Narrow Results

The aim of this paper is to identify the solvable groups with four supercharacter theories. To achieve this, the research will first identify abelian groups with four supercharacter theories. Then it is proved that the only non-abelian group with four supercharacter theories is dihedral group of order 38.

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a $\mathrm{\Delta }$-product or ∗-product from the values of its corresponding supercharacters.

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = e^h/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the -characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine -characters of π₁(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.

This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to $2$-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-representation theory of $2$-groups, as studied by the second author.

We discuss the relationship between the derived length and the number of character degrees in the restricted setting of a normally monomial p-group, G, of maximal class. We continue the Lie algebra approach implemented by Keller, Ragan, and Tims. With a number of technical results, we improve the existing bound, , to obtain, .

For a finite group $G$, we denote by $p(G)$ the minimal degree of faithful permutation representations of $G$, and denote by $q(G)$ and $c(G)$, the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the rational field $\mathbb{Q}$ and the complex field $ℂ$, respectively. In this paper, we calculate $p(G)$, $q(G)$ and $c(G)$ for the groups of order $p^5$, where $p$ is an odd prime.

We verify a conjecture proposed by Qian on character codegrees in the case of symmetric and alternating groups.

In [2], we gave algorithms to calculate c(G), q(G) and p(G) for a finite group G. In this paper, we show that for a finite p-group G, where p is a prime, q(G)=p(G). Moreover, for odd prime p, c(G)=q(G)=p(G).

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, and the non-existence of simple zeros or poles in such quotients.

In this note, we study conjectures of Artin and Langlands and derive the automorphy of all solvable groups of order at most 200, three groups excepted.