Decomposition Numbers of Sp_{4}(2^{a}) in Odd Characteristics

Donald L. White ^{∗} Department of Mathematics and Computer Science Kent State University Kent, OH 44242 white@mcs.kent.edu

August 8, 1994 (To appear in J. Algebra)

1 Introduction and Notation

Let G be the finite symplectic group Sp_{4}(q), where q is a power of a prime. The order of G is |G| = q^{4}(q^{2} + 1)(q − 1)^{2}(q + 1)^{2}. The r-decomposition numbers of G for all primes r not dividing q were determined in [9, 10, 11] in case q is odd. In this paper we consider the case where q is even.

Let q = 2^{a} be a power of 2 and let r be an odd prime dividing |G|. Then r divides exactly one of q^{2} + 1, q − 1, or q + 1. By a result of L. Puig [8], if r|q − 1, then the Brauer characters are liftable and the decomposition matrix is easily determined. If r|q^{2} + 1, then all r-blocks are cyclic and the Brauer trees follow from results of P. Fong and B. Srinivasan [4]. We include the decomposition matrices and Brauer trees for these cases for completeness. In the case r|q + 1, the decomposition matrix is determined up to one parameter in one column of the matrix for the principal block.

Computations involving characters were done using CHEVIE [6], a computer algebra package based on MAPLE [2]. The character table of G was first calculated by H. Enomoto [3]. The table was recomputed (and the misprints corrected) by F. Lübeck, and is included in the CHEVIE package. We use the CHEVIE notation for the ordinary characters of G. The correspondence between the CHEVIE notation and that of Enomoto is given in Table 1.

∗Research partially supported by the DFG as part of the research project “Algorithmic Number Theory and Algebra,” and by the Kent State University Research Council.

Table 1: Notation for Characters of Sp_{4}(q), q = 2^{a}

Degree  CHEVIE Enomoto  Index  No. of Chars.
1  χ_{1}  θ_{0}  1
q(q + 1)^{2}/2  χ_{2}  θ_{1}  1
q(q^{2} + 1)/2  χ_{3}  θ_{2}  1
q(q^{2} + 1)/2  χ_{4}  θ_{3}  1
q(q − 1)^{2}/2  χ_{5}  θ_{5}  1
q^{4}  χ_{6}  θ_{4}  1
(q + 1)(q^{2} + 1)  χ_{7}[k]  χ_{6}(k)  k ∈ T_{1}  (q − 2)/2
q(q + 1)(q^{2} + 1) χ_{8}[k]  χ_{10}(k)  k ∈ T_{1}  (q − 2)/2
(q − 1)(q^{2} + 1)  χ_{9}[k]  χ_{8}(k)  k ∈ T_{2}  q/2
q(q − 1)(q^{2} + 1) χ_{10}[k]  χ_{12}(k)  k ∈ T_{2}  q/2
(q + 1)(q^{2} + 1) χ_{11}[k]  χ_{7}(k)  k ∈ T_{1}  (q − 2)/2
q(q + 1)(q^{2} + 1) χ_{12}[k]  χ_{11}(k)  k ∈ T_{1}  (q − 2)/2
(q − 1)(q^{2} + 1) χ_{13}[k]  χ_{9}(k)  k ∈ T_{2}  q/2
q(q − 1)(q^{2} + 1) χ_{14}[k]  χ_{13}(k)  k ∈ T_{2}  q/2
(q + 1)^{2}(q^{2} + 1) χ_{15}[k, l] χ_{1}(k, l) k, l ∈ T_{1}, k ≠ l (q − 2)(q − 4)/8
(q^{4} − 1)  χ_{16}[k]  χ_{2}(k)  k ∈ R_{2}  q(q − 2)/4
(q^{4} − 1)  χ_{17}[k, l] χ_{3}(k, l) k ∈ T_{1}, l ∈ T_{2}  q(q − 2)/4
(q − 1)^{2}(q + 1)^{2} χ_{18}[k]  χ_{5}(k)  k ∈ R_{1}  q^{2}/4
(q − 1)^{2}(q^{2} + 1) χ_{19}[k, l] χ_{4}(k, l) k, l ∈ T_{2}, k ≠ l  q(q − 2)/8

The index sets T_{1}, T_{2}, R_{1}, and R_{2} in the table are defined as follows. Let T_{1} = {1, …, (q − 2)/2} and T_{2} = {1, …, q/2}. Let R_{1} be a set of class representatives for the equivalence relation on R_{1}^{∗} = {1, …, q^{2}} defined by

i ∼ j ⇔ i ≡ ±j or ± qj(mod q^{2} + 1).

Let R_{2} be a set of class representatives for the equivalence relation on

R_{2}^{∗} = {j ∈ Z : 1 ≤ j ≤ q^{2} − 1, q − 1 |̸ j, q + 1 |̸ j}

defined by

i ∼ j ⇔ i ≡ ±j or ± qj(mod q^{2} − 1).

In all cases, the characters in a given family that lie in the same r-block have the same restriction to the r-regular elements. We will therefore delete the indices from the notation when considering a particular block. The restriction of an ordinary character χ to the r-regular elements will be denoted by χ̂. We will refer to characters of projective modules as projective characters and to characters of projective indecomposable modules as projective indecomposable characters. The missing entries in the decomposition matrices are zeros.

2 The Case r|q + 1

In this section, r denotes a prime dividing q + 1. Let q + 1 = r^{d}m, where m is not divisible by r. Denote by T′_{2}, R′_{2} the set of multiples of r^{d} in T_{2}, R_{2}, respectively. Then |T′_{2}| = (m − 1)/2 and |R′_{2}| = (q − 2)(m − 1)/4.

The Sylow r-subgroup of G is isomorphic to Z_{r^{d}} ×Z_{r^{d}}, and all defect groups are either trivial, Z_{r^{d}}, or Z_{r^{d}} × Z_{r^{d}}. Thus all r-blocks not of maximal defect are cyclic. Using central characters, it is easily shown that the sets of ordinary characters defined below are the r-blocks of G of positive defect. The characters of types χ_{2}, χ_{15}, and χ_{18} are in blocks of defect 0.

Definition 2.1 Define for s, t ∈ T′_{2} (s ≠ t), u ∈ T_{1}, and v ∈ R′_{2},

B_{0} = {χ_{1}, χ_{3}, χ_{4}, χ_{5}, χ_{6}}
∪ {χ_{9}[k], χ_{10}[k], χ_{13}[k], χ_{14}[k], χ_{19}[k, l] : m|k, m|l};
B_{9}(s) = {χ_{9}[k], χ_{10}[k], χ_{19}[k, l] : m|l, k ≡ ±s(mod m)};
B_{13}(s) = {χ_{13}[k], χ_{14}[k], χ_{19}[k, l] : k ≡ ±s(mod m), l ≡ ±k(mod m)};
B_{19}(s, t) = {χ_{19}[k, l] : k ≡ ±s(mod m), l ≡ ±t(mod m)};
B_{7}(u) = {χ_{7}[u], χ_{8}[u], χ_{17}[u, l] : m|l};
B_{11}(u) = {χ_{11}[u], χ_{12}[u], χ_{16}[l] : l ≡ ±(q + 1)u (mod (q − 1)m)};
B_{16}(v) = {χ_{16}[k] : k ≡ ±v or ± qv(mod (q − 1)m)};
B_{17}(u, s) = {χ_{17}[u, l] : l ≡ ±s(mod m)}.

We first compute decomposition matrices for the r-blocks of maximal defect. Theorem 2.2 If r|q + 1, the non-cyclic r-blocks are B_{0}, B_{9}(s), B_{13}(s), and B_{19}(s, t). The decomposition matrices are as follows, with 1 ≤ α ≤ q/2.

B_{0}:

Degrees  Chars. Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5}  No. of Chars.
1  χ_{1}  1 . .  .  .  1
q(q^{2} + 1)/2  χ_{3}  1 1 .  .  .  1
q(q^{2} + 1)/2  χ_{4}  1 . 1  .  .  1
q(q − 1)^{2}/2  χ_{5}  …  1  .  1
q^{4}  χ_{6}  1 1 1  α  1  1
(q − 1)(q^{2} + 1)  χ_{9}  . . 1  1  .  (r^{d} − 1)/2
q(q − 1)(q^{2} + 1) χ_{10}  . . 1 α − 1 1  (r^{d} − 1)/2
(q − 1)(q^{2} + 1) χ_{13}  . 1 .  1  .  (r^{d} − 1)/2
q(q − 1)(q^{2} + 1) χ_{14}  . 1 . α − 1 1  (r^{d} − 1)/2
(q − 1)^{2}(q^{2} + 1) χ_{19}  … α − 2 1 (r^{d} − 1)(r^{d} − 3)/8

B_{9}(s):

Degrees  Chars. Φ_{1} Φ_{2} No. of Chars.
(q − 1)(q^{2} + 1)  χ_{9}  1 .  r^{d}
(q − 1)^{2}(q^{2} + 1) χ_{19}  . 1 r^{d}(r^{d} − 1)/2
q(q − 1)(q^{2} + 1) χ_{10}  1 1  r^{d}

B_{13}(s):

Degrees  Chars. Φ_{1} Φ_{2} No. of Chars.
(q − 1)(q^{2} + 1) χ_{13}  1 .  r^{d}
(q − 1)^{2}(q^{2} + 1) χ_{19}  . 1 r^{d}(r^{d} − 1)/2
q(q − 1)(q^{2} + 1) χ_{14}  1 1  r^{d}

B_{19}(s, t):

Degrees  Chars. Φ_{1} No. of Chars.
(q − 1)^{2}(q^{2} + 1) χ_{19}  1  r^{2d}

Proof. That the blocks listed are of maximal defect and therefore are the noncyclic blocks follows from the character degrees. Because all characters of a given family in the same block have the same restriction to the r-regular elements, the blocks B_{19}(s, t) each have one Brauer character and all decomposition numbers are 1 as claimed.

The Brauer characters χ̂_{9} and χ̂_{19} in the block B_{9}(s) form a basic set, and we have the relation χ̂_{9} + χ̂_{19} = χ̂_{10}. Since χ_{2} is a character of defect 0, the characters χ_{2} ⊗ χ_{2} and χ_{2} ⊗ χ_{4} are projective. We have the following scalar products with projective characters:

χ_{2} ⊗ χ_{4} χ_{2} ⊗ χ_{2}
χ_{9}  1  0
χ_{19}  0  1
χ_{10}  1  1

It follows that the restrictions of these projective characters to the block B_{9}(s) are indecomposable (see [10, Lemma 5.2], for example), and the decomposition matrix for B_{9}(s) is as claimed.

The Brauer characters χ̂_{13} and χ̂_{19} in the block B_{13}(s) form a basic set and we have the relation χ̂_{13} + χ̂_{19} = χ̂_{14}. The scalar products with the projective characters χ_{2} ⊗ χ_{2} and χ_{2} ⊗ χ_{3} are as follows:

χ_{2} ⊗ χ_{3} χ_{2} ⊗ χ_{2}
χ_{13}  1  0
χ_{19}  0  1
χ_{14}  1  1

The restrictions of these projective characters to the block B_{13}(s) are therefore indecomposable and the decomposition matrix for B_{13}(s) is as claimed.

We now consider the principal block B_{0}. It follows from [5, Theorem 5.1] that the restrictions of the unipotent characters χ_{1}, χ_{3}, χ_{4}, χ_{5}, and χ_{6} in B_{0} to the r-regular classes form a basic set for B_{0}. Hence there are 5 irreducible Brauer characters in B_{0}. We have the following relations:

χ̂_{9} = χ̂_{5} + χ̂_{4} − χ̂_{1}
χ̂_{10} = χ̂_{6} − χ̂_{5} − χ̂_{3}
χ̂_{13} = χ̂_{5} + χ̂_{3} − χ̂_{1}
χ̂_{14} = χ̂_{6} − χ̂_{5} − χ̂_{4}
χ̂_{19} = χ̂_{6} − 2χ̂_{5} − χ̂_{4} − χ̂_{3} + χ̂_{1}.

The corresponding decomposition numbers in each column must also satisfy these relations.

Denote by χ^{(i)} the decomposition number in column i corresponding to the ordinary character χ. Suppose χ_{6}^{(i)} = 0 for some i. Then the relations above show that χ_{10}^{(i)} = −χ_{5}^{(i)} − χ_{3}^{(i)} and χ_{14}^{(i)} = −χ_{5}^{(i)} − χ_{4}^{(i)}. Since decomposition numbers must be non-negative integers, these equations imply χ_{3}^{(i)} = χ_{4}^{(i)} = χ_{5}^{(i)} = 0. Then χ_{9}^{(i)} = −χ_{1}^{(i)}, and we have χ_{1}^{(i)} = 0. The decomposition matrix would therefore have a column of zeros, which is impossible. Hence χ_{6}^{(i)} ≥ 1 for all i. It follows that if Φ is any projective character whose scalar product with χ_{6} is 1, then the restriction of Φ to B_{0} is indecomposable (see [10, Lemma 5.3]).

Let B be a Borel subgroup of G. The order of B is |B| = q^{4}(q − 1)^{2} and is relatively prime to r, hence the induced character 1_{B}^{G} is projective. By the results of [1, §68], we have

1_{B}^{G} = χ_{1} + 2χ_{2} + χ_{3} + χ_{4} + χ_{6}.

The component of 1_{B}^{G} in B_{0} is therefore a projective indecomposable character. Denote this character by Φ_{1}. As before, the tensor product of Φ_{1} or the defect 0 character χ_{2} with any ordinary character is projective. We have the following scalar products of basic set characters with projective characters:

1_{B}^{G} χ_{2} ⊗ χ_{3} χ_{2} ⊗ χ_{4} Φ_{1} ⊗ χ_{5} χ_{2} ⊗ χ_{7}[1]
χ_{1} 1  0  0  0  0
χ_{3} 1  1  0  0  0
χ_{4} 1  0  1  0  0
χ_{5} 0  0  0  1  0
χ_{6} 1  1  1  q/2  1

The restrictions, Φ_{2} and Φ_{3}, of χ_{2}⊗χ_{3} and χ_{2}⊗χ_{4}, respectively, to the block B_{0} are therefore projective indecomposable characters. If q ≠ 2, then the character χ_{2} ⊗ χ_{7}[1] exists, and its restriction, Φ_{5}, to the principal block is a projective indecomposable character. If q = 2, so r = 3, then there are no characters of type χ_{7}. In this case, G ≅ S_{6}, and Φ_{5} can be obtained by inducing the (defect 0) character of S_{5} of degree 6 to G. Hence, in any case, the decomposition numbers in columns 1, 2, 3, and 5 are as claimed.

Finally, the projective character Φ_{1} ⊗ χ_{5} must contain the remaining projective indecomposable character Φ_{4}. The restriction of Φ_{1} ⊗χ_{5} to B_{0} is Φ_{4} +βΦ_{5}, for some non-negative integer β. We have seen that α = χ_{6}^{(4)} must be at least 1, and we now have that α + β = q/2. Hence α ≤ q/2. □

Note that if r^{d} ≠ 3 then the principal block B_{0} contains at least one character of type χ_{19}, with corresponding decomposition number α − 2. Hence if r^{d} ≠ 3, then 2 ≤ α ≤ q/2. In particular, if q = 4 and r = 5, then α = 2. On the other hand, if q = 2 (so G ≅ S_{6}) and r = 3, then there are no characters of type χ_{19} in B_{0}, and 1 ≤ α ≤ q/2 = 1, hence α = 1.

The following theorem gives the Brauer trees of the cyclic r-blocks of positive defect. This theorem also follows from results of [4].

Theorem 2.3 If r|q + 1, then the cyclic r-blocks of positive defect are B_{7}(u), B_{11}(u), B_{16}(v), and B_{17}(u, s). The Brauer trees are as follows.

[figure: B_{7}(u):; χ_{7}; χ_{8}; χ_{17}]

[figure: B_{11}(u):; χ_{11}; χ_{12}; χ_{16}]

[figure: B_{16}(v):; χ_{16}; χ_{16}]

[figure: B_{17}(u, s):; χ_{17}; χ_{17}]

Proof. That the blocks listed are the blocks of defect d and therefore are the cyclic blocks follows from the character degrees. Because all characters from a given family in the same block have the same restriction to the r-regular elements, each of the blocks B_{16}(v), B_{17}(u, s) has only one Brauer character, and the Brauer trees are as claimed.

The blocks B_{7}(u) and B_{11}(u) have 2 Brauer characters each, with the (r^{d} − 1)/2 characters of types χ_{17} and χ_{16}, respectively, corresponding to the exceptional nodes. The ordinary character of highest degree in each block (type χ_{8}, χ_{12}, respectively) cannot correspond to an end node, and therefore must correspond to the center node on the tree. □

3 The Case r|q − 1

In this section, r denotes a prime dividing q − 1. Let q − 1 = r^{d}m, where m is not divisible by r. Denote by T′_{1}, R′_{2} the set of multiples of r^{d} in T_{1}, R_{2}, respectively. Then |T′_{1}| = (m − 1)/2 and |R′_{2}| = q(m − 1)/4.

As before, the Sylow r-subgroup of G is Z_{r^{d}} × Z_{r^{d}}, and all defect groups are either trivial, Z_{r^{d}}, or Z_{r^{d}} × Z_{r^{d}}. Thus all r-blocks not of maximal defect are cyclic. The sets of ordinary characters defined below are the r-blocks of G of positive defect, as is easily verified using central characters. The characters of types χ_{5}, χ_{18}, and χ_{19} are in blocks of defect 0.

Definition 3.1 Define for s, t ∈ T′_{1} (s ≠ t), u ∈ T_{2}, and v ∈ R′_{2},

B_{0} = {χ_{1}, χ_{2}, χ_{3}, χ_{4}, χ_{6}}
∪ {χ_{7}[k], χ_{8}[k], χ_{11}[k], χ_{12}[k], χ_{15}[k, l] : m|k, m|l};
B_{7}(s) = {χ_{7}[k], χ_{8}[k], χ_{15}[k, l] : m|l, k ≡ ±s(mod m)};
B_{11}(s) = {χ_{11}[k], χ_{12}[k], χ_{15}[k, l] : k ≡ ±s(mod m), l ≡ ±k(mod m)};
B_{15}(s, t) = {χ_{15}[k, l] : k ≡ ±s(mod m), l ≡ ±t(mod m)};
B_{9}(u) = {χ_{9}[u], χ_{10}[u], χ_{17}[k, u] : m|k};
B_{13}(u) = {χ_{13}[u], χ_{14}[u], χ_{16}[l] : l ≡ ±(q − 1)u (mod (q + 1)m)};
B_{16}(v) = {χ_{16}[k] : k ≡ ±v or ± qv(mod (q + 1)m)};
B_{17}(s, u) = {χ_{17}[k, u] : k ≡ ±s(mod m)}.

We first consider the blocks of maximal defect. Theorem 3.2 If r|q − 1, the non-cyclic r-blocks are B_{0}, B_{7}(s), B_{11}(s), and B_{15}(s, t). The decomposition matrices are as follows.

B_{0}:

Degrees  Chars. Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5}  No. of Chars.
1  χ_{1}  1 ….  1
q(q + 1)^{2}/2  χ_{2}  . 1 …  1
q(q^{2} + 1)/2  χ_{3}  . . 1 . .  1
q(q^{2} + 1)/2  χ_{4}  … 1 .  1
q^{4}  χ_{6}  …. 1  1
(q + 1)(q^{2} + 1)  χ_{7}  1 1 1 . .  (r^{d} − 1)/2
q(q + 1)(q^{2} + 1) χ_{8}  . 1 . 1 1  (r^{d} − 1)/2
(q + 1)(q^{2} + 1) χ_{11}  1 1 . 1 .  (r^{d} − 1)/2
q(q + 1)(q^{2} + 1) χ_{12}  . 1 1 . 1  (r^{d} − 1)/2
(q + 1)^{2}(q^{2} + 1) χ_{15}  1 2 1 1 1 (r^{d} − 1)(r^{d} − 3)/8

B_{7}(s):

Degrees  Chars. Φ_{1} Φ_{2} No. of Chars.
(q + 1)(q^{2} + 1)  χ_{7}  1 .  r^{d}
q(q + 1)(q^{2} + 1) χ_{8}  . 1  r^{d}
(q + 1)^{2}(q^{2} + 1) χ_{15}  1 1 r^{d}(r^{d} − 1)/2

B_{11}(s):

Degrees  Chars. Φ_{1} Φ_{2} No. of Chars.
(q + 1)(q^{2} + 1) χ_{11}  1 .  r^{d}
q(q + 1)(q^{2} + 1) χ_{12}  . 1  r^{d}
(q + 1)^{2}(q^{2} + 1) χ_{15}  1 1 r^{d}(r^{d} − 1)/2

B_{15}(s, t):

Degrees  Chars. Φ_{1} No. of Chars.
(q + 1)^{2}(q^{2} + 1) χ_{15}  1  r^{2d}

Proof. It is easily verified by checking character degrees that the blocks indicated are of maximal defect and hence are the non-cyclic blocks. All characters of type χ_{15} in the block B_{15}(s, t) have the same restriction to the r-regular elements, hence there is only one Brauer character in the block and all decomposition numbers are 1.

Since r|q − 1 and r does not divide the order of the Weyl group of G, it follows from results of Puig [8] that the Brauer characters in all r-blocks are liftable (see also [7, Proposition 2.13]). In the block B_{7}(s), we have the basic set {χ̂_{7}, χ̂_{8}} and relation χ̂_{7} + χ̂_{8} = χ̂_{15}. Therefore χ_{7} and χ_{8} are irreducible modulo r. Similarly, in the block B_{11}(s), we have the basic set {χ̂_{11}, χ̂_{12}} and relation χ̂_{11} + χ̂_{12} = χ̂_{15}, hence χ_{11} and χ_{12} are irreducible modulo r. The decomposition matrices of these blocks are therefore as claimed.

By [5, Theorem 5.1], the restrictions of the unipotent characters χ_{1}, χ_{2}, χ_{3}, χ_{4}, and χ_{6} in B_{0} to the r-regular classes form a basic set for the principal block B_{0}. By [7, Proposition 2.13], these characters are irreducible modulo r. The remainder of the decomposition matrix for B_{0} follows from the relations

χ̂_{7} = χ̂_{1} + χ̂_{2} + χ̂_{3}
χ̂_{8} = χ̂_{2} + χ̂_{4} + χ̂_{6}
χ̂_{11} = χ̂_{1} + χ̂_{2} + χ̂_{4}
χ̂_{12} = χ̂_{2} + χ̂_{3} + χ̂_{6}
χ̂_{15} = χ̂_{1} + 2χ̂_{2} + χ̂_{3} + χ̂_{4} + χ̂_{6}.
□

It is possible to obtain the decomposition matrices in Theorem 3.2, up to one parameter in the principal block, by the same methods used in Theorem 2.2. For B = B_{7}(s) or B = B_{11}(s), we have χ_{5} ⊗ χ_{5} ≈_{B} Φ_{1} and χ_{5} ⊗ χ_{6} ≈_{B} (q/2)Φ_{2} (where Ψ ≈_{B} Φ means that the projective characters Ψ and Φ have the same restriction to the block B). For the principal block B_{0}, we have χ_{5} ⊗χ_{5} ≈_{B_{0}} Φ_{1}, χ_{5} ⊗ χ_{9}[1] ≈_{B_{0}} Φ_{3}, χ_{5} ⊗ χ_{13}[1] ≈_{B_{0}} Φ_{4}, χ_{5} ⊗ χ_{6} ≈_{B_{0}} (q/2)Φ_{5}. We have been unable to obtain Φ_{2} by these methods, although given Theorem 3.2, we have χ_{5} ⊗ χ_{18}[1] ≈_{B_{0}} Φ_{2} + (q/2 − 1)Φ_{5}.

The following theorem gives the Brauer trees of the cyclic r-blocks of positive defect. This theorem also follows from results of [4].

Theorem 3.3 If r|q − 1, then the cyclic r-blocks of positive defect are B_{9}(u), B_{13}(u), B_{16}(v), and B_{17}(s, u). The Brauer trees are as follows.

[figure: B_{9}(u):; χ_{9}; χ_{17}; χ_{10}]

[figure: B_{13}(u):; χ_{13}; χ_{16}; χ_{14}]

[figure: B_{16}(v):; χ_{16}; χ_{16}]

[figure: B_{17}(s, u):; χ_{17}; χ_{17}]

Proof. That the blocks listed are the blocks of defect d and therefore are the cyclic blocks follows from the character degrees. Because all characters from a given family in the same block have the same restriction to the r-regular elements, each of the blocks B_{16}(v), B_{17}(s, u) has only one Brauer character, and the Brauer trees are as claimed.

The blocks B_{9}(u) and B_{13}(u) have 2 Brauer characters each, with the (r^{d} − 1)/2 characters of types χ_{17} and χ_{16}, respectively, corresponding to the exceptional nodes. These are also the ordinary characters of highest degree in each block and cannot correspond to end nodes, hence must correspond to the center nodes on the trees. □

4 The Case r|q^{2} + 1

In this section, r denotes a prime dividing q^{2} + 1. Let q^{2} + 1 = r^{d}m, where m is not divisible by r. Denote by R′_{1} the set of multiples of r^{d} in R_{1}, so that |R′_{1}| = (m−1)/4. The Sylow r-subgroup of G is cyclic of order r^{d}, so all r-blocks are cyclic. Using this fact and the central characters, it is easily verified that the sets of ordinary characters defined below are the r-blocks of G of positive defect. All characters not listed are in blocks of defect 0.

Definition 4.1 Define for s ∈ R′_{1},

B_{0} = {χ_{1}, χ_{2}, χ_{5}, χ_{6}, χ_{18}[k] : m|k};
B_{18}(s) = {χ_{18}[k] : k ≡ ±s or ± qs(mod m)}.

Theorem 4.2 If r|q^{2} + 1, then all r-blocks are cyclic and the Brauer trees of the blocks of positive defect are as follows.

[figure: B_{0}:; χ_{1}; χ_{2}; χ_{6}; χ_{18}; χ_{5}]

[figure: B_{18}(s):; χ_{18}; χ_{18}]

Proof. All characters of type χ_{18} in the same block have the same restriction to the r-regular elements. Hence the block B_{18}(s) has one Brauer character and the Brauer tree is as shown. The (r^{d} − 1)/4 characters of type χ_{18} in B_{0} correspond to the exceptional node, and B_{0} has 4 Brauer characters.

The sum of the degrees of two adjacent nodes on the Brauer tree must be divisible by r^{d}. This partitions the ordinary characters in B_{0} into two sets, {χ_{1}, χ_{5}, χ_{6}} and {χ_{2}, χ_{18}}, such that the characters in each set can only be joined to characters in the other set. Moreover, the ordinary character in the block of highest degree, χ_{6}, cannot correspond to an end node. Hence χ_{6} must be joined to both χ_{2} and the exceptional node χ_{18}. The character χ_{3} is of defect 0, hence χ_{3} ⊗ χ_{3} and χ_{3} ⊗ χ_{9}[1] are projective. We have the following scalar products:

χ_{3} ⊗ χ_{3} χ_{3} ⊗ χ_{9}[1]
χ_{1}  1  0
χ_{2}  1  0
χ_{5}  0  1
χ_{6}  0  0
χ_{18}  0  1

These projective characters show that χ_{1} must be joined to χ_{2} and that χ_{5} must be joined to the exceptional node χ_{18}, and the Brauer tree is as claimed. □

5 The Case q Odd, Revisited

We now return to the case where q is odd, considered in [10]. Let G = Sp_{4}(q) with q odd and let r be an odd prime dividing q + 1. Notation for ordinary characters will be as in [10]. In Theorem 4.2 of [10], we were able to determine the second projective indecomposable character, Φ_{2}, only up to two parameters. All computations were done by hand, making even a small search of projective characters arising from tensor products rather difficult and generally not productive. As a result, we were unable to find a projective character containing θ_{10} (and hence Φ_{2}) that did not contain θ_{11}. Inspired by the computer calculations used in §2 above, we have been able to improve on Theorem 4.2 of [10] as follows.

Theorem 5.1 Let G = Sp_{4}(q), with q odd, and let r be an odd prime dividing q + 1. If α and β are the two parameters in the decomposition matrix of the principal block corresponding to Φ_{2}, as in Theorem 4.2 of [10], then β = 0 and α ≤ (q − 1)/2.

Proof. The projective indecomposable characters Φ_{1}, Φ_{3}, Φ_{4}, and Φ_{5} were determined in [10]. The restriction of 1_{B}^{G} to the principal block is

Φ_{1} = 1_{G} + θ_{11} + θ_{12} + θ_{13}.

Thus Φ_{1} ⊗ θ_{10} is projective, and we have the following scalar products of projective characters with basic set characters:

Φ_{1} Φ_{2} Φ_{3} Φ_{4} Φ_{5} Φ_{1} ⊗ θ_{10}
1_{G} 1 0 0 0 0  0
θ_{10} 0 1 0 0 0  2
θ_{11} 1 β 1 0 0  0
θ_{12} 1 0 0 1 0  0
θ_{13} 1 α 1 1 1  q − 1

Let Ψ be the restriction of Φ_{1} ⊗ θ_{10} to the principal block. The decomposition matrix then shows that Ψ = 2Φ_{2} + γΦ_{5}, for some non-negative integer γ. Hence β = 0, and 2α + γ = q − 1, thus α ≤ (q − 1)/2. □

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