Supercharacters and superclasses for algebra groups

By Persi Diaconis, I. Isaacs. 2007

230 citations recorded by the source corpus.

We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and the supercharacter theory we develop extends results of Carlos Andre and Ning Yan that were originally proved in the upper triangular case. This theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table. We discuss connections with the Kirillov orbit method and with Gelfand pairs, and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class, respectively. We also show that products of supercharacters are positive integer combinations of supercharacters.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/80e6963ca3b7e7017549ec3e88d64c423087ac77

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Citations in the corpus

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups (2008, background): “By definition (see, for example, [4], [7], [11], or [17]), the superclass of Um(q) which contains the element 1 + eD,φ ∈ Um(q) is the subset 1 + VD,φ = {1 + a : a ∈ VD,φ} ⊆ Um(q), and we define the superclass KD,φ of U to be the intersection KD,φ = U ∩ (1 + VD,φ) = {x ∈ U : x − 1 ∈ VD,φ}.”
Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras (2010, methodology): “Diaconis and Isaacs [16] then generalized the concept to arbitrary finite groups, and we reproduce a version of this more general definition below.”
Branching rules in the ring of superclass functions of unipotent upper-triangular matrices (2008, background): “Supercharacters satisfy a variety of nice properties, as described in [12].”
Ramanujan sums as supercharacters (2012, background): “The theory of supercharacters, of which classical character theory is a special case, was recently introduced by Diaconis and Isaacs (2008) [14] to generalize the basic characters of André [3–5].”
The Markov chain Monte Carlo revolution (2008, methodology): “The detailed use of this required new formula [4] and leads to an extension of the theory to algebra groups in joint work with Isaacs and Theme [22, 34].”
Supercharacter formulas for pattern groups (2006, methodology): “2 Supertheory versus usual character theory For η ∈ J∗, let annJ (η) = {ρ ∈ J ∗ | XφXρ ∈ ker(λη), for all φ ∈ J } annJ (η) = {ρ ∈ J ∗ | XρXφ ∈ ker(λη), for all φ ∈ J } Diaconis and Isaacs give the following characterization of which supercharacters are irreducible characters [10].”
Supercharacter Theory Constructions Corresponding to Schur Ring Products (2012, methodology): “In fact, several of the initial theorems in [7] appeared in a different form in the work of Tamaschke on Schur rings [20].”