Anchors of irreducible characters
aa r X i v : . [ m a t h . G R ] N ov ANCHORS OF IRREDUCIBLE CHARACTERS
RADHA KESSAR, BURKHARD K ¨ULSHAMMER, AND MARKUS LINCKELMANN
Abstract.
Given a prime number p , every irreducible character χ of a finitegroup G determines a unique conjugacy class of p -subgroups of G which wewill call the anchors of χ . This invariant has been considered by Barker inthe context of finite p -solvable groups. Besides proving some basic propertiesof these anchors, we investigate the relation to other p -groups which can beattached to irreducible characters, such as defect groups, vertices in the senseof J. A. Green and vertices in the sense of G. Navarro. Dedicated to the memory of J. A. Green
September 1, 20181.
Introduction
Let p be a prime number and O a complete discrete valuation ring with residuefield k = O /J ( O ) of characteristic p and field of fractions K of characteristic 0.For G a finite group, we denote by Irr( G ) the set of characters of the simple KG -modules. For χ ∈ Irr( G ), we denote by e χ the unique primitive idempotent in Z ( KG ) satisfying χ ( e χ ) = 0. The O -order O Ge χ in the simple K -algebra KGe χ is a G -interior O -algebra, via the group homomorphism G → ( O Ge χ ) × sending g ∈ G to ge χ . Since ( O Ge χ ) G = Z ( O Ge χ ) is a subring of the field Z ( KGe χ ), itfollows that O Ge χ is a primitive G -interior O -algebra. In particular, O Ge χ is aprimitive G -algebra. By the fundamental work of J. A. Green [7], it has a defectgroup. This is used in work of Barker [1] to prove a part of a conjecture of Robinson(cf. [24, 4.1, 5.1]) for blocks of finite p -solvable groups. In order to distinguish thisinvariant from defect groups of blocks and from vertices of modules, we introducethe following terminology. Definition 1.1.
Let G be a finite group and let χ ∈ Irr( G ). An anchor of χ is adefect group of the primitive G -interior O -algebra O Ge χ .By the definition of defect groups, an anchor of an irreducible character χ of G is a subgroup P of G which is minimal with respect to e χ ∈ ( O Ge χ ) GP , where( O Ge χ ) GP denotes the image of the relative trace map Tr GP : ( O Ge χ ) P → ( O Ge χ ) G .Green’s general theory in [7, §
5] implies that the anchors of χ form a conjugacyclass of p -subgroups of G .For the remainder of the paper we make the blanket assumption that K and k are splitting fields for the finite groups arising in the statements below. In a fewplaces, this assumption is not necessary; see the Remark 1.7 below. The second author gratefully acknowledges support by the DFG (SPP 1388).
Theorem 1.2.
Let G be a finite group and let χ ∈ Irr( G ) . Let B be the block of O G containing χ and let L be an O G -lattice affording χ . Let P be an anchor of χ and denote by ∆ P the image { ( x, x ) | x ∈ P } of P under the diagonal embedding of G in G × G . The following hold. (a) P is contained in a defect group of B . (b) P contains a vertex of L . (c) We have O p ( G ) ≤ P . (d) The suborder O P e χ of O Ge χ is local, and O Ge χ is a separable extension of O P e χ . (e) ∆ P is contained in a vertex of the O ( G × G ) -module O Ge χ and P × P containsa vertex of O Ge χ . Moreover, ∆ P is a vertex of O Ge χ if and only if χ is ofdefect zero. For G a finite group, we denote by IBr( G ) the set of O -valued Brauer charactersof the simple kG -modules. We denote by G ◦ the set of p ′ -elements in G , and for χ a K -valued class function on G , we denote by χ ◦ the restriction of χ to G ◦ . Theorem 1.3.
Let G be a finite group and χ ∈ Irr( G ) . Let B be the block of O G containing χ and let L be an O G -lattice affording χ . Let P be an anchor of χ . Thefollowing hold. (a) If χ ◦ ∈ IBr( G ) , then L is unique up to isomorphism, P is a vertex of L , and P × P is a vertex of the O ( G × G ) -module O Ge χ . (b) Let τ be a local point of P on O Ge χ . Then the multiplicity module of τ issimple. In particular, O p ( N G ( P τ )) = P and P is centric in a fusion system of B . (c) If B has an abelian defect group D , then D is an anchor of χ . (d) If χ has height zero, then P is a defect group of B , and P × P is a vertex ofthe O ( G × G ) -module O Ge χ . The hypothesis χ ◦ ∈ IBr( G ) in the first statement of Theorem 1.3 holds if χ is aheight zero character of a nilpotent block. If G is p -solvable, then by the Fong-Swantheorem [4, § ϕ ∈ IBr( G ) there is χ ∈ Irr( G ) such that χ ◦ = ϕ . Thefact that anchors are centric is essentially proved in the proof of [1, Theorem] as animmediate consequence of results of Kn¨orr [14], Picaronny-Puig [20], and Th´evenaz[26]; see the proof of 3.7 below for details.The next result shows that anchors are invariant under Morita equivalencesgiven by a bimodule with endopermutation source, hence in particular under sourcealgebra equivalences. Theorem 1.4.
Let G , G ′ be finite groups. Let B , B ′ be blocks of O G , O G ′ , withdefect groups D , D ′ , respectively. Suppose that B and B ′ are Morita equivalentvia a B - B ′ -bimodule M which has an endopermutation source, when viewed as an O ( G × G ′ ) -module. Let χ ∈ Irr( B ) and χ ′ ∈ Irr( B ′ ) such that χ and χ ′ correspondto each other under the Morita equivalence determined by M . Then there is anisomorphism D ∼ = D ′ sending an anchor of χ to an anchor of χ ′ . In particular, if B and B ′ are source algebra equivalent, then χ and χ ′ have isomorphic anchors. The last statement in Theorem 1.4 can be made more precise: if B , B ′ aresource algebra equivalent, then the isomorphism D ∼ = D ′ can be chosen to have anextension to a source algebra isomorphism; see Theorem 4.1 below. NCHORS 3
In [17] Navarro associated, via the theory of special characters, to each ordinaryirreducible character χ of a p -solvable group G , a G -conjugacy class of pairs ( Q, δ ),where Q is a p -subgroup of G and δ is an ordinary irreducible character of Q , whichbehave in certain ways as the Green vertices of indecomposable modules (see also[5],[2] and [3]). We call such a pair ( Q, χ ) a
Navarro vertex of χ . We prove thefollowing two results relating Navarro vertices and anchors (see Section 5 for thedefinitions). Theorem 1.5.
Let G be a finite p -solvable group and let χ ∈ Irr( G ) such that χ ◦ ∈ IBr( G ) . Let ( Q, δ ) be a Navarro vertex of χ . Then Q contains an anchor of χ .Further, if p is odd or δ is the trivial character, then Q is an anchor of χ . Theorem 1.6.
Let G be a finite group of odd order, and let χ ∈ Irr( G ) haveNavarro vertex ( Q, δ ) . Then Q is contained in an anchor of G . We give examples which show that equality does not always hold in the abovetheorem. We also give examples which show that if | G | is even, then a Navarrovertex need not be contained in any anchor of χ .Section 2 of the paper contains some basic properties of quotient orders of finitegroup algebras. In Section 3, we prove the theorems 1.2 and 1.3. In section 4 wecharacterise anchors at the source algebra level, and use this to prove Theorem 1.4.In Section 5 we prove some properties of anchors with respect to normal subgroups.Section 6 contains the proofs of Theorems 1.5 and 1.6. In Section 7 we compareanchors of O Ge χ to the defect groups of k ⊗ O O Ge χ . Remark 1.7.
The splitting field hypothesis on K and k is not needed in Theorem1.2 and the Propositions 3.1, 3.2, 3.3, and 3.5, on which the proof of Theorem 1.2is based. This hypothesis is also not needed in Theorem 4.1, stating that anchorscan be read off the source algebras of a block. Remark 1.8.
Let G be a finite group, χ ∈ Irr( G ), and b the block idempotentof the block of O G to which χ belongs; that is, b is the primitive idempotent in Z ( O G ) satisfying be χ = e χ . Let P be an anchor of χ ; that is, P is a minimalsubgroup of G such that there exists an element c ∈ ( O Ge χ ) P satisfying e χ =Tr GP ( c ). We clearly have ( O G ) P e χ ⊆ ( O Ge χ ) P , but this inclusion need not be anequality, and this is one of the main issues for calculating anchors. The spaces( O G ) P e χ and ( O Ge χ ) P have the same O -rank, since ( KG ) P e χ = ( KGe χ ) P . Anequality ( O G ) P e χ = ( O Ge χ ) P implies that P is a defect group of the block of O G to which χ belongs; see Proposition 3.9 below.2. Orders of characters
Let G be a finite group and χ ∈ Irr( G ). Then O Ge χ is an O -order in the simple K -algebra KGe χ , called the O - order of χ . In general, O Ge χ is not a subalgebraof O G , but it is an O -free quotient of O G . The map f : O G → O Ge χ sending x ∈O G to xe χ is an epimorphism of O -orders; in particular, the O -orders O Ge χ and O G/ Ker( f ) are isomorphic. Since we assume that K and k are splitting fields forall finite groups arising in this paper, we have e χ = χ (1) | G | P g ∈ G χ ( g − ) g , and as a K -algebra, KGe χ is isomorphic to the matrix algebra K χ (1) × χ (1) . In particular, KGe χ has up to isomorphism a unique simple left module M χ , and we have dim K ( M χ ) = χ (1). Since the O -rank of O Ge χ is χ (1) , it follows that k ⊗ O O Ge χ is a k -algebra KESSAR, K¨ULSHAMMER, AND LINCKELMANN of dimension χ (1) whose isomorphism classes of simple modules are in bijectionwith the set IBr( G | χ ) = { ϕ ∈ IBr( G ) : d χ,ϕ = 0 } where d χ,ϕ denotes the decomposition number attached to χ and ϕ , defined by theequation χ ◦ = P ϕ ∈ IBr( G ) d χ,ϕ ϕ . It is well-known that d χ,ϕ = χ ( i ), where i is aprimitive idempotent in O G such that O Gi is a projective cover of a simple modulewith Brauer character ϕ . In particular, d χ,ϕ = 0 if and only if ie χ = 0. The simple k ⊗ O O Ge χ -module N ϕ corresponding to ϕ ∈ IBr( G | χ ) has dimension ϕ (1). Thuswe have an isomorphism of k -algebras k ⊗ O O Ge χ /J ( k ⊗ O O Ge χ ) ∼ = Y ϕ ∈ IBr( G | χ ) k ϕ (1) × ϕ (1) . Let P ( N ϕ ) denote an O Ge χ -lattice which is a projective cover of N ϕ . Thenthe KGe χ -module K ⊗ O P ( N ϕ ) is isomorphic to M d χ,ϕ ; in particular, we haverk O ( P ( N ϕ )) = d χ,ϕ χ (1). Setting ℓ χ = | IBr( G | χ ) | , the decomposition matrix of the O -order O Ge χ is the 1 × ℓ χ -matrix∆ χ = ( d χ,ϕ : ϕ ∈ IBr( G | χ )) . Hence the Cartan matrix of O Ge χ is the ℓ χ × ℓ χ -matrix C χ = ∆ ⊤ χ ∆ χ = ( d χ,ϕ d χ,ψ : ϕ, ψ ∈ IBr( G | χ )) . By definition, C χ has rank 1. The only non-zero invariant factor of C χ isgcd( d χ,ϕ d χ,ψ : ϕ, ψ ∈ IBr( G | χ )) = gcd( d χ,ϕ : ϕ ∈ IBr( G | χ )) . If χ ◦ ∈ IBr( G ), then it is well-known that the O -order O Ge χ is isomorphic to O χ (1) × χ (1) (see e. g. [13, Prop. 4.1]), and thus the k -algebra k ⊗ O O Ge χ isisomorphic to k χ (1) × χ (1) . In this case the decomposition matrix ∆ χ is the 1 × C χ .Since Z ( KGe χ ) = Ke χ ∼ = K , we have Z ( O Ge χ ) = O e χ ∼ = O ; in particular, Z ( O Ge χ ) is a local O -order, and hence Z ( k ⊗ O O Ge χ ) is a local k -algebra, bystandard lifting theorems for central idempotents. It is obvious that Z ( k ⊗ O O Ge χ ) ⊇ k ⊗ O Z ( O Ge χ ) = k ⊗ e χ . This inclusion can be proper, or equivalently, the canonical map Z ( O Ge χ ) → Z ( k ⊗ O O Ge χ ) need not be surjective. The following example illustrates this. Example 2.1.
Let G be the dihedral group of order 8, let χ ∈ Irr( G ) with χ (1) = 2,and let p = 2. We represent G in the form G = h a, b i where a = (cid:18) − (cid:19) and b = (cid:18) (cid:19) . Then O Ge χ is isomorphic to the subalgebra Λ of O × generated by a and b . Notethat ba = − ab , so that (1 ⊗ b )(1 ⊗ a ) = (1 ⊗ a )(1 ⊗ b ) in k ⊗ O Λ. This shows that k ⊗ O O Ge χ ∼ = k ⊗ O Λ is commutative and of dimension 4.
NCHORS 5 Proofs of the theorems 1.2 and 1.3
Proposition 3.1.
Let G be a finite group, χ ∈ Irr( G ) , and let P be an anchor of χ . Let B be the block of O G containing χ , and let L be an O G -lattice affording χ .The following hold.(i) P is contained in a defect group of B .(ii) P contains a vertex of L .Proof. Let D be a defect group of B . Then there exists an element x ∈ ( O G ) D suchthat Tr GD ( x ) = 1 B . Then xe χ ∈ ( O Ge χ ) D , and Tr GD ( xe χ ) = Tr GD ( x ) e χ = 1 B e χ = e χ .Thus D contains an anchor of χ , and (i) follows. Let y ∈ ( O Ge χ ) P such thatTr GP ( y ) = e χ . Then the map η : L → L sending z to yz , is an element in End O P ( L )such that Tr GP ( η ) = id L . By Higman’s criterion, P contains a vertex of L , whencethe result. (cid:3) Proposition 3.2.
Let G be a finite group, χ ∈ Irr( G ) , and let P be an anchor of χ . Then O p ( G ) ≤ P .Proof. Set N = O p ( G ). Arguing by contradiction, suppose that P does not contain N . Then P is a proper subgroup of P N . For g ∈ N , we have g − ∈ J ( O N ) ⊆ J ( O G ). It follows that gd − d , dg − − d , and gdg − − d = gdg − − dg − + dg − − d are contained in J ( O Ge χ ) for all g ∈ N and all d ∈ O Ge χ . Let d ∈ ( O Ge χ ) P suchthat Tr GP ( d ) = e χ . By the above , we have Tr P NP ( d ) − | P N : P | d ∈ J ( O Ge χ ). Since p divides | P N : P | , it follows that x = Tr P NP ( d ) ∈ J ( O Ge χ ). Applying Tr GP N tothis element shows that e χ = Tr GP ( d ) = Tr GP N ( x ) ∈ J ( O Ge χ ), a contradiction. (cid:3) Let R be an O -order with unitary suborder S . We recall that R is called a separable extension of S if the multiplication map µ : R ⊗ S R → R sending x ⊗ y to xy for all x , y ∈ R splits as a map of R - R -bimodules. This is equivalent to thecondition that 1 R = µ ( z ) for some z ∈ ( R ⊗ S R ) R . Proposition 3.3.
Let G be a finite group and χ ∈ Irr( G ) . Let P be an anchor of χ . Then the O -order O Ge χ is a separable extension of its local suborder O P e χ .Proof. The O -order O P is local, and the map O P → O Ge χ induced by multipli-cation with e χ is a homomorphism of O -algebras with image O P e χ . Thus O P e χ isa local suborder of O Ge χ . Let T be a transversal for G/P , and let d ∈ ( O Ge χ ) P such that e χ = Tr GP ( d ) = X g ∈ T gdg − . Then the element x = P g ∈ T gde χ ⊗ g − e χ ∈ O Ge χ ⊗ O Pe χ O Ge χ is independent ofthe choice of T since gude χ ⊗ u − g − e χ = gde χ ue χ ⊗ u − e χ g − e χ = gde χ ⊗ g − e χ for g ∈ T and u ∈ P . But, for h ∈ G , hT is another transversal for G/P . Thus x = X g ∈ T hgde χ ⊗ g − h − e χ = hxh − . This shows that x ∈ ( O Ge χ ⊗ O Pe χ O Ge χ ) O Ge χ , and µ ( x ) = X g ∈ T gde χ g − e χ = Tr GP ( d ) e χ = e χ = e χ KESSAR, K¨ULSHAMMER, AND LINCKELMANN where µ : O Ge χ ⊗ O P e χ O Ge χ → O Ge χ , denotes the multiplication map sending a ⊗ b to ab . The result follows. (cid:3) Remark 3.4.
The proposition above implies that k ⊗ O O Ge χ is also a separableextension of k ⊗ O O P e χ . Note that k ⊗ O O P e χ is a homomorphic image of thegroup algebra kP . Thus, if P is cyclic, then k ⊗ O O P e χ has finite representationtype. Proposition 3.5.
Let G be a finite group and χ ∈ Irr( G ) . Let P be an anchor of χ . Then ∆ P is contained in a vertex of the O ( G × G ) -module O Ge χ and P × P contains a vertex of O Ge χ . Moreover, ∆ P is a vertex of O Ge χ if and only if χ isof defect zero.Proof. Viewing O Ge χ as an O ( G × G )-module, we have O Ge χ (∆ P ) = 0, where O Ge χ (∆ P ) is the Brauer quotient of O Ge χ (cf. [27, § O Ge χ is relatively P × G -projective. Indeed, let T be a transversal for G/P , and let d ∈ ( O Ge χ ) P suchthat e χ = Tr GP ( d ) = X g ∈ T gdg − . The map O Ge χ → O ( G × G ) ⊗ O ( P × G ) O Ge χ , ( x → X u ∈ T ( g, ⊗ dg − x ) , x ∈ O Ge χ is an O ( G × G )-module splitting of the surjective O ( G × G )-module homomorphism O ( G × G ) ⊗ O ( P × G ) O Ge χ , ( y ⊗ y ′ → yy ′ ) , y ∈ O ( G × G ) , y ′ ∈ O Ge χ proving the claim. Similarly, O Ge χ is relatively G × P -projective. Let R be avertex of O Ge χ contained in G × P and let R be a vertex of O Ge χ contained in P × G . Since R and R are G × G -conjugate, it follows that R ≤ x P × P forsome x ∈ G and hence that O Ge χ is relatively P × P = ( x − , ( x P × P )-projective.This proves the second assertion.Finally, if ∆ P is a vertex of O ( G × G ), then Res O ( G × G ) O ( G × O Ge χ is projective. Inparticular, the character of O Ge χ as a left O G -module vanishes on the p -singularelements of G . Since the character of O Ge χ is a multiple of χ , it follows that χ is of p -defect zero. Conversely, if χ is of p -defect zero, then by Proposition 3.1 wehave P = 1, hence 1 = P × P = ∆ P is a vertex of O Ge χ . (cid:3) Proof of Theorem 1.2.
Proposition 3.1 implies (a) and (b) of the theorem. Propo-sition 3.2 proves (c). Part (d) follows from Proposition 3.3 and Part (e) is provedin Proposition 3.5. (cid:3)
Proposition 3.6.
Let G be a finite group, and let χ ∈ Irr( G ) with anchor P .Suppose that χ ◦ ∈ IBr( G ) . Let L be an O G -lattice affording χ . Then L is uniqueup to isomorphism, P is a vertex of L , and P × P is a vertex of the O ( G × G ) -module O Ge χ .Proof. The hypotheses imply that the O -orders O Ge χ and O χ (1) × χ (1) are isomor-phic. Since O χ (1) is the only indecomposable O χ (1) × χ (1) -lattice, up to isomorphism, L is the only indecomposable O Ge χ -lattice, up to isomorphism. Thus the canonicalmap O Ge χ −→ End O ( L ) NCHORS 7 is an isomorphism of G -interior O -algebras, and hence these two primitive G -interior O -algebras have the same defect groups. Higman’s criterion implies that P isa vertex of L . Moreover, we have a canonical O ( G × G )-module isomorphismEnd O ( L ) ∼ = L ⊗ O L ∗ , where L ∗ is the O -dual of L . This implies that P × P is avertex of O Ge χ . (cid:3) We observe next that the multiplicity modules of the primitive G -interior O -algebras O Ge χ are simple. Background material on multiplicity modules can befound in [26, § Proposition 3.7.
Let G be a finite group and χ ∈ Irr( G ) . Let P τ be a defectpointed group of the primitive G -interior O -algebra O Ge χ . Then P is an anchor of χ and the multiplicity module of τ is simple. In particular, we have O p ( N G ( P τ )) = P , and P is centric in a fusion system of the block containing χ .Proof. The fact that P is an anchor of χ is a standard property of local pointedgroups on primitive G -algebras (see e. g. [27, (18.3)]). As noted earlier, we have( O Ge χ ) G ∼ = O . Set ¯ N = N G ( P τ ) /P ). It follows from [26, 9.1.(c), 9.3.(b)] that themultiplicity module V τ of τ is simple. The well-known Lemma 3.8 below impliesthat O p ( N G ( P τ )) = P . By results of Kn¨orr in [14], every vertex of a lattice withirreducible character χ is centric in a fusion system of the block containing χ .Centric subgroups in a fusion system are upwardly closed. Since the anchor P of χ contains a vertex of every lattice with character χ , the last statement follows.One can prove this also by applying the results of [20] directly to the G -interior O -algebra O Ge χ . (cid:3) Lemma 3.8.
Let G be a finite group, A a primitive G -algebra, and let P τ be a defectpointed group on A . If the multiplicity module of τ is simple, then O p ( N G ( P τ )) = P .Proof. Set ¯ N = N G ( P τ ) /P . The multiplicity module V τ of τ is a module over atwisted group algebra k α ¯ N for some α ∈ H ( ¯ N ; k × ). Since P τ is maximal, V τ isprojective (cf. [26, 9.1]). Since V τ is also simple by the assumptions, it follows that k α ¯ N has a block which is a matrix algebra over k . By [27, (10.5)] we have k α ¯ N ∼ = kN ′ e for some central p ′ -extension N ′ of ¯ N and some idempotent e ∈ Z ( kN ′ ). Thusthe multiplicity module corresponds to a defect zero block of kN ′ . Since O p ( N ′ ) iscontained in all defect groups of all blocks of kN ′ , it follows that O p ( N ′ ) is trivial.By elementary group theory, the canonical map N ′ → ¯ N sends O p ( N ′ ) onto O p ( ¯ N ),and hence O p ( ¯ N ) is trivial, or equivalently, O p ( N G ( P τ )) = P . (cid:3) Proof of Theorem 1.3.
Part (a) is proved in Proposition 3.6. Part (b) follows fromProposition 3.7, and (c) is an immediate consequence of (b). If χ has height zerothen χ (1) p = | G : D | p where D is a defect group of B . Then D contains a vertex Q of L , and | G : Q | p divides (rk O L ) p = χ (1) p = | G : D | p . This implies Q = D .Hence D is an anchor of χ in this case. A similar argument shows that D × D is avertex of the O ( G × G )-module O Ge χ . This proves (d). (cid:3) Proposition 3.9.
Let G be a finite group, χ ∈ Irr( G ) , and let P be an anchor of χ . If ( O G ) P e χ = ( O Ge χ ) P , then P is a defect group of the block of O G to which χ belongs.Proof. Denote by b the primitive idempotent in Z ( O G ) such that be χ = e χ . Supposethat ( O G ) P e χ = ( O Ge χ ) P . Then e χ = Tr GP ( ce χ ) = Tr GP ( c ) e χ for some c ∈ ( O Gb ) P . KESSAR, K¨ULSHAMMER, AND LINCKELMANN
Thus w = Tr GP ( c ) is not contained in J ( Z ( O Gb )), hence invertible in Z ( O Gb ).Therefore b = w − w = Tr GP ( w − c ), which implies that P contains a defect groupof b , hence is equal to a defect group of b by Theorem 1.2 (a). (cid:3) Anchors and source algebras
We show that anchors of characters in a block can be read off the source algebrasof that block, and use this to prove Theorem 1.4. As before, we refer to [27, § G be a finite group, B a block of O G , and D a defect group of B . Wedenote by Irr( B ) the subset of all χ ∈ Irr( G ) satisfying χ (1 B ) = χ (1). Let i bea source idempotent in B D ; that is, i is a primitive idempotent in B D satisfyingBr D ( i ) = 0. Then A = iBi = i O Gi is a source algebra of B . We view A as a D -interior O -algebra with the embedding of D → A × induced by multiplicationwith i . By [21, 3.5], the A - B -bimodule iB = i O G and the B - A -bimodule Bi = O Gi induce a Morita equivalence between A and B . In particular, if X is a simple K ⊗ O B -module, then iX is a simple K ⊗ O A -module, and this correspondenceinduces a bijection between Irr( B ) and the set of isomorphism classes of simple K ⊗ O A -modules. Equivalently, the map e χ ie χ is a bijection between primitiveidempotents in Z ( K ⊗ O B ) and Z ( K ⊗ O A ). If U is a B -lattice with character χ ∈ Irr( B ), then iU is an A -lattice such that K ⊗ O iU is a simple K ⊗ O A -modulecorresponding to χ . This Morita equivalence induces a bijection between the O -free quotients of B and of A . If χ ∈ Irr( B ), then the O -free quotient O Ge χ = Be χ corresponds to the O -free quotient i O Gie χ = Aie χ = Ae χ . Note that Ae χ is againa D -interior O -algebra, via the canonical surjection A → Ae χ . Note further that Ae χ is a direct summand of Be χ as an O D - O D -bimodule, since it is obtained frommultiplying Be χ on the left and on the right by the idempotent ie χ in ( Be χ ) D .The next result shows that anchors of χ can be characterised in terms of the order Ae χ . This is based on a variation of standard arguments, similar to those used in[15, § Theorem 4.1.
Let G be a finite group, B a block of O G , D a defect group of B ,and i ∈ B D a source idempotent. Set A = iBi . Let χ ∈ Irr( B ) .(i) Let Q be a p -subgroup of G such that ( Be χ )( Q ) = 0 . Then there is x ∈ G suchthat x Q ≤ D and such that ( Ae χ )( x Q ) = { } .(ii) Let Q be a subgroup of D of maximal order subject to ( Ae χ )( Q ) = { } . Then Q is an anchor of χ .Proof. We use basic properties of local pointed groups; see e. g. [27, §
18] for anexpository account of this material. Let γ be the local point of D on B containing i . Let Q be a p -subgroup of G such that ( Be χ )( Q ) = { } . Since e χ is the unitelement of Be χ , this is equivalent to Br Be χ Q ( e χ ) = 0. By considering a primitivedecomposition of 1 B in B Q it follows that there is a primitive idempotent j ∈ B Q such that Br Be χ Q ( je χ ) = 0. Then necessarily also Br BQ ( j ) = 0, because the canonicalmap B → Be χ sends ker(Br BQ ) to ker(Br Be χ Q ). Thus j belongs to a local point δ of Q on B . Since the maximal local pointed groups on B are all G -conjugate, it followsthat there is x ∈ G such that x Q δ ≤ D γ . In other words, after replacing Q δ by asuitable G -conjugate, we may assume that Q δ ≤ D γ , and hence that j ∈ A Q forsome choice of j in δ . For this choice of j , we have je χ ∈ Ae χ . Thus the condition NCHORS 9 Br Be χ Q ( je χ ) = 0 is equivalent to Br Ae χ Q ( je χ ) = 0; we use here the fact, mentionedabove, that Ae χ is a direct summand of Be χ as an O P - O P -bimodule. In particular,we have ( Ae χ )( Q ) = { } . This proves (i). For (ii), let Q be a subgroup of D suchthat ( Ae χ )( Q ) = { } and such that the order of Q is maximal with respect to thisproperty. Then ( Be χ )( Q ) = { } , and hence Q is contained in an anchor R of χ .By (i), there is x ∈ G such that x R ≤ D and such that ( Ae χ )( x R ) = { } . Themaximality of | Q | forces Q = R , whence the result. (cid:3) Theorem 4.1 implies that anchors are invariant under source algebra equiva-lences. By a result of Scott [25] and Puig [23, 7.5.1], an isomorphism betweensource algebras is equivalent to a Morita equivalence given by a bimodule with atrivial source (see also [16, §
4] for an expository account). In order to extend theinvariance of anchors to Morita equivalences given by bimodules with endopermu-tation source, we need to describe these Morita equivalences at the source algebralevel. Let G , G ′ be finite groups, and let B , B ′ be blocks of O G , O G ′ with defectgroups D , D ′ , respectively. By results of Puig in [23, § B and B ′ given by a bimodule with endopermutation source implies anidentification D = D ′ such that for some choice of source idempotents i ∈ B D , i ′ ∈ ( B ′ ) D , setting A = iBi and A ′ = i ′ B ′ i ′ , we have D -interior O -algebra isomorphisms A ′ ∼ = e ( S ⊗ O A ) e , A ∼ = e ′ ( S op ⊗ O A ′ ) e ′ , where S = End O ( V ) for some indecomposable endopermutation O D -module V with vertex D , and where e , e ′ are primitive idempotents in ( S ⊗ O A ) D , ( S op ⊗ O A ′ ) D , respectively, satisfying Br D ( e ) = 0, Br D ( e ′ ) = 0. These isomorphisms induceinverse equivalences between mod( A ) and mod( A ′ ), sending an A -module U to the A ′ -module e ( V ⊗ O U ), and an A ′ -module U ′ to the A -module e ′ ( V ∗ ⊗ O U ′ ). Here V ∗ is the O -dual of V ; note that End O ( V ∗ ) ∼ = S op as D -interior O -algebras. Proof of Theorem 1.4.
We use the notation above. Let χ ∈ Irr( B ) and χ ′ ∈ Irr( B ′ ) such that χ and χ ′ correspond to each other through the Morita equiv-alence mod( B ) ∼ = mod( A ) ∼ = mod( A ′ ) ∼ = mod( B ′ ) described above. As mentionedat the beginning of this section, the primitive idempotent in Z ( K ⊗ O A ) corre-sponding to χ is ie χ . Similarly, the primitive idempotent in Z ( K ⊗ O A ′ ) is equal to i ′ e χ ′ . The explicit description of the Morita equivalence between A and A ′ aboveimplies that we have i ′ e χ ′ = e · (1 S ⊗ ie χ ) ie χ = e ′ · (1 S op ⊗ i ′ e χ ′ )where these equalities are understood in the algebras K ⊗ O A and K ⊗ O A ′ . ByTheorem 4.1, it suffices to show that for Q a subgroup of D , we have ( Ae χ )( Q ) = { } if and only if ( A ′ e χ ′ )( Q ) = { } . It suffices to show one implication, because theother follows then from exchanging the roles of A and A ′ . Thus it suffices to showthat if ( Ae χ )( Q ) = { } , then ( A ′ e χ ′ ) ′ ( Q ) = { } . Let Q be a subgroup of D suchthat ( Ae χ )( Q ) = { } . We have ( S ⊗ O A )(1 S ⊗ ie χ ) = S ⊗ O Ae χ . Since S has a D -stable basis, it follows from [22, 5.6] that ( S ⊗ O Ae χ )( Q ) = S ( Q ) ⊗ k ( Ae χ )( Q ) = { } . Since A ′ e χ ′ is obtained from S ⊗ O Ae χ by left and right multiplication withthe idempotent e , it follows that ( A ′ e χ ′ )( Q ) = { } as required. (cid:3) Anchors and normal subgroups
We prove in this section some results on anchors of characters which are inducedfrom a normal subgroup or inflated from quotients. Since an anchor of an irreduciblecharacter χ contains a vertex of any lattice affording χ , constructing suitable latticesis one of the tools for getting lower bounds on anchors. The following is well-known(we include a proof for the convenience of the reader). Lemma 5.1.
Let G be a finite group and χ ∈ Irr( G ) . Let S be a simple kG -modulewith Brauer character ϕ such that d χϕ = 0 . Then there exists an O G -lattice L affording χ such that L has a unique maximal submodule M , and such that L/M ∼ = S .Proof. Let i be a primitive idempotent in O G such that O Gi is a projective coverof S . Since χ ( i ) = d χϕ = 0, there is an O -pure submodule L ′ of O Gi such that L = O Gi/L ′ affords χ . Since the projective indecomposable O G -module O Gi hasa unique maximal submodule and S is its unique simple quotient, it follows thatthe image, denoted M , in L of the unique maximal submodule of O Gi is the uniquemaximal submodule of L , and satisfies L/M ∼ = S . (cid:3) In [19], Plesken showed that if G is a p -group and χ is an irreducible character of G , then there exists an O G -lattice affording χ whose vertex is G . Our next resultis a slight variation on this theme. Proposition 5.2.
Let G be a finite group, N a normal subgroup of G of p -powerindex, and χ ∈ Irr( G ) . If there exists ϕ ∈ IBr( G ) of degree not divisible by | G : N | and such that d χ,ϕ = 0 , then there exists an O G -lattice L with character χ whichis not relatively O N -projective. In particular, in that case, N does not contain theanchors of χ .Proof. Let S be a simple kG -module with Brauer character ϕ of degree not divisibleby | G : N | and such that d χ,ϕ = 0. By 5.1 there exists an O G -lattice L with a uniquemaximal submodule M such that χ is the character of L and such that L/M ∼ = S .Note that the character of M is also equal to χ . We will show that one of M or L is not relatively O N -projective. Arguing by contradiction, suppose that L and M are relatively O N -projective. By Green’s indecomposability theorem, there areindecomposable O N -modules Z and U such that L ∼ = Ind GN ( Z ) and M ∼ = Ind GN ( U ).Then χ = Ind GN ( τ ), where τ is the character of Z . Since χ = Ind GN ( τ ) is irreducible,it follows that the different G -conjugates x τ of τ , with x running over a set ofrepresentatives R of G/N in G , are pairwise different. Similarly, χ = Ind GN ( τ ′ ),where τ ′ is the character of U . After replacing U by x U for a suitable element x ∈ G , we may assume that τ ′ = τ . By the above, we have Res GN ( L ) ∼ = L x ∈R x Z ,and the characters of these summands are the pairwise different conjugates x τ of τ . In particular, Res GN ( L ) has a unique O -pure summand with character τ , andthis summand is isomorphic to Z . We denote this summand abusively again by Z .Similarly, Res GN ( M ) has a unique O -pure summand, abusively again denoted by U ,with character τ . Since M ⊆ L induces an equality K ⊗ O M = K ⊗ O L , it followsthat K ⊗ O U = K ⊗ O Z . Moreover, we have U ⊆ K ⊗ O Z ∩ L = Z , where thesecond equality holds as Z is O -pure in L . Thus the inclusion M ⊆ L is obtainedfrom inducing the inclusion map U ⊆ Z from N to G . By the construction of M ,the inclusion M ⊆ L induces a map k ⊗ O M → k ⊗ O L with cokernel S . Thus we NCHORS 11 havedim k ( S ) = codim( k ⊗ O M → k ⊗ O L ) = | G : N | codim( k ⊗ O U → k ⊗ O Z ) . This contradicts the assumption that ϕ (1) is not divisible by | G : N | . Thus one of L or M is not relatively O N -projective. (cid:3) Corollary 5.3.
Let G be a finite group, N a normal subgroup of p -power index,and let χ ∈ Irr( G ) . Let P be an anchor of χ . If there exists ϕ ∈ IBr( G ) of degreeprime to p such that d χϕ = 0 , then G = P N .Proof.
Arguing by contradiction, suppose that
P N is a proper subgroup of G . Since G/N is a p -group, it follows that P N is contained in a normal subgroup M of index p in G . Then M contains every anchor of χ , hence M contains the vertices of any O G -lattice affording χ . Let ϕ ∈ IBr( G ) such that ϕ (1) is prime to p and suchthat d χ,ϕ = 0. Proposition 5.2 implies however that | G : M | = p divides ϕ (1), acontradiction. (cid:3) We record here an extension of [19, Lemma 3] which will be used in the nextsection.
Proposition 5.4.
Let G be a finite group, P a Sylow p -subgroup, and χ ∈ Irr( G ) .Suppose that Res GP ( χ ) is irreducible and that there exists an irreducible Brauer char-acter ϕ of p ′ -degree of G such that d χ,ϕ = 0 . Then there exists an O G -lattice L affording χ with vertex P . In particular, the Sylow p -subgroups of G are the anchorsof χ .Proof. Let π be a generator of J ( O ) and let S be a simple kG -module with Brauercharacter ϕ . By 5.1, there is an O G -lattice L affording χ such that k ⊗ O L hasa simple head isomorphic to S . Let N be the maximal submodule of L . Then πL ⊂ N and N is an O G -lattice affording χ . The invariant factors of the O -module L/N are either 1 or π and the number of non-trivial invariant factors of L/N equals dim k ( L/N ) = dim k ( S ). Thus the product of the invariant factorsof L/N equals π dim k S . By hypothesis, Res GP ( L ) is irreducible. If Res GP ( L ) hasvertex P , then L has vertex P . So, we may assume that Res GP ( L ) is relatively U -projective for some proper subgroup U of P . By Green’s indecomposabilitytheorem, Res GP ( L ) = Ind PU ( M ) for some O U -lattice M . By [19, Lemma 3], Res GP ( N )has vertex P , whence N has vertex P . Note that [19, Lemma 3] is stated for O a localisation of the | P | -th cyclotomic integers, but as remarked in [19, Page 235],[19, Lemma 3] remains true in our setting. The second assertion of the propositionfollows from Proposition 3.1. (cid:3) Proposition 5.5.
Let G be a finite group, N a normal subgroup, and let χ ∈ Irr( G ) such that χ = Ind GN ( τ ) for some τ ∈ Irr( N ) . Let V be an O N -lattice withcharacter τ . Suppose that the composition series of the kN -modules k ⊗ O x V , with x running over a set of representatives of G/N in G , are pairwise disjoint. Then O Ge χ ∼ = Ind GN ( O N e τ ) as G -interior O -algebras. In particular, N contains thevertices of all lattices affording χ , and N contains the anchors of χ .Proof. It suffices to show that e τ belongs to O Ge χ . Indeed, if this is true, then theassumptions on χ and τ imply that e χ = Tr GN ( e τ ), and the different conjugates of e τ appearing in Tr GN ( e τ ) are pairwise orthogonal idempotents in O Ge χ . In particular,we have e τ O Ge τ = O N e τ . It follows from [27, (16.6)] that O Ge χ ∼ = Ind GN ( O N e τ ). It remains to show that e τ belongs to O Ge χ . Let I be a primitive decompositionof 1 in O N . Let i ∈ I such that e τ i = 0. This condition is equivalent to k ⊗ O V having a composition factor isomorphic to the unique simple quotient T i of the O N -module O N i . Since the different G -conjugates of the kN -module k ⊗ O V havepairwise disjoint composition series, it follows that e x τ i = 0 for x ∈ G \ N . Thus e χ i = e τ i ∈ O Ge χ for any i ∈ I such that e τ i = 0. Taking the sum over all such i implies that e τ ∈ O Ge χ . The last statement follows from the fact that e χ =Tr GN ( e τ ) and Higman’s criterion, for instance, or directly from the fact that Ind GN induces a Morita equivalence between O N e τ and O Ge χ . (cid:3) Corollary 5.6.
Let G be a finite group, N a normal subgroup of p -power index,and let χ ∈ Irr( G ) such that χ = Ind GN ( τ ) for some τ ∈ Irr( N ) . Suppose that d χ,ϕ is either or for every ϕ ∈ IBr( G ) . Then N contains the anchors of χ .Proof. Let V be an O N -lattice affording τ . Let I be a primitive decomposition of1 in O N . By Green’s indecomposability theorem, I remains a primitive decompo-sition in O G . Let i ∈ I . We have χ ( i ) = X x x τ ( i )where x runs over a set of representatives R of G/N in G . By the assumptions onthe decomposition numbers of χ , the left side is either 1 or 0. Thus either x τ ( i ) =0 for all x ∈ R , or there is exactly one x = x ( i ) ∈ R with x τ ( i ) = 0. This impliesthat the composition series of the different G -conjugates of k ⊗ O V are pairwisedisjoint. The result follows from 5.5. (cid:3) Proposition 5.7.
Let G be a finite group, N a normal subgroup of G , and χ ∈ Irr( G ) . Suppose that χ is the inflation to G of an irreducible character ψ ∈ Irr(
G/N ) . Let P be an anchor of χ . Then P N/N is an anchor of ψ , and P ∩ N isa Sylow p -subgroup of N .Proof. Let d ∈ ( O Ge χ ) P such that Tr GP ( d ) = e χ . The assumptions imply that thecanonical map G → G/N induces a G -algebra isomorphism O Ge χ ∼ = O G/N e ψ suchthat N acts trivially on both algebras. Thus e χ = Tr GP ( d ) = | P N : P | Tr GP N ( d ).This implies that P is a Sylow p -subgroup of P N , and hence that P ∩ N is aSylow p -subgroup of N . Since the isomorphism O Ge χ ∼ = O G/N e ψ sends Tr GP N ( d )to Tr G/NP N/N ( ¯ d ), where ¯ d is the canonical image of d , it follows that P N/N containsan anchor of ψ . Using the fact that P is a Sylow p -subgroup of P N , one easilychecks that any proper subgroup of
P N/N is of the form
QN/N for some propersubgroup Q of P containing P ∩ N . The previous isomorphism implies that P N/N is an anchor of ψ . (cid:3) Example 5.8. (1) p = 2, G = S , χ (1) = 2: Then χ lies in a defect zero block of G , hence by Theorem 1.2, the trivial group is the only anchor of χ .(2) p = 2, G = S , χ (1) = 2: Then χ is inflated from the character in part (1).In this case, by Proposition 5.7 the Klein four subgroup V of S is the only anchorof χ . However, the defect groups of the block containing χ (i.e. of the principalblock of S ) are the Sylow 2-subgroups of S (i.e. dihedral groups of order 8).(3) p = 2, G = S : All irreducible characters of G except the one of degree 6are of height zero in their block. So their anchors coincide with their defect groups,by Theorem 1.3 (a) . NCHORS 13
Now let χ ∈ Irr( G ) with χ (1) = 6. Then χ is induced from an irreducible characterof the alternating group A of degree 3. Thus there exists an O G -lattice affording χ with vertex V .On the other hand, χ is labelled by the partition λ = (3 , ,
1) of 5. By the remarkon p. 511 of [28], the Specht module S λ is indecomposable, and S × S is a vertexof S λ , by Theorem 2 in [28]. Thus the anchors of χ are Sylow 2-subgroups of G ,by Theorem 1.2 (b). 6. Navarro vertices
We prove Theorems 1.5 and 1.6.
Theorem 6.1.
Let G be a finite p -solvable group. Let χ ∈ Irr( G ) and let ( Q, δ ) bea Navarro vertex of χ . Supppose that χ ◦ ∈ IBr( G ) . Then Q contains an anchor of χ . Moreover, if δ = 1 Q or if p is odd, then Q is an anchor of χ .Proof. Since χ ◦ ∈ IBr( G ), there is a unique O G -lattice L affording χ , up to iso-morphism. Moreover, k ⊗ O L is the unique simple kG -module with Brauer char-acter χ ◦ , up to isomorphism. Recall that there is a nucleus ( W, γ ) of χ such that χ = Ind GW ( γ ), and Q ∈ Syl p ( W ) (cf. [17, p. 2763]). Further, γ ∈ Irr( W ) hasa unique factorization γ = αβ where α ∈ Irr( W ) is p ′ -special and β ∈ Irr( W ) is p -special. Going over to Brauer characters, we have χ ◦ = Ind GW ( γ ◦ ) and γ ◦ = α ◦ β ◦ ;in particular, γ ◦ , α ◦ , β ◦ ∈ IBr( W ). Let R be a vertex of the unique O W -latticeaffording γ and let R be a vertex of the unique kW -module affording γ ◦ . Then,up to conjugation in W , R ≤ R ≤ Q . Since χ = Ind GW ( γ ), R is also a vertex ofthe O G -lattice affording χ and hence by Proposition 3.1 (iii), R is an anchor of χ .This proves the first assertion.Since α is p ′ -special, the p -part of the degree of γ equals the p -part of the degreeof β . Since G is p -solvable, it follows that | R | = | W | p β (1) p . Now suppose that p is odd. Since β ◦ is irreducible, by [18, Lemma 2.1], β islinear. It follows from the above that R = Q = R , proving the second assertionwhen p is odd. Since δ = Res WQ ( β ), a similar argument works when δ = 1 Q . (cid:3) Lemma 6.2.
Let G be a finite p -solvable group and χ ∈ Irr( G ) . Suppose that χ is p -special and that there exists ϕ ∈ IBr( G ) of p ′ -degree such that d χϕ = 0 . Thenthere exists an O G -lattice affording χ with vertex a Sylow p -subgroup of G .Proof. This is immediate from Proposition 5.4 and the fact that the restrictionof a p -special character of G to a Sylow p -subgroup of G is irreducible (cf. [6,Prop. 6.1]). (cid:3) The following is due to G. Navarro.
Lemma 6.3.
Let G be a finite group of odd order and χ ∈ Irr( G ) . Suppose that χ is p -special. Then the trivial Brauer character of G is a constituent of χ ◦ .Proof. By the Feit-Thompson theorem, G is solvable and hence p -solvable. Let H be a p -complement of G and let ζ be a primitive | G | p -th root of unity. By [6, Theo-rem 6.5], Q [ ζ ] is a splitting field of χ . Thus Res GH ( χ ) is a rational valued characterof odd degree. Hence, Res GH ( χ ) contains a real valued irreducible constituent, say α . By Brauer’s permutation lemma, the number of real-valued irreducible charac-ters of H equals the number of real conjugacy classes of H . Since | H | is odd, α isthe trivial character of H . By Frobenius reciprocity, χ is an irreducible constituentof Ind GH ( α ). On the other hand, since H is a p -complement of G , Ind GH ( α ) is thecharacter of the projective indecomposable O G -module corresponding to the trivial kG -module. (cid:3) Combining the two results above yields the Theorem 1.6. In fact we prove more.
Theorem 6.4.
Let G be a finite group of odd order, let χ ∈ Irr( G ) and let ( Q, δ ) be a Navarro vertex of χ . Then there exists an O G -lattice affording χ with vertex Q . In particular, Q is contained in an anchor of χ .Proof. Let (
W, γ ) be a nucleus of χ such that Q is a Sylow p -subgroup of W and δ = Res WQ ( α ), where γ = αβ , with α a p ′ -special character and β a p -specialcharacter of W (cf. [17, Sections 2,3]). By Lemma 6.3, the trivial Brauer characteris a constituent of β ◦ . By Lemma 6.2, there exists an O W -lattice X affording β and with vertex Q . Let Y be an O W -lattice affording α . Then V = Y ⊗ X is an O W -lattice affording γ . We claim that V has vertex Q . Indeed if V isrelatively R -projective, then every indecomposable sumand of Y ∗ ⊗ V is relatively R -projective. On the other hand, since α has p ′ -degree, the O W -lattice Y ∗ ⊗ Y hasa direct summand isomorphic to the trivial O W -module. Thus, Y ∗ ⊗ Y ⊗ X hasa direct summand isomorphic to X . Since Q is a vertex of X , and Q is a Sylow p -subgroup of W , it follows that R is a Sylow p -subgroup of W . Then Ind GW ( V )is an O G -lattice with character χ = Ind GW ( γ ). Clearly, Ind GW ( V ) is relatively O Q -projective. Suppose if possible that Ind GW ( V ) is relatively O R -projective for someproper subgroup R of Q , say Ind GW ( V ) is a summand of Ind GR ( X ) for some R properly contained in Q and for some O R -lattice X . By the Mackey formula, V isa summand of Ind WW ∩ x R (Res x RW ∩ x R x X ) for some x ∈ G . This is a contradiction as | x R | < | Q | . Thus Q is a vertex of Ind GW ( V ) proving the first assertion. The secondis immediate from the first and Proposition 5.4. (cid:3) I. M. Isaacs and G. Navarro provided us with an example of a p -special char-acter of a p -solvable group none of whose irreducible Brauer constituents havedegree prime to p . Proposition 5.5 can be used to prove that the anchors of theIsaacs-Navarro example, which we give below, are strictly contained in the Sylow p -subgroups of the ambient group (so in particular, these characters are not affordedby any lattice with full vertex). Example 6.5. [Isaacs -Navarro] Let p = 5 and let M be the semidirect product ofan extraspecial group of order 5 and of exponent 5, acted on faithfully by Q wherethe action is trivial on the center. Let G = M ≀ C be the wreath product of M bya cyclic group of order 5. In G , there is the normal subgroup N = M × · · · × M ,with each M i isomorphic to M . Also, there is a cyclic subgroup C of order 5 thatpermutes the M i transitively. Note that M has a 5-special character α of degree5. Let θ ∈ Irr( N ) be the product of α with trivial characters of M , M , M and M . Then θ has degree 5 and χ = θ G is 5-special of degree 25.There is a Sylow 2-subgroup S of G with the form Q × Q × ... × Q , where Q i is a Sylow 2-subgroup of M i and the Q i are permuted transitively by C . Now θ S is the product of α Q with trivial characters on the other Q i . NCHORS 15
Also, α Q is the sum of the irreducible character of degree 2 and the threenontrivial linear characters, so there is no trivial constituent. It follows that θ S hasno C -invariant irreducible constituent. The same is therefore true about χ S . Theneach 5-Brauer irreducible constituent of χ has degree divisible by 5.The construction also shows that if x ∈ G \ N , then θ and x θ have no irre-ducible Brauer constituents in common. So by Proposition 5.5, the anchors of χ are contained in N .In conjunction with Proposition 5.2, the following example provides characterswhose anchors are not contained in Navarro vertices. The construction is similarto that in the Isaacs-Navarro example above. Example 6.6.
Suppose that M = O p,p ′ ( M ), and α is an irreducible p -specialcharacter of M such that Res MO p ( M ) α is irreducible. Suppose further that thereexists a nontrivial irreducible Brauer character ϕ of M such that d α,ϕ = 0. Let β be the irreducible character of M with O p ( M ) in the kernel of β and such that β ◦ equals the dual ϕ ∗ of ϕ . Then β is p ′ -special.Let G = M ≀ C p . In G , there is the normal subgroup N = M ×· · ·× M p with each M i isomorphic to M . Let ˜ α ∈ Irr( N ) be the product of α and the trivial charactersof M , · · · , M p , let ˜ β ∈ Irr( N ) be the product of β and the trivial characters of M , · · · , M p and let ˜ ϕ be the product of ϕ with the trivial Brauer characters of M , · · · , M p . Let χ = Ind GN (˜ α ˜ β ).Since ˜ α is p -special and ˜ β is p ′ -special, by results of [6], ˜ α ˜ β is an irreduciblecharacter of N . By construction, neither ˜ α nor ˜ β is G -stable. Hence, also bygeneral results on p -factorable characters, ˜ α ˜ β is not G -stable. Since | G/N | = p , itfollows that χ is an irreducible character of G . Now, since ˜ β is not G -stable, it iseasy to see that χ is not p -factorable. On the other hand, N is a maximal normalsubgroup of G . Thus ( N, ˜ α ˜ β ) is a nucleus of G in the sense of [17], and the Sylow p -subgroups of N are the first components of the Navarro vertices of χ .We have (˜ α ˜ β ) ◦ = ˜ α ◦ ˜ β ◦ , and ˜ ϕ is an irreducible Brauer constituent of ˜ α and˜ β ◦ = ˜ ϕ ∗ . Since ˜ ϕ (1) = ϕ (1) is relatively prime to p , it follows that the trivialBrauer character of N is a constituent of (˜ α ˜ β ) ◦ . Consequently, the trivial Brauercharacter of G occurs as a constituent of χ . Thus, by Proposition 5.2, the anchorsof χ are not contained in N . 7. Lifting
Let G be a finite group and χ ∈ Irr( G ). Let P be an anchor of χ . Then k ⊗ O O Ge χ is a G -interior k -algebra. Since( k ⊗ O O Ge χ ) G = Z ( k ⊗ O O Ge χ )is a local k -algebra, it follows that k ⊗ O O Ge χ is a primitive G -interior k -algebra.Since k ⊗ O ( O Ge χ ) P ⊆ ( k ⊗ O O Ge χ ) P ,k ⊗ O O Ge χ has a defect group Q contained in P . We will see below that weoften (but not always) have equality here. If χ ◦ ∈ IBr( G ), then there is, up toisomorphism, a unique O G -lattice L affording χ , and k ⊗ O L is the unique simple kG -module with Brauer character χ ◦ , up to isomorphism. We have seen above thatin that case the G -interior O -algebra O Ge χ is isomorphic to End O ( L ). This impliesthat the G -interior k -algebra k ⊗ O O Ge χ is isomorphic to End k ( k ⊗ O L ). Thus the anchor P of χ is a vertex of L , and the defect group Q of k ⊗ O O Ge χ is a vertexof k ⊗ O L . The examples 7.1 and 7.2 below illustrate the cases where Q = P and Q < P , respectively.
Example 7.1.
Let G be the symmetric group S n , for a positive integer n . Let χ ∈ Irr( G ) such that χ ◦ ∈ IBr( G ), and let L be an O G -lattice affording χ . Weclaim that L and k ⊗ O L have the same vertices.Indeed, let λ be the partition of n labelling χ . Since the Specht lattice S λ O isan O G -lattice affording χ , the uniqueness of L implies that S λ O ∼ = L . Thus the kG -module S λk ∼ = k ⊗ O S λ O ∼ = k ⊗ O L has Brauer character χ ◦ and is thereforesimple.A result by Hemmer (cf. [8]) implies that S λk lifts to a p -permutation O G -lattice M . Then K ⊗ O M is a simple KG -module; that is, K ⊗ O M ∼ = S µK ∼ = K ⊗ O S µ O forsome partition µ of n . Moreover, S µk is isomorphic to k ⊗ O M ∼ = S λk ; in particular,we have Hom kG ( S λk , S µk ) = 0 = Hom kG ( S µk , S λk ) . Suppose first that p >
2. Then [11, Proposition 13.17] implies that λ ≥ µ and µ ≥ λ , hence µ = λ . The uniqueness of L implies that S λ O ∼ = L ∼ = M ; in particular, L is a p -permutation O G -lattice. Hence L and k ⊗ O L have the same vertices.It remains to consider the case p = 2. In this case a theorem by James andMathas (cf. [12]) implies that either λ is 2-regular, or the conjugate partition λ ′ is2-regular, or n = 4 and λ = (2 , λ is 2-regular. If µ isalso 2-regular then we certainly have λ = µ . If µ ′ is 2-regular then we have λ = µ ′ ,in a similar way. Now, arguing as in the case p >
2, we conclude that L and k ⊗ O L have the same vertices. Example 7.2.
Let p = 2, G = GL (2 ,
3) and N = SL (2 , R be the uniqueSylow 2-subgroup of N and H a complement of R in N . Let τ be the 2-dimensionalirreducible character of R and let η be the unique extension of τ to N with deter-minantal order a power of 2 (cf. Corollary (6.28) in [10]). Let χ be an extensionof η to G . Then χ is 2-special, by [9, Proposition 40.5]. Further, χ ◦ is irreducibleand equals Ind GN ( ψ ), where ψ is a linear Brauer character of N . (Note that therestriction of χ ◦ to H is a sum of two distinct irreducible Brauer characters).Thus, R is a vertex of the unique kG -module affording χ ◦ and R is contained insome (and hence every) vertex of the O G -lattice affording χ . Since χ is not inducedfrom any character of N , and G/N is a 2-group, Green’s indecomposability theoremimplies that the O G -lattice affording χ is not relatively N -projective. Hence R isproperly contained in a vertex of the O G -lattice affording χ , which is consequentlya Sylow 2-subgroup of G . Remark 7.3.
Let G be a finite p -solvable group and χ ∈ Irr( G ) such that χ ◦ ∈ IBr( G ). Let L be an O G -lattice affording χ . Suppose, as in the above examplethat a vertex P of L strictly contains a vertex R of k ⊗ O L . Let S be an O P -latticesource of L . We claim that S is not an endopermutation module. Indeed, assumethe contrary. Since P is a vertex of S and since S is endopermutation, k ⊗ O S is anindecomposable endopermutation kP -module with vertex P . On the other hand, k ⊗ O S is a direct summand of k ⊗ O L , k ⊗ O L has vertex R , and R is strictlycontained in P , a contradiction. NCHORS 17
Acknowledgements.
The authors are grateful to Susanne Danz for her helpwith the example on symmetric groups. They would also like to thank GabrielNavarro for providing us with Lemma 6.3 and Marty Isaacs and Gabriel Navarrofor providing us with Example 6.5. Work on this paper started when the secondauthor visited the City University of London in 2013, with the kind support of theDFG, SPP 1388. He is grateful for the hospitality received at the City University.
References [1] L. Barker,
Defects of Irreducible Characters of p -Soluble Groups , J. Algebra (1998),178–184.[2] J. P. Cossey, Vertex subgroups and vertex pairs in solvable groups , in: Character theoryof finite groups, pp. 17–32, Contemp. Math., 524, Amer. Math. Soc., Providence, RI,2010.[3] J. P. Cossey and M. L. Lewis,
Lifts and vertex pairs in solvable groups , Proc. Edinb.Math. Soc. (2) (2012), 143–153.[4] C. W. Curtis and I. Reiner, Methods of representation theory, Vol I, John Wiley andSons, New York, 1981.[5] C. W. Eaton, Vertices for irreducible characters of a class of blocks , J. Algebra (2005), 492-499.[6] D. Gajendragadkar,
A class of characters of finite π -separable groups , J. Algebra (1979), 237–259.[7] J. A. Green, Some remarks on defect groups , Math. Z. (1968), 133–150.[8] D. J. Hemmer,
Irreducible Specht modules are signed Young modules , J. Algebra (2006), 433-441.[9] B. Huppert, Character theory of finite groups, Walter de Gruyter, Berlin - New York,1998.[10] I. M. Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1976.[11] G. D. James, The representation theory of the symmetric groups, Springer-Verlag, Berlin,1978.[12] G. James and A. Mathas,
The irreducible Specht modules in characteristic
2, Bull. Lon-don Math. Soc. (1999), 457-462.[13] R. Kessar, S. Koshitani, M. Linckelmann On symmetric quotients of symmetric algebras ,J. Algebra (2015), 423–437.[14] R. Kn¨orr,
On the vertices of irreducible modules , Ann. of Math. (2) (1979), 487-499.[15] M. Linckelmann,
The source algebras of blocks with a Klein four defect group , J. Algebra (1994), 821–854.[16] M. Linckelmann,
On splendid derived and stable equivalences between blocks of finitegroups , J. Algebra (2001), 819–843.[17] G. Navarro,
Vertices for characters of p -solvable groups , Trans. Amer. Math. Soc. (2002), 2759-2773.[18] G. Navarro, Modularly irreducible characters and normal subgroups , Osaka J. Math (2011), 329-332.[19] W. Plesken, Vertices of irreducible lattices over p -groups , Comm. Algebra (1982),227-236.[20] C. Picaronny and L. Puig, Quelques remarques sur un th`eme de Kn¨orr , J. Algebra (1987) 69–73.[21] L. Puig,
Pointed groups and construction of characters.
Math. Z. (1981), 265–292.[22] L. Puig,
Nilpotent blocks and their source algebras , Invent. Math. (1988), 77–116.[23] L. Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks ,Progress in Math. , Birkh¨auser Verlag, Basel (1999)[24] G. R. Robinson,
Local structure, vertices, and Alperin’s conjecture , Proc. London Math.Soc. (1996), 312–330.[25] L. L. Scott, unpublished notes (1990).[26] J. Th´evenaz, Duality in G -algebras , Math. Z. (1988), 47–85.[27] J. Th´evenaz, G -algebras and modular representation theory, Clarendon Press, Oxford,1995. [28] M. Wildon, Two theorems on the vertices of Specht modules , Arch. Math. (2003),505-511. Department of Mathematics, City University London EC1V 0HB, United Kingdom
E-mail address : [email protected] Institut f¨ur Mathematik, Friedrich-Schiller-Universit¨at, 07743 Jena, Germany
E-mail address : [email protected] Department of Mathematics, City University London EC1V 0HB, United Kingdom
E-mail address ::