- Title Information
- Title
- Theory and Applications of Harish-Chandra Integrals
- Name:
Personal
- Name Part
- McSwiggen, Colin Smith
- Role
- Role Term:
Text
- creator
- Name:
Personal
- Name Part
- Menon, Govind
- Role
- Role Term:
Text
- Advisor
- Name:
Personal
- Name Part
- Zuber, Jean-Bernard
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Novak, Jonathan
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Ramanan, Kavita
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Department of Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Origin Information
- Copyright Date
- 2020
- Physical Description
- Extent
- xi, 220 p.
- digitalOrigin
- born digital
- Note:
thesis
- Thesis (Ph. D.)--Brown University, 2020
- Genre (aat)
- theses
- Abstract
- This thesis studies the integral of an exponential function over an adjoint orbit of a compact Lie group. These integrals are named after Harish-Chandra, who derived an exact formula for them in 1957, and they comprise a ubiquitous family of special functions that arise in many areas of mathematics and physics. The best known example is the Harish-Chandra--Itzykson--Zuber (HCIZ) integral over the unitary group, which has become an important object of study in random matrix theory and quantum field theory. Here we present the general theory of Harish-Chandra integrals in a unified fashion, giving five different proofs of Harish-Chandra's 1957 formula along with detailed derivations of the specific realizations of the formula for all compact classical groups. We then develop several new applications to representation theory, random matrix theory, and algebraic combinatorics. These applications include majorization inequalities for special functions related to group characters, a detailed investigation of a random matrix model that provides a probabilistic generalization of Horn's problem in linear algebra, and new formulae for Littlewood--Richardson coefficients and other tensor product multiplicities of semisimple Lie algebras in terms of the volumes of Berenstein--Zelevinsky polytopes. As indicated in the text, some chapters include joint work with Robert Coquereaux and Jean-Bernard Zuber, as well as with Jonathan Novak.
- Subject
- Topic
- Combinatorics
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01400410")
- Topic
- Applied mathematics
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01012104")
- Topic
- Mathematical physics
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01743340")
- Topic
- Representations of Lie algebras
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/01089803")
- Topic
- Random matrices
- Subject (fast)
(authorityURI="http://id.worldcat.org/fast", valueURI="http://id.worldcat.org/fast/00936132")
- Topic
- Functions, Special
- Language
- Language Term (ISO639-2B)
- English
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20200720
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations