Supercharacter Theory Research Articles

Supercharacters and mixed moments of Kloosterman sums

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

Supercharacters and the discrete Fourier, cosine, and sine transforms

ABSTRACTUsing supercharacter theory, we identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method affords a combinatorial interpretation for the matrix entries.

Supercharacter theories of semiextraspecial p-groups and Frobenius groups

In this paper, we describe all supercharacter theories of semiextraspecial p-groups and Frobenius groups. We first review the constructions which will classify these supercharacter theories. We then show that the supercharacter theories of these groups, as well as the more general Camina pairs, must have nontrivial C-normal subgroups. Next, we examine the superclass structure to show that the supercharacter theory can be constructed from supercharacter theories of subgroups of the group and its quotient groups. We can apply this to count the number of supercharacter theories for the group.

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings.

A supercharacter approach to Heilbronn sums

Various algebraic properties of Heilbronn's exponential sum can be deduced through the use of supercharacter theory, a novel extension of classical character theory due to Diaconis–Isaacs and André. This perspective yields a variety of formulas and provides a method for computing the number of solutions to Fermat-type congruences.

A Plancherel measure associated to set partitions and its limit

In recent years increasing attention has been paid on the area of supercharacter theories, especially to those of the upper unitriangular group. A particular supercharacter theory, in which supercharacters are indexed by set partitions, has several interesting properties, which make it object of further study. We define a natural generalization of the Plancherel measure, called superplancherel measure, and prove a limit shape result for a random set partition according to this distribution. We also give a description of the asymptotical behavior of two set partition statistics related to the supercharacters. The study of these statistics when the set partitions are uniformly distributed has been done by Chern, Diaconis, Kane and Rhoades.

Supercharacters for finite groups of triangular type

ABSTRACTWe construct a supercharacter theory for the finite groups of triangular type. Its special case is the supercharacter theory for algebra groups of Diaconis and Isaacs. The supercharacter analog of the Kirillov formula for irreducible characters of unipotent groups is proved.

Groups with exactly two supercharacter theories

ABSTRACTIn this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ℤ3 and S3. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).

Supercharacter theory for groups of invertible elements of reduced algebras

We construct a supercharacter theory for the group of invertible elements of a reduced algebra. For the case of the triangular group, we obtain the formula for values of supercharacters on superclasses.

Normal supercharacter theories and their supercharacters

In order to find a tractable theory to substitute for the wild character theory of the group of n×n unipotent upper-triangular matrices over a finite field Fq, André and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G. We call such supercharacter theory the normal supercharacter theory generated by S. It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions.

Supercharacter Theories Constructed by the Method of Little Groups

The “method of little groups” describes the irreducible characters of semidirect products with abelian normal subgroups in terms of the irreducible characters of the factor groups. We modify this method to construct supercharacter theories of semidirect products with abelian normal subgroups. In particular, we apply this construction to reproduce known supercharacter theories of several families of unipotent groups. We also utilize our method to construct a collection of new supercharacter theories of the unipotent upper-triangular matrices.

The Schur–Wielandt Theory for Central S-Rings

Two basic results on S-rings over an Abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. Neither of these is directly generalized to the non-Abelian case. Nevertheless, we prove that the two theorems are true for central S-rings over any group, i.e., for S-rings that are contained in the center of the group ring of that group (such S-rings arise naturally in the supercharacter theory). Extending the concept of a B-group introduced by Wielandt, we show that every Camina group is a generalized B-group, whereas simple groups, with few exceptions, cannot be of this type.

Restrictions of rainbow supercharacters

The maximal subgroup of unipotent upper-triangular matrices of the finite general linear groups is a fundamental family of p-groups. Their representation theory is well known to be wild, but there is a standard supercharacter theory, replacing irreducible representations by super-representations, that gives us some control over its representation theory. While this theory has a beautiful underlying combinatorics built on set partitions, the structure constants of restricted super-representations remain mysterious. This paper proposes a new approach to solving the restriction problem by constructing natural intermediate modules that help “factor” the computation of the structure constants. We illustrate the technique by solving the problem completely in the case of rainbow supercharacters (and some generalizations). Along the way we introduce a new q-analogue of the binomial coefficients that depend on an underlying poset.

A supercharacter theory for involutive algebra groups

If J is a finite-dimensional nilpotent algebra over a finite field k, the algebra group P=1+J admits a (standard) supercharacter theory as defined in [16]. If J is endowed with an involution σ, then σ naturally defines a group automorphism of P=1+J, and we may consider the fixed point subgroup CP(σ)={x∈P:σ(x)=x−1}. Assuming that k has odd characteristic p, we use the standard supercharacter theory for P to construct a supercharacter theory for CP(σ). In particular, we obtain a supercharacter theory for the Sylow p-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto in [7,8] for the special case of the symplectic and orthogonal groups.

The Hopf monoid on nonnesting supercharacters of pattern groups

We construct supercharacter theories for a collection of unipotent matrix groups and produce a Hopf monoid from the supercharacters. These supercharacter theories are coarser than those defined by Diaconis--Isaacs for algebra groups and have supercharacters and superclasses indexed by nonnesting labeled set partitions. We compute the supercharacter tables and describe the product and coproduct of the Hopf monoid combinatorially. We also show that this Hopf monoid is free.

Supercharacters of unipotent groups defined by involutions

We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic groups and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table.

Supercharacters, exponential sums, and the uncertainty principle

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. André. We study supercharacter theories on (Z/nZ)d induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

Super-π-Brauer Characters and Super-π-Regular Classes

Supercharacter theories for an arbitrary finite group were developed by Diaconis and Isaacs. Let G* denote the set of all π-regular elements of G. Set Iπ(G) = {χ* | χ ∈Irr(G) and χ* ≠ α* + β* for characters α, β of G}, where χ* means the restriction of χ to G*. In this paper, we consider the partitions of Iπ(G) and G* for a π-separable group G. We obtain some results which generalize those of Diaconis and Isaacs.

Supercharacters of Unipotent Groups

\textbfAbstract. We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra groups. The resulting supercharacter theories agree with those of André and Neto in the case of the unipotent orthogonal and symplectic matrices and generalize to a large collection of subgroups. In the unitary group case, we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table. \bigbreak

Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups

We consider the lattice of supercharacter theories, in the sense of Diaconis and Isaacs [5], of the cyclic group of order n. We find necessary and sufficient conditions on n for that lattice to be upper or lower semimodular.