{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/116152"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/116152","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Equivariant Schubert calculus and applications","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","abstract_has_math":false,"creators":["Robichaux, Colleen"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Yong, Alexander","Tolman, Susan","Di Francesco, Philippe","Kedem, Rinat"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-08","date_published":"2022-08","updated_at":"2026-07-22T22:24:55Z","subjects":["Schubert calculus","Combinatorial algorithms","Equivariant cohomology","Castelnuovo-Mumford regularity"],"languages":["en","eng"],"rights":["Copyright 2022 Colleen Robichaux"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/116152","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Yong, Alexander","Tolman, Susan","Di Francesco, Philippe","Kedem, Rinat"]},{"key":"dc:creator","label":"Author","values":["Robichaux, Colleen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-08","2022-06-06"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Schubert calculus","Combinatorial algorithms","Equivariant cohomology","Castelnuovo-Mumford regularity"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2022 Colleen Robichaux"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/116152"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Colleen Robichaux, accepted the attached license on 2022-06-01 at 16:34.","The student, Colleen Robichaux, submitted this Dissertation for approval on 2022-06-01 at 16:37.","This Dissertation was approved for publication on 2022-06-06 at 14:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18054 on 2022-11-15 at 17:37:31","A central problem in algebraic combinatorics is to determine a combinatorial rule to compute the Schubert calculus of generalized flag varieties. For the Grassmannian, a solution is given by the Littlewood–Richardson rule. One strategy for studying these problems is to approach them in the richer setting of equivariant cohomology. Double Schubert polynomials are polynomial representatives of the equivariant cohomology classes in the complete flag variety Fln. Specializing them gives Schubert polynomials, representatives of ordinary cohomology classes in Fln. With A. Adve and A. Yong, we give the first polynomial time algorithm to decide if a monomial coefficient of a Schubert polynomial is zero. We introduce a tableau criterion to determine this vanishing. Further we show that explicitly computing these monomial coefficients is #P-complete. Littlewood–Richardson polynomials control the equivariant Schubert calculus of the Grassmannian. With A. Adve and A. Yong, we show that deciding the vanishing of a Littlewood–Richardson polynomial is polynomial time. This generalizes the work of J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni. With H. Yadav and A. Yong, we study the equivariant Schubert calculus of isotropic flag manifolds. We use Billey’s formula to establish an explicit correspondence between these isotropic structure coefficients. Additionally we introduce shifted edge labeled tableaux that conjecturally compute structure coefficients for a specialization of the equivariant Schubert calculus of the Lagrangian Grassmannian given by D. Anderson-W. Fulton. We prove additional cases of this conjecture. These results work towards finding a rule for computing the equivariant Schubert calculus of the Lagrangian Grassmannian. Double Schubert polynomials also appear as multidegrees of matrix Schubert varieties. The K-polynomials of matrix Schubert varieties are the double Grothendieck polynomials. With J. Rajchgot, Y. Ren, A. St. Dizier, and A. Weigandt, we use these polynomials to derive an explicit combinatorial rule for the Castelnuovo–Mumford regularity of Grassmannian matrix Schubert varieties. With J. Rajchgot and A. Weigandt we generalize these results for certain Kazhdan–Lusztig varieties. We use this to prove a correction of a conjecture of M. Kummini-V. Lakshmibai-P. Sastry-C. S. Seshadri."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Equivariant Schubert calculus and applications"]}]}],"canonical_facts":{"dc:contributor":["Yong, Alexander","Tolman, Susan","Di Francesco, Philippe","Kedem, Rinat"],"dc:creator":["Robichaux, Colleen"],"dc:date":["2022-08","2022-06-06"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-11-15 without embargo terms","The student, Colleen Robichaux, accepted the attached license on 2022-06-01 at 16:34.","The student, Colleen Robichaux, submitted this Dissertation for approval on 2022-06-01 at 16:37.","This Dissertation was approved for publication on 2022-06-06 at 14:52.","DSpace SAF Submission Ingestion Package generated from Vireo submission #18054 on 2022-11-15 at 17:37:31","A central problem in algebraic combinatorics is to determine a combinatorial rule to compute the Schubert calculus of generalized flag varieties. For the Grassmannian, a solution is given by the Littlewood–Richardson rule. One strategy for studying these problems is to approach them in the richer setting of equivariant cohomology. Double Schubert polynomials are polynomial representatives of the equivariant cohomology classes in the complete flag variety Fln. Specializing them gives Schubert polynomials, representatives of ordinary cohomology classes in Fln. With A. Adve and A. Yong, we give the first polynomial time algorithm to decide if a monomial coefficient of a Schubert polynomial is zero. We introduce a tableau criterion to determine this vanishing. Further we show that explicitly computing these monomial coefficients is #P-complete. Littlewood–Richardson polynomials control the equivariant Schubert calculus of the Grassmannian. With A. Adve and A. Yong, we show that deciding the vanishing of a Littlewood–Richardson polynomial is polynomial time. This generalizes the work of J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni. With H. Yadav and A. Yong, we study the equivariant Schubert calculus of isotropic flag manifolds. We use Billey’s formula to establish an explicit correspondence between these isotropic structure coefficients. Additionally we introduce shifted edge labeled tableaux that conjecturally compute structure coefficients for a specialization of the equivariant Schubert calculus of the Lagrangian Grassmannian given by D. Anderson-W. Fulton. We prove additional cases of this conjecture. These results work towards finding a rule for computing the equivariant Schubert calculus of the Lagrangian Grassmannian. Double Schubert polynomials also appear as multidegrees of matrix Schubert varieties. The K-polynomials of matrix Schubert varieties are the double Grothendieck polynomials. With J. Rajchgot, Y. Ren, A. St. Dizier, and A. Weigandt, we use these polynomials to derive an explicit combinatorial rule for the Castelnuovo–Mumford regularity of Grassmannian matrix Schubert varieties. With J. Rajchgot and A. Weigandt we generalize these results for certain Kazhdan–Lusztig varieties. We use this to prove a correction of a conjecture of M. Kummini-V. Lakshmibai-P. Sastry-C. S. Seshadri."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/116152"],"dc:language":["en","eng"],"dc:rights":["Copyright 2022 Colleen Robichaux"],"dc:subject":["Schubert calculus","Combinatorial algorithms","Equivariant cohomology","Castelnuovo-Mumford regularity"],"dc:title":["Equivariant Schubert calculus and applications"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:55Z"}