Fusion rings with few degrees

The integral group rings for finite groups are precisely those fusion rings whose basis elements have Frobenius–Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here, we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius–Perron dimensions take exactly one value distinct from 1. Our goal is a set of results to assist in characterizing when such fusion rings are categorifiable. As proof of concept, we complete the classification of categorifiable near‐group fusion rings for an infinite collection of finite abelian groups, a task that to‐date has only been completed for three such groups.

Simple classical groups of Lie type are determined by their character degrees

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.

Let G be a finite group and let be the set of degrees of complex irreducible characters of G. An important program in character theory is to obtain the structure of G when is known. In this article, we consider a dual version of this approach. Indeed, we consider non-solvable groups with five character degrees and present a description of the set of character degrees for such groups. As an application of this result, it is shown that a finite group with five character degrees and containing no prime power must be a solvable group.

Let \(G\) be a finite group. The character degree graph of \(G\), which is denoted by \(\Gamma (G)\), is the graph whose vertices are the prime divisors of the character degrees of the group \(G\) and two vertices \(p_1\) and \(p_2\) are joined by an edge if \(p_1p_2\) divides some character degree of \(G\). In this paper we prove that the simple group \(\mathrm{PSL}(2,p^2) \) is uniquely determined by its character degree graph and its order. Let \(X_1(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. As a consequence of our results we prove that if \(G\) is a finite group such that \(X_1(G)=X_1(\mathrm{PSL}(2,p^2) )\), then \(G\cong \mathrm{PSL}(2,p^2) \). This implies that \(\mathrm{PSL}(2,p^2) \) is uniquely determined by the structure of its complex group algebra.

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

The fusion rules in [Formula: see text] for a finite group [Formula: see text] can be computed in terms of character inner products. Using an explicit formula (Theorem 3.2) for these fusion rules, we show that [Formula: see text] is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of [Formula: see text]-matrices. Furthermore, we observe that the fusion rules for Dihedral groups [Formula: see text] with [Formula: see text] odd are extremely similar to the fusion rules of type-B level-[Formula: see text] fusion algebras of Wess–Zumino–Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.

Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others.The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees.

Let G be a finite group. The degree (vertex) graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.

ABSTRACTWe define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.

We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group G on a finite vector space V under certain conditions, to a more general class of finite solvable groups G. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

We establish linear bounds for the ρ-σ conjecture for irreducible character degrees and conjugacy class sizes of any finite group.

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality holds, where is the set of irreducible character degrees of G and is the derived length of G. In this paper, we show that this inequality holds if G is a solvable Frobenius group. Also, we prove that if for some normal Sylow p-subgroup P of G, then . Moreover, we investigate this conjecture for some sets of the real-valued irreducible character degrees.

In this paper, we determine all of finite groups whose order and the largest of their irreducible character degrees are the same as $${\textit{PGL}}( 2 , p^{2} ) $$ for all odd prime numbers p. As a consequence, we show that the groups $${\textit{PGL}}( 2 , p^{2} ) $$ are uniquely determined by the structure of their complex group algebras.

In the remarkable paper (Blau, J. Algebr. 396:220–271, 2013) on fusion rings, Blau introduced a very important operation on two table algebras (the Blau-construction). It is very difficult to describe the structure of table algebras obtained by recursively applying the Blau-construction. In this paper, we define the wedge-direct sum for a sequence of table algebras. Using the wedge-direct sum of a sequence of table algebras, we are able to not only give a clear description of the structure of table algebras obtained by recursively applying the Blau-construction but also obtain the characterization and classification of a class of p-table algebras. The structure of the Bose-Mesner algebras of the wreath products and wedge products of association schemes can also be clearly described in terms of the wedge-direct sum. As an application, we get some new and known results about the wreath products and wedge products of association schemes.