No abstract

This paper classifies the indecomposable modules over a class of quasi-Frobenius algebras over a field k. These algebras have a k-basis described by a Brauer graph.The product of two of these basis elements is either another basis element or zero. The Brauer trees that arise in the study of blocks with cyclic defect group are a special case.The modules may be described as diagrams of vector spaces subject to certain relationships of commutativity and to certain composites being zero. In this interpretation the commutativity of k need not be assumed. As the motivation of this paper is the representation theory of certain finite groups, the presentation is for commutative k only. However no restriction on the characteristic or algebraic closure of k is made.A module is said to be uniserial if it has a unique composition series. A module M is said to be special if it has two uniserial submodules U and V such that Urn V is the socle of M and is simple, M/U and M/V are uniserial and U + V is the unique maximal submodule of M. In particular, uniserial and (by special convention) simple modules are special. If k is algebraically closed, finite dimensional kalgebras whose indecomposable projective modules are special can have their representation theory determined by the methods of this paper.As implied above, the modular representation theory of a block with cyclic defect group is a special case of the work in this paper. More significantly, let k be commutative, have characteristic 2 and contain a splitting field for the group G =$4,$5, A 7 or PSL(2,q), where q is an odd prime power. The indecomposable projective kG-modules are special, and so the indecomposable modular representations of kG are classified by the main theorem of this paper. Section 2 implicitly describes the structure of the indecomposable projective modules in the principal block in each ease. The non-principal blocks cause little difficulty. The details need some modification when k does not contain a splitting field.The groups mentioned above have dihedral Sylow subgroups. Indeed any simple group with dihedral Sylow subgroup is isomorphic to one of them. Thus the classification of their modular representations could, in principle, be deduced from Ringel's classification [7] (also obtained by Bondarenko [1]) of the modular

No abstract

No abstract

A d-furcated flow is a network flow whose support graph has maximum outdegree d. Take a single-sink multicommodity flow problem on any network and with any set of routing demands. Then we show that the existence of feasible fractional flow with node congestion one implies the existence of a d-furcated flow with congestion at most 1 + 1 d−1 , for d ≥ 2. This result is tight, and so the congestion gap for d-furcated flows is bounded and exactly equal to 1 + 1 d−1 . For the case d = 1 (confluent flows), it is known that the congestion gap is unbounded, namely Θ(log n). Thus, allowing single-sink multicommodity network flows to increase their maximum outdegree from one to two virtually eliminates this previously observed congestion gap.As a corollary we obtain a factor 1 + 1 d−1 -approximation algorithm for the problem of finding a minimum congestion d-furcated flow; we also prove that this problem is maxSNPhard. Using known techniques these results also extend to degree-constrained unsplittable routing, where each individual demand must be routed along a unique path.