On centers of blocks with one simple module

Abstract

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).

On the Radical of the Center of Small Symmetric Local Algebras

“…In this section we are revisiting the article [7] From now on, F will denote an algebraically closed field of characteristic 2 and B…”

“…In this section we are revisiting the article [7], especially its fourth section. The article mentioned is concerned with determining the isomorphism type of the center of a non-nilpotent 2-block B with elementary abelian defect group of order 16 and one isomorphism type of simple modules.…”

“…Some more recent examples of this principle can be found in the literature, e. g. in [4], the Morita equivalence class of a whole family of 3-blocks with elementary abelian defect groups of order 9 was determined by investigating certain 9-dimensional symmetric local algebras. In [7] the authors were able to determine the isomorphism type of the center of a class of 2-blocks with elementary abelian defect groups of order 16. A big part of the argument was, again, based on investigating symmetric local algebras of a certain type.…”

“…Moreover, this section will contain the main result of the present paper. In the final section we will apply a modification of the main result in characteristic 2 to the setting in [7]. This will allow us to considerably shorten the computations and give a more elegant proof of the main result of that article.…”

“…His approach, however, relies on the classification of finite simple groups. The approach taken in the article[7] and the thesis[6] can be seen as a classification-free attempt to determine the structure of the unique non-nilpotent 2-block with elementary abelian defect group of order 16 and one isomorphism type of simple modules.We will, in this section, prove a stronger result of our Theorem 3.2 which, on the other hand, needs some more restrictive assumptions. Using this result, we can give another more elegant proof that one of the two isomorphism types of the 182 PIERRE LANDROCK center given in [7, Proposition 2.1] cannot occur.…”