Advances in the Combinatorics of the Unipotent Upper Triangular Groups

Introduction
Motivations
Unipotent objects for GLn
Investigation of p-groups
Supercharacter theory
Outline of thesis and main results
How to read this thesis; notation warning
Preliminaries
Representation theoretic preliminaries
Basic definitions
Structure theory
Characters, class functions, and central idempotents
Inner products; common functors
Supercharacter theory
Examples
A second definition of supercharacter theory
Hopf algebras
Basic definitions
Connectedness and Takeuchi's formula
The conjugacy classes and irreducible representations of GLn
A GL-compatible Hopf algebra of unitriangular class functions
Hopf structures
Set compositions
Vector Species and Hopf Monoids
Morphisms, subspecies and submonoids
From Hopf monoids to Hopf algebras
Species of class function
Matrix Groups
Species of (super)class functions
The Hopf monoid cf(UT)
Combinatorial underpinnings
Functorial Subgroups
The downward functor Resf; coassociativity
The upward functor Inf; associativity
Compatibility
Naturality
The Hopf submonoid scf
The product
The coproduct
Verification of the Hopf submonoid
The image of the Fock functor K
The Hopf algebra of class functions
The Hopf subalgebra of normal supercharacters
Non(co)commutativity
A Hopf algebra antiautomorphism
Guay-Paquet's Hopf algebra
Induction to the general linear group
Review of cf(GL) as a Hopf algebra
Proof of Theorem 3.7.1
The unipotent combinatorics of the q-chromatic symmetric function
Preliminary material
Hopf algebras and (Quasi-)symmetric functions
Dyck paths and related objects
Supercharacter Theory
Homomorphisms between Hopf algebras of (super)class functions
The image of induction
Factoring canoCQS through cfunisupp(GL)
The chromatic quasisymmetric function
The vertical strip LLT polynomial as a GLn-character
The pseudosupercharacters
Factoring canoLLT through cf(GL)
The vertical strip LLT polynomial
The normal subgroups of the unipotent upper triangular group
Preliminaries
Nonnesting set partitions
Normal subgroups and ideals from nonnesting set partitions
Loopless binary matroids
Splices
Bindings, rows, and columns
Tight splices
Labeling tight splices
Ordering rows and columns
Lie algebra ideals
Splices and families of ideals
Labeled loopless binary matroids and ideals
Normal subgroups
Proof of Theorem 5.0.1
Further results on normal subgroups
The lattice geometry of supercharacter theories
Preliminaries
Subgroup lattices and Galois connections
Permutation Modules
Superclass function spaces
Sheaves
The lattice construction
Constructing the lattice supercharacter theory
An analogue of supernormality for non-normal subgroups
The sheaf construction
The category L( G), covers, and bases
Another decomposition of CG
The gluing theorem for bases
From bases to coverings
Examples of the finest global section over a basis
The GVZ supercharacter theory
Algebra groups
Bibliography
Unipotent representations of GLn, the Hecke algebra Hn(q), and Sn