THE WEIGHTED FUSION CATEGORY ALGEBRA AND THE q-SCHUR ALGEBRA FOR GL_{2}(q)

SEJONG PARK

Abstract. We show that the weighted fusion category algebra of the principal 2-block b_{0} of GL_{2}(q) is the quotient of the q-Schur algebra S_{2}(q) by its socle, for q an odd prime power. As a consequence, we get a canonical bijection between the set of isomorphism classes of simple kGL_{2}(q)b_{0}-modules and the set of conjugacy classes of b_{0}-weights in this case.

1. Introduction

Let k be an algebraically closed field of characteristic l > 0. Let q be a prime power which is coprime to l. Consider two k-algebras associated with GL_{n}(q), namely the weighted fusion category algebra \overline{F}(b_{0}) of the principal l-block b_{0} of GL_{n}(q) defined by Linckelmann [7] and the q-Schur algebra S_{n}(q) introduced by Dipper and James [4]. Since both are quasi-hereditary and carry informations of representations of GL_{n}(q)(Theorems 3 and 5), one may conjecture that there is a certain relation between them.

We give definitions and some properties of these algebras in Sections 2 and 3. In Section 4, we compute the Morita types of these algebras in a special case and find a relation between them:

Theorem 1. Let k be an algebraically closed field of characteristic 2 and let q be an odd prime power. Then the weighted fusion category algebra \overline{F}(b_{0}) over k of the principal 2-block b_{0} of GL_{2}(q) is Morita equivalent to the quotient of the q-Schur algebra S_{2}(q) over k by its socle.

Theorem 1 implies a canonical bijection between the set of isomorphism classes of simple kGL_{2}(q)b_{0}-modules and the set of isomorphism classes of simple \overline{F}(b_{0})- modules, which in turn is in a bijective correspondence with the set of conjugacy classes of b_{0}-weights. We discuss this canonical bijection in more detail in Section 5.

2. The weighted fusion category algebra

We summarize the construction of the weighted fusion category algebra and its basic properties, following Linckelmann [7]. We restrict our attention to the principal block case and avoid discussing the “twisting and gluing” procedure.(See [7, 4.1–4.4, 5.1])

Let k be an algebraically closed field of characteristic l > 0 and let G be a finite group. Let b_{0} be the principal l-block of G, i.e. the unique primitive idempotent in the center Z(kG) of the group algebra kG which is not contained in the augmentation ideal of kG. Fix a defect group P of b_{0}, namely a Sylow l-subgroup of G. The fusion system of the block b_{0} (on P) is the same as the fusion system of the group G (on P): it is the category F = F_{P} (G) whose objects are the subgroups of P and such that for each pair Q, R of subgroups of P the morphism set Hom_{F}(Q, R) consists of the group homomorphisms from Q to R induced by conjugations in G. It is independent of the choice of a defect group P, up to equivalence of categories.

The orbit category of F is the category \overline{F} whose objects are again the subgroups of P and such that for each pair Q, R of subgroups of P the morphism set Hom\overline{F}(Q, R) consists of the orbits of the group of inner automorphisms Inn(R) of R in Hom_{F}(Q, R).

A subgroup Q of P is called F-centric if every subgroup R of P which is Fisomorphic to Q is centric in P, i.e. C_{P} (R) = Z(R). We denote by \overline{F}^{c} the full subcategory of the orbit category \overline{F} consisting of F-centric subgroups of P.

Let k\overline{F}^{c} be the category algebra of \overline{F}^{c} over k, that is, the k-algebra whose k-basis consists of morphisms of \overline{F}^{c} and such that multiplication is induced by composition of morphisms.

Definition 2. With the above notations, let \overline{e} = ∑_{Q} \overline{e}_{Q} where Q runs over all Fcentric subgroups of P and \overline{e}_{Q} denotes the sum of all defect zero blocks of kAut\overline{F}(Q). Then the weighted fusion category algebra of the block b_{0} is the truncated algebra

\overline{F}(b_{0}) = \overline{e}k\overline{F}^{c}\overline{e}.

The significance of the weighted fusion category algebra is summarized in the following theorem:

Theorem 3 ([7, 4.5, 5.1]). Let k be an algebraically closed field of characteristic l > 0 and let b_{0} be the principal l-block of a finite group G. Then the weighted fusion category algebra \overline{F}(b_{0}) over k of the block b_{0} is quasi-hereditary and Alperin’s weight conjecture for the block b_{0} is equivalent to the equality

l(kGb_{0}) = l(\overline{F}(b_{0}))

where l(A) denotes the number of isomorphism classes of simple A-modules for a finite dimensional k-algebra A.

Remark. We refer to Alperin’s original paper [1] for the definition of weights and the statement of Alperin’s weight conjecture. Note that, for an F-centric subgroup Q of P, the defect zero blocks of kAut\overline{F}(Q) appearing in Definition 2 (if any) correspond to the b_{0}-weights having Q as their first component. Alperin’s weight conjecture is positively confirmed, among others, for finite general linear groups by Alperin and Fong [2] in odd characteristics and by An [3] in characteristic 2.

3. The q-Schur algebra

We review the definition and some basic properties of the q-Schur algebra defined by Dipper and James [4], following the presentation of Mathas [8].

Let k be a field and let q be a nonzero element of k. The Iwahori-Hecke algebra of the symmetric group Σ_{n} on n letters is the k-algebra H = H_{k,q}(Σ_{n}) whose k-basis is {T_{w} | w ∈ Σ_{n}} and such that multiplication is given by

T_{w}T_{s} = \begin{cases}T_{ws}, & if l(ws) > l(w), \\ qT_{ws} + (q − 1)T_{w}, & if l(ws) < l(w), \end{cases}

where w ∈ Σ_{n}, s = (i, i + 1) ∈ Σ_{n} for some 0 < i < n, and l(w) is the length of w.

THE WEIGHTED FUSION CATEGORY ALGEBRA AND THE q-SCHUR ALGEBRA 3

A composition of n is a sequence μ = (μ_{1}, μ_{2}, …) of nonnegative integers μ_{i} whose sum is equal to n. The height of a composition μ is the smallest positive integer d such that μ_{d+1} = μ_{d+2} = ⋯ = 0. For a composition μ of n with height d, let Σ_{μ} be the corresponding Young subgroup of Σ_{n} isomorphic to Σ_{μ_{1}} × Σ_{μ_{2}} × ⋯ × Σ_{μ_{d}}. Set m_{μ} = ∑_{w∈Σ_{μ}} T_{w} and let M^{μ} = m_{μ}H, the right H-submodule of H generated by m_{μ}.

Definition 4. Let Λ(d, n) be the set of all compositions of n with height ≤ d. Then the q-Schur algebra is the endomorphism algebra

S_{d,n}(q) = End_{H}( ⨁_{μ∈Λ(d,n)} M^{μ}).

We write S_{n}(q) = S_{n,n}(q).

The q-Schur algebra has the following properties:

Theorem 5 ([8, 4.16, 6.47]). Let k be a field and let q be a nonzero element of k. Then the q-Schur algebra S_{d,n}(q) over k is quasi-hereditary. If char k = l > 0 and q is a prime power which is coprime to l, then the decomposition matrix of kGL_{n}(q) is completely determined by the decomposition matrices of the q^{r}-Schur algebras S_{m}(q^{r}) over k for rm ≤ n.

Gruber and Hiss [6] and Takeuchi [10] give an alternative way of computing the Morita types of the q-Schur algebras.

Theorem 6 ([6], [10]). Let k be an algebraically closed field of characteristic l > 0 and let q be a prime power which is coprime to l. Let G = GL_{n}(q) and let B be the set of all upper triangular matrices in G. Then the q-Schur algebra S_{n}(q) over k is Morita equivalent to the image of the k-algebra homomorphism

kG → End_{k}(k[G/B])

sending a ∈ kG to the k-linear endomorphism of k[G/B] given by left multiplication by a on k[G/B].

4. The case G = GL_{2}(q), q odd, in characteristic 2

Let k be an algebraically closed field of characteristic 2 and let q be an odd prime power. In this case, we have

Proposition 7. The weighted fusion category algebra \overline{F}(b_{0}) over k of the principal 2-block b_{0} of GL_{2}(q) is Morita equivalent to the path algebra of the quiver

\xymatrix{^{1}• & •^{2} \ar[l]_{α}}

Proposition 8 ([5, 3.3(A)]). The q-Schur algebra S_{2}(q) over k is Morita equivalent to the path algebra of the quiver

\xymatrix{^{1}• \ar[r]^{β} & •^{2} \ar[l]^{γ}}

with relation βγ = 0.

A proof of Proposition 7 is given in Sections 4.1 and 4.2. Proposition 8 is a consequence of more general results of Erdmann and Nakano [5]. For the convenience of the reader, we sketch a proof of Proposition 8 in Section 4.3. Theorem 1 follows immediately from Propositions 7 and 8.

4.1. Proof of Proposition 7 when q ≡ 3 mod 4. Let G = GL_{2}(q) where q is a prime power such that q ≡ 3 mod 4. Let 2^{m−2} be the highest 2-power dividing q + 1 and let ξ be a primitive 2^{m−1}th root of unity in 𝔽_{q^{2}}. Note that m ≥ 4. Then the subgroup P of G generated by

x = (\begin{matrix}0 & 1 \\ 1 & a \end{matrix}) ,  t = (\begin{matrix}1 & a \\ 0 & −1 \end{matrix})  (a = ξ + ξ^{q})

is a Sylow 2-subgroup of G. One immediately checks that x and t are of order 2^{m−1} and 2, respectively, and

txt = x^{2^{m−2}−1}.

In other words, P is the semidiheral group SD_{2^{m}} of order 2^{m}.

Let F = F_{P} (G). Then the F-centric subgroups of P are as follows:

(1) C_{2} × C_{2} ≅ ⟨x^{2^{m−2}}, tx^{2i}⟩

(2) D_{2^{k}} ≅ ⟨x^{2^{m−k}}, tx^{2i}⟩ where 3 ≤ k ≤ m − 1

(3) Q_{2^{k}} ≅ ⟨x^{2^{m−k}}, tx^{2i+1}⟩ where 3 ≤ k ≤ m − 1

(4) C_{2^{m−1}} ≅ ⟨x⟩

(5) P Recall that the automorphism groups of cyclic, dihedral, semidihedral, and (generalized) quaternion 2-groups of order ≥ 4 are all nontrivial 2-groups except for

Aut(C_{2} × C_{2}) ≅ Σ_{3}, Aut(Q_{8}) ≅ Σ_{4}.

So the \overline{F}-automorphism group of an F-centric subgroup R of P of type (2), (3) with k > 3, (4), or (5) is a (possibly trivial) 2-group. If R = P, then since Aut\overline{F}(P) is also a 2′-group, we have Aut\overline{F}(P) = \{id_{P} \} and hence \overline{e}_{P} = 1. Let e_{1} = \overline{e}_{P}. If R < P, then we have Inn(R) < Aut_{F}(R) and hence Aut\overline{F}(R) is a nontrivial 2-group. Therefore \overline{e}_{R} = 0.

Also, since x^{2^{m−2}} = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ Z(G) and tx^{2i}, −tx^{2i} are G-conjugate, the \overline{F}- automorphism group of a Klein four subgroup R of P is isomorphic to C_{2}, yielding \overline{e}_{R} = 0. Thus it remains to consider the quaternion subgroups of order 8. Set

Q_{i} = ⟨x^{2^{m−3}}, tx^{2i+1}⟩, i = 0, 1, …, 2^{m−4} − 1.

First observe that all Q_{i} are P -conjugate. Indeed, for each pair of indices i, j, let k = (2^{m−3} − 1)(j − i). Then

x^{k}tx^{2i+1}x^{−k} = tx^{(2^{m−2}−1)k}x^{2i+1−k} = tx^{2j+1}.

So it suffices to consider only Q := Q_{0} = ⟨x^{2^{m−3}}, tx⟩. We have Aut_{P} (Q) ≅ D_{8} and Aut(Q) ≅ Σ_{4}. Thus Aut_{F}(Q) is either Aut_{P} (Q) or Aut(Q). Since x^{2^{m−3}} and tx are G-conjugate but automorphisms in Aut_{P} (Q) do not send x^{2^{m−3}} to tx, one finds that

Aut_{F}(Q) = Aut(Q) ≅ Σ_{4}.

(Note that this also follows from the Frobenius normal p-complement theorem.) Now Aut\overline{F}(Q) = Aut_{F}(Q)/Aut_{Q}(Q) and Aut_{Q}(Q) ≅ C_{2} × C_{2}. Thus we have

Aut\overline{F}(Q) ≅ Σ_{3}.

Since kΣ_{3} ≅ kC_{2} × M_{2}(k) as k-algebras, one finds that \overline{e}_{Q}kAut\overline{F}(Q)\overline{e}_{Q} ≅ M_{2}(k). Let e_{2} be the element of \overline{e}_{Q}kAut\overline{F}(Q)\overline{e}_{Q} which corresponds to ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) via this isomorphism.

Set A := \overline{e}k\overline{F}^{c}\overline{e} where \overline{e} = \overline{e}_{P} + \overline{e}_{Q}. Then A is Morita equivalent to \overline{F}(b_{0}). We have a decomposition as k-vector spaces

A = A_{1} ⊕ A_{2} ⊕ J

where

A_{1} = \overline{e}_{P} kAut\overline{F}(P)\overline{e}_{P} ≅ k,
A_{2} = \overline{e}_{Q}kAut\overline{F}(Q)\overline{e}_{Q} ≅ M_{2}(k),
J = \overline{e}_{P} kHom\overline{F}(Q, P)\overline{e}_{Q}.

Since J^{2} = 0 and A/J ≅ k × M_{2}(k), J is the Jacobson radical of A and there are exactly two nonisomorphic simple A-modules S_{1}, S_{2} with corresponding projective indecomposable A-modules P_{1} = Ae_{1}, P_{2} = Ae_{2}. Note that P_{1} = S_{1} ≅ k and JP_{2}/J^{2}P_{2} = kHom\overline{F}(Q, P)e_{2} ≅ S_{1}. Therefore we get the desired result.

4.2. Proof of Proposition 7 when q ≡ 1 mod 4. Let G = GL_{2}(q) where q is a prime power such that q ≡ 1 mod 4. Let 2^{m} be the highest 2-power dividing q − 1 and let η be a primitive 2^{m}th root of unity in 𝔽_{q}. Note that m ≥ 2. Then the subgroup P of G generated by

x = (\begin{matrix}η & 0 \\ 0 & 1 \end{matrix}) , y = (\begin{matrix}1 & 0 \\ 0 & η \end{matrix}) , t = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

is a Sylow 2-subgroup of G. Since x, y commute and txt = y, we see that P ≅ C_{2^{(m−1)/2}} ≀ Σ_{2}. Note that Z_{0} := Z(P) = Z(G) ∩ P = ⟨xy⟩ ≅ C_{2^{m}}.

Let F = F_{P} (G). Then the F-centric subgroups of P are as follows:

(1) ⟨x, y⟩

(2) ⟨xy, tx^{i}⟩ where η^{i} ≠ η^{2j} for any integer j

(3) ⟨xy, x^{2^{i}}, tx^{j}⟩ where 0 ≤ i ≤ m − 1, 0 ≤ j < 2^{i} Let R be an F-centric subgroup of P. If R = ⟨x, y⟩, then we have

Aut\overline{F}(R) ≅ N_{G}(R)/RC_{G}(R) = LΣ_{2}/L ≅ Σ_{2},

where L denotes the diagonal subgroup of G and Σ_{2} is viewed as the subgroup of the permutation matrices in G.

Now suppose that R is of type (2) or (3). Since Z_{0} ⊆ Z(G), elements of Z_{0} are fixed by any F-morphism. So every F-automorphism of R induces an automorphism of R/Z_{0}, giving rise to a surjective group homomorphism

Φ : Aut_{F}(R) ↠ Aut_{G/Z_{0}}(R/Z_{0}).

Note that the kernel Ker(Φ) of Φ is isomorphic to a certain subgroup of the group Hom(R, Z_{0}) whose multiplication is given by pointwise multiplication. In particular Ker(Φ) is an abelian 2-group.

If R is of type (2), then R/Z_{0} ≅ C_{2}, so Aut(R/Z_{0}) = {id_{R/Z_{0}}}. One can easily check that Ker(Φ) ≅ C_{2} in this case. Since R is abelian, it follows that Aut\overline{F}(R) ≅ C_{2}.

Suppose that R is of type (3). Then R/Z_{0} is a dihedral 2-group of order ≥ 4; it is of order 4 (i.e. a Klein four group) if and only if i = m − 1. So if i ≠ m − 1, then R/Z_{0} is a dihedral 2-group of order ≥ 8, and hence its automorphism group is a (nontrivial) 2-group. Thus Aut_{F}(R) is a 2-group. Now if R < P, then Inn(R) < Aut_{F}(R), so Aut\overline{F}(R) is a nontrivial 2-group; if R = P, then Aut\overline{F}(P) is also a 2′-group and hence Aut\overline{F}(P) = \{id_{P} \}.

Finally, let R be of type (3) with i = m − 1. There are two P -conjugacy classes among these F-centric subgroups. Indeed, for any j,

⟨xy, x^{2^{m−1}}, tx^{j}⟩ ≅ ⟨xy, x^{2^{m−1}}, tx^{j+2}⟩

because x^{−1}(tx^{j+1}y)x = tx^{j+2}. Set

R_{1} = ⟨xy, x^{2^{m−1}}, t⟩, R_{2} = ⟨xy, x^{2^{m−1}}, tx⟩.

Since R_{i}/Z_{0}(i = 1, 2) is a Klein four group, its full automorphism group is isomorphic to Σ_{3}, permuting its three nonidentity elements. Those three nonidentity elements of R_{1}/Z_{0} are all G-conjugate; in R_{2}/Z_{0}, the elements txZ_{0} and tx^{2^{m−1}+1}Z_{0} are G-conjugate but x^{2^{m−1}}Z_{0} is not G-conjugate to these two. For both i = 1, 2, we have Ker(Φ) = Inn(R_{i}) ≅ C_{2} × C_{2}. Thus

Aut\overline{F}(R_{1}) ≅ Σ_{3}, Aut\overline{F}(R_{2}) ≅ C_{2}.

Therefore we get the same quiver as in Proposition 7.

4.3. Proof of Proposition 8. Let B be the set of all upper triangular matrices in G. For u ∈ 𝔽_{q}, set

[u] := (\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) .

Also set

t := (\begin{matrix}ϵ & 0 \\ 0 & 1 \end{matrix}) , w := (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

where ϵ is a generator of the multiplicative group 𝔽_{q}^{×}. Then we have

G/B = { B, wB, [ϵ^{i}]wB }_{1≤i≤q−1}.

Let

kG → End_{k}(k[G/B])

be the k-algebra homomorphism of Theorem 6 and denote its image by S. This map is the k-linear extension of the group homomorphism

ψ : G → Σ_{G/B} ↪ GL_{k}(k[G/B])

where the first homomorphism sends g ∈ G to the permutation of G/B induced by left multiplication by g and the second inclusion sends permutations of G/B to corresponding permutation matrices. Observe that the following correspondence

\begin{matrix}B & wB & [ϵ]wB & [ϵ^{2}]wB & ⋯ & [ϵ^{q−1}]wB \\ ↕ & ↕ & ↕ & ↕ & ⋯ & ↕ \\ [\begin{matrix}1 \\ 0 \end{matrix}] & [\begin{matrix}0 \\ 1 \end{matrix}] & [\begin{matrix}ϵ \\ 1 \end{matrix}] & [\begin{matrix}ϵ^{2} \\ 1 \end{matrix}] & ⋯ & [\begin{matrix}ϵ^{q−1} \\ 1 \end{matrix}] \end{matrix}

respects the G-action on G/B by left multiplication and the natural G-action on the

projective line over 𝔽_{q}, where [\begin{matrix}u \\ v \end{matrix}] denotes the image of (\begin{matrix}u \\ v \end{matrix}) in the projective line.

Denote above elements by v_{1}, v_{2}, …,v_{q+1}, respectively, and write V = k[G/B] = kv_{1} ⊕ kv_{2} ⊕ ⋯ ⊕ kv_{q+1}. Then ψ factors through

PGL_{2}(q) ≅ G/Z(G) ↪ GL_{k}(V),

and hence

S = Im(kPGL_{2}(q) → End_{k}(V)).

V is a (q + 1)-dimensional S-module with the natural S-action. Now we find its composition series. First of all, V has an obvious 1-dimensional simple S-submodule

V_{1} = k(v_{1} + v_{2} + ⋯ + v_{q+1}).

Let us denote the elements of the quotient module V/V_{1} as

[λ_{1}, λ_{2}, …, λ_{q+1}] := λ_{1}v_{1} + λ_{2}v_{2} + … + λ_{q+1}v_{q+1} + V_{1}

with λ_{i} ∈ k. Then the (q − 1)-dimensional S-submodule V_{2} of V/V_{1} given by

V_{2} = { [λ_{1}, λ_{2}, …, λ_{q+1}] | λ_{1} + λ_{2} + … + λ_{q+1} = 0 }

is also simple because PGL_{2}(q) acts 3-transitively on { v_{1}, v_{2}, …, v_{q+1} }.(See [9, Table 1]) Let W be the inverse image in V of V_{2}. Observe that V, W are uniserial S-modules with composition series (V_{1}, V_{2}, V_{1}), (V_{2}, V_{1}), respectively. In particular, both V and W are indecomposable.

It is well known that V = k[G/B] is a projective S-module and that there are exactly two simple S-modules up to isomorphism. Then, since S is quasi-hereditary, it follows from the composition seris of V that the standard modules for V_{1} and V_{2} are V_{1} and W, respectively, and W is also projective. Therefore we conclude that S, and hence the q-Schur algebra S_{2}(q), is Morita equivalent to the path algebra of the quiver given in Proposition 8.

5. A remark on a canonical bijection of simple modules

Let k be an algebraically closed field of characteristic 2 and let q be an odd prime power. Let b_{0} be the principal 2-block of G = GL_{n}(q). The algebra homomorphism in Theorem 6 restricts to the surjective algebra homomorphism

kGb_{0} ↠ S

where S is a k-algebra which is Morita equivalent to the q-Schur algebra S_{n}(q). On the other hand, in Theorem 1 we showed that there is another surjective algebra homomorphism

S_{0} ↠ T_{0}

where S_{0} and T_{0} are, respectively, the basic algebras of the q-Schur algebra S_{n}(q) and the weighted fusion category algebra \overline{F}(b_{0}) when n = 2. Combining these two surjective algebra homomorphisms, we see that simple \overline{F}(b_{0})-modules can be viewed as simple kGb_{0}-modules when n = 2. Since we have

l(\overline{F}(b_{0})) = number of partitions of n = l(kGb_{0})

for n = 2 (in fact, for every n by An [3]), we get a canonical bijection between simple kGb_{0}-modules and simple \overline{F}(b_{0})-modules in this case. But, as mentioned in the remark after Theorem 3, there is a canonical bijection between the set of isomorphism classes of simple \overline{F}(b_{0})-modules and the set of conjugacy classes of b_{0}- weights. Thus we get a canonical bijection between the set of isomorphism classes of simple kGb_{0}-modules and the set of conjugacy classes of b_{0}-weights when n = 2.

Acknowledgements. We would like to thank Markus Linckelmann for his valuable suggestions and comments throughout writing this paper. Also we thank Radha Kessar for her comment on the canonical bijection and Will Turner for pointing us to the work of Erdmann and Nakano [5].

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University of Aberdeen, Aberdeen, UK AB24 3UE

  E-mail address: spark@maths.abdn.ac.uk