Clifford theory of characters in induced blocks

Koshitani, Shigeo; Späth, Britta

doi:10.1090/proc/12431

Cited by 18 publications

(13 citation statements)

References 13 publications

(20 reference statements)

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“…according to [KS13,2.5(b)]. This proves that the proposition, in particular (v), holds with χ 2 as χ and ψ 2 as ψ.…”

Section: 2supporting

confidence: 58%

“…I∩J ( φ)) according to Lemmas 2.4 and 2.5 of [KS13]. By Lemma 3.7 of [KS13] there is a unique character in Irr(I ∩ J | χ) with this property, hence Res A I∩J ( χ) = Res…”

Section: Clifford Theory In Covering Blocksmentioning

confidence: 94%

“…Then there exists an extension As90,18.5]. According to Theorem C(b)(1) of [KS13] there exists an extension χ 2 ∈ Irr(I) of χ to I with bl( χ 2 ) = bl(Res…”

Section: Clifford Theory In Covering Blocksmentioning

confidence: 99%

“…results from [KS13]. Section 5 recalls and establishes some useful properties of groups of Lie type A.…”

Section: Introductionmentioning

confidence: 99%

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On the inductive Alperin–McKay condition for simple groups of type A

Cabanes

Späth

2015

Journal of Algebra

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As a sequel to [CS13b], we verify the so-called inductive AM-condition introduced in [Sp12] for simple groups of type A and blocks with maximal defect. This is part of the program set up to verify the Alperin-McKay conjecture through its reduction to a problem on quasi-simple groups (see [Sp13]) but also the missing direction of Brauer's height zero conjecture (see [NS14]).Proving that the bijection in [CS13b] preserves ℓ-blocks is relatively easy (see [CS13a, 7.1]) but the additional conditions required by the inductive AM-condition need a specific work. This is explained in Section 2 below. We first give in Definition 2.1 our version of the inductive AM-condition relative to a single defect subgroup. We then show that it suffices to check it for every defect subgroup to get the inductive AM-condition. In passing we take the opportunity to give an equivalent property that may turn out easier to check than the one in [Sp13]; in particular it uses only ordinary characters, no projective representations. In Section 4, we adapt the criterion introduced in [Sp12] and already used in [CS13b] by incorporating blocks to our situation. Some of the methods make use of The second author has been supported by the Deutsche Forschungsgemeinschaft, SPP 1388 and ERC Advanced Grant 291512.. By the definition of φ this leads to bl Res H J 2 ∩H ( ψ) J 2 = bl Reswe know from [Mu13] that bl Res H J 3 ∩H ( ψ) = bl Res H J 2 ∩H ( ψ) J 3 ∩H and bl Res A J 3 ( χ 3 ) = bl Res A J 2 ( χ 3 ) J 3 .

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“…according to [KS13,2.5(b)]. This proves that the proposition, in particular (v), holds with χ 2 as χ and ψ 2 as ψ.…”

Section: 2supporting

confidence: 58%

“…I∩J ( φ)) according to Lemmas 2.4 and 2.5 of [KS13]. By Lemma 3.7 of [KS13] there is a unique character in Irr(I ∩ J | χ) with this property, hence Res A I∩J ( χ) = Res…”

Section: Clifford Theory In Covering Blocksmentioning

confidence: 94%

See 2 more Smart Citations

On the inductive Alperin–McKay condition for simple groups of type A

Cabanes

Späth

2015

Journal of Algebra

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“…Let N ≤ J ≤ L be a group with p ∤ |J : N |. According to[KS13, Theorem C(b)], there exists a character δ ∈ Irr(J ∩ H) with M ≤ ker(δ) such that bl(ψ J∩H δ) J = bl( χ J ).According to [KS13, Lemma 2.5], the character then also satisfies bl(ψ M,y δ M,y ) N,y = bl( χ N,y ) for every y ∈ L ∩ J.Since ǫ (y) is uniquely defined by Equation (1) we see that ǫ (y) = δ M,y . By the definition of ǫ this impliesǫ J∩H = δ J∩H .Accordingly ǫ E is a character for every groupE ≤ L ∩ H with p ∤ | M, E : M |.In order to apply Brauer's characterization of characters, see for example Corollary (8.12) of[Isa76], we have to consider ǫ E for every elementary group E ≤ L ∩ H that is the direct product of some p-group E p and a p ′ -group E p ′ .…”

mentioning

confidence: 99%