This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory 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 GelfandTsetlin pattern.As an application, we disprove a conjecture of Berenstein and Kirillov [1] about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each n ≥ 5 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.

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 Ar, 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 Bn, Cn, and Dn.

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope.Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

If P ⊂ R d is a rational polytope, then i P (n) := #(nP ∩ Z d ) is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P . The minimum period of i P (n) must divide D(P ) = min{n ∈ Z >0 : nP is an integral polytope}. Few examples are known where the minimum period is not exactly D(P ). We show that for any D, there is a 2-dimensional triangle P such that D(P ) = D but such that the minimum period of i P (n) is 1, that is, i P (n) is a polynomial in n. We also characterize all polygons P such that i P (n) is a polynomial. In addition, we provide a counterexample to a conjecture by T. Zaslavsky about the periods of the coefficients of the Ehrhart quasi-polynomial.

Let J and K be convex sets in R n whose affine spans intersect at a single rational point in J ∩ K, and let J ⊕ K = conv(J ∪ K). We give formulas for the generating functionof lattice points in all integer dilates of J ⊕ K in terms of σ cone J and σ cone K , under various conditions on J and K. This work is motivated by (and recovers) a product formula of B. Braun for the Ehrhart series of P ⊕Q in the case where P and Q are lattice polytopes containing the origin, one of which is reflexive.In particular, we find necessary and sufficient conditions for Braun's formula and its multivariate analogue.2000 Mathematics Subject Classification. Primary 05C15; Secondary 11P21, 52B20.