On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial time algorithm, based on counting the lattice points in polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based on the ellipsoid algorithm, to decide when the coefficients are nonzero in polynomial time for arbitrary rank. Our experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank. Using an implementation of this algorithm, we provide experimental evidence for conjectured generalizations of the saturation property of Littlewood--Richardson coefficients. One of these conjectures seems to be valid for types B_n, C_n, and D_n.Comment: 21 pages, 6 table

Vertices of Gelfand-Tsetlin Polytopes

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory \mathfrak{gl}_n \C and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.Comment: 14 pages, 3 figures, fixed attribution

Different types of integrability and their relation to decoherence in central spin models

We investigate the relation between integrability and decoherence in central spin models with more than one central spin. We show that there is a transition between integrability ensured by the Bethe ansatz and integrability ensured by complete sets of commuting operators. This has a significant impact on the decoherence properties of the system, suggesting that it is not necessarily integrability or nonintegrability which is related to decoherence, but rather its type or a change from integrability to nonintegrability.Comment: 4 pages, 3 figure

Equivalence of domains for hyperbolic Hubbard-Stratonovich transformations

We settle a long standing issue concerning the traditional derivation of non-compact non-linear sigma models in the theory of disordered electron systems: the hyperbolic Hubbard-Stratonovich (HS) transformation of Pruisken-Schaefer type. Only recently the validity of such transformations was proved in the case of U(p,q) (non-compact unitary) and O(p,q) (non-compact orthogonal) symmetry. In this article we give a proof for general non-compact symmetry groups. Moreover we show that the Pruisken-Schaefer type transformations are related to other variants of the HS transformation by deformation of the domain of integration. In particular we clarify the origin of surprising sign factors which were recently discovered in the case of orthogonal symmetry.Comment: 30 pages, 3 figure

A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

We present a multivariate generating function for all n x n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B_n of n x n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric

Coded Parity Packet Transmission Method for Two Group Resource Allocation

Gap value control is investigated when the number of source and parity packets is adjusted in a concatenated coding scheme whilst keeping the overall coding rate fixed. Packet-based outer codes which are generated from bit-wise XOR combinations of the source packets are used to adjust the number of both source packets. Having the source packets, the number of parity packets, which are the bit-wise XOR combinations of the source packets can be adjusted such that the gap value, which measures the gap between the theoretical and the required signal-to-noise ratio (SNR), is controlled without changing the actual coding rate. Consequently, the required SNR reduces, yielding a lower required energy to realize the transmission data rate. Integrating this coding technique with a two-group resource allocation scheme renders efficient utilization of the total energy to further improve the data rates. With a relatively small-sized set of discrete data rates, the system throughput achieved by the proposed two-group loading scheme is observed to be approximately equal to that of the existing loading scheme, which is operated with a much larger set of discrete data rates. The gain obtained by the proposed scheme over the existing equal rate and equal energy loading scheme is approximately 5 dB. Furthermore, a successive interference cancellation scheme is also integrated with this coding technique, which can be used to decode and provide consecutive symbols for inter-symbol interference (ISI) and multiple access interference (MAI) mitigation. With this integrated scheme, the computational complexity is signi cantly reduced by eliminating matrix inversions. In the same manner, the proposed coding scheme is also incorporated into a novel fixed energy loading, which distributes packets over parallel channels, to control the gap value of the data rates although the SNR of each code channel varies from each other

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

On a Linear Program for Minimum-Weight Triangulation

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial- time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274