invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

p blocks relative to a character of a normal subgroup abstract let g be a finite group let n g and let irr n be a g invariant character we fix a prime p and we introduce a canonical partition of irr g relative to p we call each member b of this partition a block and to each block b we naturally associate a conjugacy class of p subgroups of g n which we call the defect groups of b if n is trivial then the blocks are the brauer p blocks using blocks we can unify the gluck wolf navarro tiep theorem and brauer s height zero conjecture in a single statement which after work of b sambale turns out to be equivalent to the height zero conjecture we also prove that the k b conjecture is true if and only if every block b has size less than or equal the size of any of its defect groups thus bringing normal subgroups to the k b conjecture

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

automorphisms of groups abstract the main purpose of the present paper is to give an explicit description of the wells map of a given group extension to the case of automorphisms acting trivially on the quotient group from this we obtain some of new necessary and sufficient conditions for an automorphism of a normal subgroup of a group to extend to the group itself with trivial action on the quotient group

brauer s height conjecture for p solvable groups we complete the proof of the height conjecture for p solvable groups using the classification of finite simple groups introduction the height conjecture is the statement that a p block of a finite group has an abelian defect group if and only if all ordinary irreducible characters in the block have height zero while a proof of this conjecture for general finite groups seems remote considerable progress has been made toward proving it for p solvable groups fong 5 proved that all characters in a block with abelian defect group have height zero in a p solvable group and he proved the converse direction for the principal block 5 and for solvable groups in the case that p is the largest prime divisor of the group order 6 recently 24 8 the converse direction has been established for all solvable groups in this paper we prove the converse direction for all p solvable groups assuming the classification of finite simple groups in its general outline this paper resembles 8 where we proved the height conjecture for solvable groups the reader is assumed to have some familiarity with 8 now we state our main results the analogs of the main results of 8 theorem a suppose that n g that g n is p solvable that t e irr n and that p 1 x i w l for all x e irr g i p then the p sylow subgroups of g n are abelian theorem b let b be a p block of a p solvable group with defect group d if every ordinary irreducible character in b has height zero then d is abelian theorem c suppose that n g that g n is p solvable and that w e irr n suppose that e is an integer such that pe l does not divide x i w l for all x e irr g ip then the derived length of a p sylow subgroup of g n is at most 2e 1 theorem d let b and d be as in theorem b if every ordinary irreducible character in b has height at most e then the derived length of d is at most 2e 1 received by the editors january 10 1983 1980 mathematics subject classification primary 20c20 20c 15 1984 american mathematical society 0002 9947 84 1 00 25 per page

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

on block induction abstract block induction is a correspondence between certain p blocks of subgroups and blocks of the whole group it is a tool with the help of which one often can reduce the solution of problems in the whole group to smaller groups the literature lists several ways to define induced blocks in the sense of brauer alperin and burry green and wheeler it is well known that any two of these concepts of block induction coincide in their common domain of definition for example each of them is defined if the subgroup h of the group g satisfies dc g d h n g d for a p subgroup d of g we study some other features of blocks which are compatible with the above induction concepts in this sense and formulate them as induction concepts so that one could deal with these and the above concepts in a unified way we compare their fields of definition to the others in general and in the special cases of defect zero blocks and of p groups we give some sufficient conditions so that induction would not be defined in any sense we show that unlike in the case of brauer sense induction block induction in the sense of alperin burry is not always defined from the normalizer of a chain of p subgroups our example also shows that admissible induction and being of multiplicity one are not transitive in the last part of the paper we study the connection of the defect groups of the induced and the original block in the case where the latter group is abelian

on hilbert divisors of brauer characters abstract the hilbert divisor p a of an irreducible p brauer character of a finite group g carries deep information about respectively the module affording in 8 we conjectured that belongs to a p block of defect 0 if and only if its hilbert divisor is 1 in this note we continue our investigations

on the order of uniprimitive permutation groups one of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group g of degree n where g x an we prove i g i exp 4v n log2 n for the case when g is not doubly transitive this result is best possible apart from an o log n factor in the exponent the best result previously known was i g i el h wielandt a similar estimate for the doubly transitive case follows in a subsequent paper for rank 3 groups with subdegree p n 2 we obtain i gi exp 4 n p log2 n in the proof we develop some new combinatorial properties of coherent configurations the results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing

computer calculation of witten s 3 manifold invariant witten s 2 1 dimensional chern simons theory is exactly solvable we compute the partition function a topological invariant of 3 manifolds on generalized seifert spaces thus we test the path integral using the theory of 3 manifolds in particular we compare the exact solution with the asymptotic formula predicted by perturbation theory we conclude that this path integral works as advertised and gives an effective topological invariant

state sum invariants of compact 3 manifolds with boundary and 6j symbols the authors extend the combinatorial construction of invariants of smooth compact closed 3 manifolds as given by turaev and viro 1991 to obtain invariants of 3 manifolds and with a boundary the techniques uses quantum 6j symbols associated to the quantized universal enveloping algebra uq sl 2 c q a root of unity to give a construction of a state sum for given triangulation this state sum is then invariant under subdivisions and isotopes of the boundary their methods also lead to a simplified proof of the main result of turaev and viro that the state sum is independent of the triangulation and hence gives rise to an invariant of the manifold they use surgery to calculate the state sum for some closed manifolds their results also confirm a finding of turaev 1990 which relates this theory to the topological quantum field theory with a chern simons action in the sense of witten 1989

a new rapid acquisition technique for direct sequence spread spectrum communications this paper describes a rapid acquisition technique which can be used in direct sequence spread spectrum systems it is sequential in nature and is based on a double threshold which defines when a decision can be made the values of the thresholds change at every examination instant the procedure makes use of a bound to the partial crosscorrelation function of pn sequences which ensures that only the sequence length needs to be given as far as the pn sequence particulars are concerned a decision is made only if the probability of error associated with a decision at that point is equal to or better than a prespecified value finally it is shown that most of the computational difficulties associated with the bound as given by a recursive algorithm can be eliminated when using a low order polynomial approximation to the bound however either by using the bound or its polynomial approximation a significant improvement in the acquisition time is obtained

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

hall subgroups and fully extendible characters abstract we say that an irreducible character of a subgroup h of g is fully extendible to g if every irreducible constituent of the induced character g is actually an extension of now let n g where g n is separable for some set of primes and let h n be a hall subgroup of g n if irr h is fully extendible to g we show that a hall subgroup k n of g n must be abelian also if is an irreducible constituent of n we show that is k invariant for some hall subgroup k n of g n and furthermore is fully extendible to k conversely if irr n is extendible to k where k n is a hall subgroup of g n we show that some irreducible character of h lying over is extendible to g

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

arithmetic results on orbits of linear groups the first four authors acknowledge the support of an arc discovery grant the first author acknowledges the support of an australian research fellowship the third author acknowledges the support of a federation fellowship the fifth author acknowledges the support of the nsf grants dms 0901241 and dms 1201374

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

weight conjectures for fusion systems abstract many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local to global statements in that they predict consequences for the representations of a finite group g given data about the representations of the p local subgroups of g the local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of kulshammer puig cohomology classes motivated by conjectures in block theory we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples s f consisting of a saturated fusion system f on a finite p group s and a compatible family

topological quantum field theory and invariants of graphs for quantum groups on the basis of generalized 6j symbols we give a formulation of topological quantum field theories for 3 manifolds including observables in the form of coloured graphs it is shown that the 6j symbols associated with deformations of the classical groups at primitive even roots of unity provide examples of this construction calculational methods are developed which in particular yield the dimensions of the state spaces as well as a rather simple proof of the relation previously found by turaev and walker for the case ofu q sl 2 c between these models and corresponding ones based on the ribbon graph construction of reshetikhin and turaev

noncoherent sequential acquisition of pn sequences for ds ss communications with without channel fading studies sequential acquisition schemes of m sequences based on the sequential probability ratio test sprt and a truncated sprt tsprt with noncoherent demodulation most reported results on sequential acquisition schemes assume that independent samples are available for the decision process the assumption of independent and identically distributed i i d samples requires the integrator in the receiver to be reset periodically this introduces loss in the effective signal to noise ratio degrading the performance in the present paper on the contrary the authors two sequential schemes use continuous integration it can be shown that the likelihood ratio is monotonic consequently the tests can be easily implemented in realtime methods are proposed for designing the decision thresholds to achieve the desired false alarm and miss probabilities performances of the proposed schemes are obtained and they suggest that the tsprt is more desirable the effect of slowly varying channel fading is also investigated results show that fading does not affect the false alarm probabilities but it can drastically reduce the probability of detecting the alignment of the two pn sequences especially when the fading is severe

tensor products of quantized tilting modules letuk denote the quantized enveloping algebra corresponding to a finite dimensional simple complex lie algebral assume that the quantum parameter is a root of unity ink of order at least the coxeter number forl also assume that this order is odd and not divisible by 3 if typeg2 occurs we demonstrate how one can define a reduced tensor product on the familyf consisting of those finite dimensional simpleuk modules which are deformations of simplel and which have non zero quantum dimension this together with the work of reshetikhin turaev and turaev wenzl prove that uk f is a modular hopf algebra and hence produces invariants of 3 manifolds also by recent work of duurhus jakobsen and nest it leads to a general topological quantum field theory the method of proof explores quantized analogues of tilting modules for algebraic groups

turaev viro invariant and 3nj symbols we propose in this work a new method to construct the turaev viro state sums using a diagrammatic presentation describing surgery operation as heegaard splittings the resulting invariants can be connected with suitable 3nj symbols and we evaluate them for the lens spaces

a generalized analysis for a dual threshold sequential detection pn acquisition receiver spread spectrum techniques have become very useful in modern communications a common spread spectrum technique is to directly modulate a pseudorandom noise pn code onto the carrier systems employing this technique are called direct sequence ds systems a primary concern with direct sequence systems involves pn code synchronization acquisition of the incoming and local signals to within at least one pn symbol followed by a fine alignment tracking which must be maintained throughout transmission this paper is concerned with the pn acquisition aspect of code synchronization specifically a dual threshold sequential detection receiver is analyzed and a general expression is derived which addresses the probability that the sequential detection procedure ceases after an arbitrary number of samples this expression is needed in order to support a characteristic function approach in deriving acquisition probability as a function of time which will be the topic of a subsequent paper

invariants of spin networks embedded in three manifolds we study the invariants of spin networks embedded in a three dimensional manifold which are based on the path integral for su 2 bf theory these invariants appear naturally in loop quantum gravity and have been defined as spin foam state sums by using the chain mail technique we give a more general definition of these invariants and show that the state sum definition is a special case this provides a rigorous proof that the state sum invariants of spin networks are topological invariants we derive various results about the bf theory spin network invariants and we find a relation with the corresponding invariants defined from chern simons theory i e the witten reshetikhin turaev invariants we also prove that the bf theory spin network invariants coincide with v turaev s definition of invariants of coloured graphs embedded in 3 manifolds and thick surfaces constructed by using shadow world evaluations our framework therefore provides a unified view of these invariants

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng