The 2-blocks of defect 4

Külshammer, Burkhard; Sambale, Benjamin

doi:10.1090/s1088-4165-2013-00433-8

Cited by 15 publications

(19 citation statements)

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“…It turns out that we have already handled the non-abelian defect groups of order 16. Next we settle the elementary abelian case which is taken from [74,171]. …”

Section: Results On the Kb/-conjecturementioning

confidence: 99%

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Blocks of Finite Groups and Their Invariants

Sambale

2014

Lecture Notes in Mathematics

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155

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“…It turns out that we have already handled the non-abelian defect groups of order 16. Next we settle the elementary abelian case which is taken from [74,171]. …”

Section: Results On the Kb/-conjecturementioning

confidence: 99%

“…This was done later in [171], and we will provide the result here with a simpler proof. Proof Let D Š MNA.r; 1/ be a defect group of B, and let Q Ä D be an Fcentric, F -radical subgroup where F is the fusion system of B.…”

Section: The Case P Dmentioning

confidence: 93%

Blocks of Finite Groups and Their Invariants

Sambale

2014

Lecture Notes in Mathematics

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155

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“…For all other nontrivial subsections (u, b u ), we only know l(b u ) ≥ 1. Finally, l(B) ≥ 2 since B is centrally controlled, by Theorem 1.1 in [Külshammer and Okuyama ∼ 2000]. Applying Brauer's formula gives…”

Section: Bicyclic Defect Groupsmentioning

confidence: 87%

“…for a major subsection (z, b z ) (this holds for example if z lies in the center of the fusion system of B; see [Külshammer and Okuyama ∼ 2000]). However, this is not true in general as we see in [Külshammer and Sambale 2013, Proposition 2.1(vii)].…”

Section: It Should Be Pointed Out That Usuallymentioning

confidence: 99%

Untitled

2013

Algebra Number Theory

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We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustat ,ȃ , and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

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“…Here, p = 2 and B has elementary abelian defect group D of order 16. In [22] the numerical invariants of B have been determined. In particular, it is known that the number of irreducible ordinary characters (of height 0) of B is k(B) = k 0 (B) = 8.…”

Section: Introductionmentioning

confidence: 99%

On centers of blocks with one simple module

Landrock

Sambale

2017

Journal of Algebra

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Let G be a finite group, and let B be a non-nilpotent block of G with respect to an algebraically closed field of characteristic 2. Suppose that B has an elementary abelian defect group of order 16 and only one simple module. The main result of this paper describes the algebra structure of the center of B. This is motivated by a similar analysis of a certain 3-block of defect 2 in [Kessar, 2012].Theorem 1.1. Let B be a non-nilpotent 2-block with elementary abelian defect group of order 16 and only one irreducible Brauer character. ThenIn particular, Z(B) has Loewy length 3.The paper is organized as follows. In the second section we consider the generalized decomposition matrix Q of B. Up to certain choices there are essentially three different possibilities for Q. A result by Puig [27] (cf. [9, Theorem 5.1]) describes the isomorphism type of Z(B) (regarded over O) in terms of Q. In this way we prove that there are at most two isomorphism types for Z(B). In the two subsequent sections we apply ring-theoretical arguments to the basic algebra of B in order to exclude one possibility for Z(B). Finally, we give some concluding remarks in the last section. Our notation is standard and can be found in [24,29].for any integer n ≥ 3. Now we will distinguish between the different cases that can occur for dim F (J 2 (A)/ J 3 (A)). We note that 2 ≤ dim F (J 2 (A)/ J 3 (A)) ≤ 4. The upper bound is clear by the preceding discussion, and if dim F (J 2 (A)/ J 3 (A)) = 1, then J 2 (A) ⊆ Z(A) by Lemma 3.3 which is a contradiction to dim F Z(A) = 8. The case dim F (J 2 (A)/ J 3 (A)) = 0 leads to J 2 (A) = 0 by Nakayama's Lemma and this is clearly false.proceed by distinguishing three subcases for an F -basis of J 2 (A)/ J 3 (A). More specifically there is always a basis of J 2 (A)/ J 3 (A) given by {x 2 + J 3 (A), d + J 3 (A)} for some d ∈ {xy, xz, yz}. This, however, implies J 3 (A) = F {x 3 , x 2 z} + J 4 (A) = F {x 2 z} + J 4 (A) and J 4 (A) = F {x 3 z} + J 5 (A) = J 5 (A).Hence, J 4 (A) = 0 by Nakayama's Lemma and therefore dim F A ≤ 1 + 3 + 2 + 1 = 7, a contradiction.(3): J 2 (A) = F {x 2 , yz} + J 3 (A) = F {x 2 , zy} + J 3 (A). We may assume that xy, xz ∈ F {x 2 } + J 3 (A) since otherwise we are in one of the previous two subcases. Using this we obtain J 3 (A) = F {x 3 , zxy, x 2 z, z 2 y} + J 4 (A) = F {x 3 , xyz, x 2 z} + J 4 (A) = F {x 3 } + J 4 (A). Hence, J 2 (A) ⊆ Z(A) by Lemma 3.3, and so dim F Z(A) ≥ dim F J 2 (A) = 12, a contradiction. We have thus shown that dim F (J 2 (A)/ J 3 (A)) = 2.Case (II.2): dim F (J 2 (A)/ J 3 (A)) = 3. Again, since x 2 / ∈ J 3 (A), there is always an F -basis of J 2 (A)/ J 3 (A) of the form {x 2 +J 3 (A), d 1 +J 3 (A), d 2 +J 3 (A)} for some d 1 , d 2 ∈ {xy, xz, yz}. Hence, we can proceed by distinguishing three subcases for a basis of J 2 (A)/ J 3 (A).

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The 2-blocks of defect 4

Abstract: Abstract. We show that the major counting conjectures of modular representation theory are satisfied for 2-blocks of defect at most 4 except one possible case. In particular, we determine the invariants of such blocks.

Cited by 15 publications

References 31 publications

Blocks of Finite Groups and Their Invariants

Blocks of Finite Groups and Their Invariants

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On centers of blocks with one simple module

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