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

Cyclic Eulerian Elements

AbstractLetSnbe the symmetric group on {1,…,n} and Q[Sn] its group algebra over the rational field; we assumen≥2. π∈Snis said a descent ini, 1≤i≤n-1, if π(i)>π (i+1); moreover, π is said to have a cyclic descent if π(n)>π(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of Q[Sn], which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents

Abelian subalgebras in Z_2-graded Lie algebras and affine Weyl groups

Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable subalgebras of g_1.Comment: 21 pages, amstex file. Minor corrections. Introduction slightly expanded. To appear in IMR

The W^\hat W-orbit of ρ\rho, Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z

Let an affine Weyl group $\hat W$ act as a group of affine transformations on a real vector space V. We analyze the $\hat W$-orbit of a regular element in V and deduce applications to Kostant's formula for powers of the Euler product and to the representations of $\hat W$ as permutations of the integers.Comment: Latex, 27 pages, minor corrections, to appear in Journal of Pure and Applied Algebr

Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebras

We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian subalgebras of simple Lie algebras.Comment: Latex, 56 pages, revised version: minor corrections, Subsection 6.2 added. To appear in Advances in Mathematic

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