we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

Bott-Samelson varieties factor the flag variety G/B into a product of CP’s with a map into G/B. These varieties are mostly studied in the case in which the map into G/B is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to tran...

The cyclic polytope C(n, d) is the convex hull of any n points on the moment curve {(t, t2, . . . , td ) : t ∈ R} in Rd . For d ′ > d , we consider the fiber polytope (in the sense of Billera and Sturmfels [6]) associated to the natural projection of cyclic polytopes π : C(n, d ) → C(n, d) which ‘forgets’ the last d ′ − d coordinates. It is known that this fiber polytope has face lattice indexe...

Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in equivariant cohomology from M to a level set Φ(p) is a surjection, and we denote the kernel by Ip. When T has isolated fixed points, we show that Ip distinguishes the chambers of the moment polytope for M . In particular, counting the number of distinct ideals Ip as p varies over different chambers ...

In type A, Bott-Samelson varieties are posets in which ascending chains are flags of vector spaces. They come equipped with a map into the flag variety G/B. These varieties are mostly studied in the case in which the map into G/B is birational to the image. In this paper we study Bott-Samelsons for general types, more precisely, we study the combinatorics a fiber of the map into G/B when it is ...

To any closed irreducible G-invariant cone in the space V of a finitedimensional representation of a semisimple Lie group there corresponds a convex polytope called the Brion polytope. This is closely connected with the action of the group G on the algebra of functions on the cone, and also with the moment map. In this paper we give a description of Brion polytopes for the spaces V themselves a...

We reconstruct an n-dimensional convex polytope from the knowledge of its directional moments up to a certain order. The directional moments are related to the projection of the polytope vertices on a particular direction. To extract the vertex coordinates from the moment information we combine established numerical algorithms such as generalized eigenvalue computation and linear interval inter...

This paper describes the polytope Pk;N of i-star counts, for all i 6 k, for graphs on N nodes. The vertices correspond to graphs that are regular or as regular as possible. For even N the polytope is a cyclic polytope, and for odd N the polytope is well-approximated by a cyclic polytope. As N goes to infinity, Pk;N approaches the convex hull of the moment curve. The affine symmetry group of Pk;...

The Duistermaat-Heckman formula for their induced measure on a moment polytope is nowadays seen as the Fourier transformof the Atiyah-Bott localization formula, applied to the T -equivariant Liouville class. From this formula one does not see directly that the measure is positive, nor that it vanishes outside the moment polytope. In [Knutson99] we gave a formula for the Duistermaat-Heckman meas...