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The Gorenstein property for modular binary forms invariants
Journal of Algebra 451 (2016) 232–247
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Journal of Algebra www.elsevier.com/locate/jalgebra
The Gorenstein property for modular binary forms invariants Amiram Braun Dept. of Mathematics, University of Haifa, Haifa, 31905 Israel
a r t i c l e
i n f o
Article history: Received 29 May 2014 Available online xxxx Communicated by Luchezar L. Avramov Keywords: Gorenstein Modular invariants
a b s t r a c t Let G ⊆ SL(2, F ) be a finite group, V = F 2 the natural SL(2, F )-module, and charF = p > 0. Let S(V ) be the symmetric algebra of V and S(V )G the ring of G-invariants. We provide examples of groups G, where S(V )G is Cohen– Macaulay, but is not Gorenstein. This refutes a natural conjecture due to Kemper, Körding, Malle, Matzat, Vogel and Wiese. Let T (G) denote the subgroup generated by all transvections of G. We show that S(V )G is Gorenstein if and only if one of the following cases holds: (1) T (G) = {1G }, (2) V is an irreducible T (G)-module, (3) V is a reducible T (G)-module |T (G)|(|T (G)| − 1).
and
|G|
divides
© 2015 Elsevier Inc. All rights reserved.
1. Introduction Let G ⊂ GL(V ) be a finite group, where V is a finite dimensional vector space over a field F with charF = p ≥ 0. The following is a classical result of K. Watanabe. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2015.11.039 0021-8693/© 2015 Elsevier Inc. All rights reserved.
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Theorem. (See [4, Theorems 6.4.10], [18,19].) Suppose that (|G|, p) = 1 and G contains no pseudo-reflection. Then the following are equivalent: (1) G ⊂ SL(V ), (2) S(V )G is a Gorenstein ring. In the modular case, that is if p divides |G|, S(V )G may fail to be Cohen–Macaulay, let alone Gorenstein, thus a suitable generalization of the above to the modular case should require the Cohen–Macaulay property. In [13, Conjecture 5], Kemper et al. conjectured that one of the above implications is valid (assuming the Cohen–Macaulay property), in the modular case as well. Explicitly: Conjecture. (See [13, Conjecture 5].) Suppose G ⊂ SL(V ). Then S(V )G is Gorenstein if (and only if) it is Cohen–Macaulay. This conjecture was verified in [13] on all 1916 examples of the database (which satisfy its hypothesis). A special case of the conjecture, assuming that G contains no transvections, was confirmed in [2, Theorem B] using non-commutative methods. A commutative proof for this special case was recently given by Fleischmann–Woodcock [9]. A statement of this special case also appears in [3, Corollary 7(vi)]. However, in my opinion, the proof of [3, Lemma 5], on which it crucially depends, is incorrect. A remarkable generalization of this special case, is proved in [9], assuming that S(V )W (G) is a polynomial ring and S(V )G is Cohen–Macaulay. Then S(V )G is Gorenstein if and only if G/W (G) ⊆ SL(m/m2 ), where W (G) is the subgroup generated by all pseudo-reflections and m is the unique homogeneous maximal ideal of S(V )W (G) . Our first objective here is to provide a binary form counterexample to this conjecture. We have: Example A. Let V = F 2 be the natural SL(2, F )-module where charF = p with p ≡ 3 (mod 4). Assume that F contains a primitive 4-th root of unity ξ. Let G ⊂ SL(V ) be given by:
ξ, G = τ, σ, where τ = 0,
0 ξ −1
1, , σ= 0,
1 . 1
Then S(V )G is Cohen–Macaulay but it is not Gorenstein. Other examples for arbitrary p can be given by using item (2) of the next result. On the positive side we have the following. Let T (G) denote the subgroup generated by all transvections of G.
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Theorem B. Let F be a field with charF = p > 0 and G ⊂ SL(2, F ) a finite subgroup where p divides |G|. Let V = F 2 be the natural SL(2, F )-module. Then S(V )G is Gorenstein if and only if one of the following cases holds: (1) V is an irreducible T (G)-module, (2) |G| divides |T (G)|(|T (G)| − 1), and V is a reducible T (G)-module. Moreover in each of these cases S(V )G is either a simple singularity hypersurface (of type A) in F 3 , or non-singular. Note that when (p, |G|) = 1 and G ⊆ SL(V ), then S(V )G is Gorenstein by Watanabe’s theorem. Also, for G ⊆ GL(2, F ) the restriction (p, |G|) = 1 holds if and only if T (G) = {1}. As a consequence we get the following: Corollary C. Let G ⊂ SL(2, F ) be a finite group and V = F 2 the natural SL(2, F )-module. Then S(V )G is Gorenstein in the following cases: (1) (2) (3) (4)
G is odd, p = 2, F = Fq , a Galois field, with q = ps and s is odd, |T (G)| = 1.
The results of the present paper were conceived by non-commutative methods. However we merely use commutative methods and only apply Watanabe’s non-modular characterization for a Gorenstein ring of invariants. 2. Counterexamples Let F be a field with charF = p > 0. We shall exhibit here finite groups G, having the following properties: (1) G ⊂ SL(V ), (2) S(V )G is Cohen–Macaulay, (3) S(V )G is not Gorenstein. This settles [13, Conjecture 5] in the negative. Example 2.1. Assume that p ≡ 3 (mod 4) and that F contains a primitive 4-th root of unity ξ. Thus ξ 2 = −1 = 1 (since p is necessarily odd). Let V = F 2 be the natural SL(2, F )-module with basis {x1 , x2 }. ξ, 0 1, 1 . and σ ≡ Let τ ≡ 0, ξ −1 0, 1
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Thus τ (x1 ) = ξx1 , τ (x2 ) = ξ −1 x2 , σ(x1 ) = x1 + x2 , σ(x2 ) = x2 . We have: τ στ
−1
ξ, = 0,
0 ξ −1
1, 0,
1 1
ξ −1 , 0,
1, −1 0 1, ξ 2 = = σ p−1 . = 0, 1 0, 1 ξ
Let G = τ, σ be the subgroup of SL(2, F ) generated by {σ, τ }. Then σ is a normal subgroup of G, |σ| = p and |G| = 4p. We extend the action of G on V into an action by automorphisms on the polynomial ring S(V ) = F [x1 , x2 ]. The next result is standard and its proof is included for completeness sake. Lemma 2.2. T ≡ F [x1 , x2 ]σ = F [a, x2 ], where a = xp1 − x1 xp−1 . 2 Proof. σ(x1 ) = x1 + x2 implies that: Norm σ (x1 ) = x1 (x1 + x2 )(x1 + 2x2 ) . . . (x1 + (p − 1)x2 ) = xp1 − x1 xp−1 = a. 2 Now by Galois theory: [Q(F [x1 , x2 ]) : Q(T )] = |σ| = p. Also [Q(F [x1 , x2 ]) : Q(F [a, x2 ])] = dega = p. Consequently Q(T ) = Q(F [a, x])). Since T is integral over F [a, x2 ] and the latter is normal (being a polynomial ring), it now follows that T = F [a, x2 ]. 2 Now clearly S(V )G is a normal domain of Krull dimension 2. In particular S(V )G is Cohen–Macaulay. We shall next compute the action of τ on T : τ (a) = (τ (x1 ))p − τ (x1 )(τ (x2 ))p−1 = (ξx1 )p − (ξx1 )(ξ −1 x2 )p−1 = ξ p xp1 − ξ 2−p x1 xp−1 . 2 Now p ≡ 3 (mod 4) implies that p −(2 −p) = 2p −2 ≡ 0 (mod 4) and therefore ξ p = ξ 2−p . Consequently τ (a) = ξ p a and τ (x2 ) = ξ −1 x2 . Consider U ≡ F a + F x2 , a two dimensional F -subspace of T . Clearly S(U ) = T = F [a, x2 ] is a polynomial ring in the two variables {a, x2 }. The action of τ on T can therefore be regarded as the natural extension p of the linear action of τ|U on U , to S(U ). ξ , 0 Therefore T τ = S(U )τ|U . Now τ|U = is the matrix representing τ|U with 0, ξ −1 respect to the basis {a, x2 }. Clearly τ|U contains no pseudo-reflections, |τ|U | = 4 and (p, 4) = 1. Also det(τ|U ) = ξ p−1 = −1 (using p ≡ 3 (mod 4)). Consequently by Watanabe’s theorem S(U )τ|U = T τ is not Gorenstein. Since S(V )G = (S(V )σ )τ = T τ , the result follows.
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Remark 2.3. (1) Observe that if p is an odd prime then Fp2 , the Galois field of p2 elements, contains 2 a primitive 4-th root of unity since F× p2 is a cyclic group of order p − 1. (2) The smallest example one obtains by the previous method is for p = 3, where |G| = 4 · 3 = 12 and F = F32 . 3. Finite subgroups of SL(2, F ) Our main objective here is to find out the extent of validity of the Kemper et al. conjecture, for finite subgroups of SL(2, F ), where F is an arbitrary field of positive characteristic p. Qualitatively speaking, most of the groups do satisfy the conjecture. Our argument also shows why and how counterexamples appear. For SL(n, F ) with n > 2 such an analysis does not exist (as yet), due partially to the difficulty in analyzing irreducible primitive actions of T (G), the G-subgroup generated by all transvections. The following theorem is our main result. Theorem 3.1. Let F be an arbitrary field with charF = p > 0. Let G ⊆ SL(2, F ) be a finite subgroup and V = F 2 , the natural SL(2, F )-module. Then (i) S(V )G is Gorenstein whenever V is an irreducible T (G)-module. (ii) If V is a reducible T (G)-module and T (G) = {1G } then S(V )G is Gorenstein if and only if |G| divides |T (G)|(|T (G)| − 1). (iii) There exists a subgroup D of G, D ⊇ T (G) with |G|/|D| ≤ 2 and S(V )D is Gorenstein. (iv) In particular S(V )G is Gorenstein in the following (not mutually disjoint) cases: (1) T (G) = {1G }, (2) G is an odd order group, (3) F = Fq , where q = ps and s is odd, (4) p = 2. (v) If p divides |G| then S(V )G is a hypersurface in F 3 whenever it is Gorenstein. Our proof will proceed in steps. We shall firstly establish some reductions which will be solely used in the irreducible primitive case. The cases where T (G) acts irreducibly on V will be handled separately. The reducible case will be handled in the main course of the proof. Our first reduction result can be obtained in several ways. Lemma 3.2. Let V0 be a k-finite dimensional G-module and F ⊃ k, a field extension. Set V ≡ V0 ⊗k F , and let G act trivially on F . Then the Gorenstein property of S(V0 )G ascends to S(V )G .
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Proof. Let w be the canonical S(V0 )G -module. By [4, Exercise 3.3.31] w ⊗k F is a canonical S(V0 )G ⊗k F -module. Now the Gorenstein property of S(V0 )G amounts to S(V0 )G ∼ =w as S(V0 )G -modules and consequently S(V0 )G ⊗k F is a canonical S(V0 )G ⊗k F -module. Now clearly S(V ) = S(V0 ) ⊗k F and consequently S(V )G = S(V0 )G ⊗k F . Hence S(V )G is its own canonical module. Since S(V )G is Cohen–Macaulay by [4, Theorem 2.1.10], the Gorenstein property of S(V )G now follows. 2 Remark 3.3. Such an ascend of the unique factorization property from S(V0)G to S(V )G is not true in general. Our second reduction result is well known. Failing to find a reference for it, we provide a proof. Lemma 3.4. Let G ⊂ GL(V ) be a finite group, where V is an irreducible F -finite dimensional G-module and charF = p > 0. Then there exists a finite subfield k ⊂ F , and a k-finite dimensional faithful irreducible G-module V0 , satisfying V = V0 ⊗k F . Proof. Denote by FG the group algebra of G over F and set Ann FG V ≡ {x ∈ FG|x.V = 0} = M. Then M is a two-sided maximal ideal in FG. Clearly Fp G ⊗Fp F = FG. Let N ≡ M ∩ Fp G = Ann Fp G V . Then the exactness of (−) ⊗Fp F shows that NFG = N ⊗Fp F ⊆ M and therefore FG/M = (Fp G/N )F . That is FG/M is generated by Fp G/N and F . Therefore FG/M is a central extension of Fp G/N , implying that Fp G/N is a simple ring and its center k ≡ Z(Fp G/N ) is contained in Z(FG/M ). This shows (by [8, Cor. 3.6, p. 54]) that FG/M = Fp G/N ⊗k S, where S is the centralizer of Fp G/N in FG/M . It follows from the previous equality that F = S = Z(FG/M ), that is Fp G/N ⊗k F = FG/M . Now [Fp G/N : Fp ] < ∞ grants that [k : Fp ] < ∞, and by Wederburn’s theorem for finite division algebras Fp G/N = Mn (k). Since V can be identified with a column of Mn (F ) = FG/M , then Mn (k) ⊗k F = Mn (F ) shows that V0 ⊗k F = V , for some irreducible Fp G/N = Mn (k)-module V0 . Clearly V0 is also a faithful irreducible G-module with G ⊂ GL(V0 ). 2 We now present a special case one needs to consider. Proposition 3.5. Suppose G ⊂ SL(2, F ) is a finite group with T (G) = SL(2, Fq ), where q = ps and Fq ⊆ F . Let V = F 2 be the natural SL(2, F )-module. Then S(V )G is Gorenstein. Proof. Fix a basis {x, y} of V . Then by [17, Ex. 1, p. 104] the two Dickson invariq2
q2
ants of S(V )GL(2,Fq ) are c20 = (xy q − yxq )q−1 and c21 = xyxyq −yx −yxq . Moreover by q q [1, Theorem 8.2.1] u = xy − yx , c21 are the two generators of S(V )SL(2,Fq ) . Let
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a, b d, ∈ G be arbitrary. Then gT (G)g −1 = T (G) implies, using g −1 = c, d −c, and 1 = detg = ad − bc: g=
1, g 0,
−b a
1 −1 1 − ac, a2 g = ∈ SL(2, Fq ), 1 −c2 , 1 + ac
as well as g
1, 1,
0 −1 1 + bd, g = d2 , 1
−b2 ∈ SL(2, Fq ). 1 − bd
Consequently ac, a2 , c2 , b2 , d2 , bd ∈ Fq . We firstly assume that a = 0. Then γ ≡ ac = aac2 ∈ Fq . If b = 0 then μ ≡ db = bd b2 ∈ Fq . 1 b Therefore 1 = ad − bc = abμ − baγ = ab(μ − γ). Hence a2 (μ−γ) = a ≡ β ∈ Fq , as well as δ ≡ ad = μb a = μβ ∈ Fq . If b = 0 then we can take β = 0 and 1 = ad implies that d ad δ ≡ a = a2 = a12 ∈ Fq . So we get that b = aβ, c = aγ, d = aδ, with β, γ, δ ∈ Fq and
a, 0 g= 0, a
1, β 1, β , with h ≡ ∈ GL(2, Fq ). γ, δ γ, δ
If a = 0 then 1 = ad − bc implies that b, c = 0 and we get a similar equality with c replacing a. 2 g(c21 ) = (aIV )(h(c21 )) = (aIV )(c21 ) = aq −q c21 . Also if g1 = Consequently a1 , b1 ∈ GL(2, Fq ), then aq1 = a1 , bq1 = b1 , cq1 = c1 , dq1 = d1 imply that: c1 , d1 g1 (u) = g1 (x)g1 (y)q − g1 (y)g1 (x)q = (a1 x + b1 y)(c1 x + d1 y)q − (c1 x + d1 y)(a1 x + b1 y)q = (a1 d1 − b1 c1 )(xy q − yxq ) = det(g1 )u. Consequently, g(u) = (aIV )(h(u)) = det(h)adegu u = aq+1 a−2 u = aq−1 u, where the second to last equality follows from 1 = det(g) = a2 det(h). Let g ∈ G and let g¯ be its image in G/T (G). Now clearly S(V )G = (S(V )T (G) )G/T (G) = (S(V )SL(2,Fq ) )G/T (G) = (F [u, c21 ])G/T (G) . Observe that F [u, c21 ] = S(U ), where U = F u + F c21 , is a 2-dimensional F -vector space. Moreover g¯ acts linearly on U and extending this action to S(U ) induces the original action of g¯ on F c21 ]. Also the [u, matrix of g¯|U with respect to the basis {u, c21 } is given aq−1 , 0 2 by the matrix , implying that det g¯|U = aq −1 . Now a2 = det(h)−1 ∈ Fq 2 0, aq −q 2 and therefore [Fq (a) : Fq ] = 1 or 2. So |Fq (a)| = q or q 2 . In both cases we have aq −1 = 1. Consequently since (|G/T (G)|, p)) = 1 (any element g¯ of order p must be 1), we get by Watanabe’s theorem that S(V )G = F [u, c21 ]G/T (G) is Gorenstein. 2
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Remark 3.6. (i) A similar result with G ⊆ SL(n, F )), T (G) = SL(n, Fq ), and n > 2 does not hold in general. This is another source for counterexamples. (ii) In fact S(V )G is a hypersurface in Proposition 3.5. Indeed the argument showed 0 aq−1 , 2 on U and aq −1 = that each g ∈ G is represented by the matrix g¯|U = 2 0, aq −q 2 aq−1 aq −q = 1. Since aq−1 = bq−1 = cq−1 = dq−1 , if they are not zero, this presentation is independent of the particular choice of a. This shows that ¯ g|U |g ∈ G = G/T (G) acts
−1 as a subgroup of the unique cyclic subgroup of order qq−1 = q + 1 of F × . Let g ∈ G ξ, 0 where g¯|U with ξ k = 1. Then S(V )G = is a generator of G/T (G). So g¯|U = 0, ξ −1 F [u, c21 ]¯g = F [uk , ck21, , uc21 ], which is clearly a simple singularity hypersurface in F 3 of type Ak−1 . 2
Our next result is also connected to the irreducible primitive case analysis. It actually shows that T (G) is its own normalizer in SL(2, F ) in this case. This may already be known. Our proof establishes this by using invariant theory. Proposition 3.7. Suppose G ⊂ SL(2, F ) is a finite group with T (G) ∼ = SL(2, F5 ), T (G) ⊆ SL(2, F9 ), and F9 ⊆ F (here p = 3). Let V = F 2 be the natural SL(2, F )-module. Then G = T (G) and consequently S(V )G is a polynomial ring. 1, 0 1, , Proof. The group T (G) can be generated by the elements 2 ξ , 1 0, where ξ is the generator of the multiplicative group F× 9 . By [14, p. 79]
1 in SL(2, F9 ) 1
S(V )T (G) = F [f10 , f12 ], 10 2 6 6 2 10 12 where f10 = x91 x2 − x1 x92 , f12 = x12 1 + x 1 x 2 − x1 x 2 + x 1 x 2 − x2
(this was obtained by using INVAR). a, b 1, 1 −1 1 − ac, a2 ∈ T (G) ⊆ Let g = ∈ G. Then g g = −c2 , 1 + ac c, d 0, 1 −b2 ξ 2 1, 0 −1 1 + bdξ 2 , g ∈ T (G). This will give SL(2, F9 ). Also g 2 = 2 2 ξ , 1 d ξ , 1 − bdξ 2 2 a2 , c , ac, b2 ,d2 , bd ∈ F9 . This together with ad − bc = 1 leads, as in Proposition 3.5, to a, 0 1, β 1, β g= , where h = ∈ GL(2, F9 ). Also by 1 = det(g) = a2 det(h), 0, a γ, δ γ, δ we have a−2 = det(h) ∈ F9 . We also have, as in Proposition 3.5, that h(f10 ) = det(h)f10 . We get that g(f10 ) = a10 h(f10 ) = a10 a−2 f10 = a8 f10 . Now F [f10 , f12 ] = S(V )T (G) is g-stable, implying by degree reasoning that g(f12 ) = εf12 for some ε ∈ F . Since g(f12 ) = a12 h(f12 ), we shall next consider h(f12 ). We have h(f12 ) = (x1 + βx2 )12 + (x1 + βx2 )10 (γx1 + δx2 )2 − (x1 + βx2 )6 (γx1 + δx2 )6 + + (x1 + βx2 )2 (γx1 + δx2 )10 − (γx1 + δx2 )12 =
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= (x91 + β 9 x92 )(x31 + β 3 x32 ) + (x91 + β 9 x92 ))(x1 + βx2 )(γx1 + δx2 )2 − − (x1 + βx2 )6 (γx1 + δx2 )6 + (x1 + βx2 )2 (γx1 + δx2 )(γ 9 x91 + δ 9 x92 ) − − (γ 9 x91 + δ 9 x92 )(γ 3 x31 + δ 3 x32 ). Therefore the only contribution to the monomial x61 x62 will come from the term −(x1 + βx2 )6 (γx1 + δx2 )6 = −(x31 + β 3 x32 )2 (γ 3 x31 + δ 3 x32 )2 . So the contribution is exactly −x61 x62 (δ 3 − β 3 γ 3 )2 = −x61 x62 (δ − βγ)6 = −x61 x62 (det(h))6 = −(a−2 )6 x61 x62 = −a−12 x61 x62 . Consequently −ε = −a12 a−12 = −1. This shows that ε = 1 and g(f12 ) = f12 . 8 Also the coefficient of x12 1 in the right hand side of h(f12 ) (using γ = 1) is (1 + γ 2 − γ 6 + γ 10 − γ 12 ) = 1 + γ 2 − γ 6 + γ 2 − γ 4 = 1 − γ 2 (1 + γ 2 + γ 4 ) =1−
γ8 − γ2 γ 2 ((γ 2 )3 − 1) = 1 − = −1 γ2 − 1 γ2 − 1
12 12 and therefore the coefficient of x12 = −1. Since 1 in g(f12 ) (= f12 ) is a (−1). Hence a 2 16 4 a ∈ F9 we have a = 1 implying that a = −1. Consequently g(f10 ) = a8 f10 = f10 . If γ = 0 then a12 = 1. So a4 = 1 and again a8 = 1. Thus g acts as the identity on F [f10 , f12 ] = S(V )T (G) . Therefore by Galois theory g, T (G) = T (G) and g ∈ T (G). 2
Recall (e.g. [6, p. 261]) that V is an imprimitive G-module if V = Ind G H (W ), where W is an H-module for some proper subgroup H of G. V is a primitive G-module if V is not imprimitive. The following is our first positive result. Theorem 3.8. Let G ⊆ SL(2, F ) be a finite group and V = F 2 the natural SL(2, F )-module. Suppose that V is an irreducible primitive T (G)-module. Then S(V )G is a hypersurface in F 3 (or a polynomial ring). Proof. Clearly V is also an irreducible G-module. Therefore by Lemma 3.4, there exist an irreducible G-module V0 and a subfield k ⊂ F with [k : Fp ] < ∞ such that G ⊂ SL(2, k) and V = V0 ⊗k F . We next show that V0 is also an irreducible T (G)-module. Indeed by [5, 49.2], V0 is a completely reducible T (G)-module. If V0 = W1 ⊕ W2 , where Wi is a proper T (G)-module for i = 1, 2, then V = V0 ⊗k F = (W1 ⊗k F ) ⊕(W2 ⊗k F ), contradicting the irreducibility of V as a T (G)-module. Similarly V0 is a primitive T (G)-module. Indeed if V0 = Ind G H (W ) ≡ W ⊗kH kG, for some proper subgroup H of T (G), then V = V0 ⊗k F = (W ⊗kH kG) ⊗k F = (W ⊗k F ) ⊗FH FG = Ind G H (W ⊗k F ),
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in contradiction to the primitivity of V as a T (G)-module. Now by applying [20] or [11], there exists an element a ∈ GL(V0 , k) = GL(2, k) such that one of the following possibilities holds: aT (G)a−1 ∈ {SL(2, Fq ), Sp(2, Fq ), SU (2, Fq )} or aT (G)a−1 is isomorphic to SL(2, F5 ) regarded as a subgroup of SL(2, F9 ) ⊆ SL(2, k). Since SL(2, Fq ) = Sp(2, Fq ) = SU (2, Fq ) (e.g. [10, Proposition 3.1, Corollary 11.29]) we need to consider first aT (G)a−1 = SL(2, Fq ). Let G1 ≡ aGa−1 . Then T (G1 ) = aT (G)a−1 = SL(2, Fq ) and G1 ⊆ SL(2, k). Therefore by Proposition 3.5 S(V0 )G1 is Gorenstein. Hence S(V0 )G is Gorenstein and by Lemma 3.2 S(V )G is Gorenstein, and therefore it is a hypersurface in F 3 by Remark 3.6(ii). If aT (G)a−1 ∼ = SL(2, 5) regarded −1 as a subgroup of SL(2, F9 ) ⊆ SL(2, k), then in G1 ≡ aGa we have T (G1 ) = aT (G)a−1 and by Proposition 3.7 G1 = T (G1 ). Hence by [14, Theorem 8.1] S(V0 )G1 is a polynomial ring implying that S(V0 )G is a polynomial ring. This shows by [14, p. 62] that S(V )G is a polynomial ring. 2 Our next objective is to handle finite groups G ⊂ SL(2, F ) with T (G) acting irreducibly and imprimitively on the natural SL(2, F )-module F 2 = V . Since it is well known (e.g. [20]) that T (G) is monomial in this case, we restrict our attention to this subfamily. Recall that H ⊂ GL(V ) is called monomial if V has a basis with respect to which the matrix of each element of H has exactly one non-zero entry in each row and column. The next easy result disposes of the charF = p = 2 cases. Lemma 3.9. Let H ⊂ SL(2, F ) be a monomial group generated by transvections and p = 2. Then H = {1}. Proof. Let {y1 , y2 } be a basis of V =F 2 with respect 0, λ, 0 g ∈ H be a transvection. Then g = or δ, 0, δ λ − 1, 0 since g − IV = and so either λ = 1 or 0, δ−1
towhich H is monomial. Let λ . The first case can’t hold 0
δ = 1 but since g ∈ SL(2, F ), 0, λ −1 then δ = −λ−1 and δ = λ and we get, in both cases, that g = 1. If g = δ, 0 −1, λ , a matrix whose determinant is 2 and is non-zero since p = 2. g − IV = −λ−1 , −1 So rank(g − IV ) > 1, which is a contradiction. 2 This limits the possibilities of T (G) to the p = 2 case. Proposition 3.10. Let G ⊂ SL(2, F ) be a finite group where charF = 2. Suppose T (G) is an irreducible monomial subgroup. Then G = T (G).
Proof. Let {x1 , x2 } be the basis of V = F 2 with respect to which T (G) is irreducible and monomial. Let {g1 , . . . , gn } be the set of transvections generating T (G).
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∗, The T (G)-irreducibility of V forces n ≥ 2. So either gi = 0, 0, possibility leads, as in Lemma 3.9, to gi = 1. So gi = −1 α i , a, b g= ∈ G. Since g normalizes T (G), we have for each c, d
a, b c, d
0, αi−1 ,
αi 0
d, c,
b bdαi−1 + acαi , = a d2 αi−1 + c2 αi ,
0, ∗ 0 . The first , or ∗, 0 ∗ αi , for i = 1, . . . , n. Let 0 i = 1, . . . , n:
b2 αi−1 + a2 αi dbαi−1 + caαi
∈ T (G).
∗, 0 0, ∗ or . So this matrix must have the form 0, ∗ ∗ 0 In the first case b2 αi−1 = a2 αi , d2 αi−1 = c2 αi . So using p = 2 we get b = aαi , d = cαi , in contradiction to ad − bc = 1. In the second case bd = acαi2 , for i = 1, . . . , n. Since α1 = α2 (and therefore α12 = α22) 0, b . we get that bd = ac = 0. If a = 0, then ad − bc = 1 implies d = 0 and g = c, 0 Also g ∈ SL(2, F ) so c = b−1 . Consequently g(x1 + bx2 ) = bx2 + bcx1 = x1 + bx2 , as well as g(x1) = bx2 = x1 + (x1 + bx2 ), so the matrix of g with respect to {x1 , x1 + bx2 } is 1, 1 . That is g is a transvection and hence g ∈ T (G). 0, 1 c = 0 and ad − bc = 1 imply b = 0 and g = therefore Suppose that a = 0. Then 0, aαi a, 0 0, αi = ∈ G, for each i. However as in the . Also g −1 αi−1 , 0 a−1 α , 0 0, a−1 i 0, aαi is a transvection and therefore previous paragraph one observes that a−1 αi−1 , 0 0, αi 0, aαi ∈ T (G), as needed. 2 it is in T (G). Consequently g = a−1 αi−1 , 0 αi−1 , 0 We can now state the second possibility for the irreducible T (G)-action. Theorem 3.11. Let G ⊂ SL(2, F ) be a finite group with T (G) acting imprimitively and irreducibly on the natural module V = F 2 . Then S(V )G is a polynomial ring. Proof. We have by (e.g. [20]) that T (G) is monomial. Alternatively, by assumption and [5, Cor. (50.4)] T (G) acts transitively on V . That is V = M1 ⊕· · ·⊕Mk , T (G) acts by permutations on {M1 , . . . , Mk } and Orbit(M1 ) = {M1 , . . . , Mk }. Clearly if gMi = Mi , for i = 1, . . . , k, for each transvection g in T (G), then T (G)Mi = Mi , for i = 1, . . . , k, in violation of the transitivity. So let g be a transvection in T (G) with gMi = Mj with i = j. Say i = 1. Suppose dim F M1 > 1 and {m1 , . . . , mr } is a basis of M1 with r > 1. Then g(m1 ) − m1 , g(m2 ) − m2 ∈ (g − IV )(V ). Moreover they are non-zero and are linearly independent as can easily be seen. This violates the pseudo-reflection property of g. Consequently by Lemma 3.9 p = 2 and by Proposition 3.10 we get T (G) = G. Therefore by [16, Theorem 2.4] S(V )G is a polynomial ring. 2
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Our next two results will be used in the proof of the reducible T (G)-module case, in Theorem 3.1. Lemma 3.12. Let charF = p and ξ ∈ F a primitive m-th root of unity, where (m, p) = 1. Let W ⊂ F be an n-dimensional Fp -subspace of F with ξW ⊆ W . Then m|pn − 1. Proof. Let Cm be the cyclic group of order m and g ∈ Cm a generator. The map φ : Cm → End Fp W , where φ(g j ) ≡ ξ j IW , for each j, makes W into a Cm -module. (m, p) = 1 grants that W is a completely reducible Fp Cm -module, where the latter denotes the group algebra of Cm over Fp . Thus W = W1 ⊕ · · · ⊕ Wr , where Wi is an irreducible Fp Cm -module, for i = 1, . . . , r. Set di ≡ dim Fp Wi . The result will follow once we show that m|pdi − 1 for each i. Let φi (g) ≡ φ(g)|Wi that is φi : Cm → GL(Wi ), for each i. If φi is not injective this will give an integer k = m, k|m with ξ k w = w, for each w ∈ Wi . This will contradict the m-th primitivity of ξ. Therefore φi : Cm → GL(Wi ) makes Wi into a faithful irreducible Cm -module, for i = 1, . . . , r. Let Mi ≡ Ann Fp Cm Wi = {x ∈ Fp Cm |xWi = 0}. Then Mi is a maximal ideal in Fp Cm , and Fp Cm /Mi is a finite field extension of Fp . Also by commutativity of Fp Cm we have Fp Cm /Mi ∼ = Wi as Fp Cm -module and consequently ∼ End Fp Cm (Wi ) = Fp Cm /Mi , as algebras, for each i. By the commutativity of Cm we also have φi (g) ∈ End Fp Cm (Wi ). Since φi is an injection, it follows that Cm is isomorphic to a subgroup of the multiplicative group of Fp Cm /Mi . Consequently since |Fp Cm /Mi | = |Wi | = pdi , we get m|pdi − 1, for each i = 1, . . . , r. 2 Lemma 3.13. Let V = F 2 = F x1 + F x2 be the natural SL(2, F ) module and σi = 1, αi ∈ SL(2, F ), a transvection, for i = 1, . . . , n. Assume that {α1 , . . . , αn } are lin0, 1 early independent over Fp . Then S(V )σ1 ,...,σn = F [an , x2 ], where an is a homogeneous n polynomial (in x1 , x2 ) of degree pn , having xp1 as one of its monomials. Proof. The proof is carried out by induction. The case n = 1 appears already in Lemma 2.2. Indeed σ1 (x1 ) = x1 +α1 x2 , σ1 (x2 ) = x2 imply that a1 = xp1 −x1 (α1 x2 )p−1 ∈ S(V )σ1 . Now |σ1 | = p together with a Galois field reasoning imply F [a1 , x2 ] = S(V )σ1 . Suppose by induction that S(V )σ1 ,...,σn−1 = F [an−1 , x2 ], where deg(an−1 ) = pn−1 , n−1 and xp1 is one of the monomials of an−1 . Now σn σi = σi σn for i = 1, . . . , n − 1 imply that F [an−1 , x2 ] is σn -stable. Therefore, since σn preserve degrees, σn (an−1 ) = n−1 αan−1 + βxp2 , α, β ∈ F . Now σn (x1 ) = x1 + αn x2 , σn (x2 ) = x2 imply, by plugging n−1 into σn (an−1 ), that xp1 is one of the monomials appearing in σn (an−1 ) and therefore α = 1. We next show that β = 0. To see this observe that αn ∈ / Span Fp {α1 , . . . , αn−1 } implies σn ∈ / σ1 , . . . , σn−1 . Therefore by Galois considerations F [an−1 , x2 ] = S(V )σ1 ,...,σn−1 S(V )σ1 ,...,σn . Thus σn (an−1 ) = an−1 , as claimed.
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n−1
Finally set an ≡ apn−1 − an−1 (βxp2 )p−1 . As in the first paragraph σn (an ) = an , so an ∈ S(V )σ1 ,...,σn , F [an , x2 ] is a polynomial ring and since |σn | = p, [F (an−1 , x2 ) : F (an , x2 )] = p. Thus F (an , x2 ) = Q(S(V )σ1 ,...,σn ) and since S(V )σ1 ,...,σn is inten gral over F [an , x2 ] we get F [an , x2 ] = S(V )σ1 ,...,σn . Clearly xp1 appears as one of the monomials of an . Alternatively, as pointed out by the referee, since (degx2 )(degan ) = pn = |σ1 , . . . , σn | and an , x2 are algebraically independent, it follows from [12, Proposition 16(b)] or [7, 3.7.5] that F [an , x2 ] = S(V )σ1 ,...,σn . 2 We finally arrive at: The proof of Theorem 3.1. The case T (G) = {1G } follows from [2, Theorem B]. Therefore we may assume that T (G) = {1G }. Let V = F 2 be the natural SL(2, F )-module. If V is an irreducible T (G)-module then Theorems 3.8 and 3.11 imply that S(V )G is Gorenstein and even a hypersurface in F 3 . We may therefore assume that V is a reducible T (G)-module. Consequently there exists a basis {x1 , x2 } of V with respect to which F, F T (G) ⊆ . Since every transvection stabilizes a one dimensional subspace of V 0, F 1, F , implying it must be F x2 . Therefore every transvection of T (G) must be in 0, 1 1, F . Consequently T (G) is elementary abelian. So that T (G) ⊆ 0, 1
1, T (G) = σ1 × · · · × σn , where σi ≡ 0,
αi , for i = 1, . . . , n, 1
and {α1 , . . . , αn } are linearly independent over Fp . Set W = Fp α1 + · · · + Fp αn , so dim Fp W = n. Consequently |T (G)| = pn . a, b 1, F −1 ∈ G. Then gσi g ∈ T (G) ⊆ forces Let g = c, d 0, 1
a, b c, d
1, 0,
αi 1
d, −c,
−b 1 − acαi , = a −c2 αi ,
λ, −1 0, λ
a2 αi 1 + acαi
1, ∈ 0,
F 1
with λ, in F . Now since gσi g −1 = and therefore c = 0 and g = 1, λ2 αi ∈ T (G), for i = 1, . . . , n, we get that λ2 αi ∈ W for i = 1, . . . , n. Con0, 1 sequently by Lemma 3.12 |λ2 | divides pn − 1. We next observe that (|G|/|T (G)|, p) = 1 and that G/T (G) is isomorphic to a (cyclic) n p subgroup cyclic group H = {α ∈ F |α2(p −1) p of the = 1}. Indeed if g ∈ T (G) then 1, F , λ 1, ∈ , so λ = 1 and g = gp = ∈ T (G). This shows that 0, λ−p 0, 1 0, 1 λ, λ1 , 1 , . (|G/T (G)|, p) = 1. Say g = g1 h with h ∈ T (G), where g = g = 1 0, λ−1 0, λ−1 1
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−1 −1 λ−1 1, F 1 λ, λ1 − 1 λ = h ∈ T (G) ⊆ and λ = λ1 . Therefore 0, λ1 λ−1 0, 1 the map φ : G/T (G) → H, which is defined by φ(g(T (G)) = λ, is a well defined injective homomorphism with its image in H since |λ2 | divides pn − 1. λ, . By Lemma 3.13 we Let g¯ ≡ gT (G) be a generator of G/T (G) with g = 0, λ−1 have S(V )T (G) = S(V )σ1 ,...,σn = F [an , x2 ]. Now since gT (G)g −1 = T (G), F [an , x2 ] n is g-stable and since g preserve degrees we get g(an ) = αan + βxp2 with α, β ∈ F . n n Since an has xp1 as one of its monomials and since g(x1 ) = λx1 + x2 we get that xp1 n n n p p p appears with coefficient λ in g(an ), that is α = λ . If λ = 1 then λ = 1 and then 1, g= ∈ T (G), which was excluded. 0, 1 n n Suppose firstly that λ = −1, then λp = (−1)p = −1 (this also holds if p = 2 since Then g1−1 g =
n
λ = 1). Consequently λ2p = 1. Let b = an + n
pn
g(b) = λ an +
n βxp2
n
n λp β xp , λ2pn −1 2
then n
n n n n λp β λp β pn λ−p xp2 = λp (an + 2pn x ) = λp b. + 2pn λ −1 λ −1 2
Clearly L = F b + F x2 is a 2-dimensional vector space over F and S(L) = F [b, x2 ] = F T . With respect to the basis {b, x2 } of L, g|L is represented by the matrix [apnn, x2 ] = λ , 0 2 n . Let |T|G| (G)| = |λ| ≡ k, then |g|L | = |λ| = k and |λ | divides (p − 1) by 0, λ−1 Lemma 3.12, implying that k|2(pn − 1). n If p = 2 then (p, 2(pn − 1)) = 1, hence λip = 1 for 1 ≤ i ≤ k forces i = k. This shows that g|L has no diagonal pseudo-reflections, so since |g|L | = k and (k, p) = 1 we get by Watanabe’s theorem that S(V )G = S(L)g|L is Gorenstein if and only if k|(pn − 1), that is |T|G| (G)| |(|T (G)| − 1). This settles item (ii) for λ = −1 and p = 2. If p = 2 this forces λ = −1. Now if k is even k = 2m, then λ2m = 1, implying, since p = 2, that λm = 1, in contradiction to |λ| = k. Hence (k, 2) = 1. So k|(2n − 1), that is |T|G| (G)| |(|T (G)| − 1). Equivalently detg|L = 1. Therefore by Watanabe’s theorem S(V )G = S(V )g|L is always Gorenstein in case p = 2. This settles also item (iv)(4). G Suppose next that λ = −1. Consequently p = 2. We shall show that S(V ) is Goren−1, stein in this case. Now g = implies that g 2 ∈ T (G). Hence |T|G| (G)| = 2. 0, −1 Observe that |T (G)| = pn is odd, so |T|G| (G)| |(|T (G)| − 1) also holds in this case.
We have S(V )G = S(V )T (G),g = (S(V )T (G) )¯g = T ¯g where g¯ = gT (G). Now since |G/T (G)| = 2 and p = 2 then by H. Cartan’s theorem [15, Lemma 5.3] T ¯g ∼ = S(m/m2 )¯g , where m = (an , x2 ) is the unique graded maximal ideal of T = F [a , x ]. n 2 −1, 0 , it follows Clearly g¯ acts linearly on m/m2 and is represented by the matrix 0, −1 by Watanabe’s theorem that S(m/m2 )¯g is Gorenstein. In fact it is generated by a ¯2n , x ¯22 , a ¯n x ¯2 , where a ¯ n = an + m2 , x ¯2 = x2 + m2 , showing that S(m/m2 )¯g is a simple singularity hypersurface of type A. This settles item (ii).
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Since |λ2 | divides (pn − 1), the above reasoning shows that S(V )g ,T (G) = S(L)g|L is Gorenstein. So by taking D ≡ g 2 , T (G) item (iii) is verified. Consequently if |G| is odd, then G = g 2 , T (G) and S(V )G is therefore Gorenstein, so item (iv)(2) is verified. Given the assumption of item (iv)(3) then λ ∈ / Fp (λ2 ) implies [Fp (λ) : Fp (λ2 )] = 2 in contradiction to [F : Fp ] = s being odd. Thus λ ∈ Fp (λ2 ) and since λ2 W ⊆ W we have λW ⊆ W , that is k = |λ| divides (pn − 1), by Lemma 3.12. Now by the previous reasoning det(g|L ) = 1 and S(V )G = S(L)g|L is Gorenstein and item (iv)(3) holds. Our final task is to verify that whenever S(V )G is Gorenstein then pn it is a hyper0 λ , g|L 3 G with surface in F . If λ = −1 then S(V ) = S(L) and g|L = 0, λ−1 n det ( g|L ) = 1. Therefore λp = λ. Consequently since S(L) = F [b, x2 ] we conclude that S(L)g|L = F [bk , xk2 , bx2 ] which is clearly a hypersurface, and in fact a simple singularity (of type Ak−1 ). The case λ = −1 was dealt with before and the result therefore follows. 2 2
2
Remark 3.14. One can use the proof of the reducible case of Theorem 3.1 to give an example, for each p ≥ 5, with the following properties: |G| = 2(p − 1)p, G ⊂ SL(2, F ), F ⊇ Fp2 , S(V )G is Cohen–Macaulay but not Gorenstein, where V ≡ F 2 and G ∩ Z(GL(V )) = {1}. Acknowledgments Thanks to Yuval Ginosar for stimulating conversations. References [1] D.J. Benson, Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Note Ser., vol. 190, Cambridge University Press, Cambridge, 1993. [2] A. Braun, On the Gorenstein property for modular invariants, J. Algebra 345 (2011) 81–99. [3] A. Broer, The direct summand property in modular invariant theory, Transform. Groups 10 (1) (2005) 5–27. [4] W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math., Cambridge University Press, 1998. [5] C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley Classics Library, 1988. [6] C.W. Curtis, I. Reiner, Methods of Representation Theory, vol. I, John Wiley & Sons, New York, 1981. [7] H. Derksen, G. Kemper, Computational invariant theory, in: Invariant Theory and Algebraic Transformation Groups, I, in: Encyclopaedia Math. Sci., vol. 130, Springer-Verlag, Berlin, 2002. [8] F. DeMeyer, E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Math., vol. 181, Springer-Verlag, 1971. [9] P. Fleischmann, C. Woodcock, Relative invariants, ideal classes and quasi-canonical modules of modular rings of invariants, J. Algebra 348 (2011) 110–134. [10] L.C. Grove, Classical Groups and Geometric Algebra, Grad. Stud. Math., vol. 39, A.M.S., 2002. [11] W. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979) 347–379. [12] G. Kemper, Calculating invariant rings of finite groups over arbitrary fields, J. Symbolic Comput. 21 (1996) 351–366.
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[13] G. Kemper, E. Körding, G. Malle, B.H. Matzat, D. Vogel, G. Wiese, A database of invariant rings, Exp. Math. 10 (2001) 537–542. [14] G. Kemper, G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transform. Groups 2 (1) (1997) 57–89. [15] G.J. Leuschke, R.A. Wiegand, Cohen–Macaulay Representations, Math. Surveys Monogr., vol. 181, American Math. Society, 2012. [16] H. Nakajima, Invariants of finite groups generated by pseudoreflections in positive characteristic, Tsukuba J. Math. 3 (1979) 109–122. [17] H.D. Neusel, L. Smith, Invariant Theory of Finite Groups, Math. Surveys Monogr., vol. 94, American Mathematical Society, 2002. [18] K. Watanabe, Certain invariant subrings are Gorenstein I, Osaka J. Math. II (1974) 1–8. [19] K. Watanabe, Certain invariant subrings are Gorenstein II, Osaka J. Math. II (1974) 379–388. [20] A.E. Zalesskii, V.N. Serežkin, Linear groups generated by transvections, Math. USSR, Izv. 10 (1976) 25–46.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
The Gorenstein property for modular binary forms invariants Amiram Braun Dept. of Mathematics, University of Haifa, Haifa, 31905 Israel
a r t i c l e
i n f o
Article history: Received 29 May 2014 Available online xxxx Communicated by Luchezar L. Avramov Keywords: Gorenstein Modular invariants
a b s t r a c t Let G ⊆ SL(2, F ) be a finite group, V = F 2 the natural SL(2, F )-module, and charF = p > 0. Let S(V ) be the symmetric algebra of V and S(V )G the ring of G-invariants. We provide examples of groups G, where S(V )G is Cohen– Macaulay, but is not Gorenstein. This refutes a natural conjecture due to Kemper, Körding, Malle, Matzat, Vogel and Wiese. Let T (G) denote the subgroup generated by all transvections of G. We show that S(V )G is Gorenstein if and only if one of the following cases holds: (1) T (G) = {1G }, (2) V is an irreducible T (G)-module, (3) V is a reducible T (G)-module |T (G)|(|T (G)| − 1).
and
|G|
divides
© 2015 Elsevier Inc. All rights reserved.
1. Introduction Let G ⊂ GL(V ) be a finite group, where V is a finite dimensional vector space over a field F with charF = p ≥ 0. The following is a classical result of K. Watanabe. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2015.11.039 0021-8693/© 2015 Elsevier Inc. All rights reserved.
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Theorem. (See [4, Theorems 6.4.10], [18,19].) Suppose that (|G|, p) = 1 and G contains no pseudo-reflection. Then the following are equivalent: (1) G ⊂ SL(V ), (2) S(V )G is a Gorenstein ring. In the modular case, that is if p divides |G|, S(V )G may fail to be Cohen–Macaulay, let alone Gorenstein, thus a suitable generalization of the above to the modular case should require the Cohen–Macaulay property. In [13, Conjecture 5], Kemper et al. conjectured that one of the above implications is valid (assuming the Cohen–Macaulay property), in the modular case as well. Explicitly: Conjecture. (See [13, Conjecture 5].) Suppose G ⊂ SL(V ). Then S(V )G is Gorenstein if (and only if) it is Cohen–Macaulay. This conjecture was verified in [13] on all 1916 examples of the database (which satisfy its hypothesis). A special case of the conjecture, assuming that G contains no transvections, was confirmed in [2, Theorem B] using non-commutative methods. A commutative proof for this special case was recently given by Fleischmann–Woodcock [9]. A statement of this special case also appears in [3, Corollary 7(vi)]. However, in my opinion, the proof of [3, Lemma 5], on which it crucially depends, is incorrect. A remarkable generalization of this special case, is proved in [9], assuming that S(V )W (G) is a polynomial ring and S(V )G is Cohen–Macaulay. Then S(V )G is Gorenstein if and only if G/W (G) ⊆ SL(m/m2 ), where W (G) is the subgroup generated by all pseudo-reflections and m is the unique homogeneous maximal ideal of S(V )W (G) . Our first objective here is to provide a binary form counterexample to this conjecture. We have: Example A. Let V = F 2 be the natural SL(2, F )-module where charF = p with p ≡ 3 (mod 4). Assume that F contains a primitive 4-th root of unity ξ. Let G ⊂ SL(V ) be given by:
ξ, G = τ, σ, where τ = 0,
0 ξ −1
1, , σ= 0,
1 . 1
Then S(V )G is Cohen–Macaulay but it is not Gorenstein. Other examples for arbitrary p can be given by using item (2) of the next result. On the positive side we have the following. Let T (G) denote the subgroup generated by all transvections of G.
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Theorem B. Let F be a field with charF = p > 0 and G ⊂ SL(2, F ) a finite subgroup where p divides |G|. Let V = F 2 be the natural SL(2, F )-module. Then S(V )G is Gorenstein if and only if one of the following cases holds: (1) V is an irreducible T (G)-module, (2) |G| divides |T (G)|(|T (G)| − 1), and V is a reducible T (G)-module. Moreover in each of these cases S(V )G is either a simple singularity hypersurface (of type A) in F 3 , or non-singular. Note that when (p, |G|) = 1 and G ⊆ SL(V ), then S(V )G is Gorenstein by Watanabe’s theorem. Also, for G ⊆ GL(2, F ) the restriction (p, |G|) = 1 holds if and only if T (G) = {1}. As a consequence we get the following: Corollary C. Let G ⊂ SL(2, F ) be a finite group and V = F 2 the natural SL(2, F )-module. Then S(V )G is Gorenstein in the following cases: (1) (2) (3) (4)
G is odd, p = 2, F = Fq , a Galois field, with q = ps and s is odd, |T (G)| = 1.
The results of the present paper were conceived by non-commutative methods. However we merely use commutative methods and only apply Watanabe’s non-modular characterization for a Gorenstein ring of invariants. 2. Counterexamples Let F be a field with charF = p > 0. We shall exhibit here finite groups G, having the following properties: (1) G ⊂ SL(V ), (2) S(V )G is Cohen–Macaulay, (3) S(V )G is not Gorenstein. This settles [13, Conjecture 5] in the negative. Example 2.1. Assume that p ≡ 3 (mod 4) and that F contains a primitive 4-th root of unity ξ. Thus ξ 2 = −1 = 1 (since p is necessarily odd). Let V = F 2 be the natural SL(2, F )-module with basis {x1 , x2 }. ξ, 0 1, 1 . and σ ≡ Let τ ≡ 0, ξ −1 0, 1
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Thus τ (x1 ) = ξx1 , τ (x2 ) = ξ −1 x2 , σ(x1 ) = x1 + x2 , σ(x2 ) = x2 . We have: τ στ
−1
ξ, = 0,
0 ξ −1
1, 0,
1 1
ξ −1 , 0,
1, −1 0 1, ξ 2 = = σ p−1 . = 0, 1 0, 1 ξ
Let G = τ, σ be the subgroup of SL(2, F ) generated by {σ, τ }. Then σ is a normal subgroup of G, |σ| = p and |G| = 4p. We extend the action of G on V into an action by automorphisms on the polynomial ring S(V ) = F [x1 , x2 ]. The next result is standard and its proof is included for completeness sake. Lemma 2.2. T ≡ F [x1 , x2 ]σ = F [a, x2 ], where a = xp1 − x1 xp−1 . 2 Proof. σ(x1 ) = x1 + x2 implies that: Norm σ (x1 ) = x1 (x1 + x2 )(x1 + 2x2 ) . . . (x1 + (p − 1)x2 ) = xp1 − x1 xp−1 = a. 2 Now by Galois theory: [Q(F [x1 , x2 ]) : Q(T )] = |σ| = p. Also [Q(F [x1 , x2 ]) : Q(F [a, x2 ])] = dega = p. Consequently Q(T ) = Q(F [a, x])). Since T is integral over F [a, x2 ] and the latter is normal (being a polynomial ring), it now follows that T = F [a, x2 ]. 2 Now clearly S(V )G is a normal domain of Krull dimension 2. In particular S(V )G is Cohen–Macaulay. We shall next compute the action of τ on T : τ (a) = (τ (x1 ))p − τ (x1 )(τ (x2 ))p−1 = (ξx1 )p − (ξx1 )(ξ −1 x2 )p−1 = ξ p xp1 − ξ 2−p x1 xp−1 . 2 Now p ≡ 3 (mod 4) implies that p −(2 −p) = 2p −2 ≡ 0 (mod 4) and therefore ξ p = ξ 2−p . Consequently τ (a) = ξ p a and τ (x2 ) = ξ −1 x2 . Consider U ≡ F a + F x2 , a two dimensional F -subspace of T . Clearly S(U ) = T = F [a, x2 ] is a polynomial ring in the two variables {a, x2 }. The action of τ on T can therefore be regarded as the natural extension p of the linear action of τ|U on U , to S(U ). ξ , 0 Therefore T τ = S(U )τ|U . Now τ|U = is the matrix representing τ|U with 0, ξ −1 respect to the basis {a, x2 }. Clearly τ|U contains no pseudo-reflections, |τ|U | = 4 and (p, 4) = 1. Also det(τ|U ) = ξ p−1 = −1 (using p ≡ 3 (mod 4)). Consequently by Watanabe’s theorem S(U )τ|U = T τ is not Gorenstein. Since S(V )G = (S(V )σ )τ = T τ , the result follows.
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Remark 2.3. (1) Observe that if p is an odd prime then Fp2 , the Galois field of p2 elements, contains 2 a primitive 4-th root of unity since F× p2 is a cyclic group of order p − 1. (2) The smallest example one obtains by the previous method is for p = 3, where |G| = 4 · 3 = 12 and F = F32 . 3. Finite subgroups of SL(2, F ) Our main objective here is to find out the extent of validity of the Kemper et al. conjecture, for finite subgroups of SL(2, F ), where F is an arbitrary field of positive characteristic p. Qualitatively speaking, most of the groups do satisfy the conjecture. Our argument also shows why and how counterexamples appear. For SL(n, F ) with n > 2 such an analysis does not exist (as yet), due partially to the difficulty in analyzing irreducible primitive actions of T (G), the G-subgroup generated by all transvections. The following theorem is our main result. Theorem 3.1. Let F be an arbitrary field with charF = p > 0. Let G ⊆ SL(2, F ) be a finite subgroup and V = F 2 , the natural SL(2, F )-module. Then (i) S(V )G is Gorenstein whenever V is an irreducible T (G)-module. (ii) If V is a reducible T (G)-module and T (G) = {1G } then S(V )G is Gorenstein if and only if |G| divides |T (G)|(|T (G)| − 1). (iii) There exists a subgroup D of G, D ⊇ T (G) with |G|/|D| ≤ 2 and S(V )D is Gorenstein. (iv) In particular S(V )G is Gorenstein in the following (not mutually disjoint) cases: (1) T (G) = {1G }, (2) G is an odd order group, (3) F = Fq , where q = ps and s is odd, (4) p = 2. (v) If p divides |G| then S(V )G is a hypersurface in F 3 whenever it is Gorenstein. Our proof will proceed in steps. We shall firstly establish some reductions which will be solely used in the irreducible primitive case. The cases where T (G) acts irreducibly on V will be handled separately. The reducible case will be handled in the main course of the proof. Our first reduction result can be obtained in several ways. Lemma 3.2. Let V0 be a k-finite dimensional G-module and F ⊃ k, a field extension. Set V ≡ V0 ⊗k F , and let G act trivially on F . Then the Gorenstein property of S(V0 )G ascends to S(V )G .
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Proof. Let w be the canonical S(V0 )G -module. By [4, Exercise 3.3.31] w ⊗k F is a canonical S(V0 )G ⊗k F -module. Now the Gorenstein property of S(V0 )G amounts to S(V0 )G ∼ =w as S(V0 )G -modules and consequently S(V0 )G ⊗k F is a canonical S(V0 )G ⊗k F -module. Now clearly S(V ) = S(V0 ) ⊗k F and consequently S(V )G = S(V0 )G ⊗k F . Hence S(V )G is its own canonical module. Since S(V )G is Cohen–Macaulay by [4, Theorem 2.1.10], the Gorenstein property of S(V )G now follows. 2 Remark 3.3. Such an ascend of the unique factorization property from S(V0)G to S(V )G is not true in general. Our second reduction result is well known. Failing to find a reference for it, we provide a proof. Lemma 3.4. Let G ⊂ GL(V ) be a finite group, where V is an irreducible F -finite dimensional G-module and charF = p > 0. Then there exists a finite subfield k ⊂ F , and a k-finite dimensional faithful irreducible G-module V0 , satisfying V = V0 ⊗k F . Proof. Denote by FG the group algebra of G over F and set Ann FG V ≡ {x ∈ FG|x.V = 0} = M. Then M is a two-sided maximal ideal in FG. Clearly Fp G ⊗Fp F = FG. Let N ≡ M ∩ Fp G = Ann Fp G V . Then the exactness of (−) ⊗Fp F shows that NFG = N ⊗Fp F ⊆ M and therefore FG/M = (Fp G/N )F . That is FG/M is generated by Fp G/N and F . Therefore FG/M is a central extension of Fp G/N , implying that Fp G/N is a simple ring and its center k ≡ Z(Fp G/N ) is contained in Z(FG/M ). This shows (by [8, Cor. 3.6, p. 54]) that FG/M = Fp G/N ⊗k S, where S is the centralizer of Fp G/N in FG/M . It follows from the previous equality that F = S = Z(FG/M ), that is Fp G/N ⊗k F = FG/M . Now [Fp G/N : Fp ] < ∞ grants that [k : Fp ] < ∞, and by Wederburn’s theorem for finite division algebras Fp G/N = Mn (k). Since V can be identified with a column of Mn (F ) = FG/M , then Mn (k) ⊗k F = Mn (F ) shows that V0 ⊗k F = V , for some irreducible Fp G/N = Mn (k)-module V0 . Clearly V0 is also a faithful irreducible G-module with G ⊂ GL(V0 ). 2 We now present a special case one needs to consider. Proposition 3.5. Suppose G ⊂ SL(2, F ) is a finite group with T (G) = SL(2, Fq ), where q = ps and Fq ⊆ F . Let V = F 2 be the natural SL(2, F )-module. Then S(V )G is Gorenstein. Proof. Fix a basis {x, y} of V . Then by [17, Ex. 1, p. 104] the two Dickson invariq2
q2
ants of S(V )GL(2,Fq ) are c20 = (xy q − yxq )q−1 and c21 = xyxyq −yx −yxq . Moreover by q q [1, Theorem 8.2.1] u = xy − yx , c21 are the two generators of S(V )SL(2,Fq ) . Let
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a, b d, ∈ G be arbitrary. Then gT (G)g −1 = T (G) implies, using g −1 = c, d −c, and 1 = detg = ad − bc: g=
1, g 0,
−b a
1 −1 1 − ac, a2 g = ∈ SL(2, Fq ), 1 −c2 , 1 + ac
as well as g
1, 1,
0 −1 1 + bd, g = d2 , 1
−b2 ∈ SL(2, Fq ). 1 − bd
Consequently ac, a2 , c2 , b2 , d2 , bd ∈ Fq . We firstly assume that a = 0. Then γ ≡ ac = aac2 ∈ Fq . If b = 0 then μ ≡ db = bd b2 ∈ Fq . 1 b Therefore 1 = ad − bc = abμ − baγ = ab(μ − γ). Hence a2 (μ−γ) = a ≡ β ∈ Fq , as well as δ ≡ ad = μb a = μβ ∈ Fq . If b = 0 then we can take β = 0 and 1 = ad implies that d ad δ ≡ a = a2 = a12 ∈ Fq . So we get that b = aβ, c = aγ, d = aδ, with β, γ, δ ∈ Fq and
a, 0 g= 0, a
1, β 1, β , with h ≡ ∈ GL(2, Fq ). γ, δ γ, δ
If a = 0 then 1 = ad − bc implies that b, c = 0 and we get a similar equality with c replacing a. 2 g(c21 ) = (aIV )(h(c21 )) = (aIV )(c21 ) = aq −q c21 . Also if g1 = Consequently a1 , b1 ∈ GL(2, Fq ), then aq1 = a1 , bq1 = b1 , cq1 = c1 , dq1 = d1 imply that: c1 , d1 g1 (u) = g1 (x)g1 (y)q − g1 (y)g1 (x)q = (a1 x + b1 y)(c1 x + d1 y)q − (c1 x + d1 y)(a1 x + b1 y)q = (a1 d1 − b1 c1 )(xy q − yxq ) = det(g1 )u. Consequently, g(u) = (aIV )(h(u)) = det(h)adegu u = aq+1 a−2 u = aq−1 u, where the second to last equality follows from 1 = det(g) = a2 det(h). Let g ∈ G and let g¯ be its image in G/T (G). Now clearly S(V )G = (S(V )T (G) )G/T (G) = (S(V )SL(2,Fq ) )G/T (G) = (F [u, c21 ])G/T (G) . Observe that F [u, c21 ] = S(U ), where U = F u + F c21 , is a 2-dimensional F -vector space. Moreover g¯ acts linearly on U and extending this action to S(U ) induces the original action of g¯ on F c21 ]. Also the [u, matrix of g¯|U with respect to the basis {u, c21 } is given aq−1 , 0 2 by the matrix , implying that det g¯|U = aq −1 . Now a2 = det(h)−1 ∈ Fq 2 0, aq −q 2 and therefore [Fq (a) : Fq ] = 1 or 2. So |Fq (a)| = q or q 2 . In both cases we have aq −1 = 1. Consequently since (|G/T (G)|, p)) = 1 (any element g¯ of order p must be 1), we get by Watanabe’s theorem that S(V )G = F [u, c21 ]G/T (G) is Gorenstein. 2
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Remark 3.6. (i) A similar result with G ⊆ SL(n, F )), T (G) = SL(n, Fq ), and n > 2 does not hold in general. This is another source for counterexamples. (ii) In fact S(V )G is a hypersurface in Proposition 3.5. Indeed the argument showed 0 aq−1 , 2 on U and aq −1 = that each g ∈ G is represented by the matrix g¯|U = 2 0, aq −q 2 aq−1 aq −q = 1. Since aq−1 = bq−1 = cq−1 = dq−1 , if they are not zero, this presentation is independent of the particular choice of a. This shows that ¯ g|U |g ∈ G = G/T (G) acts
−1 as a subgroup of the unique cyclic subgroup of order qq−1 = q + 1 of F × . Let g ∈ G ξ, 0 where g¯|U with ξ k = 1. Then S(V )G = is a generator of G/T (G). So g¯|U = 0, ξ −1 F [u, c21 ]¯g = F [uk , ck21, , uc21 ], which is clearly a simple singularity hypersurface in F 3 of type Ak−1 . 2
Our next result is also connected to the irreducible primitive case analysis. It actually shows that T (G) is its own normalizer in SL(2, F ) in this case. This may already be known. Our proof establishes this by using invariant theory. Proposition 3.7. Suppose G ⊂ SL(2, F ) is a finite group with T (G) ∼ = SL(2, F5 ), T (G) ⊆ SL(2, F9 ), and F9 ⊆ F (here p = 3). Let V = F 2 be the natural SL(2, F )-module. Then G = T (G) and consequently S(V )G is a polynomial ring. 1, 0 1, , Proof. The group T (G) can be generated by the elements 2 ξ , 1 0, where ξ is the generator of the multiplicative group F× 9 . By [14, p. 79]
1 in SL(2, F9 ) 1
S(V )T (G) = F [f10 , f12 ], 10 2 6 6 2 10 12 where f10 = x91 x2 − x1 x92 , f12 = x12 1 + x 1 x 2 − x1 x 2 + x 1 x 2 − x2
(this was obtained by using INVAR). a, b 1, 1 −1 1 − ac, a2 ∈ T (G) ⊆ Let g = ∈ G. Then g g = −c2 , 1 + ac c, d 0, 1 −b2 ξ 2 1, 0 −1 1 + bdξ 2 , g ∈ T (G). This will give SL(2, F9 ). Also g 2 = 2 2 ξ , 1 d ξ , 1 − bdξ 2 2 a2 , c , ac, b2 ,d2 , bd ∈ F9 . This together with ad − bc = 1 leads, as in Proposition 3.5, to a, 0 1, β 1, β g= , where h = ∈ GL(2, F9 ). Also by 1 = det(g) = a2 det(h), 0, a γ, δ γ, δ we have a−2 = det(h) ∈ F9 . We also have, as in Proposition 3.5, that h(f10 ) = det(h)f10 . We get that g(f10 ) = a10 h(f10 ) = a10 a−2 f10 = a8 f10 . Now F [f10 , f12 ] = S(V )T (G) is g-stable, implying by degree reasoning that g(f12 ) = εf12 for some ε ∈ F . Since g(f12 ) = a12 h(f12 ), we shall next consider h(f12 ). We have h(f12 ) = (x1 + βx2 )12 + (x1 + βx2 )10 (γx1 + δx2 )2 − (x1 + βx2 )6 (γx1 + δx2 )6 + + (x1 + βx2 )2 (γx1 + δx2 )10 − (γx1 + δx2 )12 =
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= (x91 + β 9 x92 )(x31 + β 3 x32 ) + (x91 + β 9 x92 ))(x1 + βx2 )(γx1 + δx2 )2 − − (x1 + βx2 )6 (γx1 + δx2 )6 + (x1 + βx2 )2 (γx1 + δx2 )(γ 9 x91 + δ 9 x92 ) − − (γ 9 x91 + δ 9 x92 )(γ 3 x31 + δ 3 x32 ). Therefore the only contribution to the monomial x61 x62 will come from the term −(x1 + βx2 )6 (γx1 + δx2 )6 = −(x31 + β 3 x32 )2 (γ 3 x31 + δ 3 x32 )2 . So the contribution is exactly −x61 x62 (δ 3 − β 3 γ 3 )2 = −x61 x62 (δ − βγ)6 = −x61 x62 (det(h))6 = −(a−2 )6 x61 x62 = −a−12 x61 x62 . Consequently −ε = −a12 a−12 = −1. This shows that ε = 1 and g(f12 ) = f12 . 8 Also the coefficient of x12 1 in the right hand side of h(f12 ) (using γ = 1) is (1 + γ 2 − γ 6 + γ 10 − γ 12 ) = 1 + γ 2 − γ 6 + γ 2 − γ 4 = 1 − γ 2 (1 + γ 2 + γ 4 ) =1−
γ8 − γ2 γ 2 ((γ 2 )3 − 1) = 1 − = −1 γ2 − 1 γ2 − 1
12 12 and therefore the coefficient of x12 = −1. Since 1 in g(f12 ) (= f12 ) is a (−1). Hence a 2 16 4 a ∈ F9 we have a = 1 implying that a = −1. Consequently g(f10 ) = a8 f10 = f10 . If γ = 0 then a12 = 1. So a4 = 1 and again a8 = 1. Thus g acts as the identity on F [f10 , f12 ] = S(V )T (G) . Therefore by Galois theory g, T (G) = T (G) and g ∈ T (G). 2
Recall (e.g. [6, p. 261]) that V is an imprimitive G-module if V = Ind G H (W ), where W is an H-module for some proper subgroup H of G. V is a primitive G-module if V is not imprimitive. The following is our first positive result. Theorem 3.8. Let G ⊆ SL(2, F ) be a finite group and V = F 2 the natural SL(2, F )-module. Suppose that V is an irreducible primitive T (G)-module. Then S(V )G is a hypersurface in F 3 (or a polynomial ring). Proof. Clearly V is also an irreducible G-module. Therefore by Lemma 3.4, there exist an irreducible G-module V0 and a subfield k ⊂ F with [k : Fp ] < ∞ such that G ⊂ SL(2, k) and V = V0 ⊗k F . We next show that V0 is also an irreducible T (G)-module. Indeed by [5, 49.2], V0 is a completely reducible T (G)-module. If V0 = W1 ⊕ W2 , where Wi is a proper T (G)-module for i = 1, 2, then V = V0 ⊗k F = (W1 ⊗k F ) ⊕(W2 ⊗k F ), contradicting the irreducibility of V as a T (G)-module. Similarly V0 is a primitive T (G)-module. Indeed if V0 = Ind G H (W ) ≡ W ⊗kH kG, for some proper subgroup H of T (G), then V = V0 ⊗k F = (W ⊗kH kG) ⊗k F = (W ⊗k F ) ⊗FH FG = Ind G H (W ⊗k F ),
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in contradiction to the primitivity of V as a T (G)-module. Now by applying [20] or [11], there exists an element a ∈ GL(V0 , k) = GL(2, k) such that one of the following possibilities holds: aT (G)a−1 ∈ {SL(2, Fq ), Sp(2, Fq ), SU (2, Fq )} or aT (G)a−1 is isomorphic to SL(2, F5 ) regarded as a subgroup of SL(2, F9 ) ⊆ SL(2, k). Since SL(2, Fq ) = Sp(2, Fq ) = SU (2, Fq ) (e.g. [10, Proposition 3.1, Corollary 11.29]) we need to consider first aT (G)a−1 = SL(2, Fq ). Let G1 ≡ aGa−1 . Then T (G1 ) = aT (G)a−1 = SL(2, Fq ) and G1 ⊆ SL(2, k). Therefore by Proposition 3.5 S(V0 )G1 is Gorenstein. Hence S(V0 )G is Gorenstein and by Lemma 3.2 S(V )G is Gorenstein, and therefore it is a hypersurface in F 3 by Remark 3.6(ii). If aT (G)a−1 ∼ = SL(2, 5) regarded −1 as a subgroup of SL(2, F9 ) ⊆ SL(2, k), then in G1 ≡ aGa we have T (G1 ) = aT (G)a−1 and by Proposition 3.7 G1 = T (G1 ). Hence by [14, Theorem 8.1] S(V0 )G1 is a polynomial ring implying that S(V0 )G is a polynomial ring. This shows by [14, p. 62] that S(V )G is a polynomial ring. 2 Our next objective is to handle finite groups G ⊂ SL(2, F ) with T (G) acting irreducibly and imprimitively on the natural SL(2, F )-module F 2 = V . Since it is well known (e.g. [20]) that T (G) is monomial in this case, we restrict our attention to this subfamily. Recall that H ⊂ GL(V ) is called monomial if V has a basis with respect to which the matrix of each element of H has exactly one non-zero entry in each row and column. The next easy result disposes of the charF = p = 2 cases. Lemma 3.9. Let H ⊂ SL(2, F ) be a monomial group generated by transvections and p = 2. Then H = {1}. Proof. Let {y1 , y2 } be a basis of V =F 2 with respect 0, λ, 0 g ∈ H be a transvection. Then g = or δ, 0, δ λ − 1, 0 since g − IV = and so either λ = 1 or 0, δ−1
towhich H is monomial. Let λ . The first case can’t hold 0
δ = 1 but since g ∈ SL(2, F ), 0, λ −1 then δ = −λ−1 and δ = λ and we get, in both cases, that g = 1. If g = δ, 0 −1, λ , a matrix whose determinant is 2 and is non-zero since p = 2. g − IV = −λ−1 , −1 So rank(g − IV ) > 1, which is a contradiction. 2 This limits the possibilities of T (G) to the p = 2 case. Proposition 3.10. Let G ⊂ SL(2, F ) be a finite group where charF = 2. Suppose T (G) is an irreducible monomial subgroup. Then G = T (G).
Proof. Let {x1 , x2 } be the basis of V = F 2 with respect to which T (G) is irreducible and monomial. Let {g1 , . . . , gn } be the set of transvections generating T (G).
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∗, The T (G)-irreducibility of V forces n ≥ 2. So either gi = 0, 0, possibility leads, as in Lemma 3.9, to gi = 1. So gi = −1 α i , a, b g= ∈ G. Since g normalizes T (G), we have for each c, d
a, b c, d
0, αi−1 ,
αi 0
d, c,
b bdαi−1 + acαi , = a d2 αi−1 + c2 αi ,
0, ∗ 0 . The first , or ∗, 0 ∗ αi , for i = 1, . . . , n. Let 0 i = 1, . . . , n:
b2 αi−1 + a2 αi dbαi−1 + caαi
∈ T (G).
∗, 0 0, ∗ or . So this matrix must have the form 0, ∗ ∗ 0 In the first case b2 αi−1 = a2 αi , d2 αi−1 = c2 αi . So using p = 2 we get b = aαi , d = cαi , in contradiction to ad − bc = 1. In the second case bd = acαi2 , for i = 1, . . . , n. Since α1 = α2 (and therefore α12 = α22) 0, b . we get that bd = ac = 0. If a = 0, then ad − bc = 1 implies d = 0 and g = c, 0 Also g ∈ SL(2, F ) so c = b−1 . Consequently g(x1 + bx2 ) = bx2 + bcx1 = x1 + bx2 , as well as g(x1) = bx2 = x1 + (x1 + bx2 ), so the matrix of g with respect to {x1 , x1 + bx2 } is 1, 1 . That is g is a transvection and hence g ∈ T (G). 0, 1 c = 0 and ad − bc = 1 imply b = 0 and g = therefore Suppose that a = 0. Then 0, aαi a, 0 0, αi = ∈ G, for each i. However as in the . Also g −1 αi−1 , 0 a−1 α , 0 0, a−1 i 0, aαi is a transvection and therefore previous paragraph one observes that a−1 αi−1 , 0 0, αi 0, aαi ∈ T (G), as needed. 2 it is in T (G). Consequently g = a−1 αi−1 , 0 αi−1 , 0 We can now state the second possibility for the irreducible T (G)-action. Theorem 3.11. Let G ⊂ SL(2, F ) be a finite group with T (G) acting imprimitively and irreducibly on the natural module V = F 2 . Then S(V )G is a polynomial ring. Proof. We have by (e.g. [20]) that T (G) is monomial. Alternatively, by assumption and [5, Cor. (50.4)] T (G) acts transitively on V . That is V = M1 ⊕· · ·⊕Mk , T (G) acts by permutations on {M1 , . . . , Mk } and Orbit(M1 ) = {M1 , . . . , Mk }. Clearly if gMi = Mi , for i = 1, . . . , k, for each transvection g in T (G), then T (G)Mi = Mi , for i = 1, . . . , k, in violation of the transitivity. So let g be a transvection in T (G) with gMi = Mj with i = j. Say i = 1. Suppose dim F M1 > 1 and {m1 , . . . , mr } is a basis of M1 with r > 1. Then g(m1 ) − m1 , g(m2 ) − m2 ∈ (g − IV )(V ). Moreover they are non-zero and are linearly independent as can easily be seen. This violates the pseudo-reflection property of g. Consequently by Lemma 3.9 p = 2 and by Proposition 3.10 we get T (G) = G. Therefore by [16, Theorem 2.4] S(V )G is a polynomial ring. 2
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Our next two results will be used in the proof of the reducible T (G)-module case, in Theorem 3.1. Lemma 3.12. Let charF = p and ξ ∈ F a primitive m-th root of unity, where (m, p) = 1. Let W ⊂ F be an n-dimensional Fp -subspace of F with ξW ⊆ W . Then m|pn − 1. Proof. Let Cm be the cyclic group of order m and g ∈ Cm a generator. The map φ : Cm → End Fp W , where φ(g j ) ≡ ξ j IW , for each j, makes W into a Cm -module. (m, p) = 1 grants that W is a completely reducible Fp Cm -module, where the latter denotes the group algebra of Cm over Fp . Thus W = W1 ⊕ · · · ⊕ Wr , where Wi is an irreducible Fp Cm -module, for i = 1, . . . , r. Set di ≡ dim Fp Wi . The result will follow once we show that m|pdi − 1 for each i. Let φi (g) ≡ φ(g)|Wi that is φi : Cm → GL(Wi ), for each i. If φi is not injective this will give an integer k = m, k|m with ξ k w = w, for each w ∈ Wi . This will contradict the m-th primitivity of ξ. Therefore φi : Cm → GL(Wi ) makes Wi into a faithful irreducible Cm -module, for i = 1, . . . , r. Let Mi ≡ Ann Fp Cm Wi = {x ∈ Fp Cm |xWi = 0}. Then Mi is a maximal ideal in Fp Cm , and Fp Cm /Mi is a finite field extension of Fp . Also by commutativity of Fp Cm we have Fp Cm /Mi ∼ = Wi as Fp Cm -module and consequently ∼ End Fp Cm (Wi ) = Fp Cm /Mi , as algebras, for each i. By the commutativity of Cm we also have φi (g) ∈ End Fp Cm (Wi ). Since φi is an injection, it follows that Cm is isomorphic to a subgroup of the multiplicative group of Fp Cm /Mi . Consequently since |Fp Cm /Mi | = |Wi | = pdi , we get m|pdi − 1, for each i = 1, . . . , r. 2 Lemma 3.13. Let V = F 2 = F x1 + F x2 be the natural SL(2, F ) module and σi = 1, αi ∈ SL(2, F ), a transvection, for i = 1, . . . , n. Assume that {α1 , . . . , αn } are lin0, 1 early independent over Fp . Then S(V )σ1 ,...,σn = F [an , x2 ], where an is a homogeneous n polynomial (in x1 , x2 ) of degree pn , having xp1 as one of its monomials. Proof. The proof is carried out by induction. The case n = 1 appears already in Lemma 2.2. Indeed σ1 (x1 ) = x1 +α1 x2 , σ1 (x2 ) = x2 imply that a1 = xp1 −x1 (α1 x2 )p−1 ∈ S(V )σ1 . Now |σ1 | = p together with a Galois field reasoning imply F [a1 , x2 ] = S(V )σ1 . Suppose by induction that S(V )σ1 ,...,σn−1 = F [an−1 , x2 ], where deg(an−1 ) = pn−1 , n−1 and xp1 is one of the monomials of an−1 . Now σn σi = σi σn for i = 1, . . . , n − 1 imply that F [an−1 , x2 ] is σn -stable. Therefore, since σn preserve degrees, σn (an−1 ) = n−1 αan−1 + βxp2 , α, β ∈ F . Now σn (x1 ) = x1 + αn x2 , σn (x2 ) = x2 imply, by plugging n−1 into σn (an−1 ), that xp1 is one of the monomials appearing in σn (an−1 ) and therefore α = 1. We next show that β = 0. To see this observe that αn ∈ / Span Fp {α1 , . . . , αn−1 } implies σn ∈ / σ1 , . . . , σn−1 . Therefore by Galois considerations F [an−1 , x2 ] = S(V )σ1 ,...,σn−1 S(V )σ1 ,...,σn . Thus σn (an−1 ) = an−1 , as claimed.
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n−1
Finally set an ≡ apn−1 − an−1 (βxp2 )p−1 . As in the first paragraph σn (an ) = an , so an ∈ S(V )σ1 ,...,σn , F [an , x2 ] is a polynomial ring and since |σn | = p, [F (an−1 , x2 ) : F (an , x2 )] = p. Thus F (an , x2 ) = Q(S(V )σ1 ,...,σn ) and since S(V )σ1 ,...,σn is inten gral over F [an , x2 ] we get F [an , x2 ] = S(V )σ1 ,...,σn . Clearly xp1 appears as one of the monomials of an . Alternatively, as pointed out by the referee, since (degx2 )(degan ) = pn = |σ1 , . . . , σn | and an , x2 are algebraically independent, it follows from [12, Proposition 16(b)] or [7, 3.7.5] that F [an , x2 ] = S(V )σ1 ,...,σn . 2 We finally arrive at: The proof of Theorem 3.1. The case T (G) = {1G } follows from [2, Theorem B]. Therefore we may assume that T (G) = {1G }. Let V = F 2 be the natural SL(2, F )-module. If V is an irreducible T (G)-module then Theorems 3.8 and 3.11 imply that S(V )G is Gorenstein and even a hypersurface in F 3 . We may therefore assume that V is a reducible T (G)-module. Consequently there exists a basis {x1 , x2 } of V with respect to which F, F T (G) ⊆ . Since every transvection stabilizes a one dimensional subspace of V 0, F 1, F , implying it must be F x2 . Therefore every transvection of T (G) must be in 0, 1 1, F . Consequently T (G) is elementary abelian. So that T (G) ⊆ 0, 1
1, T (G) = σ1 × · · · × σn , where σi ≡ 0,
αi , for i = 1, . . . , n, 1
and {α1 , . . . , αn } are linearly independent over Fp . Set W = Fp α1 + · · · + Fp αn , so dim Fp W = n. Consequently |T (G)| = pn . a, b 1, F −1 ∈ G. Then gσi g ∈ T (G) ⊆ forces Let g = c, d 0, 1
a, b c, d
1, 0,
αi 1
d, −c,
−b 1 − acαi , = a −c2 αi ,
λ, −1 0, λ
a2 αi 1 + acαi
1, ∈ 0,
F 1
with λ, in F . Now since gσi g −1 = and therefore c = 0 and g = 1, λ2 αi ∈ T (G), for i = 1, . . . , n, we get that λ2 αi ∈ W for i = 1, . . . , n. Con0, 1 sequently by Lemma 3.12 |λ2 | divides pn − 1. We next observe that (|G|/|T (G)|, p) = 1 and that G/T (G) is isomorphic to a (cyclic) n p subgroup cyclic group H = {α ∈ F |α2(p −1) p of the = 1}. Indeed if g ∈ T (G) then 1, F , λ 1, ∈ , so λ = 1 and g = gp = ∈ T (G). This shows that 0, λ−p 0, 1 0, 1 λ, λ1 , 1 , . (|G/T (G)|, p) = 1. Say g = g1 h with h ∈ T (G), where g = g = 1 0, λ−1 0, λ−1 1
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−1 −1 λ−1 1, F 1 λ, λ1 − 1 λ = h ∈ T (G) ⊆ and λ = λ1 . Therefore 0, λ1 λ−1 0, 1 the map φ : G/T (G) → H, which is defined by φ(g(T (G)) = λ, is a well defined injective homomorphism with its image in H since |λ2 | divides pn − 1. λ, . By Lemma 3.13 we Let g¯ ≡ gT (G) be a generator of G/T (G) with g = 0, λ−1 have S(V )T (G) = S(V )σ1 ,...,σn = F [an , x2 ]. Now since gT (G)g −1 = T (G), F [an , x2 ] n is g-stable and since g preserve degrees we get g(an ) = αan + βxp2 with α, β ∈ F . n n Since an has xp1 as one of its monomials and since g(x1 ) = λx1 + x2 we get that xp1 n n n p p p appears with coefficient λ in g(an ), that is α = λ . If λ = 1 then λ = 1 and then 1, g= ∈ T (G), which was excluded. 0, 1 n n Suppose firstly that λ = −1, then λp = (−1)p = −1 (this also holds if p = 2 since Then g1−1 g =
n
λ = 1). Consequently λ2p = 1. Let b = an + n
pn
g(b) = λ an +
n βxp2
n
n λp β xp , λ2pn −1 2
then n
n n n n λp β λp β pn λ−p xp2 = λp (an + 2pn x ) = λp b. + 2pn λ −1 λ −1 2
Clearly L = F b + F x2 is a 2-dimensional vector space over F and S(L) = F [b, x2 ] = F T . With respect to the basis {b, x2 } of L, g|L is represented by the matrix [apnn, x2 ] = λ , 0 2 n . Let |T|G| (G)| = |λ| ≡ k, then |g|L | = |λ| = k and |λ | divides (p − 1) by 0, λ−1 Lemma 3.12, implying that k|2(pn − 1). n If p = 2 then (p, 2(pn − 1)) = 1, hence λip = 1 for 1 ≤ i ≤ k forces i = k. This shows that g|L has no diagonal pseudo-reflections, so since |g|L | = k and (k, p) = 1 we get by Watanabe’s theorem that S(V )G = S(L)g|L is Gorenstein if and only if k|(pn − 1), that is |T|G| (G)| |(|T (G)| − 1). This settles item (ii) for λ = −1 and p = 2. If p = 2 this forces λ = −1. Now if k is even k = 2m, then λ2m = 1, implying, since p = 2, that λm = 1, in contradiction to |λ| = k. Hence (k, 2) = 1. So k|(2n − 1), that is |T|G| (G)| |(|T (G)| − 1). Equivalently detg|L = 1. Therefore by Watanabe’s theorem S(V )G = S(V )g|L is always Gorenstein in case p = 2. This settles also item (iv)(4). G Suppose next that λ = −1. Consequently p = 2. We shall show that S(V ) is Goren−1, stein in this case. Now g = implies that g 2 ∈ T (G). Hence |T|G| (G)| = 2. 0, −1 Observe that |T (G)| = pn is odd, so |T|G| (G)| |(|T (G)| − 1) also holds in this case.
We have S(V )G = S(V )T (G),g = (S(V )T (G) )¯g = T ¯g where g¯ = gT (G). Now since |G/T (G)| = 2 and p = 2 then by H. Cartan’s theorem [15, Lemma 5.3] T ¯g ∼ = S(m/m2 )¯g , where m = (an , x2 ) is the unique graded maximal ideal of T = F [a , x ]. n 2 −1, 0 , it follows Clearly g¯ acts linearly on m/m2 and is represented by the matrix 0, −1 by Watanabe’s theorem that S(m/m2 )¯g is Gorenstein. In fact it is generated by a ¯2n , x ¯22 , a ¯n x ¯2 , where a ¯ n = an + m2 , x ¯2 = x2 + m2 , showing that S(m/m2 )¯g is a simple singularity hypersurface of type A. This settles item (ii).
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Since |λ2 | divides (pn − 1), the above reasoning shows that S(V )g ,T (G) = S(L)g|L is Gorenstein. So by taking D ≡ g 2 , T (G) item (iii) is verified. Consequently if |G| is odd, then G = g 2 , T (G) and S(V )G is therefore Gorenstein, so item (iv)(2) is verified. Given the assumption of item (iv)(3) then λ ∈ / Fp (λ2 ) implies [Fp (λ) : Fp (λ2 )] = 2 in contradiction to [F : Fp ] = s being odd. Thus λ ∈ Fp (λ2 ) and since λ2 W ⊆ W we have λW ⊆ W , that is k = |λ| divides (pn − 1), by Lemma 3.12. Now by the previous reasoning det(g|L ) = 1 and S(V )G = S(L)g|L is Gorenstein and item (iv)(3) holds. Our final task is to verify that whenever S(V )G is Gorenstein then pn it is a hyper0 λ , g|L 3 G with surface in F . If λ = −1 then S(V ) = S(L) and g|L = 0, λ−1 n det ( g|L ) = 1. Therefore λp = λ. Consequently since S(L) = F [b, x2 ] we conclude that S(L)g|L = F [bk , xk2 , bx2 ] which is clearly a hypersurface, and in fact a simple singularity (of type Ak−1 ). The case λ = −1 was dealt with before and the result therefore follows. 2 2
2
Remark 3.14. One can use the proof of the reducible case of Theorem 3.1 to give an example, for each p ≥ 5, with the following properties: |G| = 2(p − 1)p, G ⊂ SL(2, F ), F ⊇ Fp2 , S(V )G is Cohen–Macaulay but not Gorenstein, where V ≡ F 2 and G ∩ Z(GL(V )) = {1}. Acknowledgments Thanks to Yuval Ginosar for stimulating conversations. References [1] D.J. Benson, Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Note Ser., vol. 190, Cambridge University Press, Cambridge, 1993. [2] A. Braun, On the Gorenstein property for modular invariants, J. Algebra 345 (2011) 81–99. [3] A. Broer, The direct summand property in modular invariant theory, Transform. Groups 10 (1) (2005) 5–27. [4] W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math., Cambridge University Press, 1998. [5] C.W. Curtis, I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley Classics Library, 1988. [6] C.W. Curtis, I. Reiner, Methods of Representation Theory, vol. I, John Wiley & Sons, New York, 1981. [7] H. Derksen, G. Kemper, Computational invariant theory, in: Invariant Theory and Algebraic Transformation Groups, I, in: Encyclopaedia Math. Sci., vol. 130, Springer-Verlag, Berlin, 2002. [8] F. DeMeyer, E. Ingraham, Separable Algebras over Commutative Rings, Lecture Notes in Math., vol. 181, Springer-Verlag, 1971. [9] P. Fleischmann, C. Woodcock, Relative invariants, ideal classes and quasi-canonical modules of modular rings of invariants, J. Algebra 348 (2011) 110–134. [10] L.C. Grove, Classical Groups and Geometric Algebra, Grad. Stud. Math., vol. 39, A.M.S., 2002. [11] W. Kantor, Subgroups of classical groups generated by long root elements, Trans. Amer. Math. Soc. 248 (1979) 347–379. [12] G. Kemper, Calculating invariant rings of finite groups over arbitrary fields, J. Symbolic Comput. 21 (1996) 351–366.
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[13] G. Kemper, E. Körding, G. Malle, B.H. Matzat, D. Vogel, G. Wiese, A database of invariant rings, Exp. Math. 10 (2001) 537–542. [14] G. Kemper, G. Malle, The finite irreducible linear groups with polynomial ring of invariants, Transform. Groups 2 (1) (1997) 57–89. [15] G.J. Leuschke, R.A. Wiegand, Cohen–Macaulay Representations, Math. Surveys Monogr., vol. 181, American Math. Society, 2012. [16] H. Nakajima, Invariants of finite groups generated by pseudoreflections in positive characteristic, Tsukuba J. Math. 3 (1979) 109–122. [17] H.D. Neusel, L. Smith, Invariant Theory of Finite Groups, Math. Surveys Monogr., vol. 94, American Mathematical Society, 2002. [18] K. Watanabe, Certain invariant subrings are Gorenstein I, Osaka J. Math. II (1974) 1–8. [19] K. Watanabe, Certain invariant subrings are Gorenstein II, Osaka J. Math. II (1974) 379–388. [20] A.E. Zalesskii, V.N. Serežkin, Linear groups generated by transvections, Math. USSR, Izv. 10 (1976) 25–46.