Polar Root Polytopes that are Zonotopes

Let $\mathcal P_{\Phi}$ be the root polytope of a finite irreducible crystallographic root system $\Phi$, i.e., the convex hull of all roots in $\Phi$. The polar of $\mathcal P_{\Phi}$, denoted $\mathcal P_{\Phi}^*$, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we establishes which polytopes $\mathcal P_{\Phi}^*$ are zonotopes and which are not. The proof is constructive.Comment: 12 page

ad-Nilpotent ideals of a Borel subalgebra II

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q} under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.Comment: AMS-TeX file, 9 pages; revised version. To appear in Journal of Algebr

Root polytopes and Borel subalgebras

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.Comment: revised version, accepted for publication in IMR

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Museum and monument attendance and tourism flow: A time series analysis approach.

This paper takes a time series analysis approach to evaluate the directions of causality between tourism flows, on the one side, and museum and monument attendance, on the other. We consider Italy as a case study, and analyze monthly data over the period January 1996 to December 2007. All considered series are seasonally integrated, and co-integration links emerge. We focus on the error correction mechanism among co-integrated time series to detect the directional link(s) of causality. Clear-cut results emerge: generally, the causality runs from tourist flows to museum and monument attendance. The non-stationary nature of time series, their co-integration relationships, and the direction of causal links suggest specific implication for tourism and cultural policies.Tourism; Museum; Seasonal unit root; Co-integration; Causality.