spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

on the structure of specht modules in the principal block of fσ3p abstract let f be a field of characteristic p at least 5 we study the loewy structures of specht modules in the principal block of f 3 p we show that a specht module in the block has loewy length at most 4 and composition length at most 14 furthermore we classify which specht modules have loewy length 1 2 3 or 4 produce a specht module having 14 composition factors describe the second radical layer and the socle of the reducible specht modules and prove that if a specht module corresponds to a partition that is p regular and p restricted then the head of the specht module does not extend the socle

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

representation type of hecke algebras of type a in this paper we provide a complete classification of the representation type for the blocks for the hecke algebra of type a stated in terms of combinatorical data the computation of the complexity of young modules is a key component in the proof of this classification result

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

3 2 pairs of symmetric group algebras and their intermediate defect 4 blocks abstract in this paper we study 3 2 pairs of symmetric group algebras and their intermediate block in detail the aim is to understand how one block of a 3 2 pair can inherit two properties the ext quiver being bipartite and the principal indecomposable modules having a common loewy length 7 from the other we establish some sufficient conditions for this inheritance and verify these conditions for some special blocks

spectral characteristics of network redundancy many real world complex networks contain a significant amount of structural redundancy in which multiple vertices play identical topological roles such redundancy arises naturally from the simple growth processes which form and shape many real world systems since structurally redundant elements may be permuted without altering network structure redundancy may be formally investigated by examining network automorphism symmetry groups here we use a group theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks in particular we describe how a network s automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs

the ordinary quiver of a weight three block of the symmetric group is bipartite abstract suppose f is a field of characteristic p 5 and that b is a p block of the symmetric group s n of defect 3 we prove that the ext 1 quiver of b is bipartite with the bipartition being described in a simple way using the leg lengths of p hooks of partitions

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

on galois structure invariants associated to tate motives we establish the equivalence of two definitions of invariants measuring the galois module structure of k groups of rings of integers in number fields due to chinburg et al on the one hand and the authors on the other we also make some remarks concerning the possibility of yet another such definition via lichtenbaum complexes

beyond rouquier partitions we obtain closed formulas in terms of littlewood richardson coefficients for the canonical basis elements of the fock space representation of uv sl e which are labeled by partitions having locally small e quotients and arbitrary e cores we further show that upon evaluation at v 1 this gives the corresponding decomposition numbers of the q schur algebra in characteristic l where q is a primitive eth root of unity if l e and q 1 otherwise whenever l is greater than the size of each constituent of the e quotient

a proof of the mullineux conjecture a partition of a positive integer n is a sequence 1 2 m 0 of integers such that i n for a positive integer p a partition 1 2 m or its young diagram is called p regular if it does not have p or more equal parts i e if there does not exist t m p 1 with t t 1 t p 1 let f be a field of characteristic p 0 it is well known that irreducible representations of the symmetric group sn over f are naturally parametrized by pregular partitions of n cf for example 9 12 if is such a partition we denote the corresponding irreducible module by d let sgnn be the one dimensional sign representation of sn over f i e sgnn f as a vector space and g f sgn g f for any g sn f f here sgn g is just the sign of the permutation g it is clear that for any irreducible d the tensor product d sgnn is also irreducible the problem usually called the problem of mullineux is to find the p regular partition such that d sgnn d put d sgnn dn in this way a bijection bn on the set pn of p regular partitions of n is defined for each positive integer n and the problem is

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

source algebras of blocks sources of simple modules and a conjecture of feit abstract we verify a finiteness conjecture of feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups

symmetric group blocks of small defect has a specht module s and a p modular irreducible module d the decomposition number s d is de ned to be the composition multiplicity of d in s the following facts areknown about blocks of weight w a if w 0 or 1 then all the decomposition numbers of the block are 0 or 1 b if w 2 and p 2 then all the decomposition numbers of the block are 0 or 1 s2 c if w 4 then some decomposition numbers of a block can be greater than 1 even if p w moreover if w 0 1 or 2 then there is a known method for determining the decompositionnumbers r the situation for the case w 3 is still not properly understood in particular thedecomposition number s 2p 2 2p 2 p 1 1 d 3p 3 2p 1 is yet to be determined for p 5 this is just one of a collection of decomposition numbers forweight 3 which we are unable to evaluate our investigation of blocks of weight 3 grew out of an attempt to improve upon the earlier resultsof martin and russell mr by explicitly calculating the decomposition numbers this led us todiscover various errors and omissions in mr which place in doubt the claim made there that whenp w all the decomposition numbers are 0 or 1 note incidentally that if the decompositionnumbers for a given block are known to be 0 or 1 then the decomposition numbers can in principle be determined by applying schaper s theorem sch jm1 2 basic resultsby using a result of brundan and kleshchev we are able to improve upon the presentation ofseveral of the basic techniques used in s1 s2 mr for estimating decomposition numbers in orderto state these results recall that the diagram of a partition is the set of nodes i j 1 j

specht modules in the auslander reiten quiver abstract let a be a symmetric algebra over an algebraically closed field we study the position of indecomposable a modules with small socles or heads in auslander reiten components of tree class a we apply the results to specht modules when a is a block of a group algebra of a symmetric group in particular we show that if the block has weight 2 then all specht modules are quasi simple that is they lie at the ends of their components

random networks eigenvalue spectra abstract we analyze the spectra of eigenvalues for random graphs with a local tree like structure the exact equations to the spectra of networks with a local tree like structure are presented we propose a simple approximation and in the framework of effective medium approximation calculate spectra of various graphs analytically we show that spectra of locally tree like random graphs gives a good description of the spectral properties of real life networks like the internet

signed young modules and simple specht modules abstract by a result of hemmer every simple specht module of a finite symmetric group over a field of odd characteristic is a signed young module while specht modules are parametrized by partitions indecomposable signed young modules are parametrized by certain pairs of partitions the main result of this article establishes the signed young module labels of simple specht modules along the way we prove a number of results concerning indecomposable signed young modules that are of independent interest in particular we determine the label of the indecomposable signed young module obtained by tensoring a given indecomposable signed young module with the sign representation as consequences we obtain the green vertices green correspondents cohomological varieties and complexities of all simple specht modules and a class of simple modules of symmetric groups and extend the results of gill on periodic young modules to periodic indecomposable signed young modules

martin s conjecture holds for weight 3 blocks of symmetric groups abstract we prove that martin s conjecture holds for weight 3 blocks of symmetric group algebras that if these blocks have abelian defect group then their projective indecomposable modules have a common radical length 7

filtrations in rouquier blocks of symmetric groups and schur algebras we study rouquier blocks of symmetric groups and schur algebras in detail and obtain explicit descriptions for the radical layers of the principal indecomposable weyl young and specht modules of these blocks at the same time the jantzen filtrations of the weyl modules are shown to coincide with their radical filtrations we also address the conjectures of martin lascoux leclerc thibon rouquier and james for these blocks

regularity of involutory pi hopf algebras abstract let h be an involutory hopf algebra over a field of characteristic 0 if h is either a finite module over its affine center or a semiprime noetherian affine pi algebra then it is auslander regular and cohen macaulay and is a finite direct sum of prime rings

network quotients structural skeletons of complex systems a defining feature of many large empirical networks is their intrinsic complexity however many networks also contain a large degree of structural repetition an immediate question then arises can we characterize essential network complexity while excluding structural redundancy in this article we utilize inherent network symmetry to collapse all redundant information from a network resulting in a coarse graining which we show to carry the essential structural information of the parent network in the context of algebraic combinatorics this coarse graining is known as the quotient we systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of real world quotients with respect to those of their parent networks in particular we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity heterogeneity and hub vertices and communication diameter and mean geodesic distance suggesting that quotients constitute the essential structural skeletons of their parent networks we summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms

rouquier blocks in chevalley groups of type e abstract in finite chevalley groups of type e we shall find a rouquier like block algebra in non defining characteristic we construct a morita equivalence between this block and a principal block of a local subgroup as an application of the equivalence we shall determine some decomposition numbers in type e

partial synchronization and partial amplitude death in mesoscale network motifs we study the interplay between network topology and complex space time patterns and introduce a concept to analytically predict complex patterns in networks of stuart landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology these patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes the underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks in addition it directly links the stability of complete in phase synchronization to only a small subset of topological eigenvalues of a network

on the main conjecture of equivariant iwasawa theory we prove the main conjecture of equivariant iwasawa theory when 0 a word of caution theorems 1 2 3 4 5 here are theorems 1 2 5 3 4 respectively in jams the main conjecture of equivariant iwasawa theory concerns the situation where l is a xed odd prime number k k is a galois extension of totally real number elds with k q and k k nite where k k is the cyclotomic zl extension we set g g k k and k g k k s a xed nite set which will normally be suppressed in the notation of primes of k containing all primes which ramify in k and all archimedean primes and m is the maximal abelian l extension of k unrami ed outside s we set x g m k it asserts that a canonical re nement f fs of the iwasawa module x is determined by the iwasawa l function lk k lk k s of k k the data f and lk k and hom rl g qc k below have been de ned in rw2 p 562 563 1 p 558 we will not repeat the de nitions now only in 1 but brie y explain how lk k should determine f denote by g qg the completed group ring zl g of g over zl and its total ring of fractions respectively the localization sequence of k theory k1 g k1 qg k0t g k0 qg has f in k0t g the reduced norms of the wedderburn components of the semi simple algebra qg induce the map det k1 qg hom rl g qc k the iwasawa l function lk k of k k is derived from the s truncated l adic artin l functions and belongs to the above group hom the equivariant main conjecture asserts that there is a unique element k1 qg satisfying det lk k and moreover that this has f in other words is the nonabelian generalization to k k of the pseudomeasure of an abelian eld extension we acknowledge nancial support provided by dfg and nserc 1where f is denoted f

adjustment matrices for weight three blocks of iwahori hecke algebras abstract we compute the adjustment matrices for weight 3 blocks of iwahori hecke algebras of type a in characteristic 2 or 3 this enables all the decomposition numbers for weight 3 blocks to be calculated

modular lie powers and the solomon descent algebra let v be an r dimensional vector space over an infinite field f of prime characteristic p and let l n v denote the nth homogeneous component of the free lie algebra on v we study the structure of l n v as a module for the general linear group gl r f when n pk and k is not divisible by p and where r n our main result is an explicit 1 1 correspondence multiplicity preserving between the indecomposable direct summands of l k v and the indecomposable direct summands of l n v which are not isomorphic to direct summands of v n our approach uses idempotents of the solomon descent algebras and in addition a correspondence theorem for permutation modules of symmetric groups

the quasihereditary structure of the auslander dlab ringel algebra abstract given an arbitrary algebra a we may associate to it a special endomorphism algebra r a introduced by auslander dlab and ringel constructed a heredity chain for r a proving that every algebra a has an associated highest weight theory in this paper we investigate the quasihereditary structure of r a using an axiomatic approach

annihilating selmer modules we prove that for a wide class of motives the local and global non commutative tamagawa number conjectures of fukaya and kato imply explicit restrictions on the galois structure of selmer modules in the sense of bloch and kato we thereby obtain a variety of new concrete and general conjectures concerning the structures of selmer modules in several important cases we prove these conjectures

james s conjecture holds for weight four blocks of iwahori hecke algebras abstract james s conjecture suggests that in certain cases the decomposition numbers for the iwahori hecke algebra of the symmetric group over a field of prime characteristic and in particular the decomposition numbers for the symmetric group itself coincide with the decomposition numbers for a corresponding iwahori hecke algebra defined over c and hence can be computed using the llt algorithm we prove this conjecture for blocks of weight 4

equivalent blocks of finite general linear groups in non describing characteristic abstract j chuang r kessar and j rickard have proved broue s abelian defect group conjecture for many symmetric groups we adapt the ideas of kessar and chuang towards finite general linear groups represented over non describing characteristic we then describe morita equivalences between certain p blocks of gl n q with defect group c p c p as q varies see theorem 2 here p and q are coprime this generalizes work of s koshitani and m hyoue who proved the same result for principal blocks of gl n q when p 3 1 in a different way

the derived categories of some blocks of symmetric groups and a conjecture of broue abstract a well known conjecture of broue in the representation theory of finite groups involves equivalences of derived categories of blocks the aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups actually we prove for these blocks a refinement of broue s conjecture due to rickard

defect 3 blocks of symmetric group algebras abstract in this paper we study if the following properties are present in defect 3 blocks of symmetric group algebras k g n where k has characteristic p 5 the ext quiver is bipartite the p i m s have a common loewy length 7 we obtain a criteria for a defect 3 block b of k g n to inherit these properties from a defect 3 block b of k g n 1 with which it forms a 3 1 pair we also verify that the principal blocks of k g n 3 p n 4 p 1 have these properties

on main conjectures in non commutative iwasawa theory and related conjectures the main conjecture of non commutative iwasawa theory of coates fukaya kato sujatha and venjakob is shown to be equivalent modulo an appropriate torsion hypothesis to a family of main conjectures over cyclotomic zp extensions together with suitable compatibility relations between the associated abelian p adic l functions this result combines with techniques of ritter and weiss and of kakde to give a concrete strategy for deducing main conjectures for non abelian compact p adic lie extensions of arbitrarily large rank from main conjectures in the classical sense as an explicit example of this approach we combine it with a recent result of ritter and weiss to prove modulo an appropriate vanishing hypothesis on invariants the main conjecture of non commutative iwasawa theory for tate motives over compact p adic lie extensions of totally real fields of arbitrarily large rank we then deduce several interesting consequences of this result concerning a range of related conjectures including the equivariant tamagawa number conjecture 1991 mathematics subject classification primary 11s40 secondary 11r29 20c05

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

weights of multipartitions and representations of ariki koike algebras abstract we consider representations of the ariki koike algebra a q deformation of the group algebra of the complex reflection group c r s n the representations of this algebra are naturally indexed by multipartitions of n and for each multipartition we define a non negative integer called the weight of we prove some basic properties of this weight function and examine blocks of small weight

decomposition numbers for weight three blocks of symmetric groups and iwahori hecke algebras let f be a field q a non zero element of f and h n h f q g n the iwahori hecke algebra of the symmetric group g n if b is a block of h n of e weight 3 and the characteristic of f is at least 5 we prove that the decomposition numbers for b are all at most 1 in particular the decomposition numbers for a p block of g n of defect 3 are all at most 1

general runner removal and the mullineux map abstract we prove a new runner removal theorem for q decomposition numbers of the level 1 fock space of type a e 1 1 generalising earlier theorems of james mathas and the author by combining this with another theorem relating to the mullineux map we show that the problem of finding all q decomposition numbers indexed by partitions of a given weight is a finite computation

irreducible specht modules for hecke algebras of type a abstract let f be a field n a non negative integer a partition of n and s the corresponding specht module for the iwahori hecke algebra h f q s n james and mathas conjecture a necessary and sufficient condition on for s to be irreducible we prove the sufficiency of this condition in the case where f has infinite characteristic and also in the case where q 1

weight two blocks of iwahori hecke algebras of type b abstract in the representation theory of iwahori hecke algebras of type a and in particular for representations of symmetric groups the notion of the weight of a block introduced by james plays a central role richards determined the decomposition numbers for blocks of weight 2 and here the same task is undertaken for weight two blocks of iwahori hecke algebras of type b using the author s own definition of the weight of a bipartition

equating decomposition numbers for different primes abstract this paper shows that certain decomposition numbers for the iwahori hecke algebras of the symmetric groups and the q schur algebras at different roots of unity in characteristic zero are equal to prove our results we first establish the corresponding theorem for the canonical basis of the level one fock space and then apply deep results of ariki and varagnolo vasserot

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group