Abstract. Recently Klyachko [Kl] has given linear inequalities on triples (λ, µ, ν) of dominant weights of GLn(C) necessary for the the corresponding Littlewood-Richardson coefficient dim(Vλ⊗Vµ⊗Vν) GLn(C) to be positive. We show that these conditions are also sufficient, which was known as the saturation conjecture. In particular this proves Horn’s conjecture from 1962, giving a recursive system of inequalities [H]. Our principal tool is a new model of the Berenstein-Zelevinsky cone for computing Littlewood-Richardson coefficients [BZ, Ze], the honeycomb model. The saturation conjecture is a corollary of our main result, which is the existence of a particularly well-behaved honeycomb associated to regular triples (λ, µ, ν). 1. The saturation conjecture A very old and fundamental question about the representation theory of GLn(C) is the following: For which triples of dominant weights λ, µ, ν does the tensor product Vλ⊗Vµ⊗Vν of the irreducible representations with those high weights contain a GLn(C)invariant vector? Another standard, if less symmetric, formulation of the problem above replaces Vν with its dual, and asks for which ν is V ∗ ν a constituent of Vλ⊗Vµ. In this formulation one can without essential loss of generality restrict to the case that λ, µ, and ν ∗ are polynomial representations, and rephrase the question in the language of Littlewood-Richardson coeffi-cients; it asks for which triple of partitions λ, µ, ν ∗ is the Littlewood-Richardson coefficient c ν∗ λµ positive. It is not hard to prove (as we will later in this introduction) that the set of such triples (λ, µ, ν) is closed under addition, so forms a monoid. In this paper we prove that this monoid is saturated, i.e. that for each triple of dominant weights (λ, µ, ν), (VNλ⊗VNµ⊗VNν) GLn(C)> 0 for some N> 0 = ⇒ (Vλ⊗Vµ⊗Vν) GLn(C)> 0. This is of particular interest because Klyachko has recently given an answer 1 to the general question above, which in one direction was only asymptotic [Kl]:

We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higher-dimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.

A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.

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this article, we give a formula for the number of lattice points in P , in the case where P is simple, that is if there are exactly n edges through each vertex of

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum A 1 + \Delta \Delta \Delta +A r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding C (A 1 ; : : : ; A r ). In this paper we extend this correspondence in a natural way to cover also noncoherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.

The linear span of isomorphism classes of posets, has a Newtonian coalgebra structure. We observe that the ab-index is a Newtonian coalgebra map from the vector space to the algebra of polynomials in the non-commutative variables a and b. This enables us to obtain explicit formulas showing how the cd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operations also have interpretations for certain classes of permutations, including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly, we show how to compute the ab-index of the Cartesian product of two posets given the ab-indexes of each poset.

We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the Klee-Minty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the random-edge simplex algorithm on Klee-Minty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a Klee-Minty cube is exponential when all paths are taken with equal probability.

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Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h-vectors of simplicial polytopes, we show that h-vectors of pure rank-d simplicial complexes that have this property satisfy h0 ≤ h1 ≤ ·· · ≤ h [d/2] and hi ≤ hd−i for 0 ≤ i ≤ [d/2]. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex eardecomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the h-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the h-vector of a matroid complex satisfies the above two sets of inequalities. 1.