相关论文: The graphic nature of the symmetric group

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods…

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…

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are…

Group theory involves the study of symmetry, and its inherent beauty gives it the potential to be one of the most accessible and enjoyable areas of mathematics, for students and non-mathematicians alike. Unfortunately, many students never…

We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a…

In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic…

Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the…

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses…

The symmetric tensor power of graphs is introduced and its fundamental properties are explored. A wide range of intriguing phenomena occur when one considers symmetric tensor powers of familiar graphs. A host of open questions are…

We give a complete list of indecomposable characters of the infinite symmetric semigroup. In comparison with the analogous list for the infinite symmetric group, one should introduce only one new parameter, which has a clear combinatorial…

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…

We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs,…

One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010),…

This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.

Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.

A special type of conjugacy classes in symmetric groups is studied and used to answer a question about odd-degree irreducible characters

We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…

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…

In this paper we examine various properties/constructions which are known for reductive groups and we do some experiments to see to what extent they generalize to symmetric spaces.

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these…