Rationality of finite group representations and local subgroups

Journal of Algebra 533 (2019) 322–338

Contents lists available at ScienceDirect

Journal of Algebra www.elsevier.com/locate/jalgebra

Rationality of finite group representations and local subgroups Michael Geline Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA

a r t i c l e

i n f o

Article history: Received 2 January 2019 Available online 17 June 2019 Communicated by Leonard L. Scott, Jr.

a b s t r a c t We show that local fields of values and local Schur indices of irreducible characters in a block of a finite group are controlled by local subgroups. A method for computing certain Schur indices and an application for the prime 2 are given. © 2019 Elsevier Inc. All rights reserved.

Keywords: Schur index Block theory

1. Introduction Let G be a finite group. The group ring QG is semisimple and hence admits a Wedderburn decomposition as a direct sum of matrix algebras over division algebras: QG ∼ =



Mni (Δi ).

i

Each matrix algebra summand corresponds to a Galois conjugacy class of absolutely irreducible characteristic zero representations of G. Letting χ denote the character of an arbitrary representation in the ith such class, one has a (non-canonical) isomorphism of fields E-mail address: [email protected]. https://doi.org/10.1016/j.jalgebra.2019.06.006 0021-8693/© 2019 Elsevier Inc. All rights reserved.

M. Geline / Journal of Algebra 533 (2019) 322–338

323

Z(Δi ) ∼ = Q(χ) where Q(χ) denotes an extension of Q generated by the values of χ. This is a subfield of the field of |G|th roots of unity. One knows that the maximal subfields of Δi are precisely the splitting fields for χ of minimal degree over Q(χ). Their common index over Q(χ) is called the Schur index of χ over Q and denoted mQ (χ). The relation dimZ(Δi ) (Δi ) = mQ (χ)2 is well known. If F/Q is a field extension, one similarly has mF (χ), the index of the Wedderburn component of F G corresponding to χ. Let Q denote a prime (possibly archimedean) of the number field Z(Δi ) lying over the rational prime q. The completions of Z(Δi ) and Δi at Q will be denoted Z(Δi )Q and Δi Q respectively. The latter is a central simple algebra over the former. Its Schur index will be denoted mQ (χ). Note that Z(Δi )Q ∼ = Qq (χ). The theorem of Brauer, Hasse, and Noether states that, in a precise way, the division algebra Δi is determined by its completions Δi Q as Q ranges over all primes of Z(Δi ). A numerical consequence of this theory is mQ (χ) = lcmQ {mQ (χ)}. If Q is another prime lying over q, then a theorem of Benard and Schacher [4, Theorem 74.20] says that mQ (χ) = mQ (χ). This is a rather special feature of division algebras arising from group algebras of finite groups. Its proof relies on the induction theorem of Witt and Berman [4, Theorem 74.29]. Because of it, we will refer to this common index simply as mq (χ). Our purpose is to show how, for q finite, the local Schur indices mq (χ) and the local fields Qq (χ) are controlled by analogous invariants for characters of q-local subgroups. Recall that the q-local subgroups of G are the normalizers of the nontrivial q-subgroups of G. The primary tool will be Galois equivariance of the Green correspondence for indecomposable lattices. This theory will be reviewed in Section 2. The results are most naturally stated in terms of all the irreducible characters (or Brauer characters) in a given q-block. To establish this background, recall that if k is a field of characteristic q, and Q is a q-subgroup of G, one has a “Brauer morphism” of k-algebras (kG)Q

BrQ

kCG (Q)

from the Q-fixed points of kG to the group algebra kCG (Q). The Brauer morphism is here simply given by projection using the natural k-basis of kG. If b is a block idempotent of kG, then BrQ (b) is a sum of block idempotents of kCG (Q). The group NG (Q) permutes the block idempotents of kCG (Q) appearing in BrQ (b), and

324

M. Geline / Journal of Algebra 533 (2019) 322–338

orbit sums yield block idempotents of kNG (Q). These blocks of kNG (Q) will be called Brauer correspondents of b. Up to conjugation in G, there is a unique q-subgroup D which is maximal subject to BrD (b) = 0. It is called a defect group of b, and it is known that the idempotent BrD (b) consists of a single block of kNG (D) which might be thought of as a “most important” Brauer correspondent of b. In fact, it is not uncommon to refer to BrD (b) as “the” Brauer correspondent of b. From now on, let K denote an extension of Qq generated by a primitive |G|th root of unity. Let OK denote the valuation ring of K, and let k denote the residue field of K. We then have the q-modular system (K, OK , k). It is a splitting system for G and all of its subgroups. If e is any central idempotent of kG, there is a unique central idempotent of OK G whose image in kG is e. This idempotent will be denoted by eˆ. One says that an ordinary character of G lies in e if it is the character of some KGˆ e-module, and that a Brauer character lies in e if it is the Brauer character of some kGe-module. We let Irr(e) denote the set of characters afforded by simple KGˆ e-modules and IBr(e) the set of (K-valued) Brauer characters of simple kGe-modules. Generalizing slightly some definitions in [5, Section 8], we associate three fields to an  arbitrary central idempotent e of kG. Write eˆ = x∈G ax x with ax ∈ OK . Qq (e) = Qq (ax | x ∈ G) Qq (IBr(e)) = Qq (ϕ | ϕ ∈ IBr(e)) Qq (Irr(e)) = Qq (χ | χ ∈ Irr(e)). The group Gal(K/Qq ) acts on the set of q-blocks of G as well as the sets Irr(1) and IBr(1). If a field automorphism fixes an irreducible Brauer character, it must also fix the block in which it lies. Thus Qq (e) ⊆ Qq (IBr(e)). Surjectivity of Brauer’s decomposition map implies that Qq (IBr(e)) ⊆ Qq (Irr(e)). In [5, Lemma 8.6], Dade points out that if b is a block idempotent with defect group D, then Qq (b) = Qq (BrD (b)). This follows from Galois equivariance of Brauer’s first main theorem and is something of a motivation for all our results here. Note this does not require Dade’s standing assumption in [5] that D be cyclic. We now state our main results. The first shows how fields of values of characters in a block are restricted by those of characters in blocks of local subgroups. The second is a similar statement about Schur indices. Theorem 1.1. Let b be a q-block idempotent of kG, and let D be a defect group of b. Then Qq (Irr(BrD (b))) ⊆ Qq (Irr(b)) ⊆ Qq (Irr(BrQ (b)) | CD (Q) ⊆ Q ⊆ D), and Qq (IBr(BrD (b))) ⊆ Qq (IBr(b)) ⊆ Qq (IBr(BrQ (b)) | CD (Q) ⊆ Q ⊆ D)

M. Geline / Journal of Algebra 533 (2019) 322–338

325

where Q ranges over the subgroups of D which contain their own centralizers, and BrQ (b) is regarded as a sum of blocks of kNG (Q). When D is abelian, the indexing group Q assumes only the value D, so the containments in Theorem 1.1 become equalities. This generalizes Dade’s [5, Theorem 8]. Theorem 1.2. Let b be a q-block idempotent of kG, and let D be a defect group of b. Assume D = 1. Then lcm{mq (χ) | χ ∈ Irr(b)} divides lcm{mq (ζ) | 1 = Q ⊆ D, ζ ∈ Irr(BrQ (b))} where Q ranges over the nontrivial subgroups of D, and BrQ (b) is regarded as a sum of blocks of kNG (Q). If D = 1 in Theorem 1.2, then b only contains one irreducible character. It remains irreducible (mod q) hence has trivial q-local Schur index [6, Corollary IV.9.4]. It is not clear whether the subgroups Q appearing in Theorems 1.1 or 1.2 can be further restricted. For example, a Galois equivariant form of Alperin’s weight conjecture would imply that Q can be confined to the subgroups occurring as vertices of Alperin’s weights, at least regarding the statement about irreducible Brauer characters in Theorem 1.1. There is a familiar situation in which Q can be confined to D. Theorem 1.3. Let b be a q-block idempotent of kG. Let D be a defect group for b, and assume that D ∩ Dx = 1 whenever x ∈ G \ NG (D). Then Qq (Irr(b)) = Qq (Irr(BrD (b))), Qq (IBr(b)) = Qq (IBr(BrD (b))), and lcm{mq (χ) | χ ∈ Irr(b)} divides lcm{mq (ζ) | ζ ∈ Irr(BrD (b))} where BrD (b) is regarded as a block of kNG (D).

326

M. Geline / Journal of Algebra 533 (2019) 322–338

It was conjectured in [7] that every irreducible character in a 2-block with abelian defect group should have trivial 2-local Schur index. Since then, it was shown in [8] that every irreducible character of height zero in a 2-block has trivial 2-local Schur index. So the conjecture in [7] follows from the more widely known “height zero” conjecture of Brauer. The relevant direction of Brauer’s conjecture has been established using the classification of finite simple groups; see [10]. As an application of Theorems 1.2 and 1.3, we shall prove some cases of the conjecture from [7] without using the classification. Theorem 1.4. Let b be a 2-block of G. Let D be a defect group of b. Assume that D is abelian and that one of the following holds: (1) Whenever x ∈ G \ NG (D), we have D ∩ Dx = 1. (2) The inertial quotient of b has prime order and acts transitively on the non-identity elements of D. (3) For every nontrivial subgroup Q of D, the group NG (Q) is solvable. Then m2 (χ) = 1 for every irreducible character χ in b. (Recall that the inertial quotient of b is T /CG (D) where T is the stabilizer in NG (D) of any block of kCG (D) appearing in BrD (b).) Under yet stronger hypotheses, a q-local Schur index in a q-local subgroup implies the existence of a q-local Schur index for an irreducible character of the entire group. Theorem 1.5. Let b be a q-block idempotent with defect group D, and assume that NG (D) ∩ Dx = 1 whenever x ∈ G \ NG (D). Then lcm{mq (χ) | χ ∈ Irr(b)} = lcm{mq (ζ) | ζ ∈ Irr(BrD (b))} where BrD (b) is regarded as a block of kNG (D). Broué has shown in [3] that for any block b, one always has gcd{mq (χ) | χ ∈ Irr(b)} = 1. 2. Galois action, Green correspondence, and fields of values Let F be an arbitrary extension of Qq such that OF , the valuation ring of F , is a discrete valuation ring. Letting kF denote the residue field of F , we have the q-modular system (F, OF , kF ). Recall that an OF G-module L is called an OF G-lattice if L is free and finitely generated as an OF -module. Let G be a finite group of automorphisms of the field F , fixing the subfield Qq elementwise. Observe that each σ ∈ G induces a ring automorphism of OF G by acting trivially on the elements of G. This action allows us to form the “twist” Lσ of L by σ. Indeed,

M. Geline / Journal of Algebra 533 (2019) 322–338

327

Lσ has the same underlying additive group L+ as L, and the action of OF G on Lσ is given by the composite OF G

σ −1

OF G

EndZ (L+ )

where the map on the right gives the action of OF G on L. If χL is the character afforded by L, it is clear that Lσ affords the character (χL )σ defined by (χL )σ (g) = σ −1 (χL (g)) for g ∈ G. So χLσ = (χL )σ . The OF G-lattices L and Lσ may or may not be isomorphic. If Lσ ∼ = L for all σ ∈ G, the lattice L will be called G-stable. An obvious necessary condition for L to be G-stable is that χL = χLσ for all σ ∈ G. Generally, this will not be sufficient. We shall need to consider Galois action on decomposable lattices. It is clear that twisting by field automorphisms distributes over direct sums. So the following is a consequence of the Krull Schmidt theorem. Lemma 2.1. Let L be a G-stable OF G-lattice. Then G permutes the isomorphism classes of indecomposable summands occurring in a decomposition of L into indecomposable OF G-lattices. Isomorphism classes in the same orbit occur with the same multiplicity. A condition for L to be G-stable that is sufficient but not necessary is the existence of an OF0 -form for L, where F0 is a subfield of F contained in the fixed field of G. This means there exists an OF0 G-lattice L0 contained in L such that OF L0 = L. Later arguments will depend rather critically on the fact that G-stability is equivalent to the existence of forms over subfields for indecomposable projective OF G-lattices. Lemma 2.2. Let G be a finite group of automorphisms of F with fixed field F0 containing Qq . Assume also that the residue field of F is perfect. Let P be an indecomposable projective OF G-lattice. Then P is G-stable if and only if P has an OF0 -form. Proof. One direction is clear. For the other, assume that P is G-stable. Because the free OF G-lattices have forms over Zq , there exists an indecomposable projective OF0 G-lattice P0 such that P is a direct summand of the OF G-lattice OF ⊗OF0 P0 . However, G transitively permutes the indecomposable summands of OF ⊗OF0 P0 . (See [4, Theorems 30.27 and 30.33].) This implies that OF ⊗OF0 P0 ∼ = P. 2

328

M. Geline / Journal of Algebra 533 (2019) 322–338

Some results of Fröhlich, Ritter, and Weiss [13] will be used later to assert the existence of G-stable OF G-lattices affording absolutely irreducible characters (for somewhat particular choices of F ). When the characters have nontrivial Schur indices, these lattices will not admit forms over subfields. 2.1. Vertices Let L be an indecomposable OF G-lattice. Recall that a vertex for L is a subgroup V of G which is minimal subject to the condition that G ∼ IndG V (ResV (L)) = L ⊕ X

for some OF G-lattice X . The Krull Schmidt theorem and Mackey double coset formula can be used to show that the vertices of L are all conjugate in G. Moreover, if the block idempotent eˆ of OF G acts as the identity on L, each vertex of L is contained in a defect group of e. An indecomposable lattice is projective if and only if its vertex is trivial. The following is straightforward. Lemma 2.3. Let σ be a finite order automorphism of F whose fixed field contains Qq . If L is an indecomposable OF G-lattice, then the vertices of L and Lσ coincide. Not every subgroup of a defect group can occur as a vertex of an irreducible module or lattice. The most significant known restriction is a result of Knörr, which will be needed in the proof of Theorem 1.1. Theorem 2.4. Let D be a defect group for block idempotent b of kF G. Let Q0 be a subgroup of D occurring as a vertex of either an absolutely simple kF G-module in b or an OF G-lattice affording an absolutely irreducible character in b. Then some G-conjugate Q of Q0 satisfies CD (Q) ⊆ Q ⊆ D. In particular, if D is abelian, we must have Q0 = D. Proof. This is the main result of [12].

2

Finally, Theorem 1.1 will also require a known fact concerning vertices when the defect group is normal. It will suffice to state this for the q-modular system (K, OK , k) defined in Section 1. Proposition 2.5. Let D be a defect group for a block idempotent b of kG. Assume that D is normal in G. Then

M. Geline / Journal of Algebra 533 (2019) 322–338

329

(1) Every simple kG-module in b has vertex D. (2) For every absolutely irreducible character χ in b, there is a finite extension F/K such that χ is afforded by an OF G-lattice with vertex D. Proof. The first statement holds because D must act trivially on all simple kG-modules. For the second statement, Feit shows in [6, Lemma I.18.2] that if we take F/K to have sufficiently large ramification index, then χ is afforded by an OF G-lattice L with semisimple (mod q) reduction. But a vertex of L must contain the vertices of the indecomposable summands of this reduction, so the result follows from the first statement. 2 We remark that it appears to be unknown whether the second statement holds if D is not assumed normal in G. 2.2. Green correspondence Fix a nontrivial q-subgroup Q and a subgroup H of G containing NG (Q). Following Green, we introduce three sets of subgroups of G: X = X(Q, H) = {Q ∩ Qx | x ∈ G \ H} Y = Y(Q, H) = {H ∩ Qx | x ∈ G \ H} A = A(Q, H) = {Q0 ⊆ Q | Q0 G X}. The notation Q0 G X means that Q0 is not G-conjugate to any subgroup of a subgroup contained in X. Note that the elements of X are proper subgroups of Q, so Q itself is an element of A. Moreover, no subgroup in A is contained in any subgroup in Y. In the main result of [9], Green shows that if L is an indecomposable OF G-lattice with vertex belonging to A, then ∼ ResG H (L) = f L ⊕ Y

(1)

for some indecomposable OF H-lattice f L with a vertex in common with L, and an OF H-lattice Y all of whose indecomposable summands have vertices contained in subgroups in Y. In particular, ResG H (L) has a unique indecomposable summand with vertex in A. Similarly, if L is an indecomposable OF H-lattice with vertex in A, then ∼ IndG H (L) = gL ⊕ X

(2)

for some indecomposable OF G-lattice gL with vertex in common with L, and an OF G-lattice X all of whose indecomposable summands have vertices contained in subgroups in X. In particular, IndG H (L) has a unique indecomposable summand with vertex in A.

330

M. Geline / Journal of Algebra 533 (2019) 322–338

The maps f and g are inverse to each other and establish a bijection between the set of isomorphism classes of indecomposable OF G-lattices with vertex in A and the set of isomorphism classes of indecomposable OF H-lattices with vertex in A. Similar statements hold for indecomposable kF G and kF H-modules. This is the Green correspondence. The Green correspondence is related to Brauer correspondents of blocks by a version of Brauer’s second main theorem. Theorem 2.6. With notation as above, suppose H = NG (Q). If L is an indecomposable OF G-lattice or kF G-module in a block b of G with vertex in A, then f L lies in a block of NG (Q) which is Brauer correspondent to b. Proof. This is [6, Theorem III.7.8]. The case in which Q is a defect group for b appears already in [9]. Notice that Q need not be a vertex of L. 2 We turn now to Galois equivariance of the Green correspondence. Proposition 2.7. Let G be a finite group of automorphisms of F with fixed field containing Qq , and let L be an indecomposable OF G-lattice with vertex in A. Then for every σ ∈ G, we have (f L)σ ∼ = f (Lσ ). In particular, (f L)σ ∼ = f L if and only σ ∼ if L = L. In this case, writing ∼ ResG H (L) = f L ⊕ Y, as in (1), we have Y σ ∼ = Y. Proof. We have an isomorphism of OF H-lattices G σ ∼ σ ResG H (L ) = (ResH (L)) .

(3)

The only summand of the left hand side with vertex in A is f (Lσ ). The only summand of the right hand side with vertex in A is (f L)σ . So the Krull Schmidt theorem implies these lattices are isomorphic. If Lσ ∼ = L, then (f L)σ ∼ = f (Lσ ) ∼ = f L. If (f L)σ ∼ = f L, then f (Lσ ) ∼ = f L, and Lσ ∼ =L follows because the Green correspondence is a bijection. In this case, one sees from the isomorphism (3) that fL ⊕ Y ∼ = f L ⊕ Yσ, and Y σ ∼ = Y follows from Krull Schmidt. 2 We omit the proof of the corresponding statement for induction. Proposition 2.8. Let G be a finite group of automorphisms of F with fixed field containing Qq . Let L be an indecomposable OF H-lattice with vertex in A.

M. Geline / Journal of Algebra 533 (2019) 322–338

331

Then for every σ ∈ G, we have (gL)σ ∼ = g(Lσ ). In particular, (gL)σ ∼ = gL if and only σ ∼ if L = L. In this case, writing ∼ IndG H (L) = gL ⊕ X as in (2), we have X σ ∼ = X. The main theorem of Galois theory now implies the following, which is all that remains to prove Theorem 1.1. Corollary 2.9. Let L be an indecomposable OF G-lattice with vertex in A. Then the values of the character χL generate the same extension of Qq as the values of the character χf L . 2.3. Proof of Theorem 1.1 Recall that for a nontrivial q-subgroup Q of D, we regard BrQ (b) as a sum of blocks of kNG (Q). Let ζ ∈ Irr(BrD (b)). By Proposition 2.5, for some finite extension F of K, there is an OF NG (D)-lattice L affording ζ with vertex D. The Green correspondence with X(D, NG (D)), Y(D, NG (D)), and A(D, NG (D)) yields an indecomposable OF G-lattice gL with vertex D. By Corollary 2.9, we have Qq (ζ) = Qq (χL ) = Qq (χgL ). But Qq (χgL ) is contained in Qq (Irr(b)) by Theorem 2.6. Because ζ was arbitrary, we conclude Qq (Irr(BrD (b))) ⊆ Qq (Irr(b)). Now let χ ∈ Irr(b). Let F be any q-adic splitting field for χ. Choose an OF G-lattice L affording χ. By Theorem 2.4, L has a vertex V satisfying CD (V ) ⊆ V ⊆ D. Thus, the indexing group Q from Theorem 1.1 assumes the value V . Applying Green correspondence with X(V, NG (V )), Y(V, NG (V )), and A(V, NG (V )) yields an indecomposable OF NG (V )-lattice f L, and by Corollary 2.9 we have Qq (χ) = Qq (χL ) = Qq (χf L ). The second containment in Theorem 1.1 follows, again using Theorem 2.6 and letting χ vary over Irr(b). Regarding irreducible Brauer characters, the argument is almost identical. Indeed, the Green correspondence is Galois equivariant in positive characteristic as well. 2

332

M. Geline / Journal of Algebra 533 (2019) 322–338

2.4. Proof of field equalities in Theorem 1.3 If D = 1, there is nothing to prove. So assume that D = 1. The hypothesis of the theorem says that the trivial subgroup of D is the only element of X(D, NG (D)). Equivalently, every nontrivial subgroup of D belongs to A(D, NG (D)). It follows, using Theorem 2.6, that every non-projective indecomposable OK G-lattice in b has a Green correspondent which is an OK NG (D)-lattice in BrD (b). Similarly, every non-projective indecomposable kG-module in b has a Green correspondent which is a kNG (D)-module in BrD (b). The result follows as in Theorem 1.1, except it is no longer necessary to keep track of the exact vertices of the modules and lattices under consideration. We simply note that (K, OK , k) is a splitting system for both G and NG (D), and irreducible lattices and modules in b or BrD (b) cannot be projective. 2 3. Galois stable irreducible lattices and the Schur index The statements about Schur indices will require us to consider both vertices and Galois stability of lattices, particularly irreducible lattices. Let (K, OK , k) be the q-modular system from Section 1. Let χ ∈ Irr(G). Let F be a subfield of K containing Qq (χ), and let F  be a finite extension of F . Then χ is afforded by an F  G-module if and only if mq (χ) divides |F  : Qq (χ)| or equivalently whenever mF (χ) divides |F  : F |. This is a result of local class field theory and does not require F  to be contained in K or even Galois over F . Assume the divisibility holds and consider an F  G-module L which affords χ. Then L contains many OF  G-lattices. Assuming that F  /F is indeed Galois, whether L contains a Gal(F  /F )-stable OF  G-lattice is referred to as “the Fröhlich problem” by Ritter and Weiss in [13] where the following very satisfying answer is provided. Theorem 3.1. With the above notation, a Gal(F  /F )-stable lattice L ⊆ L exists if and only if mF (χ) divides the ramification index of the extension F  /F . Proof. This is [13, Lemmas 1 and 3]. Ritter and Weiss actually formulate and solve the problem for number fields. The local fields we are concerned with here form a base case for their Theorem B. 2 It will be necessary to identify suitable splitting fields for the irreducible characters under consideration. The most economical way to do this is to use the Benard-Schacher theorem [4, Theorem 74.20] which in turn depends on the Witt-Berman induction theorem [4, Theorem 74.29]. Proposition 3.2. Let G, K, and χ be as above. Let p be a prime dividing the local Schur index mq (χ). Let F be the unique field between Qq (χ) and K satisfying

M. Geline / Journal of Algebra 533 (2019) 322–338

333

p  |F : Qq (χ)|, and |K : F | = some power of p. Then mF (χ) = mq (χ)p , and there exists an extension F  /F such that (1) The extension F  /F is Galois and totally ramified. (2) The group Gal(F  /F ) has order mF (χ). Proof. The fact that mF (χ) = mq (χ)p is explained in, for example, [4, Proposition 74.35]. Let pm denote this power of p. By the Benard-Schacher theorem, Qq (χ) and hence F contain the primitive pm th roots of unity. Kummer theory then implies that if π is a uniformizer for F , we can let m F  be the field obtained by adding a root of the polynomial xp − π to F . 2 4. Global to local 4.1. Proof of Theorem 1.2 Let χ ∈ Irr(b). Let p be a prime dividing mq (χ). We must find a nontrivial q-subgroup Q and an irreducible character ζ in a q-block of kNG (Q), Brauer correspondent to b, such that mq (χ)p divides mq (ζ). Let F  /F be an extension constructed as in Proposition 3.2, so mF (χ) = mq (χ)p . We shall now consider an indecomposable OF  G-lattice L such that (1) We have Lσ ∼ = L for all σ ∈ Gal(F  /F ). (2) The multiplicity [χL , χ] ≡ 0 (mod p). (3) The lattice L is chosen to have smallest possible vertex subject to Conditions (1) and (2). By Theorem 3.1, there exist Galois stable OF  G-lattices with character exactly χ, so lattices L as desired do indeed exist. Let L be one. Assume that the vertex of L is trivial. Then L is an indecomposable projective OF  G-lattice which is Gal(F  /F )-stable. It follows from Lemma 2.2 that L has an OF -form. This implies that mF (χ) divides [χL , χ], contradicting the fact that p divides mF (χ). So the vertices of L are not trivial. Let Q denote one. The Green correspondence with X(Q, NG (Q)), Y(Q, NG (Q)), and A(Q, NG (Q)) yields an indecomposable OF  NG (Q)-lattice f L. We have ∼ IndG NG (Q) (f L) = L ⊕ X

334

M. Geline / Journal of Algebra 533 (2019) 322–338

for some OF  G-lattice X . By Proposition 2.8, X is Gal(F  /F )-stable. Also, every indecomposable summand of X has vertex smaller than Q. We now claim that [χX , χ] ≡ 0 (mod p). Assume otherwise. By Lemma 2.1, Gal(F  /F ) permutes the indecomposable summands of X . But this group is a p-group that fixes χ. It follows that X must have an indecomposable summand X0 which is Gal(F  /F )-stable and satisfies [χX0 , χ] ≡ 0 (mod p). But the vertex of X0 is smaller than Q, contradicting minimality of the vertex of L. By Frobenius reciprocity, it now follows that [χf L , ResG NG (Q) (χ)] = [χL , χ] + [χX , χ] ≡ 0 (mod p). Observe that the composite field KF  (in some algebraic closure of Qq ) is a Galois  extension of F of p-power degree. Writing both χf L and ResG NG (Q) (χ) out as Gal(KF /F ) orbit sums of absolutely irreducible characters of NG (Q), one now sees that there exists ζ ∈ Irr(NG (Q)) such that |F (ζ) : F | · [χf L , ζ] · [ResG NG (Q) (χ), ζ] ≡ 0 (mod p). Because |F (ζ) : F | is a power of p, we conclude that F (ζ) = F . Because p  [ResG NG (Q) (χ), ζ], it follows from [4, Proposition 74.35] that mF (ζ) = mF (χ). In particular, mq (χ)p divides mq (ζ). Finally, because [χf L , ζ] = 0, Theorem 2.6 implies that ζ does indeed lie in a block of kNG (Q) Brauer correspondent of b. 2 4.2. Proof of Schur index portion of Theorem 1.3 Let χ ∈ Irr(b). Let p be a prime dividing mq (χ). We must find an irreducible character ζ in the block BrD (b) of kNG (D) such that mq (χ)p divides mq (ζ). This will be simpler than the previous argument. Let F  /F be an extension constructed as in Proposition 3.2, so mF (χ) = mq (χ)p . Let L be an OF  G-lattice which is Gal(F  /F )-stable and affords the character χ. Such a lattice exists by Theorem 3.1. Again, there is nothing to prove if D = 1, so we may assume L is not projective. As in the proof of the first part of Theorem 1.3, L then has a Green correspondent OF  NG (D)-lattice f L lying in BrD (b). We have ∼ IndG NG (D) (f L) = L ⊕ X for some OF  G-lattice X which, under current assumptions, must be projective. As before, p must divide the multiplicity [χX , χ], for if it did not some indecomposable summand of X would have an OF -form and afford a character in which χ appears with p multiplicity. This would contradict the assumption that p divides mF (χ).

M. Geline / Journal of Algebra 533 (2019) 322–338

335

So Frobenius reciprocity yields [χf L , ResG NG (D) (χ)] ≡ 1 ≡ 0 (mod p), and we obtain ζ ∈ Irr(NG (D)) with |F (ζ) : F | · [χf L , ζ] · [ResG NG (D) (χ), ζ] ≡ 0 (mod p) and argue exactly as before. 2 5. Local to global 5.1. Proof of Theorem 1.5 Because our assumption is stronger than that in Theorem 1.3, we already know that the lcm on the left hand side of the statement divides that on the right. Thus, if ζ ∈ Irr(BrD (b)), and p is a prime dividing mq (ζ), it suffices to find χ ∈ Irr(b) such that mq (ζ)p divides mq (χ). Let F  /F be an extension constructed as in Proposition 3.2, with the present ζ in place of the “χ” used there. So mF (ζ) = mq (ζ)p . Let L be a Gal(F  /F )-stable OF  NG (D)-lattice affording the character ζ. Again, there is nothing to prove if D = 1, so we may assume L is not projective. Consider the Green correspondence with X(D, NG (D)), Y(D, NG (D)), and A(D, NG (D)). As before, every nontrivial subgroup of D belongs to A(D, NG (D)), so we obtain an OF  G-lattice gL with ∼ ResG NG (D) (gL) = L ⊕ Y where every indecomposable summand of the Gal(F  /F )-stable lattice Y has vertices contained in subgroups in Y(D, NG (D)). Under current assumptions, this means Y is projective. It follows that p must divide the multiplicity [χY , ζ], and Frobenius reciprocity yields [χgL , IndG NG (D) (ζ)] ≡ 1 ≡ 0 (mod p).  Finally, writing the characters χgL and IndG NG (D) (ζ) out as sums of Gal(KF /F )-orbits and using the above, we get a χ ∈ Irr(G) such that

|F (χ) : F | · [χgL , χ] · [χ, IndG NG (D) (ζ)] ≡ 0 (mod p). Because |F (χ) : F | is a p-power, we have F (χ) = F . Because p  [χ, IndG NG (D) (ζ)], it follows from [4, Proposition 74.35] that mF (ζ) = mF (χ). In particular, mq (ζ)p divides mq (χ).

M. Geline / Journal of Algebra 533 (2019) 322–338

336

Finally, because [χgL , χ] = 0, Theorem 2.6 implies that χ does indeed lie in the block b. 2 5.2. Cyclic blocks If D is cyclic, then the equality in Theorem 1.5 holds without any hypothesis on the embedding of D in G. This is weaker than the character by character analysis performed in [1], but we shall not need to distinguish between “exceptional” and “non-exceptional” characters as is done there. Proposition 5.1. Let b be a q-block of G with a cyclic defect group D. Then lcm{mq (χ) | χ ∈ Irr(b)} = lcm{mq (ζ) | ζ ∈ Irr(BrD (b))}. Proof. The technique of Theorems 1.3 and 1.5 combines well with the usual use of Green correspondence for cyclic blocks. Let Q be the unique subgroup of D of order p. It follows from [5, Lemmas 1.3 and 1.4] that BrQ (b) consists of a single block of kNG (Q). We have NG (D) ⊆ NG (Q) and use the Green correspondence with X(D, NG (Q)), Y(D, NG (Q)), and A(D, NG (Q)). Relevant “error terms” in induced and restricted modules between G and NG (Q) are projective. An argument given in [6, Lemma VII.1.5] shows an indecomposable lattice in b has a Green correspondent lying in BrQ (b) and conversely. We may thus argue as in Theorems 1.3 and 1.5, obtaining lcm{mq (χ) | χ ∈ Irr(b)} = lcm{mq (ζ) | ζ ∈ Irr(BrQ (b))}. It remains to pass from the block BrQ (b) of NG (Q) to the block BrD (b) of NG (D). One way to do this is to consider the blocks of Zq NG (Q) and Zq NG (D) which contain BrQ (b) and BrD (b) respectively when scalars are extended from Zq to OK . These blocks over Zq are known to be Morita equivalent (over Zq ). For example, a result of Rouquier [14, Theorem 10.1] shows that the blocks over OK are Morita equivalent. Then a more recent result of Kessar and Linckelmann [11, Theorem 6.5] implies that this Morita equivalence descends to a Morita equivalence of the blocks over Zq . 2 6. Height zero characters of local subgroup The theorems of Section 1 are meant to reduce the calculation of local Schur indices of irreducible characters to those of irreducible characters of local subgroups. We indicate here how to find local Schur indices of height zero irreducible characters in blocks with normal defect group. This uses a result of Benard [1] which is itself based on an idea of Berman [2].

M. Geline / Journal of Algebra 533 (2019) 322–338

337

Proposition 6.1. Let H be a finite group with a q-block b having normal defect group D. Let ζ ∈ Irr(b) have height zero, and let λ be an irreducible constituent of ResH D (ζ). Then λ(1) = 1, and if IG = {h ∈ H | λh = λσ for some σ ∈ Gal(Qq (λ)/Qq )}, then ζ = IndH IG (ψ) for some ψ ∈ Irr(IG ). Moreover, mq (ζ) = |Qq (ζ, ϕ) : Qq (ζ)|, for any irreducible Brauer character ϕ appearing in the (mod q) reduction of ψ. Proof. It is well known that, because ζ has height zero, we have λ(1) = 1. A standard argument from Clifford theory asserts that there is ψ ∈ Irr(IG ) satisfying IndH IG (ψ) = ζ, and that for any such we have Qq (ζ) = Qq (ψ)

(4)

mq (ζ) = mq (ψ).

(5)

and

(Clifford theory arguments of this kind work over global fields as well.) Now ψ lies in a block of IG with defect group D. Moreover, ker(λ) is normal in IG , and we have ker(λ) ⊆ ker(ψ). So we can regard ψ as in irreducible character of IG / ker(λ). This does not affect its field of values or Schur index. Finally, ψ lies in a block of IG / ker(λ) with defect group D/ ker(λ). Because λ(1) = 1, this defect group is cyclic. So [1, Sections 6 and 7] implies that mq (ψ) = |Qq (ψ, ϕ) : Qq (ψ)| where ϕ is any irreducible Brauer character appearing in the (mod q) reduction of ψ. The result now follows from Equations (4) and (5). 2 6.1. Proof of Theorem 1.4 Assume that m2 (χ) = 1 for some χ ∈ Irr(b). In the first case, by Theorem 1.3, there exists ζ ∈ Irr(BrD (b)) with m2 (ζ) = 1. The height of ζ is zero because D is abelian. Let ψ, λ, and IG be as in Proposition 6.1. Regarded as an irreducible character of IG / ker(λ), the character ψ lies in a 2-block with cyclic defect. Such blocks are nilpotent. This implies that ψ remains irreducible (mod 2), which in turn implies m2 (ψ) = 1. But m2 (ψ) = m2 (ζ), a contradiction.

338

M. Geline / Journal of Algebra 533 (2019) 322–338

In the second case, by Theorem 1.2, there is a nontrivial subgroup Q of D and an irreducible character ζ of NG (Q) in BrQ (b) with m2 (ζ) = 1. But this block of NG (Q) has abelian defect group D. If Q = D, one argues as in the first case. If Q = D, the assumption implies that the block containing ζ is nilpotent. It follows that ζ remains irreducible (mod 2), so has trivial 2-local index here as well. In the third case, use Theorem 1.2 as before. But now NG (Q) is a solvable group, so the main result of [7] implies the irreducible characters in BrQ (b) have trivial 2-local Schur index. 2 Acknowledgment The author is grateful to Madhav Nori and Akaki Tikaradze for several helpful discussions over a long period of time. References [1] Mark Benard, Schur indices and cyclic defect groups, Ann. of Math. (2) 103 (1976) 283–304. [2] S.D. Berman, Representations of finite groups over an arbitrary field and over rings of integers, Izv. Akad. Nauk SSR Ser. Mat. 30 (1966) 69–132, Amer. Math. Soc. Transl. Ser. 2 64 (1967) 147–215. [3] Michel Broué, Certains invariants entiers d’un p-bloc, Math. Z. 154 (1977) 283–286. [4] W. Charles Curtis, Reiner Irving, Methods of Representation Theory. Vol. I and II. With Applications to Finite Groups and Orders, Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1981. [5] E.C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966) 20–48. [6] Walter Feit, The Representation Theory of Finite Groups, North Holland, 1982. [7] M. Geline, Schur indices and Abelian defect groups, J. Algebra 319 (10) (2008) 4140–4146. [8] Michael Geline, David Gluck, Local Schur indices, blocks, and character values, Bull. Lond. Math. Soc. 42 (2010) 707–712. [9] J.A. Green, A transfer theorem for modular representations, J. Algebra 1 (1964) 73–84. [10] Radha Kessar, Gunter Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2) 178 (1) (2013) 321–384. [11] Radha Kessar, Markus Linckelmann, Descent of equivalences and character bijections, in: Geometric and Topological Aspects of the Representation Theory of Finite Groups, in: Springer Proc. Math. Stat., vol. 242, Springer, Cham, 2018, pp. 181–212. [12] Reinhard Knörr, On the vertices of irreducible modules, Ann. of Math. (2) 110 (3) (1979) 487–499. [13] Jürgen Ritter, Alfred Weiss, Galois action on integral representations, J. Lond. Math. Soc. (2) 46 (1992) 411–431. [14] Raphaël Rouquier, The derived category of blocks with cyclic defect groups, in: Steffen König, Alexander Zimmermann (Eds.), Derived Equivalences for Group Rings, in: Lecture Notes in Mathematics, vol. 1685, Springer, 1998, pp. 199–220.