Quasi-projective characters in a block Wolfgang Willems Burkhard K - PowerPoint PPT Presentation

Quasi-projective characters in a block Wolfgang Willems Burkhard K¨ ulshammer’s 60th birthday, Jena, July 22 -25, 2015 1

Essen, 198? 2

Marseille, in front of Notre Dame de la Garde, 1986? 3

(A. Zalesski) 1. Introduction Def. a) An ordinary character Λ of a finite group G is called quasi-projective if � Λ = a ϕ Φ ϕ with a ϕ ∈ Z ϕ ∈ IBr p ( G ) where Φ ϕ denotes the ordinary character associated to the projective cover of the module afforded by ϕ . b) A p -Brauer character Φ is called quasi-projective if � a ϕ Φ ϕ ) ◦ . Φ = ( ϕ ∈ IBr p ( G ) 4

Def. We call a quasi-projective character Λ (resp. Φ) indecomposable if there is no splitting Λ = Λ 1 + Λ 2 (resp.Φ = Φ 1 + Φ 2 ) with Λ i (resp.Φ i ) � = 0 and quasi-projective character. Remark. An indecomposable quasi-projective charac- ter belongs to a block. 5

To be brief we put Iqp( B ) = set of indecomposable quasi-projective ordi- nary characters of the p -block B (call that: Hilbert basis for the decomp. matrix of B ) IBqp( B ) = set of indecomposable quasi-projective Brauer characters of B . (call that: Hilbert basis for the Cartan matrix of B ) 6

Example. G = A 5 , p = 2 , B 0 the principal block. Irr( B 0 ) = { χ 1 , χ 2 , χ 3 , χ 5 } degrees: 1 , 3 , 3 , 5 IBr 2 ( B 0 ) = { β 1 , β 2 , β 3 } degrees: 1 , 2 , 2 | Iqp( B 0 ) | = 4 : Φ 1 − Φ 3 = 1 + χ 2 , Φ 1 − Φ 2 = 1 + χ 3 , Φ 2 = χ 2 + χ 5 , Φ 3 = χ 3 + χ 5 7

| IBqp( B 0 ) | =6: (3Φ 1 − 2Φ 2 − 2Φ 3 ) ◦ = 4 β 1 (2Φ 2 − Φ − 1) ◦ = 2 β 2 (2Φ 3 − Φ 1 ) ◦ = 2 β 3 (Φ 1 − Φ 3 ) ◦ = 2 β 1 + β 2 (Φ 1 − Φ 3 ) ◦ = 2 β 1 + β 3 (Φ 2 + Φ 3 − Φ 1 ) ◦ = β 2 + β 3 8

Example. G = PSL (2 , 7) , p = 7 , B 0 the principal block. Irr( B 0 ) = { χ 1 , χ 2 , χ 3 , χ 4 , χ 5 } degrees: 1 , 3 , 3 , 6 , 8 IBr 7 ( B 0 ) = { β 1 , β 2 , β 3 } degrees: 1 , 3 , 5 | Iqp( B 0 ) | = 5: 1 + χ 4 , 1 + χ 2 + χ 3 , χ 4 + χ 5 , χ 2 + χ 3 + χ 5 | IBqp( B 0 ) | = 11: 7 β 1 , 7 β 2 , 7 β 3 , β 1 + 4 β 3 , β 2 + 5 β 3 , 4 β 1 + β 2 , β 1 + β 2 + 2 β 3 , 2 β 1 + β 3 , 2 β 2 + 3 β 3 , β 1 + 2 β 2 , 3 β 2 + β 3 9

Example. G = McL, p = 2 , B 0 the principal block | Irr( B 0 ) | = 18 , | IBr 2 ( B 0 ) = 8 | Iqp( B 0 ) | = 38 = 2 . 19 | IBqp( B 0 ) | = 8304 = 2 4 . 3 . 173 10

Problems. 1. What is the meaning of (indecomposable) quasi- projective? 2. What can we say about Iqp( B ) or IBqp( B )? 3. Is there a reasonable good function in terms of B which bounds | Iqp( B ) | or | IBqp( B ) | ? 11

2. Hilbert bases Let D, C denote the decomposition resp.Cartan matrix of a block B . Quasi-projective characters � � � a ϕ Φ ϕ = ( d χϕ a ϕ ) χ. ϕ ∈ IBr ( B ) χ ∈ Irr ( B ) ϕ ∈ IBr ( B ) � �� � =( Da ) χ ≥ 0 a ϕ Φ ϕ ) ◦ = � � � ( ( c ψϕ a ϕ ) ψ. ϕ ∈ IBr ( B ) ψ ∈ IBr ( B ) ϕ ∈ IBr ( B ) � �� � =( Ca ) ψ ≥ 0 12

Hilbert basis of a matrix A ∈ ( Z ) k,l Def. cone ( A ) = { x ∈ R l | Ax ≥ 0 } Facts. a) (Gordon 1873, Hilbert 1890) cone ( A ) is generated by a finite so-called integral Hilbert- basis; i.e., ∃ h 1 , . . . , h t ∈ cone ( A ) ∩ Z l s.t. any c ∈ cone ( A ) ∩ Z l can be written as c = � t i =1 a i h i with a i ∈ N 0 . 13

b) (van der Corput, 1931) If ker A = 0, then a minimal integral Hilbertbasis is unique. Denote them by H A . c) If ker A = 0, then A H A are the indecomposable vec- tors in A ( cone ( A ) ∩ Z l ). 14

Applications. • D H D = Iqp( B ) C H C = IBqp( B ) • Explicit computations. Software package 4ti2 (Hemmecke, K¨ oppe, Malkin, Walter) 15

3. Indecomp. quasi-projective ordinary characters. • quasi-projective character = p -vanishing character | G | p | Λ(1) if Λ is quasi-projective • • χ ∈ Irr( B ) quasi-projective ⇒ B of defect zero • | Iqp( B ) | ≥ | IBr( B ) | 16

A Brauer character ϕ is called quasi-liftable if Def. there exists an ordinary character χ such that χ ◦ = bϕ with b ∈ N Navarro, 10.16) If Λ = � (cf. ϕ a ϕ Φ ϕ is a Lemma. quasi-projective character and ϕ is quasi-liftable, then a ϕ ≥ 0. (If χ ◦ = nϕ , then na ϕ = (Λ , nϕ ) ◦ = (Λ , χ ) ≥ 0 . ) Example. G = 2 F 4 (2) ′ ˙ 2 and p = 2. There exists a non-liftable β ∈ IBr( G ) and χ, ψ ∈ Irr( G ) χ ◦ = 2 β ψ ◦ = 3 β. such that and 17

Theorem. Equivalent are: a) Iqp( B ) = { Φ ϕ | ϕ ∈ IBr( B ) } . b) Every β ∈ IBr( B ) is quasi-liftable. Proof. b) ⇒ a) Navarro’s Lemma or   0 0 n 1 ... 0 0     • D =   0 0 n l   ∗ Da ≥ 0 ( a ∈ Z l ) ⇒ a ≥ 0 • 18

a) ⇒ b) Suppose that β ∈ IBr( B ) is not quasi-liftable. • For each χ ∈ Irr ( B ) with d χ,β � = 0 there exists a β � = ψ ∈ IBr( B ) with d χ,ψ � = 0. • b = max { d χ,β | χ ∈ Irr( B ) } Λ = − Φ β + b � ϕ � = β Φ ϕ • (Λ , χ ) = − d χ,β + b � • ϕ � = β d χ,ϕ ≥ 0 • Λ quasi-projective, not projective character. 19

Question. Are the following equivalent? a) Iqp( B ) = { Φ ϕ | ϕ ∈ IBr p ( B ) } . b) Each ϕ ∈ IBr p ( B ) is quasi-liftable. c) Each ϕ ∈ IBr p ( B ) is liftable. 20

Theorem. Let B be a block with a cyclic defect group > 1. By χ 0 we denote the sum of exceptional irreducible characters of B (if such characters exist). Furthermore let Irr 0 ( B ) be the set consisting of χ 0 and all the non-exceptional irreducible characters of B . Then � Λ = a ϕ Φ ϕ ∈ Iqp( B ) ϕ ∈ IBr p ( B ) if and only if Λ = χ + ψ for χ, ψ ∈ Irr 0 ( B ) where the distance between χ and ψ in the Brauer tree is odd. 21

Example. G = PSL (2 , 17), p = 17, B 0 the principal 17-block 20 = | Iqp( B 0 ) | �≤ | δ ( B 0 ) | = 17 Question. How to bound | Iqp( B ) | in terms of invariants of B ? 22

If all ϕ ∈ IBr( B ) are quasi-liftable, then • Iqp( B )= { Φ ϕ | ϕ ∈ IBr( B ) } . • l ( B ) = 1 ⇒ Iqp( B ) = { Φ ϕ } • Does l ( B ) = 2 imply | Iqp( B ) | = 2? ( G = 2 .A 8 , p = 3 , block #5) 23

Let Λ = � ϕ ∈ IBr ( B ) a ϕ Φ ϕ = � χ ∈ Irr ( B ) b χ χ ∈ Iqp( B ). Minkowski 1896: cone ( D ) = cone ( { a 1 , . . . , a m | 0 � = a i ∈ Z d } ), � k � where m ≤ l − 1 a i are solutions of l − 1 linearily independent equa- • tions of Dx = 0. H D ⊆ { a 1 , . . . a m } ∪ • { a ∈ cone ( D ) ∩ Z d | a = � i λ i a i , λ i ∈ [0 , 1) } 24

Ewald/Wessel ’91: • If l ≥ 2, then � a i � ∞ | a ϕ | ≤ ( l − 1) max i 25

4. Indecomp. quasi-projective Brauer characters Theorem. Let d ( B ) denote the defect of the block B . a) For each ϕ ∈ IBr p ( B ) there is a minimal p -power, say p a ( ϕ ) such that p a ( ϕ ) ϕ ∈ IBqp( B ) where a ( ϕ ) ≤ d ( B ). b) There exists ϕ ∈ IBr p ( B ) with a ( ϕ ) = d ( B ) Consequence. If ϕ ∈ IBr( B ), then � p a ( ϕ ) ϕ ( x ) for x a p ’-element, ϕ ( x ) = 0 otherwise. is a generalized character of B . 26

Example. G = A 5 , p = 2, B 0 the principal block elementary divisors: 4,1,1 2 a ( ϕ ) for ϕ ∈ IBr( B 0 ): 4,2,2 Question. Does a ( ϕ ) = 0 for ϕ ∈ IBr( B ) imply that B is of defect 0? Question. Can one characterize blocks B with | IBqp( B ) | = | IBr( B ) | ? 27

Fact. We always have a ( ϕ ) ≥ d ( B ) − ht ( ϕ ) where ht ( ϕ ) = ν p ( ϕ (1)) − ν p ( | G | ) + d ( B ). Question. Is a ( ϕ ) = d ( B ) − ht ( ϕ ) , if G is p -solvable and ϕ ∈ IBr( B )? Example. G = McL, p = 2 , ϕ ∈ IBr( B 0 ) of degree 7 . 2 9 . | G | 2 = 2 7 • • a ( ϕ ) = | d ( B ) − ht ( ϕ ) | = 2 28

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Happy Birthday 30