MOD 2 COHOMOLOGY OF THE UNITARY GROUPS U_{3}(2^{n})

D. J. BENSON AND S. MARTIN

1. Introduction

Let n ≥ 2 be a natural number, and put q = 2^{n}. Let 𝔽_{q^{2}} be the field of q^{2} elements, with Frobenius automorphism F : α ↦ α^{2} of order 2n. Define the 3-dimensional unitary group GU_{3}(q) of order q^{3}(q^{3} + 1)(q^{2} − 1)(q + 1), as the set of linear transformations of a 3-dimensional space V over 𝔽_{q^{2}} which leave a given non-degenerate hermitian form f invariant. We choose an 𝔽_{q^{2}}-basis of V with respect to which f has matrix

J = [ \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ] ,

so that the Sylow 2-normaliser is upper triangular. Thus, a matrix M is in GU_{3}(q) if and only if MJF^{n}(M^{−1}) = J.

Let G = SU_{3}(q) be the subgroup of GU_{3}(q) consisting of the matrices of determinant one. Then G has order q^{3}(q^{3} + 1)(q^{2} − 1) and Z(G) has order one or three according as n is even or odd. The quotient G/Z(G) = P SU_{3}(q) we shall write as U_{3}(q).

Let k be an algebraically closed field of characteristic 2. Simple kGmodules are constructed as follows. The algebraic group SL_{3}(k) has four irreducibles in the restricted fundamental region, namely k, V, V^{∗} and the (first) Steinberg module of dimension 8. Applying the Steinberg Twisted Tensor Product Theorem, G has q^{2} irreducible modules.

This paper is a contribution to the study of the cohomology of G over k. We shall be concerned with calculating the dimensions of some groups Ext_{kG}^{1}(M, M′) with M, M′ irreducible kG-modules. We also give quite a lot of information about modules with three, four and five composition factors, both for the group G, and for the Sylow 2-normaliser in G.

We should also mention at this point that Peter Sin [12] has independently calculated Ext^{1} between simple kG-modules (as well as modules for some other groups of Lie rank one [13]), by totally different methods. It is also worth pointing out that the complexities of the simple kG-modules have been computed by Fleischmann [5, Theorem 4.1], though we do not need his result here.

As an illustration, in case n = 2 we get the simple group U_{3}(4) of order 62,400. This group has sixteen irreducible modules. Namely, there is the trivial module, 1; there are four three dimensional modules: 3_{1}, the natural

M′

M 1 3_{1} 3_{2} 3_{3} 3_{4} 8_{1} 8_{2} 9_{1} 9_{2} 9_{3} 9_{4} 24_{1} 24_{2} 24_{3} 24_{4} 64 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 3_{1} 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 3_{2} 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 3_{3} 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 3_{4} 0 1 1 0 0 0 0 0 1 0 0 0 0 1 0 8_{1} 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 8_{2} 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 9_{1} 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 9_{2} 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 9_{3} 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 9_{4} 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 24_{1} 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 24_{2} 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 24_{3} 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 24_{4} 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 64  0

Table 1. dim_{k}Ext_{U_{3}(4)}^{1}(M, M′)

three-dimensional module for U_{3}(4), together with 3_{2} = 3_{1}^{F^{2}} = 3_{1}^{∗}, 3_{3} = 3_{1}^{F^{3}}, 3_{4} = 3_{1}^{F}; two 8-dimensional modules 8_{1} = 3_{1} ⊗ 3_{2} − 1, 8_{2} = 3_{3} ⊗ 3_{4} − 1; four 9-dimensional modules 9_{1} = 3_{2} ⊗ 3_{4}, 9_{2} = 3_{1} ⊗ 3_{3}, 9_{3} = 3_{1} ⊗ 3_{4}, 9_{4} = 3_{2} ⊗ 3_{3}; four 24-dimensional modules 24_{1} = 3_{3} ⊗ 8_{1}, 24_{2} = 3_{4} ⊗ 8_{1}, 24_{3} = 3_{2} ⊗ 8_{2}, 24_{4} = 3_{1} ⊗ 8_{2}; and finally the (second) Steinberg module of dimension 64. The methods of this paper give most of the Ext^{1} groups between these simple modules. The remaining Ext^{1} groups are easily obtained by algebraic group techniques (tensoring with the Steinberg module—see Section 4), and the complete answer is given in Table 1.

Character tables, and other facts about U_{3}(4) are to be found in the AT-LAS [3] and in the appendix. For general information about G and other classical groups, see Gorenstein [6] and Humphreys [7]; for the conventions on diagrams for modules, see the paper of Benson and Carlson [1]. Our notation throughout is standard, with one exception: if a module M is uniserial with composition factors A_{1}, A_{2}, …, A_{r} in that order from the top, we write M = Uni(A_{1}, A_{2}, …, A_{r}).

2. Modules for the Sylow 2-normaliser N of G

Let N be the group of upper triangular matrices in G, of order q^{3}(q^{2} − 1). Then N is the semidirect product of a Sylow 2-subgroup P of order q^{3} consisting of the upper unitriangular matrices and a cyclic group H of order q^{2} − 1 of diagonal matrices of the form diag(a, a^{q−1}, a^{−q}). The matrices in N have the form

[ \begin{matrix}a & b & c \\ & a^{q−1} & a^{−1}b^{q} \\ & & a^{−q} \end{matrix} ]

with a^{−1}c^{q} + a^{−q}c = b^{q+1}.

The group P consists of the matrices in N with diagonal entries equal to 1, and is a special 2-group of order q^{1+2}. In other words Z(P) = Φ(P) = P′ is elementary abelian of order q and P/Z(P) is elementary abelian of order q^{2}. Also, every involution in P lies in Z(P).

Since P has trivial intersection with its distinct conjugates, if M is any kG-module, then the restriction map of cohomology gives an isomorphism

H^{i}(G, M) ≅ H^{i}(N, M ↓_{N}) ≅ H^{i}(P, M ↓_{P})^{N/P} .

Since H^{i}(G, Hom_{k}(M, M′)) ≅ Ext_{kG}^{i}(M, M′) we have the following:

Proposition 2.1. If M and M′ are kG-modules,

Ext_{kG}^{i}(M, M′) ≅ Ext_{kN}^{i}(M ↓_{N}, M′ ↓_{N}).

Let ε be the 1-dimensional representation of kH spanned by the first basis vector in V. In what follows we shall write ( i) instead of ε^{i}, the tensor i’th power of ε. Note that this only depends on i modulo q^{2} − 1. The next result is immediate from the form of the matrices for N.

Theorem 2.2. Let V denote the natural 3-dimensional module for G. Then

V ↓_{N} = Uni((1), (q − 1), (−q)).

The module V ↓_{N} appearing in Theorem 2.2 is uniserial of length 3. We now aim to classify all uniserial kN-modules. To do this, we begin by examining the structure of the projective indecomposable kN-modules.

Let k = k_{N} be the trivial kN-module, with projective cover P (k). To describe the Loewy structure of this module, we use the functorial description of Jennings Theorem [9] given in Quillen [11]. Namely, for any p-group P, let Γ_{r}(P) be the r’th dimension subgroup of P; this is defined as the lowest descending central series with Γ_{1}(P) = P and the property that the p’th power of an element of Γ_{r}(P) is in Γ_{rp}(P). Then the graded abelian group Jen_{∗}(P) = ⨁_{r⩾1} k ⊗_{𝔽_{p}} (Γ_{r}(P)/Γ_{r+1}(P)) is a p-restricted Lie algebra with respect to commutators and p’th powers. The Jennings-Quillen theorem states that its restricted universal enveloping algebra UJen_{∗}(P) is isomorphic to the associated graded algebra gr_{∗}(kP) of the radical filtration of the group algebra kP. Since H is a group acting on P, it also acts on UJen_{∗}(P), and the isomorphism between this and gr_{∗}(kP) is an isomorphism of kH-modules.

In our case, this shows that gr_{∗}(kP) ≅ UJen_{∗}(P) is a gr_{∗}(kN)-module isomorphic to gr_{∗}(P (k)). The dimension subgroups of P are given by Γ_{1}(P) = P, Γ_{2}(P) = Z(P) and Γ_{r}(P) = 1 for r ⩾ 3. Let V = Γ_{1}(P)/Γ_{2}(P) and X = Γ_{2}(P)/Γ_{3}(P) as kH-modules. Since there is a kN-module Uni((1), (q − 1)) obtained from the top two Loewy layers of V ↓_{N}, it follows by tensoring with the module (−1) and applying the Frobenius, that there are also kN-modules of the form Uni((0), (2^{n+j} − 2^{j+1})) for 0 ⩽ j ⩽ 2n − 1. These must therefore be quotients of P (k), and so we have

V ≅ ⨁_{0⩽j⩽2n−1} V_{j}

where V_{j} = (2^{n+j} − 2^{j+1}).

Now X is a summand of Λ^{2}(V) = ⨁_{0⩽i<j⩽2n−1} V_{ij} where V_{ij} = V_{i} ⊗ V_{j}. Since X has dimension n and is invariant under F, we have

X ≅ ⨁_{0⩽j⩽n−1} V_{j,n+j}.

Now applying the Jennings-Quillen theorem we obtain the following.

Theorem 2.3. The first three Loewy layers of the kN-module P (k) are

k ; V ; Λ^{2}(V) ⊕ X ; ⋮

Thus the third Loewy layer consists of the modules V_{ij} with multiplicity one if i − j ≢ n (mod 2n), and multiplicity two if i − j ≡ n (mod 2n).

In the case i − j ≢ n (mod 2n), the module V_{ij} in the third Loewy layer extends the modules V_{i} and V_{j} in the second Loewy layer.

In the case i − j ≡ n (mod 2n), one of the two copies of V_{ij} extends V_{i} in the second Loewy layer, and the other extends V_{j}.

In particular, the only uniserial modules of length three with k as the top composition factor, are of the form Uni(k, V_{i}, V_{n+i}). All kN-modules of length three may be obtained from these by tensoring with a one-dimensional module.

Proof. Most of this has already been proved in the preceding discussion. To complete the proof, suppose there is a uniserial module of the form Uni(k, V_{i}, V_{ij}). By dualising and tensoring with V_{ij}, we see that there is also a uniserial module of the form Uni(k, V_{j}, V_{ij}). Thus V_{ij} occurs with multiplicity two in the third Loewy layer, and we are in the case where i − j ≡ n (mod 2n).

Terminology: We say that a module is of diamond shape if it has Loewy length 3 and has a unique top and a unique bottom composition factor.

Corollary 2.4. All kN-modules of diamond shape with k as the top composition factor have the form

[figure: k; V_{i}; V_{j}; V_{ij}]

for i − j ≡ n (mod 2n). All kN-modules of diamond shape may be obtained from these by tensoring with a one-dimensional module.

The next step is to look at modules of pentagonal shape and also “Y” shaped modules.

Corollary 2.5. There do not exist modules having diagrams of the following shapes:

[figure: •; •; •; •; •; •; •; •]

Proof. Obviously the second diagram is dual to the first. If a module with the first diagram exists, observe that both length 3 uniserial submodules have the same composition factors, by the last statement of Theorem 2.3. However, the same theorem also shows that there is no two-dimensional Ext^{1} group between simple kN-modules.

Lemma 2.6. There is a kN-module of pentagonal shape, namely:

[figure: (1); (q − 1); (2q − 3); (−q); (−2)]

Proof. Using Theorem 2.2, we find that the representation is given by the matrices

[ \begin{matrix}a & b & c & a^{1−q}b^{q} & a^{−1}bc \\ & a^{q−1} & a^{−1}b^{q} & 0 & a^{−1}c^{q} \\ & & a^{−q} & 0 & a^{−q−1}b \\ & & & a^{2−2q} & a^{−2q}b^{2} \\ & & & & a^{−2} \end{matrix} ] .

Remark: Observe, by Corollary 2.4 that there is a diamond

[figure: (1); (2 − 2q); (2q − 3); (−2)] and so there are is also a pentagon [figure: (1); (q − 1); (2 − 2q); (−q); (−2)] .

All pentagons may be obtained from one of the above two pentagonal modules by applying powers of F and tensoring with one-dimensional modules. The two families of modules so obtained are dual to each other.

Lemma 2.7. There do not exist uniserial kN-modules of length 4.

Proof. By Theorem 2.3, if there is such a module, then there exists one of the form

Uni((1), (q − 1), (−q), (−2)).

Consider the pullback M of this module with the pentagon of Lemma 2.6.

[figure: (1); (q − 1); M; (2q − 3); (−q); (−2); (1); (q − 1); (1); (−q); (q − 1); (−2); (−q)]

Then M has a uniserial quotient of the form Uni((1), (2q−3), (−2)). However, such a module does not exist by Theorem 2.3.

3. Extensions between simples of small dimension

By Proposition 2.1, when computing Ext^{1} groups between simple modules for G, it suffices to work with restrictions of these kG-modules to N. This technique is used extensively below.

Proposition 3.1. The trivial kG-module does not extend V.

Proof. By Proposition 2.1, it is enough to show that

Ext_{kN}^{1}((0), Uni((1), (q − 1), (−q))) = {0}.

This follows from Lemma 2.8 and the fact that

Ext_{kN}^{1}((0), (q − 1)) = Ext_{kN}^{1}((0), (−q)) = {0}.

Corollary 3.2. The trivial kG-module does not extend any three-dimension-al simple kG-module.

Proof. This comes from Proposition 3.1, after applying the Frobenius map.

Proposition 3.3. A given three-dimensional simple module for G extends precisely two isomorphism types of three-dimensional kG-modules; namely the modules obtained by applying F^{n+1} and F^{n−1}.

Proof. Again it suffices to work with restrictions of modules to N. By application of the Frobenius map, and tensoring with one-dimensionl modules, it is enough to consider extensions of V ↓_{N(P)} by other uniserial modules of length 3. By the results of Section 2, we then have exactly two possibilities for the extensions, namely:

[figure: (1); Y =; (q − 1); (2q); (−q); (2 − 2q); (−2); (1); Y′ =; (q − 1); (\frac{q}{2}); (−q); (\frac{q^{2}}{2} − \frac{q}{2}); (−\frac{q^{2}}{2} )]

Now show existence of such modules by exhibiting Y as a pushout, and Y′ as a pullback, as in the proof of Lemma 2.8.

Proposition 3.4. Let V ⊗ V^{∗} = k ⊕ W. Then the trivial kG-module does not extend any eight-dimensional simple kG-module F^{j}(W).

Proof. It suffices to show that k does not extend W. By Proposition 3.3, and the fact that k does not extend itself, we have

Ext_{kG}^{1}(k, W) = Ext_{kG}^{1}(k, k ⊕ W) ≅ Ext_{kG}^{1}(k, V ⊗ V^{∗})
≅ Ext_{kG}^{1}(V, V) = {0}.

A similar proof deals with:

Proposition 3.5. Let F^{i}(V) ⊗ F^{j}(V) with i ≠ j, n + j, be an arbitrary nine-dimensional simple kG-module. Then

dim_{k}Ext_{kG}^{1}(k, F^{i}(V) ⊗ F^{j}(V)) = \{ \begin{matrix}1, & if |i − j| ≡ 2 (mod 2n) \\ 0, & otherwise \end{matrix}

Proposition 3.6. W ↓_{N} has the following diagram:

[figure: (q + 1); (2q − 1); (2 − q); (0); (0); (q − 2); (1 − 2q); (−1 − q)]

Proof. The tensor

(V ⊗ V^{∗})↓_{N}= ( \begin{matrix}(1) \\ (q − 1) \\ (−q) \end{matrix} ) ⊗ ( \begin{matrix}(q) \\ (1 − q) \\ (−1) \end{matrix} )

is a direct sum of the 1-dimensional module (0) and the given module.

Proposition 3.7. There is no extension between any three-dimensional simple kG-module and any eight-dimensional simple kG-module.

Proof. By Proposition 2.1, we work at the level of N. The plan will be to consider extensions of three-dimensional uniserial kN-modules (see Theorem 2.3) with W ↓_{N}. Using the results 2.3 through to 2.8 we can eliminate all cases except the following two modules. We either have

[figure: (q + 1); (2q − 1); (2 − q); (1 − q − \frac{q^{2}}{2} ); (0); (0); (−\frac{q^{2}}{2} − \frac{q}{2}); (q − 2); (1 − 2q); (−q − \frac{q}{2}); (−1 − q)]

or we have 2n possible extensions

[figure: (q + 1); (2q − 1); (2 − q); (−2^{n+j} + 2^{j+1}); (0); (0); (−2^{n+j} + 2^{j+1} + 1 − 2q); (q − 2); (1 − 2q); (−2^{n+j} + 2^{j+1} − 1 − q); (−1 − q)]

In the first case, the extending three-dimensional module is not the restriction of a three-dimensional kG-module, because the top factor must be a power of two. In the second set of cases, none of the extending three-dimensional modules are restrictions for the same reason.

Corollary 3.8. The trivial kG-module does not extend any 24-dimensional simple kG-module.

Proof. This follows from Proposition 3.7 since every 24-dimensional simple kG-module is a tensor product of a three-dimensional and an eightdimensional simple kG-module.

Proposition 3.9. Given any three-dimensional simple kG-module there is a unique nine-dimensional simple kG-module with which it has a non-vanishing Ext^{1} group. Namely, F^{n+j+1}(V) extends F^{n+j}(V) ⊗ F^{j+1}(V) once.

Proof. The restriction of every nine-dimensional simple kG-module to N is of the form

( \begin{matrix}(2^{i}) \\ (2^{n+i} − 2^{i}) \\ (−2^{n+i}) \end{matrix} ) ⊗ ( \begin{matrix}(2^{j}) \\ (2^{n+j} − 2^{j}) \\ (−2^{n+j}) \end{matrix} )

with i ≠ j. By repeated use of 2.3 to 2.8, and swapping i and j if necessary, we see that the only way of extending this module by a uniserial of length three is as follows.

[figure: (2^{j}+2^{i}); (2^{n+j}+2^{n+i+1}−2^{j}); (2^{j}+2^{n+i}−2^{i}); (2^{n+j}−2^{j}+2^{i}); (−2^{n+j}+2^{n+i+1}); (2^{j}−2^{n+i}); (2^{n+j}−2^{j}+2^{n+i}−2^{i}); (−2^{n+j}+2^{i}); (2^{n+i+1}−2^{j+1}); (−2^{n+j}−2^{j}−2^{n+i}); (2^{n+j}+2^{n+i}−2^{i}); (−2^{n+j}−2^{n+i})]

This can only happen when i = j or i = n+j −1. The first case is impossible, since we do not get a nine-dimensional simple. The latter case is possible, and by applying the Frobenius, we may assume, without loss of generality, that the extension is of the form:

[figure: (2 + q); (2q); (3 − q); (3q − 2); (−2q + 2); (1); (q − 1); (−q); (−2); (2q − 3); (1 − 3q); (−2q − 1)]

We need to show that such an extension exists. We can construct this as a pullback once we have produced a module of the form

[figure: (2q); (1); (3q − 2); (−2q + 2); Z =; (q − 1); (−q); (2q − 3); (1 − 3q); (−2); (−2q − 1)]

Consider the tensor products

Z′ = \begin{matrix}(0) \\ (2 − 4q) \end{matrix} ⊗ [figure: (2q); (3q − 2); (1); (q − 1); (2q − 3)] = [figure: (2q); (1); (3q − 2); (−2q + 2); (q − 1); (−q); (2q − 3); (1 − 3q); (3 − 4q); (−2q − 1)]

Z′′ = \begin{matrix}(0) \\ (1 − 2q) \end{matrix} ⊗ \begin{matrix}(2q) \\ (2 − 2q) \\ (−2) \end{matrix} = [figure: (2q); (1); (2 − 2q); (3 − 4q); (−2); (−2q − 1)]

and

Z′′′ = \begin{matrix}(0) \\ (1 − 2q) \end{matrix} ⊗ \begin{matrix}(2q) \\ (2 − 2q) \end{matrix} = [figure: (2q); (1); (−2q + 2); (3 − 4q)]

Now form the pullback M as follows:

\xymatrix{M \ar[r] \ar[d] & Z′′ \ar[d] \\ Z′ \ar[r] & Z′′′}

where M is a module of the following form:

M = [figure: (2q); (1); (3q − 2); (−2q + 2); (q − 1); (−q); (2q − 3); (1 − 3q); (3 − 4q); (−2); (−2q − 1); (−2q − 1)]

Note that this diagram is not a valid diagram according to the rules of [1], because there are two linearly dependent extensions of (3 − 4q) by (−2q − 1) at the bottom. However, after quotienting out the diagonal copy of (−2q −1) indicated by the dotted line, both these extensions disappear, and there is now a submodule (3 − 4q). On quotienting out this submodule, we obtain a module Z of the required form.

Proposition 3.10. The only uniserial kG-modules of length three whose composition factors are three-dimensional simple modules, are all Frobenius powers of Uni(V, F^{n+1}(V), V).

Proof. By Proposition 3.3, the only extensions of V by another three-dimen-sional simple are extensions by F^{n+1}(V) or by F^{n−1}(V) and these extensions have diagrams

[figure: •; •; •; •; •; •; •; •; •; •; •; •]

Extending the modules with the above diagrams by a further three-dimen-sional simple, we get the following diagrams:

[figure: •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

The second, third and fourth of these diagrams contradict Corollary 2.5; the first case gives the required result.

Remark: By restricting to N, it is easy to see that

V ⊗ V = Uni(Λ^{2}(V), F (V), Λ^{2}(V))

which is the image under F^{n} of the module given in Proposition 3.10.

Corollary 3.11. There do not exist modules of diamond shape for U_{3}(q) consisting solely of three-dimensional simple kU_{3}(q)-modules.

Proof. By Proposition 3.3 if there is a module of diamond shape, it would have to be some Frobenius of a module with diagram

[figure: V; F^{n+1}(V); F^{n−1}(V); V]

If this exists, then pulling back with the module of Proposition 3.10, we obtain a module Uni(V, F^{n−1}(V), V). But this contradicts Proposition 3.10.

For similar reasons we have

Corollary 3.12. There do not exist uniserial kU_{3}(q)-modules of length four, pentagons, or Y-shaped modules, whose factors consist solely of three-dimen-sional simple modules.

4. U_{3}(4) and the algebraic group

In this section, we give an outline of the algebraic group techniques used to complete Table 1 in the introduction, giving dim_{k}Ext_{U_{3}(4)}^{1}(M, M′), for M and M′ simple. We refer to Jantzen [8] for background material on representations of algebraic groups.

Let G be the algebraic group SL_{3}(k), with Borel subgroup B, and let φ be the composite of the graph automorphism (that is, transpose inverse on matrices) with the square of the Frobenius map, F^{2}. The group U_{3}(4) is equal to the fixed points G^{φ}. We denote by V the natural three-dimensional module for G, and by St the second Steinberg module, of dimension 64, which restricts to the Steinberg module for U_{3}(4). Since this restriction is projective, the restriction of V ⊗ St is also projective, and comparing characters, we see that it is the projective cover of the 24-dimensional simple module V ⊗F (W).

Using ↓_{B} and ↑^{G} to denote restriction and algebraic induction between G and B, we have

V ⊗ St = (V ↓_{B} ⊗λ)↑^{G}

where λ is the one-dimensional module for B corresponding to the highest weight of St. Now V ↓_{B} has a filtration by one-dimensional modules with weights ω_{0}, ω_{1} − ω_{0}, −ω_{1}, where ω_{0} and ω_{1} are the fundamental dominant weights. Thus, V ↓_{B} ⊗λ has a filtration by one-dimensional modules with weights 4ω_{0} + 3ω_{1}, 2ω_{0} + 4ω_{1}, 3ω_{0} + 2ω_{1}. Finally, since all these weights are well inside the restricted fundamental region, induction is exact with respect to this filtration, and so V ⊗ St has a filtration with filtered quotients (from the top) the dual Weyl modules H^{0}(4ω_{0}+3ω_{1}), H^{0}(2ω_{0}+4ω_{1}), H^{0}(3ω_{0}+2ω_{1}). The reader is referred to Wang Jian-Pan [14] and also Donkin [4] and Mathieu [10] for the application of this technique more generally.

Now, using Jantzen’s Sum Formula [8, II.8] to get a filtration of these dual Weyl modules, and also using the fact that the dual of V ⊗ St is its image under the graph automorphism of G, we obtain the diagram for V ⊗ St as in Figure 1.

Restricting to the finite group U_{3}(4), and using the labelling we have given in the introduction, together with the information we already have on Ext_{U_{3}(4)}^{1} from Section 3, we see that the projective cover of 24_{4} has diagram as given in Figure 2.

The second Loewy layer of this module tells us dim_{k}Ext_{U_{3}(4)}^{1}(M, M′) for M a 24-dimensional simple, and M′ an arbitrary simple. Using the general formula

Ext^{1}(A, B ⊗ C) ≅ Ext^{1}(A ⊗ B^{∗}, C),

we obtain the remaining Ext_{U_{3}(4)}^{1} groups in Table 1 apart from self-extensions of eight-dimensional simple modules, and extensions between a nine-dimen-sional simple module and its image under F^{2}. We now consider these cases.

Proposition 4.1. Ext_{U_{3}(4)}^{1}(8_{1}, 8_{1}) = {0}.

Proof.

Ext_{U_{3}(4)}^{1}(8_{1}, 8_{1}) = Ext_{U_{3}(4)}^{1}(3_{1} ⊗ 3_{2}, 3_{1} ⊗ 3_{2}) = Ext_{U_{3}(4)}^{1}(3_{1} ⊗ 3_{1}, 3_{2} ⊗ 3_{2})

= Ext_{U_{3}(4)}^{1}(Uni(3_{2}, 3_{4}, 3_{2}), Uni(3_{1}, 3_{3}, 3_{1})),

(see the remark after Proposition 3.10). By Proposition 3.10 there cannot be any extension.

To deal with Ext_{U_{3}(4)}^{1}(9_{1}, 9_{2}) we require two elementary lemmas:

Lemma 4.2. (1) Ext_{U_{3}(4)}^{1}(3_{2}, Uni(3_{2}, 3_{4}, 3_{2})) = {0}; (2) dim_{k}Ext_{U_{3}(4)}^{1}(3_{3}, Uni(3_{2}, 3_{4}, 3_{2})) = 1. In this case the only extension is

[figure: 3_{2}; 3_{3}; 3_{4}; 3_{2}]

Proof. (1) By Proposition 3.3, 3_{2} extends 3_{4} only once, and does not extend 3_{2}.

(2) By tensoring we have that

Ext_{U_{3}(4)}^{1}(3_{3}, Uni(3_{2}, 3_{4}, 3_{2})) = Ext_{U_{3}(4)}^{1}(3_{3}, 3_{1} ⊗ 3_{1})

= Ext_{U_{3}(4)}^{1}(3_{3} ⊗ 3_{2}, 3_{1}) = Ext_{U_{3}(4)}^{1}(9_{4}, 3_{1}),

[figure: V ⊗ F (W); F (V^{∗}); F^{2}(V); F (V) ⊗ F^{2}(V^{∗}); F (V^{∗}); F (V) ⊗ V^{∗}; V^{∗} ⊗ F^{2}(V^{∗}); F^{2}(V); F (V^{∗}); V^{∗} ⊗ F (V^{∗}) ⊗ F^{2}(V); F (V) ⊗ V^{∗}; V ⊗ F (W); F (V^{∗}); V^{∗} ⊗ F^{2}(V^{∗}); F^{2}(V); F (V) ⊗ V^{∗}; F (V) ⊗ F^{2}(V^{∗}); F (V^{∗}); F^{2}(V); F (V^{∗}); V ⊗ F (W)]

Figure 1. Diagram for V ⊗ St

and the latter space has dimension one by Proposition 3.9. Lemma 4.3. Ext_{U_{3}(4)}^{1}(Uni(3_{2}, 3_{3}), Uni(3_{2}, 3_{4}, 3_{2})) = {0}. Proof. The short exact sequence

0 → 3_{3} → Uni(3_{2}, 3_{3}) → 3_{2} → 0

gives rise to the long exact sequence ⋯ → Hom(Uni(3_{2}, 3_{3}), Uni(3_{2}, 3_{4}, 3_{2})) → Hom(3_{2}, Uni(3_{2}, 3_{4}, 3_{2})) → Ext_{U_{3}(4)}^{1}(3_{3}, Uni(3_{2}, 3_{4}, 3_{2})) → Ext_{U_{3}(4)}^{1}(Uni(3_{2}, 3_{3}), Uni(3_{2}, 3_{4}, 3_{2}))

→ Ext_{U_{3}(4)}^{1}(3_{2}, Uni(3_{2}, 3_{4} 3_{2})) → ⋯

[figure: 24_{4}; 3_{3}; 3_{2}; 3_{3}; 9_{3}; 9_{1}; 3_{2}; 8_{1}; 1; 9_{2}; 3_{3}; 3_{4}; 24_{4}; 9_{1}; 3_{1}; 3_{3}; 3_{4}; 1; 9_{2}; 3_{2}; 8_{1}; 9_{3}; 9_{1}; 3_{3}; 3_{2}; 3_{3}; 24_{4}]

Figure 2. Diagram for projective cover of 24_{4}

The first and last groups above are zero, and the second and third are onedimensional by Lemma 4.2. So the second map is an isomorphism and the lemma is proved.

Proposition 4.4. Ext_{U_{3}(4)}^{1}(9_{1}, 9_{2}) = {0}.

Proof.

Ext_{U_{3}(4)}^{1}(9_{1}, 9_{2}) = Ext_{U_{3}(4)}^{1}(3_{2} ⊗ 3_{4}, 3_{1} ⊗ 3_{3}) = Ext_{U_{3}(4)}^{1}(3_{4} ⊗ 3_{4}, 3_{1} ⊗ 3_{1})

= Ext_{U_{3}(4)}^{1}(Uni(3_{3}, 3_{2}, 3_{3}), Uni(3_{2}, 3_{4}, 3_{2}))

By Lemma 4.3, any non-trivial extension has to come from

Ext_{U_{3}(4)}^{1}(3_{3}, Uni(3_{2}, 3_{4}, 3_{2}))

and hence must be

[figure: 3_{2}; 3_{3}; 3_{4}; 3_{2}; 3_{2}; 3_{3}]

By Corollary 3.12 this is impossible.

5. Appendix

In Tables 2 to 4 below we give the 2-modular character table, decomposition matrix, and Cartan matrix for U_{3}(4).

We begin by reminding the reader of the ATLAS ([3]) notation, namely:

1. zn = e^{\frac{2πi}{n}}

2. ∗m = image of adjacent irrationality under the Galois automorphism z ↦ (zn)^{m}

3. ∗∗ = ∗(−1)

4. (−)&m = (−) + (−) ∗ m

5. d13 = z13&3&9.

Table 2: 2-modular character table of U_{3}(4)

1A  3A  5A  B∗∗  C∗2  D∗3  5E  F∗ 13A
1  1  1  1  1  1  1  1  1
3_{1}  0  z5+2&2  ∗∗  ∗2  ∗3  −b5  ∗  d13
3_{2}  0  ∗∗  z5+2&2  ∗3  ∗2  −b5  ∗  ∗∗
3_{3}  0  ∗3  ∗2  z5+2&2  ∗∗  ∗  −b5 ∗8
3_{4}  0  ∗2  ∗3  ∗∗  z5+2&2  ∗  −b5 ∗5
8_{1}  −1  4+2b5  4+2b5  ∗  ∗  −b5  ∗ 1−b13
8_{2}  −1  ∗  ∗  4+2b5  4+2b5  ∗  −b5  ∗
9_{1}  0 −z5+2&2−2&4 ∗∗  ∗2  ∗3  −1  −1 ∗8
9_{2}  0  ∗∗ −z5+2&2−2&4 ∗3  ∗2  −1  −1 ∗5
9_{3}  0  ∗3  ∗2 −z5+2&2−2&4 ∗∗  −1  −1 ∗∗
9_{4}  0  ∗2  ∗3  ∗∗ −z5+2&2−2&4 −1  −1 −1−d13
24_{1}  0 4z5+2&2−2&4 ∗∗  ∗2  ∗3  −1  −1 ∗∗
24_{2}  0  ∗∗ 4z5+2&2−2&4 ∗3  ∗2  −1  −1 1−d13
24_{3}  0  ∗3  ∗2 4z5+2&2−2&4 ∗∗  −1  −1 ∗5
24_{4}  0  ∗2  ∗3  ∗∗ 4z5+2&2−2&4 −1  −1 ∗8
64  1  4  4  4  4  −1  −1 −1
B∗∗  C∗5  D∗8  15A  B∗∗  C∗2  D∗8
1  1  1  1  1  1  1
∗∗  ∗5  ∗8  z5−&2  ∗∗  ∗2  ∗3
d13  ∗8  ∗5  ∗∗  z5−&2  ∗3  ∗2
∗5  d13  ∗∗  ∗3  ∗2  z5−&2  ∗∗
∗8  ∗∗  d13  ∗2  ∗3  ∗∗  z5−&2
1−b13  ∗  ∗  1−b5  1−b5  ∗  ∗
∗  1−b13  1−b13  ∗  ∗  1−b5  1−b5
∗5 −1−d13  ∗∗  2z5+2&2+&4  ∗∗  ∗2  ∗3
∗8  ∗∗  −1−d13  ∗∗  2z5+2&2+&4  ∗3  ∗2
−1−d13 ∗8  ∗5  ∗3  ∗2  2z5+2&2+&4  ∗∗
∗∗  ∗5  ∗8  ∗2  ∗3  ∗∗  2z5+2&2+&4
1−d13  ∗8  ∗5 −2z5−&2−2&4 ∗∗  ∗2  ∗3
∗∗  ∗5  ∗8  ∗∗ −2z5−&2−2&4 ∗3  ∗2
∗8  ∗∗  1−d13  ∗3  ∗2 −2z5−&2−2&4 ∗∗
∗5  1−d13  ∗∗  ∗2  ∗3  ∗∗ −2z5−&2−2&4
−1  −1  −1  1  1  1  1

Table 3: 2-modular decomposition matrix of U_{3}(4)

1 3 3 3 3 8 8 9 9 9 9 24 24 24 24 64
1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
12 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0
13 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
13 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
13 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
13 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
39 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0
39 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0
52 1 0 1 2 0 0 0 1 0 1 0 0 0 0 1 0
52 1 1 0 0 2 0 0 0 1 0 1 0 0 1 0 0
52 1 0 2 0 1 0 0 0 1 1 0 1 0 0 0 0
52 1 2 0 1 0 0 0 1 0 0 1 0 1 0 0 0
64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
65 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0
65 0 1 2 1 1 0 1 0 0 1 1 1 0 0 0 0
65 0 2 1 1 1 0 1 0 0 1 1 0 1 0 0 0
65 0 1 1 1 2 1 0 1 1 0 0 0 0 1 0 0
65 0 1 1 2 1 1 0 1 1 0 0 0 0 0 1 0
75 1 0 2 2 1 1 0 1 1 1 0 0 0 0 1 0
75 1 2 0 1 2 1 0 1 1 0 1 0 0 1 0 0
75 1 1 2 0 2 0 1 0 1 1 1 1 0 0 0 0
75 1 2 1 2 0 0 1 1 0 1 1 0 1 0 0 0

Table 4: 2-modular Cartan matrix of U_{3}(4)

1 3 3 3 3 8 8 9 9 9 9 24 24 24 24 64
1 16 12 12 12 12 4 4 8 8 8 8 2 2 2 2
3 12 26 14 18 18 6 8 10 8 8 14 2 6 4 1
3 12 14 26 18 18 6 8 8 10 14 8 6 2 1 4
3 12 18 18 26 14 8 6 14 8 10 8 1 4 2 6
3 12 18 18 14 26 8 6 8 14 8 10 4 1 6 2
8 4 6 6 8 8 6 1 6 6 2 2 0 0 2 2
8 4 8 8 6 6 1 6 2 2 6 6 2 2 0 0
9 8 10 8 14 8 6 2 10 6 4 4 0 2 2 3
9 8 8 10 8 14 6 2 6 10 4 4 2 0 3 2
9 8 8 14 10 8 2 6 4 4 10 6 3 2 0 2
9 8 14 8 8 10 2 6 4 4 6 10 2 3 2 0
24 2 2 6 1 4 0 2 0 2 3 2 3 0 0 0
24 2 6 2 4 1 0 2 2 0 2 3 0 3 0 0
24 2 4 1 2 6 2 0 2 3 0 2 0 0 3 0
24 2 1 4 6 2 2 0 3 2 2 0 0 0 0 3
64  1

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