Blocks with nonabelian defect groups which have cyclic subgroups of index p

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M _n+1(p) of order p ⁿ⁺¹ and exponent p ⁿ for n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M _n+1(p) is normal in G. The proof is independent of the classification of finite simple groups.     

Blocks with trivial intersection defect groups

We show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups, or is Morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non-abelian simple groups. In particular we classify all blocks of automorphism groups of non-abelian simple groups whose defect groups are non-cyclic and intersect pairwise trivially. A consequence is that Donovans conjecture holds for blocks whose defect groups intersect pairwise trivially.in final form: 14 January 2003Mathematics Subject Classification (2000): 20C20This research was supported in part by the Marsden Fund of New Zealand via grant UOA 810.     

Blocks with dihedral defect groups in solvable groups

Mathematische Zeitschrift -     

Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate k _i (B) and l _i (B) for an arbitrary nonnegative integer i.     

Blocks and defect groups of monomial groups

In this paper the p-block structure and the defect groups of finite monomial groups are determined.     

Chordal Coxeter groups

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a chord. As an application, we construct counterexamples to two conjectures concerning the isomorphism problem for Coxeter groups.      

Rigidity of Coxeter groups

Let be a Coxeter group acting properly discontinuously and cocompactly on manifolds and such that the fixed point sets of finite subgroups are contractible. Let be a -homotopy equivalence which restricts to a -homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then is -homotopic to a -homeomorphism.     

Trivial source modules in blocks with cyclic defect groups

We shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non-identity p-elements in finite groups instead of doing huge calculation. The method is only for a case where defect groups are cyclic. Nevertheless, it works well at least when we want to prove Broué’s abelian defect group conjecture for blocks which have elementary abelian defect groups of order p ².     

Symmetric generation of Coxeter groups

We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.      

Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ? ^N . We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space of holomorphic functions on with reproducing kernel equal to the Dunkl-kernel. The definition and properties of extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrödinger model and in the Fock model. Further, we prove a branching decomposition of as a unitary -module and a general version of Hecke's formula for the Dunkl transform.     

Coxeter groups,Coxeter monoids and the Bruhat order

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.