Much can be learned about a finite group from its character table, but sometimes that table can be difficult to compute. Supercharacter theories are generalizations of character theory defined by P. Diaconis and I.M. Isaacs, in which certain (possibly reducible) characters called supercharacters take the place of the irreducible characters, and a certain coarser partition of the group takes the place of the conjugacy classes.
We present five new ways to construct new supercharacter theories out of supercharacter theories already known to exist, including a direct product, a lattice-theoretic join, two products over normal subgroups, and a duality for supercharacter theories of abelian groups.
In 2008, Diaconis annd 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.
In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are $\mathbb{Z}_3$ and $S_3$. We also show that the only nonsolvable group with two supercharacter theories is ${\rm Sp} (6,2)$.
A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions.The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.
In this paper, we study the superscharacter 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 nonidentity superclass of size $1$ or a nonprincipal 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 supercharac…
The Jordan-Hölder Theorem is a general term given to a collection of theorems about maximal chains in suitably nice lattices. For example, the well-known Jordan-Hölder type theorem for chief series of finite groups has been rather useful in studying the structure of finite groups. In this paper, we present a Jordan-Hölder type theorem for supercharacter theories of finite groups, which generalizes the one for chief series of finite groups.
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 cla…
The supercharacter theories of $C_p \times C_2 \times C_2$ were classified in the language of Schur rings by Evdokimov, Kovács, and Ponomarenko in [EKP16]. It was shown that every nontrivial supercharacter theory of $C_p \times C_2 \times C_2$ can be constructed as a wedge product, a direct product, or is generated by automorphisms. We use this classification to give a precise count of the distinct supercharacter theories of $C_p\times C_2\times C_2$ and describe when a supercharacter theory can be constructed by more than one method. We also present an alternative proof of the classification …
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 GLd(Zn) on Zdn. In this paper we take Zdp , p prime, and by the action of certain subgroups of GLd(Zp) we find supercharacter table of Zdp .
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 $\mathsf{S}$, when $\mathsf{S}$ is the algebra group supercharacter …
We construct supercharacter theories for the finite groups constructed by parabolic constraction from simple groups of $A,B,C,D$ Lie types. In terms of rook placements in the root systems we classify supecharacters and superclasses.
The concept of a supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and exemplified with a particular 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. Our construction can be understood as a cruder version of Kirillov's orbit method and a generalisation of Diaconis and Isaacs construction for finite…
