rate compatible puncturing of low density parity check codes in this correspondence we consider puncturing of low density parity check ldpc codes for additive white gaussian noise awgn channels we show that good puncturing patterns exist and that the puncturing can be performed in a rate compatible fashion furthermore rate compatible puncturing results in a small loss of performance with respect to threshold namely the punctured code is good in terms of threshold across a range of rates when compared with the optimal codes for each rate this allows one to implement a single mother encoder and decoder that is good across a wide range of rates

mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

sum power iterative water filling for multi antenna gaussian broadcast channels in this correspondence we consider the problem of maximizing sum rate of a multiple antenna gaussian broadcast channel bc it was recently found that dirty paper coding is capacity achieving for this channel in order to achieve capacity the optimal transmission policy i e the optimal transmit covariance structure given the channel conditions and power constraint must be found however obtaining the optimal transmission policy when employing dirty paper coding is a computationally complex nonconvex problem we use duality to transform this problem into a well structured convex multiple access channel mac problem we exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the mac which can easily be mapped to the optimal bc policies

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

on achievable rates and complexity of ldpc codes over parallel channels bounds and applications a variety of communication scenarios can be modeled by a set of parallel channels upper bounds on the achievable rates under maximum likelihood ml decoding and lower bounds on the decoding complexity per iteration of ensembles of low density parity check ldpc codes are presented the communication of these codes is assumed to take place over statistically independent parallel channels where the component channels are memoryless binary input and output symmetric the bounds are applied to ensembles of punctured ldpc codes where the puncturing patterns are either random or possess some structure our discussion is concluded by a diagram showing interconnections between the new theorems and some previously reported results

reliable channel regions for good binary codes transmitted over parallel channels we study the average error probability performance of binary linear code ensembles when each codeword is divided into j subcodewords with each being transmitted over one of j parallel channels this model is widely accepted for a number of important practical channels and signaling schemes including block fading channels incremental redundancy retransmission schemes and multicarrier communication techniques for frequency selective channels our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior e g turbo and low density parity check ldpc codes for a given good code ensemble we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum likelihood ml decoding to construct reliable regions we study a modifed 1961 gallager bound for parallel channels by allowing codeword bits to be randomly assigned to each component channel the average parallel channel gallager bound is simplified to be a function of code weight enumerators and channel assignment rates special cases of this bound average union bhattacharyya ub shulman feder sf simplified sphere ss and modified shulman feder msf parallel channel bounds allow for describing reliable channel regions using simple functions of channel and code spectrum parameters parameters describing the channel are the average parallel channel bhattacharyya noise parameter the average channel mutual information and parallel gaussian channel signal to noise ratios snrs code parameters include the union bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble reliable channel regions of repeat accumulate ra codes for parallel binary erasure channels becs and of turbo codes for parallel additive white gaussian noise awgn channels are numerically computed and compared with simulation results based on iterative decoding in addition an examp

channel quantization using constellation based codebooks for multiuser mimo ofdm in this paper we propose clustered quantization techniques for multiuser multi input multi output mimo orthogonal frequency division multiplexing ofdm using constellation based codebooks constellation based codebooks provide scalability and efficient codeword search capability which are key features for practical multiuser mimo ofdm systems with a large number of antennas the proposed clustered quantization scheme quantizes consecutive subcarriers into a single codeword that minimizes aggregated quantization errors we base our new clustering techniques on two constellation based quantization methods namely equal magnitude angular quantization emaq and squared lattice angular quantization new efficient codebook search algorithms are proposed for the clustered quantization in addition we propose new constellations to guarantee different users quantize channels into distinct codewords one is a rotated m psk constellation suitable for randomly distributed user scenarios and the other is a random phase equal magnitude rpem constellation suitable for ill conditioned user scenarios thus full spatial multiplexing gain can be achievable even with small number of users finally a near sphere codeword search algorithm is proposed for the rpem in simulations the proposed clustered quantization shows up to 50 higher throughput compared to conventional fixed pilot channel quantization also we show our new constellations for emaq improve throughput almost 35 compared to the standard emaq

some complete intersection symplectic quotients in positive characteristic invariants of a vector and a covector abstract given a linear action of a group g on a k vector space v we consider the invariant ring k v v g where v is the dual space we are particularly interested in the case where v f q n and g is the group u n of all upper unipotent matrices or the group b n of all upper triangular matrices in gl n f q in fact we determine f q v v g for g u n and g b n the result is a complete intersection for all values of n and q we present explicit lists of generating invariants and their relations this makes an addition to the rather short list of doubly parametrized series of group actions whose invariant rings are known to have a uniform description

parity check density versus performance of binary linear block codes new bounds and applications the moderate complexity of low density parity check ldpc codes under iterative decoding is attributed to the sparseness of their parity check matrices it is therefore of interest to consider how sparse parity check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity this issue was addressed by sason and urbanke and it is revisited in this paper the remarkable performance of ldpc codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum likelihood ml decoding and the additional loss which is imposed by the suboptimality of the decoder these issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes and lower bounds on the asymptotic density of their parity check matrices as a function of the gap between their achievable rates and the channel capacity these bounds are valid under ml decoding and hence they are valid for any suboptimal decoding algorithm the new bounds improve on previously reported results by burshtein and by sason and urbanke and they hold for the case where the transmission takes place over an arbitrary memoryless binary input output symmetric mbios channel the significance of these information theoretic bounds is in assessing the tradeoff between the asymptotic performance of ldpc codes and their decoding complexity per iteration under message passing decoding they are also helpful in studying the potential achievable rates of ensembles of ldpc codes under optimal decoding by comparing these thresholds with those calculated by the density evolution technique one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms

limited feedback unitary precoding for spatial multiplexing systems multiple input multiple output mimo wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity spatial multiplexing is a common space time modulation technique for mimo communication systems where independent information streams are sent over different transmit antennas unfortunately spatial multiplexing is sensitive to ill conditioning of the channel matrix precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter which is often not realistic this correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter the index of the optimal precoder is conveyed from the receiver to the transmitter over a low delay feedback link criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a rayleigh fading matrix channel the design criteria are shown to be equivalent to packing subspaces in the grassmann manifold using the projection two norm and fubini study distances simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback

space division multiple access with a sum feedback rate constraint on a multiantenna broadcast channel simultaneous transmission to multiple users by joint beamforming and scheduling is capable of achieving high throughput which grows double logarithmically with the number of users the sum rate for channel state information csi feedback however increases linearly with the number of users reducing the effective uplink capacity to address this problem a novel space division multiple access sdma design is proposed where the sum feedback rate is upper bounded by a constant this design consists of algorithms for csi quantization threshold based csi feedback and joint beamforming and scheduling the key feature of the proposed approach is the use of feedback thresholds to select feedback users with large channel gains and small csi quantization errors such that the sum feedback rate constraint is satisfied despite this constraint the proposed sdma design is shown to achieve a sum capacity growth rate close to the optimal one moreover the feedback overflow probability for this design is found to decrease exponentially with the difference between the allowable and the average sum feedback rates numerical results show that the proposed sdma design is capable of attaining higher sum capacities than existing ones even though the sum feedback rate is bounded

capacity achieving ldpc codes through puncturing the performance of punctured low definition parity check ldpc codes under maximum likelihood ml decoding is studied in this correspondence via deriving and analyzing their average weight distributions awds and the corresponding asymptotic growth rate of the awds in particular it is proved that capacity achieving codes of any rate and for any memoryless binary input output symmetric mbios channel under ml decoding can be constructed by puncturing some original ldpc code with small enough rate moreover it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process these results show high potential for puncturing to be used in designing capacity achieving codes and in rate compatible coding under any mbios channel

ldpc code design considerations for non uniform channels irregular low density parity check ldpc code design for parallel sub channels with different qualities is investigated such channels appear in many communication systems e g orthogonal frequency division multiplexing systems when channel knowledge is available at both the transmitter and receiver following the literature we consider allotted ldpc codes which carefully assign different parts of the code to sub channels to reduce the number of design parameters and allow for efficient design semi regular allotted codes have been suggested we first formulate the design of semi regular codes as a mixed integer linear programming relaxing the semi regularity constraint broadens the search space which results in improved codes and also a more efficient design via linear programming while information theoretic results suggest that having channel state information for a fixed power assignment does not change the capacity we show that under non optimal decoding or when the maximum degree allowed in the code is small allotted codes significantly outperform conventional ones finally for the case that neither side has the channel knowledge thus capacity loss is inevitable we see that the reduced capacity can still be approached by ldpc codes

a new generalisation of macdonald polynomials we introduce a new family of symmetric multivariate polynomials whose coefficients are meromorphic functions of two parameters q t and polynomial in a further two parameters u v we evaluate these polynomials explicitly as a matrix product at u v 0 they reduce to macdonald polynomials while at q 0 u v s they recover a family of inhomogeneous symmetric functions originally introduced by borodin

instanton counting on blowup i 4 dimensional pure gauge theory we give a mathematically rigorous proof of nekrasov s conjecture the integration in the equivariant cohomology over the moduli spaces of instantons on 4 gives a deformation of the seiberg witten prepotential for n 2 susy yang mills theory through a study of moduli spaces on the blowup of 4 we derive a differential equation for the nekrasov s partition function it is a deformation of the equation for the seiberg witten prepotential found by losev et al and further studied by gorsky et al

on decoding of low density parity check codes over the binary erasure channel this paper investigates decoding of low density parity check ldpc codes over the binary erasure channel bec we study the iterative and maximum likelihood ml decoding of ldpc codes on this channel we derive bounds on the ml decoding of ldpc codes on the bec we then present an improved decoding algorithm the proposed algorithm has almost the same complexity as the standard iterative decoding however it has better performance simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm we also provide some graph theoretic properties of different decoding algorithms of ldpc codes over the bec which we think are useful to better understand the ldpc decoding methods in particular for finite length codes

parity check density versus performance of binary linear block codes over memoryless symmetric channels we derive lower bounds on the density of parity check matrices of binary linear codes which are used over memoryless binary input output symmetric mbios channels the bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit error probability vanishes for every mbios channel we construct a sequence of ensembles of regular low density parity check ldpc codes so that an upper bound on the asymptotic density of their parity check matrices scales similarly to the lower bound the tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right regular ldpc codes which was introduced by shokrollahi and which is known to achieve the capacity of this channel under iterative message passing decoding we show that this sequence of ensembles is asymptotically optimal in a sense to be defined in this paper strengthening a result of shokrollahi finally we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs and study their performance limitations over mbios channels the latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error correction codes

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

multi user mimo with limited feedback using alternating codebooks accurate channel information at the transmitter is crucial to multi user mimo performance unfortunately the bandwidth of the control channel by which the feedback is conveyed is often limited an important issue is how to improve multi user mimo performance with minimal feedback conventional feedback techniques focus on improving the quantized codebook performance using various quantization criteria in this paper instead of trying to optimize a single codebook we apply multiple alternating codebooks to effectively reduce the multi user mimo quantization error in addition to the reduction provided by the single codebook optimization techniques that is we use an existing quantization codebook design methodology to create not one but multiple such similar codebooks the codebooks are alternated at each feedback instance creating a larger virtual codebook with the same number of feedback bits as the single smaller codebook simulation results show that significant performance gain in multi user mimo systems is obtained via the alternating codebook scheme

moduli of mckay quiver representations i the coherent component for a finite abelian group g gl n k we describe the coherent component y of the moduli space m of stable mckay quiver representations this is a not necessarily normal toric variety that admits a projective birational morphism formula obtained by variation of geometric invariant theory quotient as a special case this gives a new construction of nakamura s g hilbert scheme hilbg that avoids the typically highly singular hilbert scheme of g points in formula to conclude we describe the toric fan of y and hence calculate the quiver representation corresponding to any point of y

on universal properties of capacity approaching ldpc code ensembles this paper is focused on the derivation of some universal properties of capacity approaching low density parity check ldpc code ensembles whose transmission takes place over memoryless binary input output symmetric mbios channels properties of the degree distributions graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information theoretic bounds these bounds are expressed in terms of the target block bit error probability and the gap in rate to capacity most of the bounds are general for any decoding algorithm and some others are proved under belief propagation bp decoding proving these bounds under a certain decoding algorithm validates them automatically also under any suboptimal decoding algorithm a proper modification of these bounds makes them universal for the set of all mbios channels which exhibit a given capacity bounds on the degree distributions and graphical complexity apply to finite length ldpc codes and to the asymptotic case of an infinite block length the bounds are compared with capacity approaching ldpc code ensembles under bp decoding and they are shown to be informative and are easy to calculate finally some interesting open problems are considered

coding for parallel channels gallager bounds and applications to turbo like codes the transmission of coded communication systems is widely modeled to take place over a set of parallel channels this model is used for transmission over block fading channels rate compatible puncturing of turbo like codes multicarrier signaling multilevel coding etc new upper bounds on the maximum likelihood ml decoding error probability are derived in the parallel channel setting we focus on the generalization of the gallager type bounds and discuss the connections between some versions of these bounds the tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes repeat accumulate ra codes and some of their recent variations e g punctured accumulate repeat accumulate codes the bounds on the decoding error probability of an ml decoder are compared to computer simulations of iterative decoding the new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels this improvement is exemplified for relatively short block lengths and it is pronounced when the block length is increased in the asymptotic case where we let the block length tend to infinity inner bounds on the attainable channel regions of modern coding techniques under ml decoding are obtained based solely on the asymptotic growth rates of the average distance spectra of these code ensembles

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

remarks on the derived mckay correspondence for hilbert schemes of points and tautological bundles we study the images of tautological bundles on hilbert schemes of points on surfaces and their wedge powers under the derived mckay correspondence the main observation of the paper is that using a derived equivalence differing slightly from the standard one considerably simplifies both the results and their proofs as an application we obtain shorter proofs for known results as well as new formulae for homological invariants of tautological sheaves in particular we compute the extension groups between wedge powers of tautological bundles associated to line bundles on the surface

a graded representation model for macdonald s polynomials we define doubly graded sn modules rmu for which we conjecture that the multiplicities of irreducible representations in various bi degrees are given by the macdonald coefficients klambdamu assuming one fundamental conjecture the modules rmu can be given several equivalent definitions which we discuss we prove the conjectures in various special cases

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

limited feedback unitary precoding for orthogonal space time block codes orthogonal space time block codes ostbcs are a class of easily decoded space time codes that achieve full diversity order in rayleigh fading channels ostbcs exist only for certain numbers of transmit antennas and do not provide array gain like diversity techniques that exploit transmit channel information when channel state information is available at the transmitter though precoding the space time codeword can be used to support different numbers of transmit antennas and to improve array gain unfortunately transmitters in many wireless systems have no knowledge about current channel conditions this motivates limited feedback precoding methods such as channel quantization or antenna subset selection this paper investigates a limited feedback approach that uses a codebook of precoding matrices known a priori to both the transmitter and receiver the receiver chooses a matrix from the codebook based on current channel conditions and conveys the optimal codebook matrix to the transmitter over an error free zero delay feedback channel a criterion for choosing the optimal precoding matrix in the codebook is proposed that relates directly to minimizing the probability of symbol error of the precoded system low average distortion codebooks are derived based on the optimal codeword selection criterion the resulting design is found to relate to the famous applied mathematics problem of subspace packing in the grassmann manifold codebooks designed by this method are proven to provide full diversity order in rayleigh fading channels monte carlo simulations show that limited feedback precoding performs better than antenna subset selection

the singularities of the principal component of the hilbert scheme of points abstract haiman proved the remarkable result that the isospectral hilbert scheme of points in the plane is normal cohen macaulay and gorenstein this implies the n conjecture and the positivity conjecture for the kostka macdonald coefficients in addition he conjectured that the isospectral hilbert scheme over the principal component of hilb n c d is cohen macaulay for any d n 1 we provide a counterexample to the conjecture

vertex operators and hall littlewood symmetric functions abstract we consider vertex operators on space v with a parameter t their components form an associative algebra which is a generalization of the clifford algebra a distinguished orthogonal basis of v is proved to be the hall littlewood symmetric functions we show that kostka foulkes polynomials or certain kazhdan lusztig polynomials for the affine weyl group of type a are matrix coefficients on the space v we also obtain certain generating functions for the product of hall littlewood functions and the kostka foulkes polynomials

successive maximization for systematic design of universally capacity approaching rate compatible sequences of ldpc code ensembles over binary input output symmetric memoryless channels a systematic construction of capacity achieving low density parity check ldpc code ensemble sequences over the binary erasure channel bec has been proposed by saeedi et al based on a method here referred to as successive maximization sm in sm the fraction of degree i nodes are successively maximized starting from i 2 with the constraint that the ensemble remains convergent over the channel in this paper we propose sm to design universally capacity approaching rate compatible ldpc code ensemble sequences over the general class of binary input output symmetric memoryless biosm channels this is achieved by first generalizing the sm method to other biosm channels to design a sequence of capacity approaching ensembles called the parent sequence the sm principle is then applied to each ensemble within the parent sequence this time to design rate compatible puncturing schemes as part of our results we extend the stability condition which was previously derived for degree 2 variable nodes to other variable node degrees as well as to the case of rate compatible codes consequently we prove that using the sm principle one is able to design universally capacity achieving rate compatible ldpc code ensemble sequences over the bec unlike the previous results in the literature the proposed sm approach is naturally extendable to other biosm channels the performance of the rate compatible schemes designed based on our method is comparable to those designed by optimization

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

on certain graded sn modules and the q kostka polynomials abstract we derive here a number of properties of the q kostka polynomials k q in particular we obtain a very accessible proof that these polynomials have non negative integer coefficients other monotonicity properties are also derived these results are obtained by studying certain graded s n modules r which afford a character that may be expressed in terms of the k q certain nesting properties of the r which correspond to the dominance order of partitions then translate themselves into combinatorial inequalities involving the k q the modules r have been given an elementary presentation by deconcini and procesi invent math 64 1981 203 219 as rather simple quotients of the polynomial ring q x 1 x 2 x n we show here that their basic properties may also be derived in an entirely elementary manner

nonuniform error correction using low density parity check codes this correspondence introduces a framework to design and analyze low density parity check ldpc codes over nonuniform channels we study ldpc codes for channels with nonuniform noise distributions rate adaptive coding and unequal error protection first we propose a technique to design ldpc codes for volume holographic memory vhm systems for which the noise distribution is nonuniform we show that the proposed coding scheme has an easy design procedure and results in efficient codes for holographic memories an important property of the proposed technique is the design of the codes that have a low error floor and low variable node degrees while maintaining performance close to the shannon limit we then show that punctured ldpc codes can be studied as a special case of our design methodology for nonuniform channels finally we propose a method to generate ldpc codes that can provide unequal error protection in addition to having a good overall performance moreover the highly protected bits can be decoded without requiring the entire word to be decoded

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

on universal ldpc code ensembles over memoryless symmetric channels a design of robust error correcting codes that achieve reliable communication over various channels is of great theoretical and practical interest such codes are termed universal this paper considers the universality of low density parity check ldpc code ensembles over families of memoryless binary input output symmetric mbios channels universality is considered both under belief propagation bp and maximum likelihood ml decoding for the bp decoding case we derive a density evolution based analytical method for designing ldpc code ensembles that are universal over various families of mbios channels we also derive a necessary condition for universality of ldpc code ensembles under bp decoding this condition is used to provide bounds on the universally achievable fraction of capacity these results enable us to provide conditions for reliable unreliable communications under bp decoding that are based on the bhattacharyya parameter of the channel for the ml decoding case we prove that properly selected regular ldpc code ensembles are universally capacity achieving for the set of equi capacity mbios channels and extend this result to punctured regular ldpc code ensembles

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

capacity approaching bandwidth efficient coded modulation schemes based on low density parity check codes we design multilevel coding mlc and bit interleaved coded modulation bicm schemes based on low density parity check ldpc codes the analysis and optimization of the ldpc component codes for the mlc and bicm schemes are complicated because in general the equivalent binary input component channels are not necessarily symmetric to overcome this obstacle we deploy two different approaches one based on independent and identically distributed i i d channel adapters and the other based on coset codes by incorporating i i d channel adapters we can force the symmetry of each binary input component channel by considering coset codes we extend the concentration theorem based on previous work by richardson et al see ibid vol 47 p 599 618 feb 2001 and kavc spl caron ic spl acute et al see ibid vol 49 p 1636 52 july 2003 we also discuss the relation between the systems based on the two approaches and show that they indeed have the same expected decoder behavior next we jointly optimize the code rates and degree distribution pairs of the ldpc component codes for the mlc scheme the optimized irregular ldpc codes at each level of mlc with multistage decoding msd are able to perform well at signal to noise ratios snr very close to the capacity of the additive white gaussian noise awgn channel we also show that the optimized bicm scheme can approach the parallel independent decoding pid capacity as closely as does the mlc pid scheme simulations with very large codeword length verify the accuracy of the analytical results finally we compare the simulated performance of these coded modulation schemes at finite codeword lengths and consider the results from the perspective of a random coding exponent analysis

capacity achieving ensembles for the binary erasure channel with bounded complexity we present two sequences of ensembles of nonsystematic irregular repeat accumulate ira codes which asymptotically as their block length tends to infinity achieve capacity on the binary erasure channel bec with bounded complexity per information bit this is in contrast to all previous constructions of capacity achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap in rate to capacity the new bounded complexity result is achieved by puncturing bits and allowing in this way a sufficient number of state nodes in the tanner graph representing the codes we derive an information theoretic lower bound on the decoding complexity of randomly punctured codes on graphs the bound holds for every memoryless binary input output symmetric mbios channel and is refined for the binary erasure channel

hall littlewood polynomials and vector bundles on the hilbert scheme abstract let e be the bundle defined by applying a polynomial functor to the tautological bundle on the hilbert scheme of n points in the complex plane by a result of haiman 5 the cech cohomology groups h i e vanish for all i 0 it follows that the equivariant euler characteristic with respect to the standard two dimensional torus action has nonnegative integer coefficients in the torus variables z 1 z 2 because they count the dimensions of the weight spaces of h 0 e we derive a formula for this euler characteristic using residue formulas for the euler characteristic coming from the description of the hilbert scheme as a quiver variety 13 14 we evaluate this expression using jing s hall littlewood vertex operator with parameter z 1 7 and a new vertex operator formula given in proposition 1 below we conjecture that the summand in this formula is a polynomial in z 1 with nonnegative integer coefficients a special case of which was known to lascoux and schutzenberger 8