endotrivial modules for finite groups of lie type a in nondefining characteristic let g be a finite group such that mathop text sl nolimits n q subseteq g subseteq mathop text gl nolimits n q and z be a central subgroup of g in this paper we determine the group t g z consisting of the equivalence classes of endotrivial k g z modules where k is an algebraically closed field of characteristic p such that p does not divide q the results in this paper complete the classification of endotrivial modules for all finite groups of untwisted lie type a initiated earlier by the authors

the classification of endo trivial modules this paper settles the classification of all endo trivial modules for a finite p group by treating the remaining case of a p group having maximal elementary abelian subgroups of rank 2 it is shown that the modules constructed by alperin using relative syzygies provide the complete list of endo trivial modules in this case

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

universal deformation rings and generalized quaternion defect groups abstract we determine the universal deformation rings r g v of certain mod 2 representations v of a finite group g which belong to a 2 modular block of g whose defect groups are isomorphic to a generalized quaternion group d we show that for these v a question raised by the author and chinburg concerning the relation of r g v to d has an affirmative answer we also show that r g v is a complete intersection even though r g n v need not be for certain normal subgroups n of g which act trivially on v

universal deformation rings and tame blocks let k be an algebraically closed field of positive characteristic and let g be a finite group suppose b is a block of kg of infinite tame representation type we find all finitely generated kg modules v that belong to b and whose endomorphism ring is isomorphic to k and determine the universal deformation ring r g v for each of these modules

the dade group of a p group let p be a prime number this paper solves the question of the structure of the group d p of endo permutation modules over an arbitrary finite p group p that was open after dade s original papers in 1978 19 20 and it gives a proof of the conjectures proposed in 4 and 10 this leads to a presentation of d p by explicit generators and relations generalizing the presentation obtained by dade when p is abelian

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

the dade group of a metacyclic p group the dade group d p of a finite p group p formed by equivalence classes of endo permutation modules is a finitely generated abelian group its torsion free rank equals the number of conjugacy classes of non cyclic subgroups of p and it is conjectured that every non trivial element of its torsion subgroup d t p has order 2 or also 4 in case p 2 the group d t p is closely related to the injectivity of the restriction map res t p rightarrow prod e t e where e runs over elementary abelian subgroups of p and t p denotes the group of equivalence classes of endo trivial modules which is still unknown for almost extra special groups p odd as metacyclic p groups have no almost extra special section we can verify the above conjecture in this case finally we compute the whole dade group of a metacyclic p group

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

modules of constant jordan type we introduce the class of modules of constant jordan type for a finite group scheme g over a field k of characteristic p 0 this class is closed under taking direct sums tensor products duals heller shifts and direct summands and includes endotrivial modules it contains all modules in an auslander reiten component which has at least one module in the class highly non trivial examples are constructed using cohomological techniques we offer conjectures suggesting that there are strong conditions on a partition to be the jordan type associated to a module of constant jordan type

finding local community structure in networks although the inference of global community structure in networks has recently become a topic of great interest in the physics community all such algorithms require that the graph be completely known here we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time this algorithm runs in time o k2d for general graphs when d is the mean degree and k is the number of vertices to be explored for graphs where exploring a new vertex is time consuming the running time is linear o k we show that on computer generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge as an application we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer

universal deformation rings and dihedral blocks with two simple modules abstract let k be an algebraically closed field of characteristic 2 and let w be the ring of infinite witt vectors over k suppose g is a finite group and b is a block of kg with a dihedral defect group d such that there are precisely two isomorphism classes of simple b modules we determine the universal deformation ring r g v for every finitely generated kg module v which belongs to b and whose stable endomorphism ring is isomorphic to k the description by erdmann of the quiver and relations of the basic algebra of b is usually only determined up to a certain parameter c which is either 0 or 1 we show that r g v is isomorphic to a subquotient ring of wd for all v as above if and only if c 0 giving an answer to a question raised by the first author and chinburg in this case moreover we prove that c 0 if and only if b is morita equivalent to a principal block

quantitative function for community detection we propose a quantitative function for community partition i e modularity density or d value we demonstrate that this quantitative function is superior to the widely used modularity q and also prove its equivalence with the objective function of the kernel k means both theoretical and numerical results show that optimizing the new criterion not only can resolve detailed modules that existing approaches cannot achieve but also can correctly identify the number of communities

affirmative answer to a question of feit as he points out if p is abelian the affirmative answer follows from dade s classification 4 our purpose here is to show that the general case reduces to the abelian one which answers affirmatively feit s question more generally in 4 dade determines the cap group associated with a finite abelian p group showing in particular that this group is finitely generated our main result here cf theorem 2 2 below implies that finite generation is still true in the non abelian case cf corollary 2 4 below this result and the main steps of a proof were already announced in 17 section81 and here we are just discharging ourselves of an old debt actually as we said in 7 section81 our result would follow from a complete classification of endo permutation modules

the classification of torsion endo trivial modules the main purpose of this paper is to show that if g is a finite p group which is not cyclic quaternion or semi dihedral then the group of endo trivial g modules is detected on restriction to elementary abelian p subgroups of rank 2 consequently the torsion subgroup of the group of endo trivial modules is trivial for any finite p group except in the known cases of cyclic quaternion and semi dihedral groups this requires a large amount of group cohomology recent bounds for the number of bocksteins in serre s theorem carlson s recent theorem expressing serre s theorem in terms of modules and finally the theory of varieties attached to modules a detection theorem for the torsion subgroup of the dade group of all endo permutation modules is deduced from the main theorem the structure of this torsion subgroup can then be fully described when p is odd using the results of bouc thevenaz

endotrivial modules for p solvable groups we determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p we also prove that our result would hold for p solvable groups provided a conjecture can be proved about the case of p nilpotent groups

on endotrivial modules for lie superalgebras abstract let g g 0 g 1 be a lie superalgebra over an algebraically closed field k of characteristic 0 an endotrivial g module m is a g supermodule such that hom k m m k ev p as g supermodules where k ev is the trivial module concentrated in degree 0 and p is a projective g supermodule in the stable module category these modules form a group under the operation of the tensor product we show that for an endotrivial module m the syzygies n m are also endotrivial and for certain lie superalgebras of particular interest we show that 1 k ev and the parity change functor actually generate the group of endotrivials additionally for a broader class of lie superalgebras for a fixed n we show that there are finitely many endotrivial modules of dimension n

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

modularity density of network community divisions the problem of dividing a network into communities is extremely complex and grows very rapidly with the number of nodes and edges that are involved in order to develop good algorithms to identify optimal community divisions it is extremely beneficial to identify properties that are similar for most networks we introduce the concept of modularity density the distribution of modularity values as a function of the number of communities and find strong indications that the general features of this modularity density are quite similar for different networks the region of high modularity generally has very low probability density and occurs where the number of communities is small the properties and shape of the modularity density may give valuable information and aid in the search for efficient algorithms to find community divisions with high modularities

connected components of the category of elementary abelian p subgroups we determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p group g for an odd prime p namely it is p if g has rank at least 3 and it is p 1 if g has rank 2 more precisely if g has rank 2 there are exactly 1 2 p 1 or possibly 3 classes for some 3 groups of maximal nilpotency class

endotrivial modules for the general linear group in a nondefining characteristic suppose that g is a finite group such that mathrm sl n q subseteq g subseteq mathrm gl n q and that z is a central subgroup of g let t g z be the abelian group of equivalence classes of endotrivial k g z modules where k is an algebraically closed field of characteristic p not dividing q we show that the torsion free rank of t g z is at most one and we determine t g z in the case that the sylow p subgroup of g is abelian and nontrivial the proofs for the torsion subgroup of t g z use the theory of young modules for mathrm gl n q and a new method due to balmer for computing the kernel of restrictions in the group of endotrivial modules

p groups with maximal elementary abelian subgroups of rank 2 abstract let p be an odd prime number and g a finite p group we prove that if the rank of g is greater than p then g has no maximal elementary abelian subgroup of rank 2 it follows that if g has rank greater than p then the poset e g of elementary abelian subgroups of g of rank at least 2 is connected and the torsion free rank of the group of endotrivial kg modules is one for any field k of characteristic p we also verify the class breadth conjecture for the p groups g whose poset e g has more than one component

picard groups in triangular geometry and applications to modular representation theory for a tensor triangulated p category k with spectrum spc k we construct an injective group homomorphism h 1 spc k m 1 p pic k 1 p where pic k is the group of invertible objects of k in modular representation theory we prove that this homomorphism induces a rational isomorphism between the picard group of the projective support variety and the group of endotrivial representations

torsion free endotrivial modules let g be a finite group and let t g be the abelian group of equivalence classes of endotrivial kg modules where k is an algebraically closed field of characteristic p we investigate the torsion free part tf g of the group t g and look for generators of tf g we describe three methods for obtaining generators each of them only gives partial answers to the question but we obtain more precise results in some specific cases we also conjecture that tf g can be generated by modules belonging to the principal block and we prove the conjecture in some cases

the stable category and invertible modules for infinite groups abstract we construct a well behaved stable category of modules for a large class of infinite groups we then consider its picard group which is the group of invertible or endotrivial modules we show how this group can be calculated when the group acts on a tree with finite stabilisers

relative projectivity and relative endotrivial modules this dissertation is concerned with modular representation theory of finite groups and more precisely with the study of classes of representations which we shall term relative endotrivial modules given a prime number p a finite group g of order divisible by p we shall say that a kg module m is endotrivial relatively to the kg module v if its endomorphism algebra endk m is isomorphic as a kg module to a direct sum of a trivial module and another module which is projective relatively to v i e in short endk m k v projective more accurately in the first part of the text projectivity relative to kg modules is used to define groups of relative endotrivial modules which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules however in order to achieve this goal we first need to develop the theory of projectivity relative to modules in particular with respect to standard group operations such as induction restriction and inflation then for finite groups having a cyclic sylow p subgroup using the structure of the group t g of endotrivial modules described in mt07 we give a complete classification of the groups of relative endotrivial modules we also study the case of groups that have a sylow p subgroup isomorphic to a klein group c2 c2 as well as the case of p nilpotent groups in a second part of the text it is shown how our new groups of relative endotrivial modules provide a natural context to generalise the dade group of a p group p to an arbitrary finite group the classification of endo permutation modules and the complete description of the structure of the dade group d p was completed in 2004 by s bouc with bou06 this adventure had started about 25 years earlier with the first papers and results by e dade in dad78a and dad78b in 1978 and the final classification was in fact achieved through the non effortless combined work of several co authors between 1998 and 2004 including s bouc j carlson n mazza and j thevenaz it is most interesting to note that crucial building pieces for this classification are indeed the endotrivial modules which are particular cases of endo permutation modules yet for an arbitrary finite group g no satisfying equivalent group structure to the dade group on a class of kg modules has been defined so far with the goal to fill this gap we turn the problem upside down in some sense and show how one can regard an endo permutation module as an endotrivial module of course not in the ordinary sense but in the relative sense this shall enable us to endow a set of isomorphism classes of endo p permutation modules with a group structure similar to that of the dade group we shall call this new group the generalised dade group of the group g explicitly compute its structure and show how it is closely related to that of the g stable points of the dade group of a sylow p subgroup of g

the group of endo permutation modules the group d p of all endo permutation modules for a finite p group p is a finitely generated abelian group we prove that its torsion free rank is equal to the number of conjugacy classes of non cyclic subgroups of p we also obtain partial results on its torsion subgroup we determine next the structure of q otimes d viewed as a functor which turns out to be a simple functor s e q indexed by the elementary group e of order p 2 and the trivial out e module q finally we describe a rather strange exact sequence relating q otimes d p q otimes b p and q otimes r p where b p is the burnside ring and r p is the grothendieck ring of qp modules

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

stable auto equivalences for local symmetric algebras abstract we construct nontrivial auto equivalences of stable module categories for elementary local symmetric algebras over a field k these auto equivalences are modeled after the spherical twists of seidel and thomas and the p n twists of huybrechts and thomas which yield auto equivalences of the derived category of coherent sheaves on a variety for group algebras of p groups in characteristic p we recover many of the auto equivalences corresponding to endo trivial modules we also obtain analogous auto equivalences for local algebras of dihedral and semi dihedral type which are not group algebras