assume that rank(𝔤) = 2. If p ≥ h where h is the Coxeter number then there exists a regular weight λ. Therefore, by Theorem 3.11, u(𝔤, χ) has infinite representation type. The Coxeter number h = 3, 4, and 7 for A_{2}, B_{2} and G_{2}, respectively. The remaining primes left with the assumptions (A1)-(A3) are p = 2 for A_{2}, p = 3 for B_{2} and p = 5 for G_{2}. In these cases there are no regular weights in Λ_{χ} so rank(𝔤) − rank(Φ_{λ}) ≤ 1 for all λ ∈ Λ_{χ}. Therefore, u(𝔤, χ) has finite representation type in these cases by Theorem 4.2. _{□}

5 Examples

In order to demonstrate how Theorem 4.2 can be applied, we will consider the following examples for 𝔤 of type A_{2} and A_{4} when p = 2.

5.1 Let 𝔤 = 𝔰𝔩_{3}(k), χ be a regular nilpotent character for 𝔤 when p = 2. The enveloping algebra u(𝔤, χ) has two blocks. The weights in X_{1}(T) are (0, 0), (0, 1), (1, 0), and (1, 1). with W ·(0, 0) = {(0, 0), (0, 1), (1, 0)} and W ·(1, 1) = {(1, 1)}. The block B_{(1,1)} is semisimple with Φ_{(1,1)} = Φ. For the other block B_{(0,0)} the subroot system Φ_{(0,0)} is of type A_{1}. The projective cover P_{χ}(0, 0) for Z_{χ}(0, 0) has three composition factors and is uniserial.

5.2 Let 𝔤 = 𝔰𝔩_{5}(k), χ be a regular nilpotent character for 𝔤 when p = 2. The W -orbits of the weights in X_{1}(T) are given by

W · (1, 1, 1, 1) = {(1, 1, 1, 1)}
W · (0, 0, 0, 0) = {(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1), (0, 0, 0, 1)}
W · (1, 1, 0, 0) = {(1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1), (1, 0, 0, 1), (1, 0, 1, 0), (0, 1, 0, 1),
(1, 1, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1), (1, 1, 0, 1)}

with Φ_{(1,1,1,1)} of type A_{4}, Φ_{(0,0,0,0)} of type A_{3}, and Φ_{(1,1,0,0)} of type A_{1} × A_{2}. According to Theorem 4.2, the block B_{(1,1,1,1)} is semisimple; B_{(0,0,0,0)} has finite representation type; and B_{(1,1,0,0)} has infinite representation type. The projective cover P_{χ}(0, 0, 0, 0) is uniserial with five composition factors and the projective cover P_{χ}(1, 1, 0, 0) is not uniserial with ten composition factors.

6 Appendix: Tables

root system β (highest short root)
A_{n}  β^{∨} = α_{1}^{∨} + α_{2}^{∨} + ⋯ + α_{n}^{∨}
B_{n}  β^{∨} = 2α_{1}^{∨} + 2α_{2}^{∨} + ⋯ + 2α_{n−1}^{∨} + α_{n}^{∨}
C_{n}  β^{∨} = α_{1}^{∨} + 2α_{2}^{∨} + ⋯ + 2α_{n}^{∨}
D_{n}  β^{∨} = α_{1}^{∨} + 2α_{2}^{∨} + ⋯ + 2α_{n−2}^{∨} + α_{n−1}^{∨} + α_{n}^{∨}
E_{6}  β^{∨} = α_{1}^{∨} + 2α_{2}^{∨} + 2α_{3}^{∨} + 3α_{4}^{∨} + 2α_{5}^{∨} + α_{6}^{∨}
E_{7}  β^{∨} = 2α_{1}^{∨} + 2α_{2}^{∨} + 3α_{3}^{∨} + 4α_{4}^{∨} + 3α_{5}^{∨} + 2α_{6}^{∨} + α_{7}^{∨}
E_{8}  β^{∨} = 2α_{1}^{∨} + 3α_{2}^{∨} + 4α_{3}^{∨} + 6α_{4}^{∨} + 5α_{5}^{∨} + 4α_{6}^{∨} + 3α_{7}^{∨} + 2α_{8}^{∨}
F_{4}  β^{∨} = 2α_{1}^{∨} + 4α_{2}^{∨} + 3α_{3}^{∨} + 2α_{4}^{∨}
G_{2}  β^{∨} = 2α_{1}^{∨} + 3α_{2}^{∨}

\begin{matrix}E_{6} & |S′| & |T′| & M & S \\ p = 7 & 3 & 4 & 7 & \{1, 5, 6\}, \{1, 3, 6\}, \{1, 2, 6\} \\ p = 5 & 2 & 3 & 5 & \{1, 2, 3, 6\}, \{1, 3, 5, 6\} (*), \{1, 2, 5, 6\} \end{matrix}

E_{7}  |S′| |T′| M S
p = 13 5 6  14 {6, 7}, {1, 7}, {2, 7}
5 6  13 {2, 6}, {1, 6}, {1, 3}, {5, 7}, {3, 7}
p = 11 4 5  12 {2, 6, 7}, {1, 6.7}, {1, 2, 7}
4 5  11 {5, 6, 7}, {1, 5, 7}, {2, 5, 7}, {1, 3, 6},
{3, 6, 7}, {1, 3, 7}, {2, 3, 7}
p = 7 3 4  10 {1, 3, 6, 7}
3 4  9 {2, 5, 6, 7}, {1, 5, 6, 7}, {1, 2, 5, 7},
{2, 3, 6, 7}, {1, 3, 6, 7}, {1, 2, 3, 7}
3 4  8 {1, 2, 5, 6}, {1, 2, 3, 6}, {2, 4, 6, 7}, {1, 4, 6, 7}, {1, 2, 4, 7}}
3 4  7 {1, 2, 4, 6} (*), {4, 5, 6, 7}, {1, 4, 5, 7} (*),

{2, 4, 5, 7}, {3, 4, 6, 7} (*), {1, 3, 4, 7}, {2, 3, 4, 7}

2 3, 4 7 {1, 2, 5, 6, 7}, {1, 2, 3, 6, 7}
p = 5 2 3  7 {1, 2, 5, 6, 7}, {1, 2, 3, 6, 7}
2 3  6 {1, 2, 4, 6, 7} (*)
2 3  5 {2, 4, 5, 6, 7}, {1, 4, 5, 6, 7}, {1, 2, 4, 5, 7} (*),

{2, 3, 4, 6, 7} (*), {1, 3, 4, 6, 7} (*), {1, 2, 3, 4, 7}

E_{8}  |S′| |T′| M S
p = 23 6 7  24 {1, 2}, {1, 7}, {2, 8}, {7, 8}
6 7  23 {1, 3}, {1, 6}, {2, 7}, {3, 8}, {6, 8}
p = 19 5 6  22 {1, 2, 8}, {1, 7, 8}
5 6  21 {2, 7, 8}, {1, 2, 7}, {1, 3, 8}, {1, 6, 8}
5 6  20 {6, 7, 8}, {1, 6, 7}, {1, 2, 6}, {2, 6, 8},

{1, 2, 3}, {2, 3, 8}, {1, 2, 7}, {2, 7, 8}

5 6  19 {1, 4, 8}, {5, 7, 8}, {1, 5, 7}, {2, 5, 8}, {1, 2, 5},
4 5, 6  {1, 2, 7, 8}
p = 17 4 5  19 {1, 2, 7, 8}
18 {1, 6, 7, 8}, {1, 2, 6, 8}, {1, 3, 7, 8}, {1, 2, 3, 8}
17 {2, 6, 7, 8}, {1, 2, 6, 7}, {2, 3, 7, 8},

{1, 2, 3, 7}, {1, 3, 6, 8}

p = 13 3 4  15 {1, 2, 6, 7, 8}, {1, 2, 3, 7, 8}
3 4  14 {1, 2, 3, 6, 8}, {1, 3, 6, 7, 8}, {1, 2, 5, 6, 8}
3 4  13 {2, 3, 6, 7, 8}, {1, 2, 3, 6, 7}, {1, 5, 6, 7, 8},

{1, 2, 5, 6, 8}, {1, 3, 5, 7, 8}, {1, 2, 3, 5, 8}, {1, 2, 4, 7, 8}

p = 11 3 4  15 {1, 2, 6, 7, 8}, {1, 2, 3, 7, 8}
3 4  14 {1, 2, 3, 6, 8}, {1, 3, 6, 7, 8}, {1, 2, 5, 6, 8}
3 4  13 {2, 3, 6, 7, 8}, {1, 2, 3, 6, 7}, {1, 5, 6, 7, 8},

{1, 2, 5, 6, 8}, {1, 3, 5, 7, 8}, {1, 2, 3, 5, 8}, {1, 2, 4, 7, 8}

3 4  12 {1, 4, 6, 7, 8}, {1, 2, 4, 6, 8} (∗),

{1, 3, 4, 7, 8}, {1, 2, 3, 4, 8}

3 4  11 {2, 4, 6, 7, 8}, {1, 2, 4, 6, 7} (*), {2, 3, 4, 7, 8},

{1, 3, 4, 6, 8}, {1, 2, 4, 5, 8}, {1, 2, 3, 4, 7}, {1, 4, 5, 7, 8} (*)

2 3, 4 11 {1, 2, 3, 6, 7, 8}
F_{4}  |S′| |T′| M S
p = 7 3 4 7 {2}
2 3 7 {1, 4}

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Department of Mathematics and Statistics, Utah St. University, Logan, UT. 84322, U.S.A.

   E-mail address: nakano@sunfs.math.usu.edu

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada

   E-mail address: pollackd@post.queensu.ca