#### Degree

Doctor of Philosophy

#### Program

Mathematics

#### Supervisor

Lex Renner

#### Abstract

It is well-known that the Eulerian polynomials, which count permutations in S_n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the Sn -orbit for a generic weight in the weight lattice of Sn . Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this thesis we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections J = {sn , sn−1 , · · · , sn−k+1 } for some k with respect to the An root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) which comes naturally from the theory of algebraic monoids. Using effectively the cross section lattice of a reductive monoid and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincar e polynomial of X(J) in terms of the Eulerian polynomials. Furthermore, we compute explicitely the Poincar ́ polynomial of X(J) using the method of descent systems in the case of (W, S) finite Weyl group of type An and J combinatorially smooth of the following forms: 1. J = {s_1 , s_4 , s_5 , · · · , s_n } ⊂ S 2. J = {s_4 , s_5 , · · · , s_n } ⊂ S.

#### Recommended Citation

Golubitsky, Letitia Mihaela, "Descent Systems, Eulerian Polynomials and Toric Varieties" (2011). *Electronic Thesis and Dissertation Repository.* Paper 134.

http://ir.lib.uwo.ca/etd/134