Vectors Of Weight Research Articles

Composantes PRV généralisées et chemins de Littelmann

We give a sufficient condition for a Littelmann path to represent a vector of extremal weight of an integrable irreducible highest weight representation of a symmetrisable Kac-Moody algebra. Thanks to this condition we present, in a more general context, an alternative proof of recent result by Boris Pasquier, Nicolas Ressayre and the author of this article on the existence of generalized PRV components.

SPACES OF QUASI-EXPONENTIALS AND REPRESENTATIONS OF THE YANGIAN Y (glN)

We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace $V(b)_\lambda \subset V(b)$ of weight $\lambda$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b)_\lambda$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b)^{sing}_\lambda$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b)^{sing}_\lambda$ with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map. These results we announced earlier in relation with a description of the quantum equivariant cohomology of the cotangent bundle of a partial flag variety and a description of commutative subalgebras of the group algebra of a symmetric group.

A combinatorial problem in the class of binary vectors of given weight

The notion of l-step is introduced in the class of binary vectors of given weight, which is a structural characteristic of a vector. Formulas for calculating the powers of sets of vectors of this class with certain structural properties related to the notion of l-step are derived.

T.P.51 Metformin reduces weight and BMI in Duchenne muscular dystrophy patients on long term glucocorticoid therapy

Abstract Obesity is a significant problem in Duchenne muscular dystrophy (DMD) due to glucocorticoid (GC) therapy and motor decline. Excessive weight gain can further impair mobility, increase risk for diabetes and cardiopulmonary disease, and affect quality of life. Metformin improves weight and insulin resistance in obesity and type 2 diabetes. To determine if metformin reduces weight and BMI in DMD patients on long term GC who have excessive weight gain and insulin resistance. This was a retrospective case series of DMD boys on daily GC therapy who were treated with metformin for excessive weight gain and insulin resistance. Primary outcomes were rate of weight gain and BMI pre and post starting metformin. Weight and BMI measurements were collected 1 year prior, at initiation of metformin, and 6 and 12 months post. Generalized linear models for the vector of weight and BMI measurements over time were fit using generalized estimating equations. Forty-five DMD patients (mean age 12.7 ± 3.1 y) were studied. Patients had insulin resistance by glucose tolerance testing. Mean rate (±SE) of weight gain decreased from 7.5 ± 1.0 pre to −0.2 ± 1.7 kg/y ( p N = 29), rate of weight gain decreased from 8.3 ± 1.5 pre to −0.6 ± 2.5 kg/y post ( p N = 16), rate of weight gain decreased from 6.1 ± 0.7 pre to 1.0 ± 1.3 kg/y post ( p 2 /y post ( p

Universal graded Specht modules for cyclotomic Hecke algebras

The graded Specht module Sλ for a cyclotomic Hecke algebra comes with a distinguished generating vector zλ∈Sλ, which can be thought of as a ‘highest weight vector of weight λ’. This paper describes the defining relations for the Specht module Sλ as a graded module generated by zλ. The first three relations say precisely what it means for zλ to be a highest weight vector of weight λ. The remaining relations are homogeneous analogues of the classical Garnir relations. The homogeneous Garnir relations, which are simpler than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.

Robust characterizations of k‐wise independence over product spaces and related testing results

AbstractA discrete distribution D over Σ1 ×··· ×Σn is called (non‐uniform) k ‐wise independent if for any subset of k indices {i1,…,ik} and for any z1∈Σ,…,zk∈Σ, PrX∼D[X···X = z1···zk] = PrX∼D[X = z1]···PrX∼D[X = zk]. We study the problem of testing (non‐uniform) k ‐wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k ‐wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}n. For the non‐uniform case, we give a new characterization of distributions being k ‐wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496–505) on uniform k ‐wise independence over the Boolean cubes to non‐uniform k ‐wise independence over product spaces. Our results yield natural testing algorithms for k ‐wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

Counting matrices over finite fields having a given number of rows of unit weight

We present a formula enumerating matrices over a finite field of a given rank and a given number of rows of unit weight, i.e., each having a single nonzero entry. We also determine the number of subspaces of a given dimension containing a given number of vectors of unit weight.

Minimum Words of Codes from Affine Planes

We show that there are non-desarguesian affine planes of order 16 for which the binary codes have vectors of minimum weight that are not the incidence vectors of lines. This is in contrast to the desarguesian case and answers an open question as to the nature of the minimum words of the code of a non-desarguesian affine plane. Further, we show that all the nontranslation planes of order 16 have hull of minimum weight smaller than 32, in fact containing words of weight 24. Most of these words of weight 24 yield words of weight 16 in the binary code of some affine plane of order 16 that are not the incidence vectors of affine lines. The search has also shown that all the non-desarguesian planes of order at most 16 are not tame. These results are also in contrast to what is known in the desarguesian case. The results are mainly by computer, using Magma.

Rectangular low level case of modular branching problem for [formula omitted

In this paper, we find an explicit combinatorial criterion for the existence of a nonzero GL n − 1 ( K ) -high weight vector of weight ( λ 1 , … , λ i − 1 , λ i − d , λ i + 1 , … , λ n − 1 ) , where d < char ( K ) and K is an algebraically closed field, in the irreducible rational GL n ( K ) -module L n ( λ 1 , … , λ n ) with highest weight ( λ 1 , … , λ n ) . For this purpose, new modular lowering operators are introduced.

Representations of certain non-rational vertex operator algebras of affine type

In this paper we study a series of vertex operator algebras of integer level associated to the affine Lie algebra A ℓ ( 1 ) . These vertex operator algebras are constructed by using the explicit construction of certain singular vectors in the universal affine vertex operator algebra N l ( n − 2 , 0 ) at the integer level. In the case n = 1 or l = 2 , we explicitly determine Zhu's algebras and classify all irreducible modules in the category O . In the case l = 2 , we show that the vertex operator algebra N 2 ( n − 2 , 0 ) contains two linearly independent singular vectors of the same conformal weight.

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers.

Determination of weight enumerators of binary extremal self-dual [formula omitted] codes

We determine the weight enumerators for which there is a binary extremal self-dual [ 42 , 21 , 8 ] code. In particular, the number β of the vectors of weight 5 in the shadow satisfies β ∈ { 0 , 1 , … , 22 , 24 , 26 , 28 , 32 , 42 } . We also give a classification of extremal self-dual [ 42 , 21 , 8 ] codes with β ∈ { 24 , 26 , 28 , 32 , 42 } .

Parametrization of self-dual codes by orthogonal matrices

We study the orthogonal group O m of m × m matrices over the field of two elements and give applications to the theory of binary self-dual codes. We show that O 2 n acts transitively on the self-dual codes of length 2 n. The subgroup O 2 n ( 1 ) , consisting of all elements in O 2 n having every row with weight congruent to 1 mod 4, acts transitively on the set of doubly even self-dual codes of length 2 n. A factorization theorem for elements of O m leads to a result about generator matrices for self-dual codes, namely if G = [ I | A ] is such a generator matrix with I= identity, then the number of rows of G having weight divisible by 4 is a multiple of 4. This generalizes the known result that a self-dual doubly-even code exists only in lengths divisible by 8. The set of inequivalent self-dual codes is shown to be in one-to-one correspondence with the H − P m double cosets in O m for a certain subgroup H . The analogous correspondence is given for doubly-even codes and H ( 1 ) − P m double-cosets in O m ( 1 ) for a certain a group H ( 1 ) . Thus the classification problem for self-dual codes is equivalent to a classification of double-cosets. The subgroups of O m generated by the permutation matrices and one transvection are determined in the Generator Theorem. The study of certain transvections leads to two results about doubly-even self-dual codes: (a) every such a code with parameters [ 2 n , n , d ] with d ⩾ 8 is obtained by applying a transvection to a doubly-even code with parameters [ 2 n , n , d − 4 ] which has some special properties related to a vector of weight 6; (b) every such code with minimum distance at least 16 is a neighbor of a singly-even, self-dual code which has a single word of minimum weight 6. A construction is given for such singly-even codes of length 2 n based on the existence of codes of length 2 n − 6 having special properties.

Intersection theorems under dimension constraints

In Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0 ⩽ t , k ⩽ n determine or estimate the maximum number of ( 0 , 1 ) -vectors in a k-dimensional subspace of the Euclidean n-space R n , such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the ( 0 , 1 ) -vectors of the same weight ω . The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used. In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21]. We also consider a diametric problem under dimension constraint. In Part II, we study the restricted case and solve the problem for t = 1 and k < 2 ω , and also for any fixed 1 ⩽ t ⩽ ω and k large.

Minimal blocks of binary even-weight vectors

Odd circuits are minimal 1-blocks over GF(2) and the odd circuit of size 2 t + 1 can be represented by the vectors of Hamming weight 2 t in a (2 t + 1)-dimensional vector space over GF(2). This is the tip of an iceberg. Let f(2 t, k, 2) be the maximum number of binary k-dimensional column vectors such that for all s, 1 ⩽ s ⩽ t, no 2 s columns sum to the zero vector. If k = 2, k = 3, k = 4, or k ⩾ 5 and 2 t is sufficiently large (for example, 2 t ⩾ 2 k − k + 1 suffices), then the set of vectors of weight 2 t in a ( f(2 t, k, 2) + 2 t −1)-dimensional vector space over GF(2) is a minimal k-block over GF(2).

The Lie theory of the Rankin–Cohen brackets and allied bi-differential operators

The Lie theoretic nature of the Rankin–Cohen brackets is here uncovered. These bilinear operations, which, among other purposes, were devised to produce a holomorphic automorphic form from any pair of such forms, are instances of SL ( 2 , R ) -equivariant holomorphic bi-differential operators on the upper half-plane. All of the latter are here characterized and explicitly obtained, by establishing their one-to-one correspondence with singular vectors in the tensor product of two sl ( 2 , C ) Verma modules. The Rankin–Cohen brackets arise in the generic situation where the linear span of the singular vectors of a given weight is one-dimensional. The picture is completed by the special brackets which appear for the finite number of pairs of initial lowest weights for which the above space is two-dimensional. Explicit formulæ for basis vectors in both situations are obtained and universal Lie algebraic objects subsuming all of them are exhibited. A few applications of these results and Lie theoretic approach are then considered. First, a generalization of the latter yields Rankin–Cohen type brackets for Hilbert modular forms. Then, some Rankin–Cohen brackets are shown to intertwine the tensor product of two holomorphic discrete series representations of SL ( 2 , R ) with another such representation occurring in the tensor product decomposition. Finally, the sought for precise relationship between the Rankin–Cohen brackets and Gordan's transvection processes of the nineteenth century invariant theory is unveiled.

Iterating lowering operators

For an algebraically closed base field of positive characteristic, an algorithm to construct some non-zero GL ( n - 1 ) -high weight vectors of irreducible rational GL ( n ) -modules is suggested. It is based on the criterion proved in this paper for the existence of a set A such that S i , n ( A ) f μ , λ is a non-zero GL ( n - 1 ) -high weight vector, where S i , n ( A ) is Kleshchev's lowering operator and f μ , λ is a non-zero GL ( n - 1 ) -high weight vector of weight μ of the costandard GL ( n ) -module ∇ n ( λ ) with highest weight λ .

Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules

Let W(\varpi_k) be the finite-dimensional irreducible module over a quantized affne algebra U'_q(\mathfrak g) with the fundamental weight \varpi_k as an extremal weight. We show that its crystal B(W(\varpi_k)) is isomorphic to the Demazure crystal B^−(−Λ_0 + \varpi_k) . This is derived from the following general result: for a dominant integral weight λ and an integral weight µ , there exists a unique homomorphism U^-q(\mathfrak g)(u_λ \otimes u_µ) → V(λ + µ) that sends u_λ \otimes u_µ to u_{λ+µ} . Here V(λ) is the extremal weight module with λ as an extremal weight, and u_λ \in V(λ) is the extremal weight vector of weight λ .

On Weight Distributions of Shifts of Goethals-like Codes

We study weight distributions of shifts of codes from a well-known family: the 3-error-correcting binary nonlinear Goethals-like codes of length n e 2m, where m ≥ 6 is even. These codes have covering radius ρ e 6. We know the weight distribution of any shift of weight i e 1, 2, 3, 5, or 6. For a shift of weight 4, the weight distribution is uniquely defined by the number of leaders in this shift, i.e., the number of vectors of weight 4. We also consider the weight distribution of shifts of codes with minimum distance 7 obtained by deleting any one position of a Goethals-like code of length n.

Extremal self-dual [40,20,8] codes with covering radius 7

We construct new extremal self-dual [40,20,8] codes with covering radius 7. It is also shown that the vectors of a fixed weight in a coset of weight 4 n+2 in an extremal doubly even self-dual code of length 24 n+16 such that the coset has no vector of weight 4 n+4 form a 1-design.