8,894 research outputs found
Pieri rule for the affine flag variety
We prove the affine Pieri rule for the cohomology of the affine flag variety
conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on
the affine nilHecke ring that is motivated by Kostant and Kumar's work on the
equivariant cohomology of the affine flag variety. We show that the cap
operators for Pieri elements are the same as Pieri operators defined by Berg,
Saliola and Serrano. This establishes the affine Pieri rule.Comment: 14 pages, Fixed typo
Centrally symmetric polytopes with many faces
We present explicit constructions of centrally symmetric polytopes with many
faces: first, we construct a d-dimensional centrally symmetric polytope P with
about (1.316)^d vertices such that every pair of non-antipodal vertices of P
spans an edge of P, second, for an integer k>1, we construct a d-dimensional
centrally symmetric polytope P of an arbitrarily high dimension d and with an
arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at
least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of
P, and third, for an integer k>1 and a>0, we construct a centrally symmetric
polytope Q with an arbitrary large number N of vertices and of dimension
d=k^{1+o(1)} such that least (1 - k^{-a}){N choose k} k-subsets of the set of
vertices span faces of Q.Comment: 14 pages, some minor improvement
- …