squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof new measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states one is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state s extension and takes the infimum over such extensions additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related the second one is based on taking classical extensions this scheme is generalized which enables to construct measures of entanglement based on the mixed convex roof of a quantity which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states as one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key the findings are finally related to analogous results in classical key agreement

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

universal presheaves on group geometries and modular representations abstract we extend the ideas of sheaf homology on buildings m a ronan and s d smith j algebra 96 1985 319 346 to more general geometries with transitive automorphism groups the main technical result is the construction of a universal extension of certain partial sheaves this guarantees a rich supply of sheaves on such geometries computation of zero homology then affords representations of the group we consider various applications of the technique for example to modular irreducibles for sporadic simple groups

on the cohomology of the finite chevalley groups abstract let g be a group of lie type let k be an algebraically closed field and let v be a faithful irreducible kg module the aim of this paper is to provide bounds for dim h 1 g v as a function of dim v

circular shift linear network codes with arbitrary odd block lengths circular shift linear network coding lnc is a class of vector lnc with low encoding and decoding complexities and with local encoding kernels chosen from cyclic permutation matrices when l is a prime with primitive root 2 it was recently shown that a scalar linear solution over gf 2 l 1 induces an l dimensional circular shift linear solution at rate l 1 l in this paper we prove that for arbitrary odd l every scalar linear solution over gf 2 m l where m l refers to the multiplicative order of 2 modulo l can induce an l dimensional circular shift linear solution at a certain rate based on the generalized connection we further prove that for such l with m l beyond a threshold every multicast network has an l dimensional circular shift linear solution at rate phi l l where phi l is the euler s totient function of l an efficient algorithm for constructing such a solution is designed finally we prove that every multicast network is asymptotically circular shift linearly solvable

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

the group ring ofsl2 p2 over thep adic integers abstract this paper describes the ring theoretic structure of the group rings ofsl2 p2 over thep adic integers

unipotent elements tilting modules and saturation the results in this paper were motivated by work of serre 19 where semisimplicity results for representations of arbitrary groups in positive characteristic were established a key ingredient in the arguments was the notion of saturation where one embeds a unipotent element of prime order of a simple algebraic group in a 1 dimensional unipotent subgroup one of our goals here is to show that saturation can be achieved rather generally and to establish a uniqueness result for the resulting unipotent subgroups as a by product we show for a simple algebraic group in good characteristic a unipotent element of prime order is contained in a particularly nice subgroup of type a1 and that the lie algebra of the ambient algebraic group usually affords a tilting module for this subgroup the tilting decompositions are applied to show that the centralizers of the unipotent element a corresponding 1 dimensional unipotent group and its lie algebra all coincide and are closely related to the centralizer of the a1 subgroup we also establish a convenient factorization of the centralizer of the unipotent element results for finite groups of lie type are also obtained

vector network coding based on subspace codes outperforms scalar linear network coding this paper considers vector network coding solutions based on rank metric codes and subspace codes the main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network the multicast networks considered in this paper have one source with h messages and the vector solution is over a field of size q with vectors of length t for a given network let the smallest field size for which the network has a scalar linear solution be q s then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be q s q t in this contribution the achieved gap is q h 2 t 2 h o t for any q geq 2 and any even h geq 4 if h geq 5 is odd then the achieved gap of the alphabet size is q h 3 t 2 h 1 o t previously only a gap of size one had been shown for networks with a very large number of messages these results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks for three messages we also show an advantage of vector network coding while for two messages the problem remains open several networks are considered all of them are generalizations and modifications of the well known combination networks the vector network codes that are used as solutions for those networks are based on subspace codes particularly subspace codes obtained from rank metric codes some of these codes form a new family of subspace codes which poses a new research problem

secret key agreement by public discussion from common information the problem of generating a shared secret key s by two parties knowing dependent random variables x and y respectively but not sharing a secret key initially is considered an enemy who knows the random variable z jointly distributed with x and y according to some probability distribution p sub xyz can also receive all messages exchanged by the two parties over a public channel the goal of a protocol is that the enemy obtains at most a negligible amount of information about s upper bounds on h s as a function of p sub xyz are presented lower bounds on the rate h s n as n to infinity are derived for the case in which x x sub 1 x sub n y y sub 1 y sub n and z z sub 1 z sub n result from n independent executions of a random experiment generating x sub i y sub i and z sub i for i 1 n it is shown that such a secret key agreement is possible for a scenario in which all three parties receive the output of a binary symmetric source over independent binary symmetric channels even when the enemy s channel is superior to the other two channels

irrationality of motivic zeta functions let k0 varq 1 l denote the grothendieck ring of q varieties with the lefschetz class inverted we show that there exists a k3 surface x over q such that the motivic zeta function x t n symnx tn regarded as an element in k0 varq 1 l t is not a rational function in t thus disproving a conjecture of denef and loeser

1 cohomology of chevalley groups abstract we relate the 1 cohomology of a finite chevalley group to induced modules and finite geometry as a corollary we get the 1 cohomology of most adjoint modules

a geometric presentation of the 26 dimensional module for f4 q abstract the fixed point sheaf of the group f 4 q acting on its 26 dimensional module v is described it is shown that the 0 homology module is isomorphic to v this gives a presentation of v by geometric generators and relations

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

bounds on entanglement distillation and secret key agreement for quantum broadcast channels the squashed entanglement of a quantum channel is an additive function of quantum channels which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication this quantity has led to an upper bound of log 1 eta 1 eta on the capacity of a pure loss bosonic channel for such a task where eta is the average fraction of photons that make it from the input to the output of the channel the purpose of this paper is to extend these results beyond the single sender single receiver setting to the more general case of a single sender and multiple receivers a quantum broadcast channel we employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel along the way developing several new properties of these measures we apply our results to the case of a pure loss broadcast channel with one sender and two receivers

cyclic and quasi cyclic ldpc codes on constrained parity check matrices and their trapping sets this paper is concerned with construction and structural analysis of both cyclic and quasi cyclic codes particularly low density parity check ldpc codes it consists of three parts the first part shows that a cyclic code given by a parity check matrix in circulant form can be decomposed into descendant cyclic and quasi cyclic codes of various lengths and rates some fundamental structural properties of these descendant codes are developed including the characterization of the roots of the generator polynomial of a cyclic descendant code the second part of the paper shows that cyclic and quasi cyclic descendant ldpc codes can be derived from cyclic finite geometry ldpc codes using the results developed in the first part of the paper this enlarges the repertoire of cyclic ldpc codes the third part of the paper analyzes the trapping set structure of regular ldpc codes whose parity check matrices satisfy a certain constraint on their rows and columns several classes of finite geometry and finite field cyclic and quasi cyclic ldpc codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances consequently their error floor performances are dominated by their minimum distances

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

classical locally finite tits chamber systems of rank 3 a chamber system v 9 j zi is over some index set z is a set of chambers together with partitions in z of v if ze z and c e v then bj is the join of all partitions fijzi e z and d c is the element of fij containing c d c is again a chamber system over j with fij restricted to z we say q is connected if d c furthermore 111 is the rank of v the most important examples of chamber systems are those obtained from groups if g is a group b a subgroup of g and xi ie z are subgroups of g containing b then 5 g b xi icl has as chambers the cosets bg g e g two such cosets bg bh being i adjacent if and only if xi g xih then g acts chamber transitively on with kernel b n bg and is connected if and only if g x 1 i e z furthermore each chamber system with transitive automorphism group g can be obtained in this way by setting b g and xi gd cr a rank 2 chamber system u hi fi is called a classical generalized ma gon mq 3 if is isomorphic to g b pi p where g is an essentially simple rank 2 group of lie type in the sense of cl b is a bore1 subgroup and p pj are the two maximal parabolic subgroups of g containing b here md is given by the relation vu wj wiw g 1 defined by the weyl group of g a rank 2 chamber system is a generalized digon if is isomorphic to v g pin p pi p where pi p are different proper subgroups of g satisfying g pipj pjpi a connected chamber system over z is called a classical tits chamber system if for each ce and i j c z di j c is either a generalized digon or a classical generalized m gon for some mi e n is focally finite if all d c are finite for motivation to consider such classical locally finite tits chamber systems see 15 and 16 if is a classical tits chamber system then the diagram d z is defined in 9 0021 8693 89 3 00

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

distillation of secret key and entanglement from quantum states we study and solve the problem of distilling a secret key from quantum states representing correlation between two parties alice and bob and an eavesdropper eve via oneway public discussion we

design of quantum ldpc codes from quadratic residue sets we design classes of quantum low density parity check ldpc codes called quasi cyclic stabilizer qcs codes from conventional qc ldpc codes the proposed qcs codes belong to the family of non calderbank shor steane stabilizer codes the qc ldpc codes are self orthogonal with respect to the symplectic inner product sip and are constructed from submatrices of nonorthogonal latin squares via array dispersion the latin squares are constructed using quadratic non residue sets of prime modulus p where p 4n pm 1 for p 4n 1 two constructions namely type i a and type i b qcs codes are proposed based on matrix superposition and matrix concatenation respectively for p 4n 1 type ii qcs codes are proposed based on permutations of a base matrix we show that the parity check matrix for type i b and type ii qcs codes is self orthogonal with respect to the sip for all orders of circulant permutation matrix this resulting in ensembles of qcs codes characterized by a single base matrix we show that the minimum distance of type ii qcs codes can be lower bounded by the minimum distance of the qc ldpc codes simulation results show that the proposed qcs codes outperform some codes in the literature with a noteworthy low error floor below 10 7 over quantum depolarizing channels

a geometric presentation for the adjoint module of sl3 k abstract let k be a field the points and lines of the projective plane p k 3 determine corresponding subspaces in the space of 3 3 traceless matrices the adjoint module for the group sl 3 k we show that these subspaces conversely determine the module as their direct limit modulo derivations of the field k

multipartite quantum correlations and local recoverability characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory more recently however the total correlation or multipartite information measure has been helpful in accomplishing this goal especially with the multipartite symmetric quantum msq discord piani et al 2008 phys rev lett 100 090502 doi 10 1103 physrevlett 100 090502 1 and the conditional entanglement of multipartite information cemi yang et al 2008 phys rev lett 101 140501 doi 10 1103 physrevlett 101 140501 2 here we apply a recent and significant improvement of strong subadditivity of quantum entropy fawzi renner 2014 in order to develop these quantities further in particular we prove that the msq discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems furthermore we prove that the cemi is a faithful entanglement measure i e it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state along the way we provide an operational interpretation of the msq discord in terms of the partial state distribution protocol which in turn as a special case gives an interpretation for the original discord quantity finally we prove an inequality that could potentially improve upon the fawzi renner inequality in the multipartite context but it remains an open question to determine whether this is so 1 http dx doi org 10 1103 physrevlett 100 090502 2 http dx doi org 10 1103 physrevlett 101 140501

complete reducibility in good characteristic let g be a simple algebraic group of exceptional type over an algebraically closed field of characteristic p 0 a closed subgroup h of g is called g completely reducible g cr if whenever h is contained in a parabolic subgroup p of g it is contained in a levi subgroup of p in this paper we determine the g conjugacy classes of non g cr simple connected subgroups of g when p is good for g for each such subgroup x we determine the action of x on the adjoint module l g and the connected centraliser of x in g as a consequence we classify all non g cr reductive subgroups of g and we additionally show the existence of and classify the following restricted non g cr semisimple subgroups non g cr reductive subgroups with trivial connected centraliser and subgroups which are maximal among reductive subgroups of g

a class of non linearly solvable networks for each positive composite integer m a network is constructed which is solvable over an alphabet of size m but is not solvable over any smaller alphabet these networks have no linear solutions over any module alphabets and are not asymptotically linearly solvable over any finite field alphabets the networks capacities are all shown to equal one and their linear capacities are all shown to be bounded away from one for all finite field alphabets in addition if m is a non power of prime composite number then such a network is not solvable over any prime power size alphabet

polynomial time algorithms for multicast network code construction the famous max flow min cut theorem states that a source node s can send information through a network v e to a sink node t at a rate determined by the min cut separating s and t recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re encode the information they receive we demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed we give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity we extend these algorithms to integer capacities and to codes that are tolerant to edge failures

linear network coding over rings part ii vector codes and non commutative alphabets in part i we studied linear network coding over finite commutative rings and made comparisons to the well studied case of linear network coding over finite fields here we consider the more general setting of linear network coding over finite possibly non commutative rings and modules we prove the following results regarding the linear solvability of directed acyclic networks over various finite alphabets for any network the following are equivalent i vector linear solvability over some field ii scalar linear solvability over some ring and iii linear solvability over some module analogously the following are equivalent a scalar linear solvability over some field b scalar linear solvability over some commutative ring and c linear solvability over some module whose ring is commutative whenever any network is linearly solvable over a module a smallest such module arises in a vector linear solution for that network over a field if a network is scalar linearly solvable over some non commutative ring but not over any commutative ring then such a non commutative ring must have size at least 16 and for some networks this bound is achieved an infinite family of networks is demonstrated each of which is scalar linearly solvable over some non commutative ring but not over any commutative ring whenever p is prime and 1 le k le 6 if a network is scalar linearly solvable over some ring of size p k then it is also k dimensional vector linearly solvable over the field mathrm gf p but the converse does not necessarily hold this result is extended to all k ge 1 when the ring is commutative

entanglement of the antisymmetric state we analyse the entanglement of the antisymmetric state in dimension d d and present two main results first we show that the amount of secrecy that can be extracted from the state is low more precisely the distillable key is bounded by o 1 d second we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state entanglement cost is larger than a constant independent of d the second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant finally we note that the regularised relative entropy of entanglement is asymptotically continuous in the state elementary and advanced facts from the representation theory of the unitary group including the concept of plethysm play a central role in the proofs of the main results

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

subgroups of typea1containing semiregular unipotent elements keywords simple algebraic groups connected simple subgroups unipotent elements exceptional groups of lie type reference ctg article 1997 001doi 10 1006 jabr 1997 7099 record created on 2008 12 16 modified on 2017 05 12

irreducible a1 subgroups of exceptional algebraic groups abstract a closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup in this paper we classify all irreducible a 1 subgroups of exceptional algebraic groups g consequences are given concerning the representations of such subgroups on various g modules for example the conjugacy classes of irreducible a 1 subgroups are determined by their composition factors on the adjoint module of g

an application of model theory to real and p adic algebraic groups we use quantifier elimination for r and qp to give elementary proofs of some basic results on algebraic groups g over these fields in particular i the commutator subgroup is closed and in the real case semialgebraic and ii if g is simple then any homomorphism of g into a compact topological group is continuous van de waerden math z 36 1933 780 786 a borel and j tits ann of math 2 97 1973 499 571

broue s conjecture for the nonprincipal block of sl 2 q with full defect abstract m broue gives an important conjecture which is called broue s abelian defect group conjecture this conjecture says that a p block where p is a prime number of a finite group with an abelian defect group is derived equivalent to its brauer correspondent in the normalizer of the defect group in this paper we prove that this conjecture is true for the nonprincipal block of sl 2 p n for a positive integer n

network coding a computational perspective in this work we study the computational perspective of network coding focusing on two issues first we address the computational complexity of finding a network code for acyclic multicast networks second we address the issue of reducing the amount of computation performed by network nodes in particular we consider the problem of finding a network code with the minimum possible number of encoding nodes i e nodes that generate new packets by performing algebraic operations on packets received over incoming links we present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph g v e in time 0 e kh v k2h2 h4k3 k h where k is the number of destinations and h is the number of packets our algorithm improves the best known running time for network code construction in addition our algorithm guarantees that the number of encoding nodes in the obtained network code is upper bounded by 0 h3k2 next we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes we prove that in the majority of settings this problem is np hard however we show that if h o 1 k o 1 and the underlying communication graph is acyclic then there exists an algorithm that solves this problem in polynomial time

variations on a theme of steinberg abstract the steinberg tensor product theorem is a fundamental tool for studying irreducible representations of simple algebraic groups over fields of positive characteristic this paper is concerned with extending the result replacing the target group sl v by an arbitrary simple algebraic group

on asymptotic continuity of functions of quantum states a useful kind of continuity of quantum states functions in asymptotic regime is so called asymptotic continuity in this letter we provide general tools for checking if a function possesses this property first we prove equivalence of asymptotic continuity with so called robustness under admixture this allows us to show that relative entropy distance from a convex set including a maximally mixed state is asymptotically continuous subsequently we consider arrowing a way of building a new function out of a given one the procedure originates from constructions of intrinsic information and entanglement of formation we show that arrowing preserves asymptotic continuity for a class of functions so called subextensive ones the result is illustrated by means of several examples

circular shift linear network coding we study a class of linear network coding lnc schemes called circular shift lnc whose encoding operations consist of only circular shifts and bit wise additions formulated as a special vector linear code over gf 2 an l dimensional circular shift linear code of degree delta restricts its local encoding kernels to be the summation of at most delta cyclic permutation matrices of size l we show that on a general network for a certain block length l every scalar linear solution over gf 2 l 1 can induce an l dimensional circular shift linear solution with 1 bit redundancy per edge transmission consequently specific to a multicast network such a circular shift linear solution of an arbitrary degree delta can be efficiently constructed which has an interesting complexity tradeoff between encoding and decoding with different choices of delta by further proving that circular shift lnc is insufficient to achieve the exact capacity of certain multicast networks we show the optimality of the efficiently constructed circular shift linear solution in the sense that its 1 bit redundancy is inevitable finally both theoretical and numerical analysis imply that with increasing l a randomly constructed circular shift linear code has linear solvability behavior comparable to a randomly constructed permutation based linear code but has shorter overheads

a bound for the dimension of ho fv of modules v λi λj for al q abstract the fixed point sheaf of the group a l q acting on the module v for i j is studied it is shown that for i j and i j l l 1 or 1 2 the dimension of the 0 homology module is naturally bounded in dimension by dim v i dim v j

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

on the geometry of the adjoint representation of a chevalley group abstract we prove that the adjoint module of a chevalley group not of type c l has a presentation by long root subalgebras subject to certain relations determined by the minimal parabolic subgroups

a generalization of dual symmetry and reciprocity for symmetric algebras abstract slicing a module into semisimple ones is useful to study modules loewy structures provide a means of doing so to establish the loewy structures of projective modules over a finite dimensional symmetric algebra over a field f the landrock lemma is a primary tool the lemma and its corollary relate radical layers of projective indecomposable modules to radical layers of the f duals of those modules dual symmetry and to socle layers of those modules reciprocity we generalize these results to an arbitrary finite dimensional algebra a our main theorem which is the same as the landrock lemma for finite dimensional symmetric algebras relates radical layers of projective indecomposable modules p to radical layers of the a duals of those modules and to socle layers of injective indecomposable modules p where is the nakayama functor a key tool to prove the main theorem is a pair of adjoint functors which we call socle functors and capital functors

additive fast fourier transforms over finite fields we present new additive fast fourier transform fft algorithms based on taylor expansions over finite fields of characteristic two the new algorithms improve previous approaches by wang and zhu 1988 cantor 1989 and von zur gathen and gerhard 1996

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

algebraic algorithms for vector network coding we develop new algebraic algorithms for scalar and vector network coding in vector network coding the source multicasts information by transmitting vectors of length l while intermediate nodes process and combine their incoming packets by multiplying them with l l coding matrices that play a similar role as coding coefficients in scalar coding we start our work by extending the algebraic framework developed for multicasting over graphs by koetter and medard to include operations over matrices we build on this generalized framework to provide a new approach for both scalar and vector code design which attempts to minimize the employed field size and employed vector length while selecting the coding operations our algorithms also lead as a special case to network code designs that employ structured matrices

primitive permutation groups with a regular subgroup abstract this paper starts the classification of the primitive permutation groups g such that g contains a regular subgroup x we determine all the triples g x with soc g an alternating or a sporadic or an exceptional group of lie type further we construct all the examples g x with g a classical group which are known to us our particular interest is in the 8 dimensional orthogonal groups of witt index 4 we determine all the triples g x with soc g p 8 q in order to obtain all these triples we also study the almost simple groups g with g p 2 n 1 q the case g u n q is started in this paper and finished in b baumeister primitive permutation groups of unitary type with a regular subgroup bull belg math soc 112 5 2006 657 673 a group x is called a burnside group or short a b group if each primitive permutation group which contains a regular subgroup isomorphic to x is necessarily 2 transitive in the end of the paper we discuss b groups

on vector linear solvability of multicast networks vector linear network coding lnc is a generalization of the conventional scalar lnc such that the data unit transmitted on every edge is an l dimensional vector of data symbols over a base field gf q vector lnc enriches the choices of coding operations at intermediate nodes and there is a popular conjecture on the benefit of vector lnc over scalar lnc in terms of alphabet size of data units there exist single source multicast networks that are vector linearly solvable of dimension l over gf q but not scalar linearly solvable over any field of size q leq q l this paper introduces a systematic way to construct such multicast networks and subsequently establish explicit instances to affirm the positive answer of this conjecture for infinitely many alphabet sizes p l with respect to an arbitrary prime p on the other hand this paper also presents explicit instances with the special property that they do not have a vector linear solution of dimension l over gf 2 but have scalar linear solutions over gf q for some q where q can be odd or even this discovery also unveils that over a given base field a multicast network that has a vector linear solution of dimension l does not necessarily have a vector linear solution of dimension l l

sheaf homology and complete reducibility abstract in the study of finite simple groups by means of the geometries provided by their local subgroups problems of structure often reduce to questions about modular representations in finite characteristic of particular interest are modules spanned by fixed points of sylow groups including high weight modules for chevalley groups for which the homology methods of ronan and smith 6 can be very useful this note presents an elementary sufficient condition for splitting of some reducible modules of this type among the applications we generalize a number of results appearing in recent work of timmesfeld 11 which provided the original inspiration for this analysis some of the ideas had appeared in a special case in 10

quasi cyclic ldpc codes an algebraic construction rank analysis and codes on latin squares quasi cyclic ldpc codes are the most promising class of structured ldpc codes due to their ease of implementation and excellent performance over noisy channels when decoded with message passing algorithms as extensive simulation studies have shown in this paper an approach for constructing quasi cyclic ldpc codes based on latin squares over finite fields is presented by analyzing the parity check matrices of these codes combinatorial expressions for their ranks and dimensions are derived experimental results show that with iterative decoding algorithms the constructed codes perform very well over the awgn and the binary erasure channels