representations of the symmetric group are reducible over simply transitive subgroups if the characteristic of f is zero this problem has been solved by saxl 16 an important feature of saxl s result is that a groupg as in the problem above is either 2 transitive or fixes a point i e is contained in some n 1 the caseg n 1 can then be settled using the branching rule and induction on the other hand an explicit list of k transitive groups for k 2 is available which can be used to complete the proof in characteristic zero see 16 for more details from now on we assume that p 0 in this case the problem is important for determining maximal subgroups of finite classical groups 1 13 however the situation is nowmore complicated for example to determine the pairs g d as above withg an one needs the mullineux conjecture 7 2 ifg is intransitive then up to a conjugation it is contained in a standard young subgroup of the form n k k the irreducible restrictions from n to n 1 have been described in 14 see also 11 6 in 11 it is also

multiplicity free subgroups of reductive algebraic groups abstract we introduce the notion of a multiplicity free subgroup of a reductive algebraic group in arbitrary characteristic this concept already exists in the work of kramer for compact connected lie groups we give a classification of reductive multiplicity free subgroups and as a consequence obtain a simple proof of a theorem of kleshchev

hadamard type inequalities for s convex functions in this paper we establish some new inequalities for differentiable functions based on concavity and s convexity we also prove several hadamard type inequalities for products of two convex and s convex functions

two new theorem on mappings uniformly continuous and convex with applications to quadrature rules and means an interesting connection exists between convex functions and uniformly continuous functions this linkage leads to some interesting functional inequalities over the open interval which works well in quadrature rules and means in this paper we establish two new theorem which connect the hermite hadamard type functions

a theorem on mappings with bounded derivatives with applications to quadrature rules and means in this paper we establish a new inequality of theorem 2 appl math lett 13 2000 19 dragomir s integral inequality for functions whose derivatives are bounded this has immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid mid point some natural applications to special means of real numbers are given for several recent results concerning dragomir s integral inequality see appl math lett 13 2000 19 where further references are listed

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

generalized degrees freedom of noncoherent mimo channels with asymmetric link strengths we study the generalized degrees of freedom gdof of block fading noncoherent multiple input multiple output mimo channels with asymmetric distributions of link strengths and a coherence time of t symbol durations we derive the optimal signaling structure for communication for the asymmetric mimo channel which is distinct from that for the mimo channel with independent and identically distributed i i d links we extend the existing results for the single input multiple output simo channel with i i d links to the asymmetric case proving that selecting the statistically best antenna is gdof optimal using the gdof result for the simo channel we prove that for t 1 the gdof is zero for mimo channels with arbitrary link strengths we show that selecting the statistically best antenna is gdof optimal for the multiple input single output miso channel we also derive the gdof for the 2 times 2 mimo channel with different exponents in the direct and cross links in this setting we show that it is always necessary to use both the antennas to achieve the gdof in contrast to the results for the 2 times 2 mimo channel with i i d links we show that having weaker crosslinks gives gdof gain compared to the case with i i d links for the noncoherent mimo channel with i i d links the traditional method of training each transmit antenna independently is degrees of freedom dof optimal whereas we observe that for the asymmetric 2 times 2 mimo channel the traditional training is not gdof optimal we extend this observation to a larger m times m mimo channel by demonstrating a strategy that can achieve larger gdof than a traditional training based method

irreducible tensor products over alternating groups abstract we classify irreducible tensor products of modular representations of the alternating group over an algebraically closed field of characteristic p 5

on multipath fading channels at high snr a noncoherent multipath fading channel is considered where neither the transmitter nor the receiver is cognizant of the realization of the path gains but both are cognizant of their statistics it is shown that if the delay spread is large in the sense that the variances of the path gains decay exponentially or slower then capacity is bounded in the signal to noise ratio snr for such channels capacity does not tend to infinity as the snr tends to infinity in contrast if the variances of the path gains decay faster than exponentially then capacity is unbounded in the snr it is further demonstrated that if the number of paths is finite then at high snr capacity grows double logarithmically with the snr and the capacity pre loglog defined as the limiting ratio of capacity to loglog snr as the snr tends to infinity is 1 irrespective of the number of paths the results demonstrate that at high snr multipath fading channels with an infinite number of paths cannot be approximated by multipath fading channels with only a finite number of paths the number of paths that are needed to approximate a multipath fading channel typically depends on the snr and may grow to infinity as the snr tends to infinity

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

on the high snr capacity of noncoherent networks we obtain the first term in the high signal to noise ratio snr asymptotic expansion of the sum rate capacity of noncoherent fading networks i e networks where the transmitters and receivers while fully cognizant of the fading law have no access to the fading realization this term is an integer multiple of log log snr with the coefficient having a simple combinatorial characterization it can be interpreted as the effective number of parallel channels that can be supported by the network i e as the maximal number of point to point single user scalar channels that can be supported by the network in a manner that will allow with proper power allocation negligible cross interference the results hold irrespective of whether the transmitters can cooperate or must operate in an multiple access regime irrespective of whether feedback from the receivers to the transmitters is available or not and irrespective of whether the receivers can cooperate or not

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

the fading number of memoryless multiple input multiple output fading channels in this correspondence we derive the fading number of multiple input multiple output mimo flat fading channels of general not necessarily gaussian regular law without temporal memory the channel is assumed to be noncoherent i e neither receiver nor transmitter have knowledge about the channel state but they only know the probability law of the fading process the fading number is the second term after the double logarithmic term of the high signal to noise ratio snr expansion of channel capacity hence the asymptotic channel capacity of memoryless mimo fading channels is derived exactly the result is then specialized to the known cases of single input multiple output simo multiple input single output miso and single input single output siso fading channels as well as to the situation of gaussian fading

on 2 modular representations of the symmetric groups abstract let g gl n denote the general linear group of invertible n n matrices with entries in an algebraically closed field f of characteristic 2 we prove the existence of certain composition factors in the restriction to gl n k gl k of some polynomial g modules as a consequence we resolve the final case of the problem addressed by jantzen and seitz of when an irreducible modular representation of the symmetric group s n remains irreducible upon restriction to the young subgroup s n k s k

on restrictions of modular spin representations of symmetric and alternating groups let f be an algebraically closed field of characteristic p and h be an almost simple group or a central extension of an almost simple group an important problem in representation theory is to classify the subgroups g of h and fh modules v such that the restriction v g is irreducible for example this problem is a natural part of the program of describing maximal subgroups in finite classical groups in this paper we investigate the case of the problem where h is the schur s double cover n or s n

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

representations of the alternating group which are irreducible over subgroups ii nsf dms 1161094 dms 0901241 dms 1201374 humboldt foundation simons foundation 204181 simons foundation fellowship 305247

on restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups i 1994 on restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups ii communications in algebra vol 22 no 15 pp 6175 6208

shannon theoretic approach to a gaussian cellular multiple access channel we obtain shannon theoretic limits for a very simple cellular multiple access system in our model the received signal at a given cell site is the sum of the signals transmitted from within that cell plus a factor spl alpha 0 spl les spl alpha spl les 1 times the sum of the signals transmitted from the adjacent cells plus ambient gaussian noise although this simple model is scarcely realistic it nevertheless has enough meat so that the results yield considerable insight into the workings of real systems we consider both a one dimensional linear cellular array and the familiar two dimensional hexagonal cellular pattern the discrete time channel is memoryless we assume that n contiguous cells have active transmitters in the one dimensional case and that n sup 2 contiguous cells have active transmitters in the two dimensional case there are k transmitters per cell most of our results are obtained for the limiting case as n spl rarr spl infin the results include the following 1 we define c sub n c spl circ sub n as the largest achievable rate per transmitter in the usual shannon theoretic sense in the one and two dimensional cases respectively assuming that all signals are jointly decoded we find expressions for limn spl rarr spl infin c sub n and limn spl rarr spl infin c spl circ sub n 2 as the interference parameter spl alpha increases from 0 c sub n and c spl circ sub n increase or decrease according to whether the signal to noise ratio is less than or greater than unity 3 optimal performance is attainable using tdma within the cell but using tdma for adjacent cells is distinctly suboptimal 4 we suggest a scheme which does not require joint decoding of all the users and is in many cases close to optimal

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

some remarks on branching rules and tensor products for algebraic groups let f be an algebraically closed field of characteristic p 0 in bk1 bk2 we have revealed and exploited various relations between the branching rules from gln f to its levi subgroups on one hand and decompositions of tensor products over gln f itself on the other for example if l is some irreducible rational gln f module and v is the natural gln f module there is a close relationship between the highest weight vectors relative to gln 1 f in the restriction l gln 1 f and the highest weight vectors relative to gln f in the tensor product l v in this paper we obtain more results in this direction some of which are valid for an arbitrary type to describe our main results we adopt standard lie theoretic notation let g be a connected reductive algebraic group over f as in j r denotes the root system of g with respect to a fixed maximal torus t r r denotes the set of positive roots determined by a choice of borel subgroup b containing t and 1 r is the corresponding base for r denote the highest long root of r by 0 and the longest element of the weyl group w ng t t by w0 we write x t for the character group hom t f y t for the cocharacter group hom f t and let be the natural pairing x t y t z for r denotes the corresponding coroot in y t and x t denotes the set x t i 0 i 1 of dominant weights all g modules are assumed to be rational for x t we have the g modules l and which are the irreducible the standard or weyl and the costandard gmodules with highest weight let dist g be the algebra of distributions of g as in j i 7 which is generated by dist t and the divided power root generators x y n for r n 1 write x i x n i yi y n i for i 1 if g is semisimple and simply connected which we may assume for the proofs dist g coincides with the hyperalgebra of g arising from the chevalley construction we note that any g module is a dist g module in a natural way see j i 7 11 ii 1 20 given a weight x t and a g module m m will denote the weight space of m if in addition x t is a dominant weight we define