Related papers: A supercharacter approach to Heilbronn sums
We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the…
The spin analogues of several classical concepts and results for Hecke algebras are established. A Frobenius type formula is obtained for irreducible characters of the Hecke-Clifford algebra. A precise characterization of the trace…
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…
Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum…
We describe the supercharacter theories of the semidirect product of H and K, $H\rtimes K$ in terms of the supercharacter theories of the direct product of H and K in the case when both H and K are Abelian groups. To do this we introduce…
In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the…
The purpose of the present paper is to investigate a hypergroup arising from irreducible characters of a compact group G and a closed subgroup of G with finite index. The convolution of this hypergroup is introduced by inducing irreducible…
We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in…
We prove an identity between three infinite families of polynomials which are defined in terms of `bosonic', `fermionic', and `one-dimensional configuration' sums. In the limit where the polynomials become infinite series, they give…
A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order.…
We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$.…
Rota-Baxter algebras and Atkinson's method are powerful tools for the factorization of characters on Hopf algebras. The theory of real resummation discovered by J. Ecalle and known as \textit{well-behaved averages theory} can be…
We enumerate generalizations of the superintegrability property $<character>\ \sim {\rm character}$ and illuminate possible general structures behind them. We collect variations of original formulas available up to date and emphasize the…
A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.
The theory of differential characters is developed completely from a de Rham - Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular…
We prove that an innocent looking inequality implies the Riemann Hypothesis and show a way to approach this inequality through sums of Legendre symbols.
Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}/n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of…
We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally…
There are two main constructions of supercharacter theories for a group $ G $. The first, defined by Diaconis and Isaacs, comes from the action of a group $A$ via automorphisms on our given group $G$. The second, defined by Hendrickson, is…
An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown…