Irreducible Characters Research Articles

On the number of solutions of some Frobenius type equations in finite groups

Let G be a finite group with automorphism group Aut ( G ) and g ∈ G . For any subgroup H of Aut ( G ) , define a map ξ H : G → C by ξ H ( g ) = | { ( x , α ) ∈ G × H : [ x , α ] = g } | , where [ x , α ] : = x − 1 α ( x ) is called the autocommutator of x and α in G. In this paper, we show that ξ H is a character of G when H is any subgroup of Aut ( G ) containing Inn ( G ) and H = Autcent ( G ) , where Inn ( G ) is the inner automorphism group of G and Autcent ( G ) is the group of all the central automorphisms of G. In these cases expressions for ξ H are also obtained in terms of irreducible characters of G. Our results generalize a classical result of Frobenius regarding the number of solutions of the commutator equation in finite groups. As applications of our results, we compute g-autocommuting probabilities of some finite groups and obtained certain bounds for g-autocommuting probability of G in terms of commuting and autocommuting probabilities.

Weights for 𝜋-partial characters of 𝜋-separable groups

Abstract The aim of this paper is to confirm a conjecture of Isaacs and Navarro from 1995, which asserts that, for any π ′ \pi^{\prime} -subgroup 𝑄 of a 𝜋-separable group 𝐺, the number of π ′ \pi^{\prime} -weights of 𝐺 with 𝑄 as the first component is larger than or equal to the number of irreducible 𝜋-partial characters of 𝐺 with 𝑄 as their vertex. We also give a sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the 𝜋-version of the Alperin weight conjecture.

Kernels of minimal characters of solvable groups

Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G, then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.

A new characterization of E 8(p) via its vanishing elements

Let G be a finite group, and g ∈ G . Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo ( N ) = Vo ( G ) is isomorphic to G. In this paper, we investigate the V-recognizability of E 8 ( p ) , where p is a prime number. As an application, among the 610 primes p with p < 10000 and p ≡ 0 , 1 , 4 ( mod 5 ) , we obtain that the method is always valid for confirming the V-recognizability of E 8 ( p ) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589.

FINITE GROUPS WHOSE CHARACTER CODEGREES ARE CONSECUTIVE INTEGERS

Abstract Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers.

Finite Groups Containing No Blocks of Defect Zero

The irreducible characters of finite groups are always contained in blocks of defects which are nonnegative integers. Even though blocks always exist in finite groups, it is not the case that blocks of defect zero would always exist as well. Blocks of defect zero contain only one irreducible ordinary character each of defect zero and the defect group of blocks of defect zero is always the trivial subgroup of a finite group. Some finite groups do not have characters of defect zero and hence no blocks of defect zero either. The object in this paper is to study finite groups containing no blocks of defect zero. Finite abelian groups and p-groups will serve as special cases in this regard, with all blocks of finite abelian groups being of full/highest defect. We shall also determine an upper bound for the number of blocks in finite groups which contain no blocks of defect zero.

On Gluck’s conjecture for wreath product type groups

Abstract A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

Odd-order groups with small average number of zeros of characters

We define, and denote by anz ( G ) , the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz ( G ) ≤ 2 .

On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p-groups

For a finite group G, we denote by c ( G ) , the minimal degree of a faithful representation of G by quasi-permutation matrices over C . For an irreducible character χ of G, the codegree of χ is defined as cod ( χ ) = | G / ker ( χ ) | / χ ( 1 ) . In this article, we establish equality between c ( G ) and a Q ≥ 0 -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c ( G ) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.

The prime-power hypothesis on the codegrees of irreducible characters

For a character χ of a finite group G, the number cod ( χ ) = [ G : Ker χ ] χ ( 1 ) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod ( χ ) ≠ cod ( ϕ ) , the greatest common divisor of cod ( χ ) and cod ( ϕ ) is a prime-power, then either G is solvable or G / Sol ( G ) is isomorphic to one of the groups Alt 5 or PSL 2 ( 8 ) , where Sol ( G ) is the solvable radical of the group G.

On the identities and cocharacters of the algebra of 3 × 3 matrices with orthosymplectic superinvolution

Let M1,2(F) be the algebra of 3×3 matrices with orthosymplectic superinvolution ⁎ over a field F of characteristic zero. We study the ⁎-identities of this algebra through the representation theory of the group Hn=(Z2×Z2)∼Sn. We decompose the space of multilinear ⁎-identities of degree n into the sum of irreducibles under the Hn-action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the ⁎-polynomial identities of M1,2(F) up to degree 3.

The Knutson Index of the representation ring

In this paper, we study if, for a given simple module over a Hopf algebra, there exists a virtual module such that their tensor product is the regular module. This is related to a conjecture by Donald Knutson, later disproved and refined by Savitskii, stating that for every irreducible character of a finite group, there exists a virtual character such that their tensor product is the regular character. We also introduce the Knutson Index as a measure of Knutson's Conjecture failure, discuss its algebraic properties and present counter-examples to Savitskii's Conjecture.

On groups associated with the affine subgroups of Sp2n(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Sp_{2n}(2)$$\\end{document}

The symplectic group Sp2n(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Sp_{2n}(2)$$\\end{document} has an affine maximal subgroup of structure ASpn=22n-1:Sp2n-2(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)$$\\end{document} which is a split extension of an elementary abelian 2-group N=22n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2^{2n-1}$$\\end{document} by G=Sp2n-2(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=Sp_{2n-2}(2)$$\\end{document}. The vector space N=22n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2^{2n-1}$$\\end{document} and its dual N∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N^{*}$$\\end{document} are not equivalent as 2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2n-1$$\\end{document} dimensional G-modules over GF(2). Therefore, a split extension of the form G¯n=N∗:Sp2n-2(2)≇N:Sp2n-2(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)\ cong N{:}Sp_{2n-2}(2)$$\\end{document} exists. In this paper, it will be shown that G¯n≅Aut(22n-2:Sp2n-2(2))=22n-2:Sp2n-2(2):2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{G}_n\\cong \ ext {Aut}(2^{2n-2}{:}Sp_{2n-2}(2))= \\left( 2^{2n-2}{:}Sp_{2n-2}(2)\\right) {:} 2$$\\end{document} for n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 3$$\\end{document}. Moreover, the ordinary irreducible characters of G¯n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{G}_n$$\\end{document} are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr(G¯5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\overline{G}_5)$$\\end{document} of the group G¯5=29:Sp8(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{G}_5=2^9{:}Sp_{8}(2)$$\\end{document} which is associated with the affine subgroup ASp5=29:Sp8(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ASp_5=2^9{:}Sp_{8}(2)$$\\end{document} of Sp10(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Sp_{10}(2)$$\\end{document}.

Infinite Families of Congruences Modulo 5 for the Number of Irreducible Characters of the Alternating Groups

In 1919, Ramanujan proved red three congruences for the partition function p(n) which denotes the number of partitions of n. The partition function p(n) can be understood as the number of irreducible characters of the symmetric group Sn . Recently, Nath and Sellers, and Xia established a number of congruences modulo 2, 3, and 5 for the numbers of spin characters of S ̂ n and A ̂ n , where S ̂ n and A ̂ n are the double covering group of Sn and the double covering group of the alternating group An , respectively. Motivated by their work, we establish infinite families of congruences modulo 5 for a(n), which denotes the number of irreducible characters of the alternating group An . In particular, we prove some strange congruences modulo 5 for a(n). For example, we prove that for k ≥ 0 , a ( 95 × 73 2 k + 1 24 ) ≡ − 3 k ( mod 5 ) .

Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let [Formula: see text] be a finite group and [Formula: see text] the set of all irreducible complex characters of [Formula: see text]. Let [Formula: see text] be the set of all irreducible complex character degrees of [Formula: see text] and denote by [Formula: see text] the set of all primes which divide some character degree of [Formula: see text]. The character-prime graph [Formula: see text] associated to [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and there is an edge between two distinct primes [Formula: see text] and [Formula: see text] if and only if the product [Formula: see text] divides some character degree of [Formula: see text]. We show that the finite nonabelian simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are uniquely determined by their degree-patterns and orders.

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Abstract Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}\in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $\mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $\mu _{w,U(n)}$.

Factorization of classical characters twisted by roots of unity: II

Fix natural numbers n⩾1, t⩾2 and a primitive tth root of unity ω. In previous work with A. Ayyer (2022) [5], we studied the factorization of specialized irreducible characters of GLtn, SO2tn+1, Sp2tn and O2tn evaluated at elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n. In this work, we extend the results to the groups GLtn+m(0⩽m⩽t−1), SO2tn+3, Sp2tn+2 and O2tn+2 evaluated at similar specializations: (1) for the GLtn+m case, we set the first tn elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n and the remaining m to y,ωy,…,ωm−1y; (2) for the other three families, the same specializations but with m=1. The main results of this paper are a characterization of partitions for which these characters vanish and a factorization of nonzero characters into those of smaller classical groups. Our motivation is the conjectures of Wagh and Prasad (2020) [18] relating the irreducible representations of Spin2n+1 and SL2n, SL2n+1 and Sp2n as well as Spin2n+2 and Sp2n. Our proofs use the Weyl character formulas and the beta-sets of t-core partitions. Lastly, we give a bijection to prove that there infinitely many t-core partitions for which these characters are nonzero.

Odd degree rational characters and the order of rational elements in finite groups

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by P. H. Tiep and H. P. Tong-Viet.

Characters of [formula omitted] and vertex operators

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GLn(Fq). Green's theory of GLn(Fq) is recovered and enhanced under the realization of the Grothendieck ring of representations RG=⨁n≥0R(GLn(Fq)) as two isomorphic Fock spaces associated to two infinite-dimensional F-equivariant Heisenberg Lie algebras hˆF‾ˆq and hˆF‾q, where F is the Frobenius automorphism of the algebraically closed field F‾q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space RG as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example.

Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in [4].