Supercharacter Theory Research Articles

Visual aspects of Gaussian periods and analogues

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties. In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations — one relating to supercharacter theory, the other relating to class field theory — along with discussions and visual examples of each.

Normal Supercharacter Theories Of The Dicyclic Groups

The supercharacter theory was developed by P. Diaconis and I. M. Isaacs as a natural generalization of the classical ordinary character theory. There are different constructions for finding the supercharacter theories of a finite group. Supercharacter theories of many finite groups, such as cyclic groups, Frobenius groups, dihedral groups, elementary abelian $p$-groups, and Camina groups, etc. are studied with different constructions. One of the constructions uses normal subgroups. In this paper, we consider dicyclic groups and find some of their normal supercharacter theories and some automorphic supercharacter theories in special cases.

Shintani descent for standard supercharacters of algebra groups

Let [Formula: see text] be a finite-dimensional nilpotent algebra over a finite field [Formula: see text] with q elements, and let [Formula: see text]. On the other hand, let [Formula: see text] denote the algebraic closure of [Formula: see text], and let [Formula: see text]. Then [Formula: see text] is an algebraic group over [Formula: see text] equipped with an [Formula: see text]-rational structure given by the usual Frobenius map [Formula: see text], and [Formula: see text] can be regarded as the fixed point subgroup [Formula: see text]. For every [Formula: see text], the nth power [Formula: see text] is also a Frobenius map, and [Formula: see text] identifies with [Formula: see text]. The Frobenius map restricts to a group automorphism [Formula: see text], and hence it acts on the set of irreducible characters of [Formula: see text]. Shintani descent provides a method to compare F-invariant irreducible characters of [Formula: see text] and irreducible characters of [Formula: see text]. In this paper, we show that it also provides a uniform way of studying supercharacters of [Formula: see text] for [Formula: see text]. These groups form an inductive system with respect to the inclusion maps [Formula: see text] whenever [Formula: see text], and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain “superdual algebra” which encodes information about the supercharacters of [Formula: see text] for [Formula: see text].

Supercharacter theories of the dicyclic groups

In this paper, all supercharacter theories of dicyclic groups are described. These supercharacter theories are constructed by using the supercharacter theories of dihedral groups of order 4n which are classified by Lamar.

Supercharacters for parabolic contractions of finite groups of 𝐴,𝐵,𝐶,𝐷 Lie types

Supercharacter theories are constructed for the finite groups obtained by parabolic contraction from simple groups of A , B , C , D A,B,C,D Lie types. Supercharacters and superclasses are classified in terms of rook placements in root systems.

Supercharacters of discrete algebra groups

Abstract The supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and illustrated with a construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with a characterization of standard supercharacters of the group of upper unitriangular matrices over an algebraically closed field of prime characteristic.

English

We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artin's conjecture regarding the holomorphy of Artin $L$-functions.

Toward a classification of the supercharacter theories of Cp × Cp

AbstractIn this paper, we study the supercharacter theories of elementary abelian $p$-groups of order $p^{2}$. We show that the supercharacter theories that arise from the direct product construction and the $\ast$-product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$-group of order $p^{2}$ that has a non-identity superclass of size $1$ or a non-principal linear supercharacter must come from either a $\ast$-product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2,\, 3,\, 5$, and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$.

Supercharacter Table of Certain Finite Groups

‎Supercharacter theory is developed by P‎. ‎Diaconis and I‎. ‎M‎. ‎Isaacs as a natural generalization of the classical ordinary character theory‎. ‎Some classical sums of number theory appear as supercharacters which are obtained by the action of certain subgroups of GL_d(Z_n) on Z_n^d‎. ‎In this paper we take Z_p^d‎, ‎p prime‎, ‎and by the action of certain subgroups of GL_d(Z_p) we find supercharacter table of Z_p^d‎.

A supercharacter theory based on the degrees of irreducible characters

A supercharacter theory based on the degrees of irreducible characters

An algorithm for computing a supercharacter theory generated from a given partition

Let [Formula: see text] be a finite group. The set of all supercharacter theories of [Formula: see text] forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of [Formula: see text], with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet of two supercharacter theories.

Eighth power moment of Kloosterman sum, supercharacters, and elliptic curves

We here use a supercharacter theory of $${\mathbb {F}}_p^2$$ to express the eighth power moment of Kloosterman sum in terms of traces of Frobenius endomorphism of certain families of elliptic curves.

Supercharacter theory via the group determinant

Frobenius is generally considered the creator of character theory of finite groups. This achievement came from the study of the group determinant, which is the determinant of a matrix coming from the regular representation. We generalize several of Frobenius’ results about the group determinant and use them to find a new formulation of supercharacter theory in terms of factorizations of the group determinant.

A categorification of the Malvenuto–Reutenauer algebra via a tower of groups

There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a categorification supplies canonical bases, inner products, and a natural avenue to prove positivity results. Recent ideas in supercharacter theory have made fashioning the representation theory of a tower of groups into a Hopf structure more tractable. This paper applies such a program to the Malvenuto–Reutenauer Hopf algebra. In particular, we design functors on the representation theory of a tower of p-groups that realize the Hopf structure of the Malvenuto–Reutenauer algebra in such a way that its well-known fundamental basis corresponds to a supercharacter basis.

Supercharacter theories for Sylow p-subgroups of the Ree groups

We determine a supercharacter theory for Sylow p -subgroups G 2 s y l 2 ( 3 2 m + 1 ) of the Ree groups G 2 2 ( 3 2 m + 1 ) , calculate the conjugacy classes of G 2 s y l 2 ( 3 2 m + 1 ) , and establish the character table of G 2 s y l 2 ( 3 ) .

Finding supercharacter theories on character tables

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the coarsest partition that belongs to a supercharacter theory. Our constructions emphasize the duality between superclasses and supercharacters. An algorithm is presented to find all supercharacter theories on a given character table. The algorithm is used to compute the number of supercharacter theories for some nonabelian simple groups with up to 26 conjugacy classes.

An algorithm for constructing all supercharacter theories of a finite group

In 2008 , Diaconis and Isaacs introduced the notion of a supercharacter theory of a finite group in which supercharacters replace with irreducible characters and superclasses by conjugacy classes. In this paper, we introduce an algorithm for constructing supercharacter theories of a finite group by which all supercharacter theories of groups containing up to 14 conjugacy classes are calculated.

The structure of normal lattice supercharacter theories

The character theory of finite groups has numerous basic questions that are often already quite involved: enumeration of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all the important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.

Supercharacter theories for algebra group extensions

Abstract We construct a few supercharacter theories for finite semidirect products where the normal subgroup is of algebra group type. In the case of algebra groups, these supercharacter theories coincide with those of P. Diaconis and I. M. Isaacs. For the parabolic subgroups of GL ⁢ ( n , 𝔽 q ) {\mathrm{GL}(n,\mathbb{F}_{q})} , the supercharacters and superclasses are classified.

An analog of nilpotence arising from supercharacter theory

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of J can be described in terms of certain central series for the algebra group G=1+J defined by S, when S is the algebra group supercharacter theory defined by Diaconis–Isaacs.