on line generalized steiner problem the generalized steiner problem gsp is defined as follows we are given a graph with non negative edge weights and a set of pairs of vertices the algorithm has to construct minimum weight subgraph such that the two nodes of each pair are connected by a path off line gsp approximation algorithms were given in agarwal et al siam j comput 24 3 1995 440 and goemans and williamson siam j comput 24 2 1995 296 we consider the on line gsp in which pairs of vertices arrive on line and are needed to be connected immediately we show that the online min cost i e greedy strategy for this problem has o log2 n competitive ratio the previous best algorithm was o nlog n competitive workshop on algorithms and data structures 1993 pp 622 633 following this work a different non greedy algorithm has been shown to achieve an o log n competitive ratio proceedings of the 29th acm symposium on theory of computing 1997 pp 344 353 we also consider the network connectivity leasing problem which is a generalization of the gsp here edges of the graph can be either bought or leased for different costs we provide simple randomized algorithm based on on line generalized steiner algorithms whose competitive ratio is within a constant factor of the best competitive algorithm for the on line gsp

generalized blocks for symmetric groups we study via character theoretic methods an l analogue of the modular representation theory of the symmetric group for an arbitrary integer l 2 we find that many of the invariants of the usual block theory ie when l is prime generalize in a natural fashion to this new context

algebras with non periodic bounded modules abstract we study weakly symmetric special biserial algebra of infinite representation type we show that usually some socle deformation of such an algebra has non periodic bounded modules the exceptions are precisely the algebras whose brauer graph is a tree with no multiple edges if the algebra has a non periodic bounded module then its hochschild cohomology cannot satisfy the finite generation property fg introduced in 9

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

character tables and defect groups abstract let b be a block of a finite group g with defect group d we prove that the exponent of the center of d is determined by the character table of g in particular we show that d is cyclic if and only if b contains a large family of irreducible p conjugate characters more generally for abelian d we obtain an explicit formula for the exponent of d in terms of character values in small cases even the isomorphism type of d is determined in this situation moreover it can read off from the character table whether d d 4 where d denotes the commutator subgroup of d we also propose a new characterization of nilpotent blocks in terms of the character table

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

on vertices of simple modules for symmetric groups of small degrees abstract we present some algorithmic methods for the computation of vertices of indecomposable and simple modules over group algebras in prime characteristic furthermore we apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3 respectively with one exception

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

a tale of two methods we describe two widely used methods for the design and analysis of approximation algorithms the primal dual schema and the local ratio technique we focus on the creation of both methods by revisiting two results by bar yehuda and even the linear time primal dual approximation algorithm for set cover and its local ratio interpretation we also follow the evolution of the two methods by discussing more recent studies

elementary divisors of the shapovalov form on the basic representation of kac moody algebras we calculate the invariant factors of the standard lattice vz inside the basic representation of untwisted kac moody algebras of ade type this in turn gives the invariant factors of cartan matrices for symmetric groups in prime characteristic as well as for the iwahori hecke algebra at roots of unity

blocks of group algebras are derived simple a derived version of maschke s theorem for finite groups is proved the derived categories bounded or unbounded of all blocks of the group algebra of a finite group are simple in the sense that they admit no nontrivial recollements this result is independent of the characteristic of the base field

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

the 2 blocks of the covering groups of the symmetric groups abstract let n be a double cover of the finite symmetric group s n of degree n i e n has a central involution z such that n z s n an irreducible character of n is called ordinary or spin according to whether it has z in its kernel or not the purpose of this paper is to determine the distribution of the spin characters of n into 2 blocks the methods applied here are essentially different from those applied to previous questions of this type we also discuss some consequences of our main result for the decomposition numbers an analogue of james well known result for the decomposition numbers of the symmetric groups is proved providing also a generalization of a theorem of benson ben theorem 1 2 in section 1 we present the background for our results and give some preliminaries in section 2 we give an explicit formula for the number of spin characters in a 2 block we also prove a result about the weight of a block containing a given non self associate spin character which will be important for the proof of our theorem on the 2 block distribution of spin characters section 3 presents some fundamental combinatorial concepts used in sections 4 and 5 the theorem concerning the spin characters in a given 2 block is proved in section 4 and in section 5 we present our results on the decomposition numbers

approximation algorithms for nonuniform buy at bulk network design buy at bulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale the objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs we present approximation algorithms for buy at bulk network design problems with costs on both edges and nodes of an undirected graph our main result is the first poly logarithmic approximation ratio for the non uniform problem that allows different cost functions on each edge and node the ratio we achieve is o log 4 h where h is the number of demand pairs in addition we present an o log h approximation for the single sink problem poly logarithmic ratios for some related problems are also obtained our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees we believe that this presents a simple yet useful general technique for network design problems

on graded cartan invariants of symmetric groups and hecke algebras we consider graded cartan matrices of the symmetric groups and the iwahori hecke algebras of type a at roots of unity these matrices are mathbb z v v 1 valued and may also be interpreted as gram matrices of the shapovalov form on sums of weight spaces of a basic representation of an affine quantum group we present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring mathbb z v v 1 this proves and generalizes a conjecture of ando suzuki yamada on the invariants of these matrices over mathbb q v v 1 and also generalizes the first author s recent proof of the kulshammer olsson robinson conjecture over mathbb z

brauer trees for the schur cover of the symmetric group abstract we determine the brauer trees of the faithful blocks of weight 1 of the schur covers of the symmetric group and the alternating group in odd characteristic

online covering with ell _q l q norm objectives and applications to network design we consider fractional online covering problems with ell q norm objectives as well as its dual packing problems the problem of interest is of the form min f x ax ge 1 x ge 0 where f x sum e c e vert x s e vert q e is the weighted sum of ell q norms and a is a non negative matrix the rows of a i e covering constraints arrive online over time we provide an online o log d log rho competitive algorithm where rho frac a max a min and d is the maximum of the row sparsity of a and max s e this is based on the online primal dual framework where we use the dual of the above convex program our result is nearly tight even in the linear special case and it expands the class of convex programs that admit online algorithms we also provide two applications where such convex programs arise as relaxations of discrete optimization problems for which our result leads to good online algorithms in particular we obtain an improved online algorithm by two logarithmic factors for non uniform buy at bulk network design and a poly logarithmic competitive ratio for throughput maximization under ell p norm capacities

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

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

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

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

block theory branching rules and centralizer algebras this paper is one in a series 8 11 exploring the algebra kgh the centralizer in the group algebra kg of the subalgebra kh where k is a field of characteristic not zero g is a finite group and h is a subgroup of g all these papers search for theorems similar to alperin s weight conjecture 2 the immediate goal is to find results that relate information about blocks of kgh or simple modules over kgh to p local information the ultimate goal is to gain insight into alperin s conjecture see the introductions to 10 and 11 for a detailed description of the program

a general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

spin block inclusions abstract in this paper we classify all block equalities and all nontrivial block inclusions for spin blocks of the double covers of the symmetric groups at different odd primes more generally we describe for an odd integer s 1 explicitly when an s block of bar partitions is contained in a t block of bar partitions for t 1 an odd integer or t 4 the question of block equality leads to the study of s t cores in the case of primes they label spin characters which are of defect 0 for different primes and therefore represent a block equality we enumerate these cores and show that there is a unique maximal one

spectral duality for block cohomology abstract throughout this paper k denotes an algebraically closed field of characteristic p 0 g denotes a finite group b denotes a p block of g corresponding to the indecomposable algebra summand kb of kg d denotes a defect group of b it is unique up to conjugacy in g p denotes the p rank of d that is the maximum of the ranks of its elementary abelian p subgroups and s t denote kb modules a graded polynomial subalgebra r of h g k is specified by taking p algebraically independent canonical homogeneous elements which are certain chern classes to be its generators a spectral sequence is constructed with ext r ext kb s t r as its e2 term and with ext kb t s with a displaced grading as its abutment the latter is thus naturally filtered a brief account of the relevant algebraic topology commutative algebra and homological algebra is given

buy at bulk network design with protection we consider approximation algorithms for buy at bulk network design with the additional constraint that demand pairs be protected against a single edge or node failure in the network in practice the most popular model used in high speed telecommunication networks for protection against failures is the so called 1 1 model in this model two edge or node disjoint paths are provisioned for each demand pair we obtain the first nontrivial approximation algorithms for buy at bulk network design in the 1 1 model for both edge and node disjoint protection requirements our results are for the single cable cost model which is prevalent in optical networks more specifically we present a constant factor approximation for the single sink case and a polylogarithmic factor approximation for the multicommodity case these results are of interest for practical applications and also suggest several new challenging theoretical problems

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

restricted 2 factor polytopes the optimal k restricted 2 factor problem consists of finding in a complete undirected graph k n a minimum cost 2 factor subgraph having degree 2 at every node with all components having more than k nodes the problem is a relaxation of the well known symmetric travelling salesman problem and is equivalent to it when k n 1 we study the k restricted 2 factor polytope we present a large class of valid inequalities called bipartition inequalities and describe some of their properties some of these results are new even for the travelling salesman polytope for the case k 3 the triangle free 2 factor polytope we derive a necessary and sufficient condition for such inequalities to be facet inducing

on the number of simple modules in a block of a finite group abstract we prove that if b is a p block with non trivial defect group d of a finite p solvable group g then l b p r where r is the sectional rank of d we remark that there are infinitely many p blocks b with non abelian defect groups and l b p r 1 we conjecture that the inequality l b p r holds for an arbitrary p block with defect group of sectional rank r we show this to hold for a large class of p blocks of various families of quasi simple and nearly simple groups

on cartan matrices and lower defect groups for covering groups of symmetric groups we determine the elementary divisors of the cartan matrices of spin p blocks of the covering groups of the symmetric groups when p is an odd prime as a consequence we also compute the determinants of these cartan matrices and in particular we confirm a conjecture by brundan and kleshchev that these determinants depend only on the weight but not on the sign of the block

character correspondences in finite general linear unitary and symmetric groups let r 0 be a prime integer and let g be a finite general linear group gl n q a unitary group u n q with q qg or a symmetric group s n of degree n the partition of the irreducible ordinary characters of s n into r blocks of s n was given by brauer s and robinson s solution of nakayama s conjecture see 7 p 245 in their fundamental paper 5 fong and srinivasan have recently classified the r blocks of gl n q and u n q for all primes r 2 with r q 1 using these classifications of the r blocks of g we show in this article that there is a natural one to one correspondence b between the irreducible characters of height zero of an r block b of g with defect group r and the irreducible characters of height zero of the brauer correspondent b of b in n ng r theorem 4 10 in particular ko b ko b where ko b denotes the number of all irreducible characters of b with height h t z 0 therefore alperin s conjecture on the numbers of irreducible characters of height zero is verified for all general linear unitary and symmetric groups if g s n then theorem 4 10 also holds for r 2 in order to establish the character correspondence three other correspondences are studied the product of which is b in sect 1 we construct for every r block b of g with defect group r a subgroup g of g with a sylow r subgroup r such that there is a natural height preserving one to one correspondence t between the set irr b of all irreducible characters of b and the set irr 0 of all irreducible characters of the principal r block o of d reduction theorems 1 9 and 1 10 the map respects the geometric conjugacy classes of characters if 0 denotes the principal r block of n nd and if b is the brauer correspondent of b in n n6 r then by theorems 3 8 and 3 10 the block ideals o and b are morita equivalent have the same decomposition numbers and there is a natural height preserving one to one correspondence a between irr bo and irr b

boundedness and periodicity of modules over qf rings be a minimal projective resolution m is called bounded if the lengths of the kernels ker di have a common upper bound and m is called periodic if some ker di is isomorphic to m in 1 p 7761 alperin posed the following question is every bounded kg module periodic alperin answered this question in the affirmative in case k is algebraic over its prime field l theorem 11 without any restriction on k eisenbud settled alperin s question in 6 theorem 9 21 in section 1 of the present paper we look at an arbitrary qf ring r instead of a group algebra and we give a condition which forces a bounded r module m to be periodic before we will formulate our result we make some preliminary remarks let m and x be modules over a ring r with the yoneda product for multiplication the abelian group extx m m oi extk m m becomes a graded ring which we call the extension ring of m and the abelian group ext m x becomes a graded right extr m m module we recall some classical examples of extension rings let g be a finite group and let k be a field on which g acts trivially then the cohomology ring h g k is the extension ring ext k k more generally let m be a finitely generated kg module then the ring hom m m is a g ring and the cohomology ring h g hom m m with cup product is canonically isomorphic to the extension ring extx m m with yoneda product these rings have been intensively studied see carlson 3 for details and for further references other well known exampes of extension rings arise in the homological algebra of local rings namely the extension

approximation algorithms for problems combining facility location and network design we present approximation algorithms for integrated logistics problems that combine elements of facility location and transport network design we first study the problem where opening facilities incurs opening costs and transportation from the clients to the facilities incurs buy at bulk costs and provide a combinatorial approximation algorithm we also show that the integer programming formulation of this problem has small integrality gap we extend the model to the version when there is a bound on the number of facilities that may be opened

dynamic algorithms via the primal dual method abstract we develop a dynamic version of the primal dual method for optimization problems and apply it to obtain the following results 1 for the dynamic set cover problem we maintain an o f 2 approximately optimal solution in o f log m n amortized update time where f is the maximum frequency of an element n is the number of sets and m is the maximum number of elements in the universe at any point in time 2 for the dynamic b matching problem we maintain an o 1 approximately optimal solution in o log 3 n amortized update time where n is the number of nodes in the graph

generalised cartan invariants of symmetric groups kulshammer olsson and robinson developed an analogue of modular representation theory of symmetric groups where is not necessarily a prime they gave a conjectural combinatorial description for invariant factors of the cartan matrix in this context we confirm their conjecture by proving a more precise blockwise version of the conjecture due to bessenrodt and hill

primitive central idempotents of finite group rings of symmetric groups let p be a prime we denote by s n the symmetric group of degree n by an the alternating group of degree n and by f p the field with p elements an important concept of modular representation theory of a finite group g is the notion of a block the blocks are in one to one correspondence with block idempotents which are the primitive central idempotents of the group ring f q g where q is a prime power here we describe a new method to compute the primitive central idempotents of f q g for arbitrary prime powers q and arbitrary finite groups g for the group rings f p s n of the symmetric group we show how to derive the primitive central idempotents of f p s n p from the idempotents of f p s n improving the theorem of osima for symmetric groups we exhibit a new subalgebra of f p s n which contains the primitive central idempotents the described results are most efficient for p 2 in an appendix we display all primitive central idempotents of f 2 s n and f 4 a n for n 50 which we computed by this method

cartan determinants and shapovalov forms we compute the determinant of the gram matrix of the shapovalov form on weight spaces of the basic representation of an affine kac moody algebra of ade type possibly twisted as a consequence we obtain explicit formulae for the determinants of the cartan matrices of p blocks of the symmetric group and its double cover and of the associated hecke algebras at roots of unity

on restriction of characters to defect groups abstract we put forward a blockwise version of a recent conjecture of 6 on finite groups let b be a p block of a finite group g with defect group d let irr b be a character with positive height in this note we conjecture that the number of distinct linear constituents of the restriction d is 0 or at least p we prove that this is indeed the case for various classes of finite groups and p blocks

homological embeddings for preprojective algebras for a fixed finite dimensional algebra a we study representation embeddings of the form mod b rightarrow mod a such an embedding is called homological if it induces an isomorphism on all ext groups and weakly homological if only ext 1 is preserved in case a is a preprojective algebra of dynkin type we give an explicit classification of all weakly homological and homological embeddings furthermore we show that for self injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the tachikawa conjecture

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

cartan invariants and central ideals of group algebras abstract in this paper we investigate general properties of cartan invariants of a finite group g in characteristic 2 one of our results shows that the cartan matrix of g in characteristic 2 contains an odd diagonal entry if and only if g contains a real element of 2 defect zero we also apply these results to 2 blocks of symmetric groups and to blocks with normal or abelian defect groups the second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks

block separations and inclusions abstract we investigate the separation of characters by blocks at different primes and the inclusions of q blocks in p blocks viewed as sets of characters and use these notions to prove results on the structure of the corresponding groups in particular we provide a new criterion for the nilpotence of a finite group g based on the separation by principal blocks and we show that a condition on block unions has strong structural consequences

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

lp rounding approximation algorithms for stochastic network design we study the steiner tree problem and the single cable single sink network design problem under a two stage stochastic model with recourse and finitely many scenarios in these models some edges are purchased in a first stage when only probabilistic information about the second stage is available in the second stage one of a finite number of specified scenarios is realized which results in the set of terminals becoming known and the opportunity to purchase additional edges under an inflated cost function to augment the first stage solution we provide constant factor approximation algorithms for these problems by rounding the linear relaxation of ip formulations of the problems our algorithms involve solving the linear relaxation first followed by a primal dual routine that is guided by the lp solution we also show that because our bounds are local the cost of each component is bounded by its cost in the lp solution we are able to obtain bounds that guard against a form of downside risk