Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

Abstract

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

To Appear in Advances in Mathematics (2012), 23 pages

References in corpus (15)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Noncommutative symmetric functions
• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables
• Polynomial realizations of some trialgebras
• The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree
• Algebraic Structures on Grothendieck Groups of a Tower of Algebras
• Superinduction for pattern groups
• Coloured peak algebras and Hopf algebras
• Combinatorial methods of character enumeration for the unitriangular group
• Combinatorial Hopf algebras and Towers of Algebras - Dimension, Quantization, and Functoriality
• Combinatorial Hopf algebras and Towers of Algebras
• Heisenberg characters, unitriangular groups, and Fibonacci numbers
• Iterative character constructions for algebra groups
• Supercharacters and pattern subgroups in the upper triangular groups

Cited by in corpus (25)

• Hopf Algebras in Combinatorics
• Hopf monoids from class functions on unitriangular matrices
• The Hopf monoid on nonnesting supercharacters of pattern groups
• Hopf Algebras and Markov Chains
• Antipode and Primitive elements in the Hopf Monoid of Super Characters
• Supercharacters of unipotent groups defined by involutions
• Combinatorial methods of character enumeration for the unitriangular group
• A supercharacter table decomposition via power-sum symmetric functions
• Strong forms of linearization for Hopf monoids in species
• The combinatorics of ${\mathrm{GL}}_n$ generalized Gelfand--Graev characters
• Iterative character constructions for algebra groups
• Asymptotic results for Representation Theory
• Restriction and induction for supercharacters of finite groups of triangular type
• A characteristic map for the symmetric space of symplectic forms over a finite field
• Closed expressions for averages of set partition statistics
• Supercharacter theory for groups of invertible elements of reduced algebras
• Towards a supercharacter theory of the parabolic subgroups
• Supercharacters for Normal Supercharacter Theory
• Symmetric functions in noncommuting variables in superspace
• Shell Tableaux: A set partition analogue of vacillating tableaux
• Supercharacters of unipotent and solvable groups
• Positive Self-Dual Hopf Algebras of Galois Characters
• Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes
• Strong forms of self-duality for Hopf monoids in species
• A unipotent realization of the chromatic quasisymmetric function