scholarly journals Weights of Partitions and Character Zeros

10.37236/1862 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Christine Bessenrodt ◽  
Jørn B. Olsson
Keyword(s):  
Irreducible Character ◽  
Prime Order ◽  
Alternating Groups ◽  
Non Linear

We classify partitions which are of maximal $p$-weight for all odd primes $p$. As a consequence, we show that any non-linear irreducible character of the symmetric and alternating groups vanishes on some element of prime order.

scholarly journals GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

2016 ◽  
Vol 224 (1) ◽  
pp. 93-167 ◽  
Author(s):  
JAY TAYLOR
Keyword(s):  
Finite Field ◽  
Wave Front ◽  
Irreducible Character ◽  
Prime Order ◽  
Algebraic Closure ◽  
Characteristic Functions ◽  
Rational Structure ◽  
Reductive Algebraic Group ◽  
Wave Front Set ◽  
Frobenius Endomorphism

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

On the Largest Irreducible Character Degree of a Finite Solvable Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 925-927 ◽  
Author(s):  
M. H. Jafari
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Irreducible Character ◽  
Prime Order ◽  
Character Degree ◽  
Finite Solvable Group ◽  
A Finite Group ◽  
Irreducible Character Degree

Let b(G) denote the largest irreducible character degree of a finite group G. In this paper, we prove that if G is a solvable group which does not involve a section isomorphic to the wreath product of two groups of equal prime order p, and if b(G) < pn, then |G:Op(G)|p < pn.

scholarly journals Degrees and rationality of characters in the principal block of $$A_n$$

2020 ◽  
Author(s):  
Eugenio Giannelli ◽  
Elena Meini
Keyword(s):  
Recent Result ◽  
Irreducible Character ◽  
Alternating Groups ◽  
Irreducible Characters ◽  
Principal Block

Abstract Given two primes p and q, we study degrees and rationality of irreducible characters in the principal p-block of $${\mathfrak {S}}_n$$ S n and $${\mathfrak {A}}_n$$ A n , the symmetric and alternating groups. In particular, we show that such a block always admits an irreducible character of degree divisible by q. This extends and generalizes a recent result of Giannelli–Malle–Vallejo.

Groups with Relatively Few Non-Linear Irreducible Characters

1968 ◽  
Vol 20 ◽  
pp. 1451-1458 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
General Question ◽  
Complex Numbers ◽  
Irreducible Characters ◽  
Non Linear ◽  
A Finite Group

In (4), Seitz characterized those finite groups which have exactly one non-linear irreducible character (over the complex numbers). In this paper we are concerned with the general question of what can be deduced about a finite group G if the number of its non-linear irreducible characters m(G) is given. In particular, does the assumption that m(G) is in some sense small when compared with the order |G| impose any restrictions on the structure of G?

scholarly journals Character degrees and nilpotence class in p-groups

1994 ◽  
Vol 57 (1) ◽  
pp. 76-80 ◽  
Author(s):  
Michael C. Slattery
Keyword(s):  
Irreducible Character ◽  
Character Degrees ◽  
Nilpotence Class ◽  
The Family ◽  
Non Linear

AbstractWork of Isaacs and Passman shows that for some sets X of integers, p-groups whose set of irreducible character degrees is precisely X have bounded nilpotence class, while for other choices of X, the nilpotence class is unbounded. This paper presents a theoren which shows some additional sets of character degrees which bound nilpotence class within the family of metabelian p-groups. In particular, it is shown that is the non-linear irreducible character degrees of G lie between pa and pb, where a ≤ b ≤ 2a − 2, then the nilpotence class of G is bounded by a function of p and b − a.

B. The Non-Linear Calculations for Pulsating Stars

1967 ◽  
Vol 28 ◽  
pp. 105-176
Author(s):  
Robert F. Christy
Keyword(s):  
Major Part ◽  
Afternoon Session ◽  
Discussion Session ◽  
Pulsating Stars ◽  
Fourth Symposium ◽  
Non Linear ◽  
Considerable Discussion ◽  
Informal Approach ◽  
Preceding Session

(Ed. note: The custom in these Symposia has been to have a summary-introductory presentation which lasts about 1 to 1.5 hours, during which discussion from the floor is minor and usually directed at technical clarification. The remainder of the session is then devoted to discussion of the whole subject, oriented around the summary-introduction. The preceding session, I-A, at Nice, followed this pattern. Christy suggested that we might experiment in his presentation with a much more informal approach, allowing considerable discussion of the points raised in the summary-introduction during its presentation, with perhaps the entire morning spent in this way, reserving the afternoon session for discussion only. At Varenna, in the Fourth Symposium, several of the summaryintroductory papers presented from the astronomical viewpoint had been so full of concepts unfamiliar to a number of the aerodynamicists-physicists present, that a major part of the following discussion session had been devoted to simply clarifying concepts and then repeating a considerable amount of what had been summarized. So, always looking for alternatives which help to increase the understanding between the different disciplines by introducing clarification of concept as expeditiously as possible, we tried Christy's suggestion. Thus you will find the pattern of the following different from that in session I-A. I am much indebted to Christy for extensive collaboration in editing the resulting combined presentation and discussion. As always, however, I have taken upon myself the responsibility for the final editing, and so all shortcomings are on my head.)

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