characteristic free decomposition of skew schur functors this note is a generalization of our thesis brandeis university 1984 whose material can be found in 2 our aim is to show that by the methods presented there one can produce a characteristic free filtration of the skew schur functor l f whose associated graded module is the direct sum of schur modules l f one would expect from the characteristic zero theory as in 2 notations and facts are freely borrowed from l

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

comments on the drinfeld realization of the quantum affine superalgebra and its hopft algebra structure by generalizing the reshetikhin and semenov tian shansky construction to supersymmetric cases we obtain the drinfeld current realization for the quantum affine superalgebra we find a simple coproduct for the quantum current generators and establish the hopf algebra structure of this super current algebra

koszul duality and extensions of exponential functors we study koszul duality in the category of strict polynomial functors we compute koszul duals of various functors and apply these results to the problem of calculating ext groups between exponential functors the main application is a full description of the ext groups between twisted exterior and divided powers and between twisted symmetric and divided powers which completes the program started in v franjou e friedlander a scorichenko a suslin general linear and functor cohomology over finite fields ann of math 150 2 1999 663 728

vertex operators of uq gl n n and highest weight representations of uq gl 2 2 we determine the exchange relations of the level one q vertex operators of the quantum affine superalgebra uq gl n n we study in detail the level one irreducible highest weight representations of uq gl 2 2 and compute the characters and supercharacters associated with these irreducible modules

characteristic free representation theory of the general linear group ii homological considerations in our first paper of this series cl we indicated how we were led to consider resolutions of schur and weyl modules of a particular form in order to prove the existence of these resolutions we were forced to enlarge the family of skew shapes to a class j containing new shapes whose corresponding schur and weyl modules had heretofore not been studied with these shapes in hand we proved in 1 the existence of some fundamental exact sequences and described how from these exact sequences we could use a mapping cone construction to build up the resolutions we were seeking for this mapping cone construction we needed maps and to provide maps we needed projectivity of tensor products of divided powers this led to the study of the schur algebra and its decomposition into orthogonal idempotents in sections 1 and 2 we review the information about the schur algebra that we need to carry out our program fortunately there is a very clear and detailed exposition of this subject in the notes of j a green s from which we borrowed very heavily in fact the main function of the first two sections is to condense and translate into our notation and terminology the relevant sections of green s notes in sections 3 and 4 we define the family of shapes j that we will study

resolutions of determinantal ideals the submaximal minors for many years there has been considerable interest in finding a resolution of the ideal generated by the minors of order p of a generic m x n matrix to put the problem more precisely suppose r is a commutative ring and x are variables with 1 i m and 1 j q n if we let r r x be the polynomial ring over r then we have the generic matrix x and we may form the ideal zp in r generated by the p x p minors of this matrix the problem then is to find an explicit free resolution of the ideal zp over the ring r it was proved by eagon and hochster lo that r i has a resolution of length m p l n p l but their proof consisted in showing that the ideal zp is perfect it did not provide a construction of the resolution in fact it is not known whether the betti numbers of the ideal zp depend on the characteristic of the ground ring r in 121 lascoux succeeded in giving an explicit resolution provided that the ground ring r contained the field q of rational numbers his construction rests heavily on the theory of schur functors and the fact that in characteristic zero the schur functors are the irreducible representations of the general linear group over the integers however the construction breaks down despite the fact that one can define the schur functors over an arbitrary commutative ring see 1 2 14 151 for various constructions of schur functors in analyzing the work of lascoux and its subsequent reworking by nielsen 131 some basic facts seemed to clamor for attention one was that within a resolution of r z there appeared to be two types of boundary maps one of degree 1 and one of degree p the maps of degree 1 were maps between sums of schur functors of fixed durfee square k see section 2 for definitions while the maps of degree p were from sums of schur functors of

multiuser random coding techniques for mismatched decoding this paper studies multiuser random coding techniques for channel coding with a given possibly suboptimal decoding rule for the mismatched discrete memoryless multiple access channel an error exponent is obtained that is tight with respect to the ensemble average and positive within the interior of lapidoth s achievable rate region this exponent proves the ensemble tightness of the exponent of liu and hughes in the case of maximum likelihood decoding an equivalent dual form of lapidoth s achievable rate region is given and the latter is shown to immediately extend to channels with infinite and continuous alphabets in the setting of single user mismatched decoding similar analysis techniques are applied to a refined version of superposition coding which is shown to achieve rates at least as high as standard superposition coding for any set of random coding parameters

mixing of ground states in vertex models we consider the analogue of the six vertex model constructed from alternating spin n 2 and spin m 2 lines where we identify the transfer matrix and the space on which it acts in terms of the representation theory of we diagonalize the transfer matrix and compute the s matrix we give a trace formula for local correlation functions when n 1 the one point function of a spin m 2 local variable for the alternating lattice with a particular ground state is given as a linear combination of the one point functions of the pure spin m 2 model with different ground states the mixing ratios are calculated exactly and are expressed in terms of irreducible characters of and the deformed virasoro algebra

a short proof of a theorem of m hashimoto abstract recently m hashimoto announced that the resolutions of determinantal ideals depend on the characteristic of the base field the purpose of this note is to give a short proof of this result one should stress that even though the method we use is different from hashimoto s his result provided us with the clue where the additional relation exists

on odd symplectic schur functions abstract proctor defined combinatorially a family of laurent polynomials called odd symplectic schur functions indexed by pairs c where is partition andcis a column length of a conjecture of proctor invent math 92 1988 307 332 includes the statement that the odd symplectic schur functions are actually characters ofsp 2n 1 c the purpose of the present note is to prove this

a level one representation of the quantum affine superalgebra uqmn a level one representation of the quantum affine superalgebra uqmn and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields character formulas of level one irreducible highest weight modules of uq are conjectured

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

characteristic free representation theory of the general linear group on etudie systematiquement les z formes des representations rationnelles du groupe lineaire generale

a smooth space of tetrahedra abstract we construct a smooth symmetric compactification of the space of all labeled tetrahedra in p 3

acyclicity of schur complexes and torsion freeness of schur modules abstract let r be a noetherian commutative ring and m a r module with pd r m 1 that has rank necessary and sufficient conditions were provided in 1 for an exterior power k m to be torsion free when m is an ideal of r similar necessary and sufficient conditions were provided in 2 for a symmetric power s k m to be torsion free we extend these results to a broad class of schur modules l m en route for any map of finite free r modules f g we also study the general structure of the schur complexes l and provide necessary and sufficient conditions for the acyclicity of any given l by computing explicitly the radicals of the ideals of maximal minors of all its differentials

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

green lazarsfeld condition for toric edge ideals of bipartite graphs abstract previously ohsugi and hibi gave a combinatorial description of bipartite graphs g whose toric edge ideal i g is generated by quadrics showing that every cycle of g of length at least 6 must have a chord this corresponds to the green lazarsfeld condition n 1 in this paper we investigate the higher syzygies of i g and give combinatorial descriptions of the green lazarsfeld conditions n p of toric edge ideals of bipartite graphs for all p 1 in particular we show that i g is linearly presented i e satisfies condition n 2 if and only if the bipartite complement of g is a tree of diameter at most 3 we also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the green lazarsfeld conditions

commutation relations of vertex operators for uq sl m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n we consider commutation relations and invertibility relations of vertex operators for the quantum affine superalgebra uq sl m n by using bosonization we show that vertex operators give a representation of the graded zamolodchikov faddeev algebra by direct computation invertibility relations of type ii vertex operators for n m are very similar to those of type i for m n

on the reliability function of the discrete memoryless relay channel bounds on the reliability function for the discrete memoryless relay channel are derived using the method of types two achievable error exponents are derived based on partial decode forward and compress forward which are well known superposition block markov coding schemes the derivations require combinations of the techniques involved in the proofs of csiszar korner marton s packing lemma for the error exponent of channel coding and marton s type covering lemma for the error exponent of source coding with a fidelity criterion the decode forward error exponent is evaluated on sato s relay channel from this example it is noted that to obtain the fastest possible decay in the error probability for a fixed effective coding rate one ought to optimize the number of blocks in the block markov coding scheme assuming the blocklength within each block is large an upper bound on the reliability function is also derived using ideas from haroutunian s lower bound on the error probability for point to point channel coding with feedback

on the resolutions of the powers of the pfaffian ideal x is a generic skew symmetric matrix and we let s stand for the polynomial ring r x r z the pfaffians of the 2k order principal submatrices of x generate an ideal of s the pfalfian ideal denoted by i it is well known that i is a generically perfect gorenstein prime ideal of grade 3 in fact a prototype of all the ideals of this kind cf b e 31 hence i has a finite free s resolution of length 3 a resolution which looks the same regardless of the ring r

semi invariants of quivers and saturation for littlewood richardson coefficients q is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities in particular the set q is saturated i e if n q then also q these results when applied to a special quiver q tn n n and to a special dimension vector show that the gln module v appears in v v if and only if the partitions and satisfy an explicit set of inequalities this gives new proofs of the results of klyachko 7 3 and knutson and tao 8 the proof is based on another general result about semi invariants of quivers theorem 1 in the paper 10 schofield defined a semi invariant cw for each indecomposable representation w of q we show that the semi invariants of this type span each weight space in si q this seems to be a fundamental fact connecting semi invariants and modules in a direct way given this fact the results on sets of weights follow at once from the results in another paper of schofield 11

koszul modules and green s conjecture we prove a strong vanishing result for finite length koszul modules and use it to derive green s conjecture for every g cuspidal rational curve over an algebraically closed field mathbf k with text char mathbf k 0 or text char mathbf k ge frac g 2 2 as a consequence we deduce that the general canonical curve of genus g satisfies green s conjecture in this range our results are new in positive characteristic whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by voisin our strategy involves establishing two key results of independent interest 1 we describe an explicit characteristic independent version of hermite reciprocity for mathfrak s mathfrak l 2 representations 2 we completely characterize in arbitrary characteristics the non vanishing behavior of the syzygies of the tangential variety to a rational normal curve

the case of the submaximal minors a finishing touch let f and g be any two finitely generated free modules over the commutative ring r of ranks m and n respectively m 2 n say let r denote the symmetric algebra s f g r is a polynomial ring r x with 1 6 i m 1 j n each x corresponds to fi gj where j and g denote bases of f and g respectively take the modules f r 0 r f and g r or0 g finitely generated and free over r and let h and gj also denote by abuse of notation the obvious r bases of f and g a distinguished element in hom fo g r is the map cp defined by cp fi g x let q also denote the corresponding map g f obtained via the canonical isomorphism

complexes associated with trace and evaluation another approach to lascoux s resolution i l th syzygy hence the length of a minimal free resolution of d is m p 1 n 1 although lascoux described explicitly maps between the consecutive syzygies he was unable to prove that this leads to a complex proofs of his lemmas 3 7 1 and 3 7 2 in s are incomplete we illustrate essential dif ficulties on the example of 1 x 1 minors in this case the lascoux s con struction gives the rth syzygy as c r lif lxg lascoux describes the differential as being composed from the maps la f lxg l fq l g for p c 1 j u i1 1 each of those maps is specified up to scalar and he does not say how to adjust them in order to get a complex even in the case p 1 when the rth syzygy equals a f g isomorphism a f g cll i l f lxg is quite complicated and it is difficult to compute these scalars roberts in his preprint 123 gave more down to earth construction of a complex but he was unable to prove that it is a resolution of d except in some special cases the purpose of this paper is to overcome these difficulties by presenting still another approach that allows us to get a complex and to prove easily that it is a minimal free resolution of d here are some essential features of our method the basic idea can be explained in case

on the structure of free resolutions of length 3 the sources of structure theory of finite free resolutions go back to hilbert in his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal this theorem was then extended by burch to arbitrary commutative rings there are various motivations for extending this results one of them is a variety of applications of the hilbert burch theorem in deformation theory and algebraic geometry cf for example s another reason comes from serre s conjecture one would like to understand in terms of resolutions why the lifting property of grothendieck does not hold in 1974 buchsbaum and eisenbud in bel introduced so called structure theorems various factorisations which occur for acyclic complexes of arbitrary length then hochster in h generalized the hilbert burch theorem to arbitrary free complexes of length 2 his results were then improved in br pw still the structure of complexes of length bigger than 2 is not understood the first steps in this direction are the results of b and the construction of a generic ring for complexes of length 3 of type 1 rz 1 1 given in pw in this paper i construct a ring which i believe is the generic ring for complexes of length 3 the construction involves new structure theorems and a new idea the notion of defects of structure theorems the paper is organized as follows section 1 is introductory i give there the background from representation theory and the cohomology groups of a lie algebra section 2 contains the proof of the new structure theorems we need the construction of the generic ring r and of the defect lie algebra l in section 3 i show that genericity of r follows from vanising of the homology of a certain family of complexes over u l the enveloping algebra of l in section 4 i apply the results of section 2 to study multi

bosonization of vertex operators for the face model we present a free boson realization of the vertex operators and their duals for the solvable sos lattice model of type we discuss a possible connection with the calculation of correlation functions

a straightening algorithm for row convex tableaux abstract we produce a new basis for the schur and weyl modules associated to a row convex shape d the basis is indexed by a new class of straight tableaux which we introduce by weakening the usual requirements for standard tableaux spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis for skew shapes this algorithm specializes to the classical straightening law the new straight basis is used to produce bases for flagged schur and weyl modules to provide groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row convex representations systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear lie superalgebra as well as to the general linear group

error exponent for multiple access channels upper bounds the problem of bounding the reliability function of a multiple access channel mac is studied two new upper bounds on the error exponent of a two user discrete memoryless dm mac are derived the first bound sphere packing is an upper bound on the exponent of the average probability of error and is the first bound of this type that is zero outside the capacity region and thus results in a tighter sphere packing exponent when compared with the tightest known exponent derived by haroutunian the second bound minimum distance is an upper bound on the exponent of the maximal as opposed to average probability of error to obtain this bound first an upper bound on the minimum bhattacharyya distance between codeword pairs is derived for a certain class of two user dm macs an upper bound on the exponent of maximal probability of error is derived as a consequence of the upper bound on the minimum bhattacharyya distance we analytically evaluate the sphere packing bound for uniform composition codes for an additive and nonsymmetric channel and show that it is tight near the boundary of the capacity region i e equal to the random coding lower bound

addendum specht modules for column convex diagrams characteristic free results for weyl modules abstract it is shown that the results in reiner and shimozono 12 have characteristic free analogues for weyl modules

regularity and cohomology of determinantal thickenings we consider the ring s c xij of polynomial functions on the vector space cm n of complex m n matrices we let gl glm c gln c and consider its action via row and column operations on cm n and the induced action on s for every gl invariant ideal i s and every j 0 we describe the decomposition of the modules extsj s i s into irreducible gl representations for any inclusion i j of gl invariant ideals we determine the kernels and cokernels of the induced maps extsj s i s extsj s j s as a consequence of our work we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals and in particular we determine which powers have a linear minimal free resolution as another consequence we characterize the gl invariant ideals i s for which the induced maps extsj s i s hij s are injective in a different direction we verify that kodaira vanishing as described in work of bhatt blickle lyubeznik singh zhang holds for determinantal thickenings

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

bases for coordinate rings of conjugacy classes of nilpotent matrices abstract a general conjecture is given for an explicit basis of the coordinate ring of the closure of the conjugacy class of a nilpotent matrix this conjecture is proven when the partition given by the transpose jordan type of the nilpotent matrix is a hook or has two parts

primary ideals associated to the linear strands of lascoux s resolution and syzygies of the corresponding irreducible representations of the lie superalgebra gl m n abstract we investigate equivariant koszul duality between primary ideals i a b of s s m m n associated to rectangular young diagrams a b and the corresponding atypical irreducible mixed supertensor representations x a b of gl m n in characteristic zero we show i a b to be h 0 s x a b of the koszul dual s x a b and we compute all the higher cohomology of s x a b to be direct sums of i a r b r with multiplicities being given by coefficients of gauss polynomials utilizing this we are able to describe the equivariant syzygies of x a b over s m m n and determine the homological dimension of i a b over s

level one highest weight representation of uq sl n 1 and bosonization of the multicomponent super t j model we study the level one irreducible highest weight representations of the quantum affine superalgebra uq sl n 1 and calculate their characters and supercharacters we obtain bosonized q vertex operators acting on the irreducible uq sl n 1 modules and derive the exchange relations satisfied by the vertex operators we give the bosonization of the multicomponent super t j model by using the bosonized vertex operators

on invariant ideals associated to classical groups abstract we study the g 1 g 2 invariant ideals of the coordinate ring of the variety of nullforms associated to a pair of classical groups over a field of characteristic zero

minimal free resolutions of determinantal ideals and irreducible representations of the lie superalgebra gl m n over k in the standard coordinate functions x on m k the i j m n homogeneous ideal of s generated by the p p minors of the m n matrix x is a prime ideal denoted i whose zero set is the determinani j p tal subvariety of m k consisting of matrices of rank less than p this m n subvariety is preserved under the natural action of the algebraic group gl k gl k on the affine space m k by row and column m n m n operations and the ideal i is preserved under the corresponding group p action on the coordinate algebra s the only prime ideals of s that are invariant under the action of gl k gl k are the ideals i for m n p

graph decomposition a new key to coding theorems a new and simple method is proposed for finding good encoders both for channels and for sources with side information this method relies on the continuous version of a graph decomposition result of lovasz the presently known best exponential error bounds for both problems follow in a unified manner with an improvement on the source coding bound the previous bounds for universal codes of the authors and marton are also improved

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

mismatched decoding error exponents second order rates and saddlepoint approximations this paper considers the problem of channel coding with a given possibly suboptimal maximum metric decoding rule a cost constrained random coding ensemble with multiple auxiliary costs is introduced and is shown to achieve error exponents and second order coding rates matching those of constant composition random coding while being directly applicable to channels with infinite or continuous alphabets the number of auxiliary costs required to match the error exponents and second order rates of constant composition coding is studied and is shown to be at most two for independent identically distributed random coding asymptotic estimates of two well known non asymptotic bounds are given using saddlepoint approximations each expression is shown to characterize the asymptotic behavior of the corresponding random coding bound at both fixed and varying rates thus unifying the regimes characterized by error exponents second order rates and moderate deviations for fixed rates novel exact asymptotics expressions are obtained to within a multiplicative 1 o 1 term using numerical examples it is shown that the saddlepoint approximations are highly accurate even at short block lengths

when does depth stabilize early on in this paper we study graded ideals i in a polynomial ring s such that the numerical function k depth s ik is constant we show that if i the rees algebra of i is cohen macaulay ii the cohomological dimension of i is not larger than the projective dimension of s i and iii the k algebra generated by some homogeneous generators of i is a direct summand of s then depth s ik is constant all the ideals with constant depth function discovered by herzog and vladoiu in 11 satisfy the criterion given above in the contest of square free monomial ideals there is a chance that a converse of the previous fact holds true

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

a converse coding theorem for mismatched decoding at the output of binary input memoryless channels an upper bound on the maximal transmission rate over binary input memoryless channels provided that the decoding decision rule is given is derived if the decision rule is equivalent to the maximum likelihood decoding matched decoding then the bound coincides with the channel capacity otherwise mismatched decoding it coincides with a known lower bound

on achievable rates and error exponents for channels with mismatched decoding the problem of characterizing achievable rates and error exponents for discrete memoryless channels with mismatched decoding is addressed a mismatched cognitive multiple access channel is introduced and an inner bound on its capacity region is derived using two alternative encoding methods 1 superposition coding and 2 random binning the inner bounds are derived by analyzing the average error probability of the code ensemble for both methods and by a tight characterization of the resulting error exponents random coding converse theorems are also derived a comparison of the achievable regions shows that in the matched case random binning performs as well as superposition coding i e the region achievable by random binning is equal to the capacity region the achievability results are further specialized to obtain a lower bound on the mismatch capacity of the single user channel by investigating a cognitive multiple access channel whose achievable sum rate serves as a lower bound on the single user channel s capacity while the achievable rate presented here may not improve the rate achieved by lapidoth s scheme in optimizing over the parameters of the random coding scheme it can improve the achieved rate for given parameters and thereby may reduce the computational complexity required to find a good code

homotopy equivalence of two families of complexes an explicit homotopy equivalence is established between two families of complexes both of which generalize the classical koszul complex

bifunctor cohomology and cohomological finite generation for reductive groups let g be a reductive linear algebraic group over a field k let a be a finitely generated commutative k algebra on which g acts rationally by k algebra automorphisms invariant theory states that the ring of invariants ag h0 g a is finitely generated we show that in fact the full cohomology ring h g a is finitely generated the proof is based on the strict polynomial bifunctor cohomology classes constructed in 22 we also continue the study of bifunctor cohomology of gl 1

the q deformed supersymmetric t j model with a boundary the q deformed supersymmetric t j model on a semi infinite lattice is diagonalized by using the level one vertex operators of the quantum affine superalgebra u q sl 2 1 we give the bosonization of the boundary states we give an integral expression for the correlation functions of the boundary model and derive the difference equations which they satisfy

complexes which arise from a matrix and a vector resolutions of divisors on certain varieties of complexes abstract consider the polynomial ring r r0 x y where r0 is a normal domain and x1 g and yg f are matrices of indeterminates the r ideal j i1 xy imin f g y defines a variety of complexes over r0 the divisor class group of r j is isomorphic to cl r0 z i where i is an ideal of r j generated by appropriately chosen lower order minors of y we produce the minimal r free resolution of i i for all integers i 1 if f is greater than or equal to g then j is a generic residual intersection of the generic grade g complete intersection i1 x the resolutions that we produce in this case are in many ways analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three gorenstein ideals

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

the monoidal structure on strict polynomial functors abstract the category of strict polynomial functors inherits an internal tensor product from the category of divided powers to investigate this monoidal structure we consider the category of representations of the symmetric group s d which admits a tensor product coming from its hopf algebra structure it is classical that there exists a functor f from the category of strict polynomial functors to the category of representations of the symmetric group our main result is that this functor f is monoidal in addition we study the relations under f between projective strict polynomial functors and permutation modules and the link to symmetric functions

affine strict polynomial functors and formality abstract we introduce the notion of an affine strict polynomial functor we show how this concept helps to understand homological behavior of the operation of frobenius twist in the category of strict polynomial functors over a field of positive characteristic we also point out for an analogy between our category and the category of representations of the group of algebraic loops on g l n

a counter example to the mismatched decoding converse for binary input discrete memoryless channels this paper studies the mismatched decoding problem for binary input discrete memoryless channels an example is provided for which an achievable rate based on superposition coding exceeds the csiszar korner hui rate thus providing a counter example to a previously reported converse result both numerical evaluations and theoretical results are used in establishing this claim

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

vertex representations of quantum affine algebras we construct vertex representations of quantum affine algebras of ade type which were first introduced in greater generality by drinfeld and jimbo the limiting special case of our construction is the untwisted vertex representation of affine lie algebras of frenkel kac and segal our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations in the case of the quantum affine algebra of type a we introduce vertex operators corresponding to all the roots and determine their commutation relations this provides an analogue of a chevalley basis of the affine lie algebra unk n in the basic representation

the minimal free resolution of the migliore peterson rings in the case that the reflexive sheaf has even rank abstract letrbe a commutative noetherian ring and f gbe a homomorphism of freer modules where rank f fand rank g g fix an elementbg 1 g 1fand a generator g for gg the module action of f on fproduces the elementb1 g g bg 1 inf letjdenote the image ofb1 f r assume that gradej f g which is the largest grade possible and is attained in the generic case the idealjmay be interpreted as the defining ideal of the degeneracy locus of a regular section of a rankf greflexive sheaf it may also be interpreted as the order ideal of an element in a second syzygy module of rankf g also jmay be interpreted as the defining ideal for the symmetric algebra of a module of projective dimension two migliore and peterson have studied the idealjunm which is the unmixed part ofj under geometric hypotheses they have shown thatr junmis a cohen macaulay ring and they have resolved this ring furthermore iff gis odd thenjunmis a gorenstein ideal and is not equal toj on the other hand iff gis even thenjunm j in the present paper we produce the resolution ofr jby freer modules in the case thatf gis even and f g 2 is a unit inr our resolution is minimal whenever the data are local or homogeneous our resolution is built from the differential graded algebra f x1 xg d where the restriction ofdto f is the koszul complex associated tob1 f rand the degree two divided power variablesx1 xghave been adjointed in order to kill the cycles g 1f the acyclicity lemma is used to prove exactness ifg 1 then the idealjis equal to the huneke ulrich almost complete intersection ideali1 yx whereyis a 1 fmatrix andxis anf falternating matrix the resolution of this ideal is already known

level one representations and vertex operators of quantum affine superalgebra uq gl n n level one representations of the quantum affine superalgebra uq gl n n associated with the appropriate nonstandard system of simple roots and q vertex operators intertwining operators associated with the level one modules are constructed explicitly in terms of free bosonic fields

explicit generalized pieri maps abstract the purpose of this paper is to describe explicitly in a characteristic free setting certain maps between skew weyl modules for the general linear group as an application we indicate briefly how our generalized pieri maps are utilized to obtain new results concerning nonzero homomorphisms between certain skew weyl modules over fields of positive characteristics in particular we obtain a new proof of the special case of the carter payne theorem considered in this paper

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

remark on the branching theorem and supersymmetric algebras supersymmetric algebras have already proved useful in giving trans parent proofs of a number of basic results of representation theory 3 7 9 1 l l 51 specifically the technique of introducing virtual variables which may have a different signature than the signature of variables to be delt with often cuts down the amount of computation furthermore the extension of results of representation theory to the superalgebraic setting sheds new light and permits us to establish natural correspondences that were formally missing in this note we carry out this program by deriving a superalgebraic version of the branching rules for the representations of the general linear group while the statement of supersymmetric branching rules is in all respects similar to the ordinary one and differing from it in our allowing variables of two signature the proof yields a useful dividend namely a simple combinatorial construction of a canonical basis for the decomposi tion of a restriction of a representation as an application we give a supersymmetric generalization of pieri s formula as well as a proof of this formula which is perhaps as short as it can be whittled down to this application has been inspired by some recent work of bofli s 61 we have benefited from the pioneering work of berele and regev 3 who were first to state such a supersymmetric extension of branching rules as well as from the insights of balentekin and bars 2 255

local cohomology with support in ideals of maximal minors and sub maximal pfaffians abstract we compute the gl equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix as well as of those with support in the ideal of 2 n 2 n pfaffians of a 2 n 1 2 n 1 generic skew symmetric matrix as an application we characterize the cohen macaulay modules of covariants for the action of the special linear group sl g on g m the main tool we develop is a method for computing certain ext modules based on the geometric technique for computing syzygies and on matlis duality