{"id":{"repo_id":"uwo","oai_identifier":"oai:uwo.scholaris.ca:20.500.14721/37919"},"canonical_url":"https://search.dev.ndltd.org/etd/uwo/oai:uwo.scholaris.ca:20.500.14721/37919","repository":{"repo_id":"uwo","name":"Western University","base_url":"https://uwo.scholaris.ca/server/oai/request"},"display":{"title":"Descent Systems, Eulerian Polynomials and Toric Varieties","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Golubitsky, Letitia Mihaela"],"institution":"The University of Western Ontario","degree_name":"Ph D","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Lex Renner"],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-04-12","date_published":"2011-04-12","updated_at":"2026-07-27T21:56:13Z","subjects":["algebraic monoids","cross section lattice","toric varieties","betti numbers","descent systems","eulerian polynomials","semisimple algebraic group"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14721/37919","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Lex Renner"]},{"key":"dc:creator","label":"Author","values":["Golubitsky, Letitia Mihaela"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-10T21:44:40Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-07-10T21:44:40Z"]},{"key":"dc:date.issued","label":"Date","values":["2011-04-12"]},{"key":"dc:publisher","label":"Institution","values":["The University of Western Ontario"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph D"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["algebraic monoids","cross section lattice","toric varieties","betti numbers","descent systems","eulerian polynomials","semisimple algebraic group"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14721/37919"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Descent Systems, Eulerian Polynomials and Toric Varieties"]}]}],"canonical_facts":{"dc:contributor.advisor":["Lex Renner"],"dc:creator":["Golubitsky, Letitia Mihaela"],"dc:date.accessioned":["2025-07-10T21:44:40Z"],"dc:date.available":["2025-07-10T21:44:40Z"],"dc:date.issued":["2011-04-12"],"dc:description":["The thesis cover page in the PDF document includes references to Western University’s previous institutional repository platform, known as Scholarship@Western, and links to that platform (beginning with ir.lib.uwo.ca). In citing or referring to this thesis, use the DOI or handle from this page instead. Sample citation: Author name, \"Thesis title.\" (Year). Western University Open Repository. https://doi.org/10.71858/123456."],"dc:description.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."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14721/37919"],"dc:language.iso":["en_ca"],"dc:publisher":["The University of Western Ontario"],"dc:subject":["algebraic monoids","cross section lattice","toric varieties","betti numbers","descent systems","eulerian polynomials","semisimple algebraic group"],"dc:title":["Descent Systems, Eulerian Polynomials and Toric Varieties"],"dc:type":["thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["Ph D"]},"updated_at":"2026-07-27T21:56:13Z"}