Advances in the Combinatorics of the Unipotent Upper Triangular Groups

Description

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