2026-09-22T20:56:04Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/125535 2025-02-06 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-02-04 without embargo terms The student, Shiliang Gao, accepted the attached license on 2024-06-24 at 15:12. The student, Shiliang Gao, submitted this Dissertation for approval on 2024-06-24 at 15:20. This Dissertation was approved for publication on 2024-06-26 at 16:09. DSpace SAF Submission Ingestion Package generated from Vireo submission #20867 on 2025-02-04 at 21:03:54 Three topics in combinatorial representation theory Gao, Shiliang 2024-06-26 Yong, Alexander Kedem, Rinat Di Francesco, Philippe Haboush, William Tensor Product Multiplicities Weight Multiplicities Standard Monomial Theory eng This thesis contains results from three projects concerning combinatorial representation theory. The first project studies Newell-Littlewood numbers. They are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for classical Lie groups. They are the structure coefficients of the Koike-Terada basis of the ring of symmetric functions. We give a systematic study concerning the combinatorics of Newell-Littlewood numbers. The second project concerns the Kostka coefficients. They are the weight space dimensions of irreducible representations of complex semisimple Lie algebras. They can be described as the number of lattice points in a Berenstein-Zelevinsky polytope. We give a root-theoretic formula for the dimension of a Berenstein-Zelevinsky polytope, in classical Lie types. Equivalently, the formula computes the degree of the stretched Kostka quasi-polynomial. The third project concerns the standard monomial theory of positroid varieties. Positroid varieties are subvarieties of the Grassmannian Gr(k,n) introduced by Knutson-Lam-Speyer, motivated by the study of total positivity. We study standard monomial theory on positroid varieties. We give an explicit alternative characterization of the standard monomials of positroid varieties under the Hodge degeneration (after work of Knutson-Lam-Speyer), and a first Gr\"obner basis for defining ideals of positroid varieties. As an application, we show that promotion and evacuation on rectangular-shaped semistandard tableaux biject standard monomials of a positroid variety with those of its cyclic shifts and w_0-reflection, respectively. 2024-08 Thesis https://hdl.handle.net/2142/125535 Copyright 2024 Shiliang Gao Mathematics Mathematics University of Illinois at Urbana-Champaign Ph.D. Dissertation