Principal 2-block Research Articles

The group of splendid Morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups

Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \begin{align*} \calT(B)\cong \Out_P(A)\rtimes \Out(P,\calF), \end{align*} where $\Out(P,\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\calF$ of $G$ on $P$ and $\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\times$.

Splendid Morita equivalences for the principal 2-blocks of 2-dimensional general linear groups in non-defining characteristic

Splendid Morita equivalences for the principal 2-blocks of 2-dimensional general linear groups in non-defining characteristic

Height zero conjecture with Galois automorphisms

We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô–Michler theorem for the prime 2, again with Galois automorphisms. We close, this time for odd primes p $p$ , with a new characterisation of p $p$ -closed groups via the decomposition numbers of certain characters.

On the Alperin–McKay conjecture for 2‐blocks of maximal defect

In this paper, we show that the Alperin–McKay conjecture holds for 2-blocks of maximal defect. A major step in the proof is the verification that principal 2-block of groups of Lie type in odd characteristic is AM-good for the prime 2.

Splendid Morita equivalences for principal 2-blocks with dihedral defect groups

Given a dihedral 2-group P of order at least 8, we classify the splendid Morita equivalence classes of principal 2-blocks with defect groups isomorphic to P. To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.

Eigenvalues of Cartan matrices of principal 2-blocks with abelian defect groups

Let G be a finite group with an abelian Sylow 2-subgroup P. Let C B be the Cartan matrix of the principal 2-block B of G. We show that the Frobenius–Perron eigenvalue ρ ( B ) of C B is a rational integer if and only if B and its Brauer correspondent block b of N G ( P ) are Morita equivalent by using a classification of finite simple groups with an abelian Sylow 2-subgroup. In this case, we can take the Brauer character table Φ b of b as a unimodular eigenvector matrix U B of C B over a complete discrete valuation ring R.

Basic sets of Brauer characters of finite groups of Lie type, III

LetG(F q ) be a finite classical group whereq is odd and the centre ofG is connected. We show that there exists a set of irreducible characters ofG(F q ) such that the corresponding matrix of scalar products with the characters of Kawanaka’s generalized Gelfand-Graev representations is square unitriangular. This uses in an essential way Lusztig’s theory of character sheaves. As an application we prove that there exists an ordinary basic set of 2-modular Brauer characters and that the decomposition matrix of the principal 2-block ofG(F q ) has a lower unitriangular shape.

Basic Sets of Brauer Characters of Finite Groups of Lie Type II

LetG(Fq) be a finite classical group whereq is odd and the centre ofG is connected. We show that there exists a set of irreducible characters ofG(Fq) such that the corresponding matrix of scalar products with the characters of Kawanaka’s generalized Gelfand-Graev representations is square unitriangular. This uses in an essential way Lusztig’s theory of character sheaves. As an application we prove that there exists an ordinary basic set of 2-modular Brauer characters and that the decomposition matrix of the principal 2-block ofG(Fq) has a lower unitriangular shape.

On the number of irreducible characters in a 2-block

Both of these conjectures have been verified for D cyclic by Dade [9] or 1 D 1 <pz by Brauer and Feit [7] (for proof see Brauer [3, II]). In addition, the first conjecture has been verified if the inertial index of the block is 1 (see Brauer [4, IV]), while the second conjecture has been partly verified for psolvable groups by Fong. In one of several discussions I had with R. Brauer in 1974, he suggested to find a verification of these conjectures in the case of a principal 2-block with H, @ Z, @ Z, as defect group without involving the classification of groups with this Sylow 2-subgroup, from which the conjectures may quite easily be checked to hold. In [ 181, Ward was forced to do this for groups of Ree-type anyway, but the computations were quite involved, and the very special properties of these groups, including their order, were used heavily. In Theorem 2 of Section 2, we prove that the first conjecture holds in general for any defect group of order (less than or) equal to 8, while Theorem 8 of Section 3 states that the second conjecture hold if the block in addition is assumed to be principal. We shall not depend in our work on the classification of these groups. In particular we give an alternate and shorter proof of the result in [ 181 mentioned above without using any other property

Principal 2-Blocks of the Simple Groups of Ree Type

Principal 2-Blocks of the Simple Groups of Ree Type

The principal 2-blocks of finite groups with abelian Sylow 2-subgroups

The principal 2-blocks of finite groups with abelian Sylow 2-subgroups

Principal 2-blocks of the simple groups of Ree type

The decomposition numbers in characteristic 2 of the groups of Ree type are determined, as well as the Loewy and socle series of the indecomposable projective modules. Moreover, we describe the Green correspondents of the simple modules. As an application of this and similar works on other simple groups with an abelian Sylow 2-subgroup, all of which have been classified apart from those considered in the present paper, we show that the Loewy length of an indecomposable projective module in the principal block of any finite group with an abelian Sylow 2-subgroup of order 2 n {2^n} is bounded by max { 2 n + 1 , 2 n } \max \{ 2n\, + \,1,\,{2^n}\} . This bound is the best possible.

On the Vertices of Irreducible Modules

The same result holds if D is a vertex of an irreducible FG-module Me B. This second assertion generalizes some known results: For p-solvable groups G, it has been shown by Hamernik and Michler [10], that for any irreducible module M in a block B of FG, a vertex of M contains-up to conjugation-the center of a defect group of B. In [13], Michler has shown that all irreducible modules in a block with cyclic defect group A have A as a vertex. A further result in this context was given by Landrock and Michler ([11], Theorem 3.7) as a by-product of their analysis of the principal 2-blocks of the smallest Janko group and the groups of Ree-type. The proof here uses properties of the Green functor to reduce to the case D < G. Then one has to study direct summands of a module induced from a normal subgroup. This is done by inspection of the ring of endomorphisms (Sections 1 and 2). Using these results, the existence of certain B-pairs (see Definition 0.3 below) is shown in Section 3. The results mentioned at the beginning follow then from work done by Brauer and Olsson. The techniques used in the first sections yield a criterion for an ordinary irreducible character X to be of height 0: If F is algebraically closed and M an R-module affording X, then the character X belonging to the p-block B is of height 0 if and only if a vertex of M is a defect group of B and p does not divide the R-rank of a source of M. The proof is given in Section 4.

On characters in the principal 2-block, II

AbstractLet k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v)–X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C2, PSL(2, 2n) or pΣL(2, 52a+1), where n≥2 and a ≥ 1.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20

On characters in the principal 2-block

AbstractLet k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

On Ree’s series of simple groups

Ree recently discovered a series of finite simple groups related to the simple Lie algebra of type (G2) [5; 6 ] . We have determined the irreducible characters of these groups. In this work, we do not use the actual definition of Ree's groups, but only the properties (l)-(5) given below. Since these are sufficient to determine the bulk of the character tables of these groups, it is our hope that they actually characterize the groups completely. The five properties used are (1) If G is one of these groups, the order of G is even and the 2Sylow subgroup of G is elementary Abelian of order 8. (2) G has no normal subgroup of index 2. (3) There is an involution / (an element of order 2) in G such that if C(J) is the centralizer of / in G and (J ) is the group generated by / , then C(J)/(J) is isomorphic to LF(2, q), the linear fractional group in two variables over a field of q elements. Here we restrict q by q^3 (mod 8) and q^27. I t follows that C(J) is the direct product of (ƒ) and a subgroup F of G which is isomorphic to LF(2, q). In F there is an element R of order (q — l)/2f and we also have (4) If R5*l, then C(R)QC(J). Finally, there is an element S of F with order (q+l)/2. Let J' = S* where / = ( g + l ) / 4 . Then S generates the commutator subgroup of C(J, the centralizer of (J , J'). We have (5) F o r S 2 , C(S*)QC(J). This condition can actually be replaced by the weaker condition (5*) Let A €N(J, J'), the normalizer in G of (ƒ, / '>, but A $ C(J, J)Let A = 1. Then A does not commute with S. From these conditions on G we derive a number of results. The approach is almost entirely by means of characters, both ordinary and modular. Condition (4) leads to two families of exceptional characters related to the classes of R( 5^1) and JR (T^J). These characters are the characters of 2-defect 1. Results of Brauer [ l ; 2; 3] then lead to two possibilities for the principal 2-block (the 2-block containing the character which is 1 everywhere) ; one contains seven 1 This note is a summary of the results obtained in the author's Harvard University dissertation of the same title, written under the direction of Richard Brauer. 2 This research was supported by the National Science Foundation.

On Ree’s series of simple groups

Ree recently discovered a series of finite simple groups related to the simple Lie algebra of type (G2) [5; 6 ] . We have determined the irreducible characters of these groups. In this work, we do not use the actual definition of Ree's groups, but only the properties (l)-(5) given below. Since these are sufficient to determine the bulk of the character tables of these groups, it is our hope that they actually characterize the groups completely. The five properties used are (1) If G is one of these groups, the order of G is even and the 2Sylow subgroup of G is elementary Abelian of order 8. (2) G has no normal subgroup of index 2. (3) There is an involution / (an element of order 2) in G such that if C(J) is the centralizer of / in G and (J ) is the group generated by / , then C(J)/(J) is isomorphic to LF(2, q), the linear fractional group in two variables over a field of q elements. Here we restrict q by q^3 (mod 8) and q^27. I t follows that C(J) is the direct product of (ƒ) and a subgroup F of G which is isomorphic to LF(2, q). In F there is an element R of order (q — l)/2f and we also have (4) If R5*l, then C(R)QC(J). Finally, there is an element S of F with order (q+l)/2. Let J' = S* where / = ( g + l ) / 4 . Then S generates the commutator subgroup of C(J, the centralizer of (J , J'). We have (5) F o r S 2 , C(S*)QC(J). This condition can actually be replaced by the weaker condition (5*) Let A €N(J, J'), the normalizer in G of (ƒ, / '>, but A $ C(J, J)Let A = 1. Then A does not commute with S. From these conditions on G we derive a number of results. The approach is almost entirely by means of characters, both ordinary and modular. Condition (4) leads to two families of exceptional characters related to the classes of R( 5^1) and JR (T^J). These characters are the characters of 2-defect 1. Results of Brauer [ l ; 2; 3] then lead to two possibilities for the principal 2-block (the 2-block containing the character which is 1 everywhere) ; one contains seven 1 This note is a summary of the results obtained in the author's Harvard University dissertation of the same title, written under the direction of Richard Brauer. 2 This research was supported by the National Science Foundation.