Three topics in combinatorial representation theory

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Gao, Shiliang

Contributors dc:contributor

• Yong, Alexander
• Kedem, Rinat
• Di Francesco, Philippe
• Haboush, William

• Tensor Product Multiplicities
• Weight Multiplicities
• Standard Monomial Theory

Statement dc:rights

• Copyright 2024 Shiliang Gao

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/125535

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F125535