Paola Cellini

Cyclic Eulerian Elements

LetSnbe the symmetric group on {1,?,n} and QSn its group algebra over the rational field; we assumen?2. ??Snis said a descent ini, 1?i?n-1, if ?(i)? (i+1); moreover, ? is said to have a cyclic descent if ?(n)?(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 QSn, 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.

Root Polytopes and Borel Subalgebras

Let

Abelian subalgebras in ℤ2-graded Lie algebras and affine Weyl groups

\Phi

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

be a finite crystallographic irreducible root system and

A Representation of Even Permutations

\mathcal P_{\Phi}

Algebraic Hypermap Morphisms

be the convex hull of the roots in

A CHARACTERIZATION OF TOTAL REFLECTION ORDERS

\Phi

A Root System Criterion for Fully Commutative and Short Braid-Avoiding Elements in Affine Weyl Groups

. We give a uniform explicit description of the polytope

Shape of Curves and Surfaces: the Combinatorics

\mathcal P_{\Phi}

High order elements and number of roots of unity in a finite group

, 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.