Consider a connected compact Lie group K acting on a connected Hamiltonian manifold M . A measure for the complexity of M is half the dimension of the symplectic reductions of M , and it is natural to study Hamiltonian manifolds with low complexity first, starting with the case of complexity zero, the so-called multiplicity free manifolds (see [GS] or [MiFo]). It has been a longstanding problem...

We define the Barbasch-Evens-Magyar varieties. show they are isomorphic to smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely symmetric orbit closures, thereby giving resolutions of singularities certain cases. Our definition parallels [P.~Magyar '98]'s construction Bott-Samelson [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces ...

In Ehrhart theory, the h∗-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal h∗-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, w...

An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, the adjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordina...

Universal polytope is the polytope de ned as the convex hull of the characteristic vectors of all triangulations for a given point con guration. The equality system de ning this polytope was found, but the system of inequalities are not known yet. Larger polytopes, corresponding to linear programming relaxations, have been used in practice. We show that (1) the universal polytope, the polytope ...

The Birkhoff polytope Bn is the convex hull of all n× n permutation matrices in Rn×n. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group G 6 GL(d,R). We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the face lattice of ...

Abstract We survey the computation of polytope volumes by algorithms Normaliz to which Lawrence algorithm has recently been added. It enabled us master volume computations for polytopes from social choice in dimension 119. This challenge required a sophisticated implementation algorithm.