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For a finite group
$G$
, let
$\unicode[STIX]{x1D6E5}(G)$
denote the character graph built on the set of degrees of the irreducible complex characters of
$G$
. In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph
$\unicode[STIX]{x1D6E5}(G)$
of a finite group
$G$
is a nonbipartite Hamiltonian graph.
Given a finite group
$\text{G}$
and a field
$K$
, the faithful dimension of
$\text{G}$
over
$K$
is defined to be the smallest integer
$n$
such that
$\text{G}$
embeds into
$\operatorname{GL}_{n}(K)$
. We address the problem of determining the faithful dimension of a
$p$
-group of the form
$\mathscr{G}_{q}:=\exp (\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q})$
associated to
$\mathfrak{g}_{q}:=\mathfrak{g}\otimes _{\mathbb{Z}}\mathbb{F}_{q}$
in the Lazard correspondence, where
$\mathfrak{g}$
is a nilpotent
$\mathbb{Z}$
-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of
$\mathscr{G}_{p}$
is a piecewise polynomial function of
$p$
on a partition of primes into Frobenius sets. Furthermore, we prove that for
$p$
sufficiently large, there exists a partition of
$\mathbb{N}$
by sets from the Boolean algebra generated by arithmetic progressions, such that on each part the faithful dimension of
$\mathscr{G}_{q}$
for
$q:=p^{f}$
is equal to
$fg(p^{f})$
for a polynomial
$g(T)$
. We show that for many naturally arising
$p$
-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.
Let
$G$
be a
$p$
-group and let
$\unicode[STIX]{x1D712}$
be an irreducible character of
$G$
. The codegree of
$\unicode[STIX]{x1D712}$
is given by
$|G:\,\text{ker}(\unicode[STIX]{x1D712})|/\unicode[STIX]{x1D712}(1)$
. If
$G$
is a maximal class
$p$
-group that is normally monomial or has at most three character degrees, then the codegrees of
$G$
are consecutive powers of
$p$
. If
$|G|=p^{n}$
and
$G$
has consecutive
$p$
-power codegrees up to
$p^{n-1}$
, then the nilpotence class of
$G$
is at most 2 or
$G$
has maximal class.
Geoffrey Robinson conjectured in 1996 that the
$p$
-part of character degrees in a
$p$
-block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes
$p\neq 2$
for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.
Let
$G$
be a group,
$p$
be a prime and
$P\in \text{Syl}_{p}(G)$
. We say that a
$p$
-Brauer character
$\unicode[STIX]{x1D711}$
is monolithic if
$G/\ker \unicode[STIX]{x1D711}$
is a monolith. We prove that
$P$
is normal in
$G$
if and only if
$p\nmid \unicode[STIX]{x1D711}(1)$
for each monolithic Brauer character
$\unicode[STIX]{x1D711}\in \text{IBr}(G)$
. When
$G$
is
$p$
-solvable, we also prove that
$P$
is normal in
$G$
and
$G/P$
is nilpotent if and only if
$\unicode[STIX]{x1D711}(1)^{2}$
divides
$|G:\ker \unicode[STIX]{x1D711}|$
for all monolithic irreducible
$p$
-Brauer characters
$\unicode[STIX]{x1D711}$
of
$G$
.
Let
$G$
be a finite group and let
$\text{Irr}(G)$
be the set of all irreducible complex characters of
$G$
. Let
$\unicode[STIX]{x1D70C}(G)$
be the set of all prime divisors of character degrees of
$G$
. The character degree graph
$\unicode[STIX]{x1D6E5}(G)$
associated to
$G$
is a graph whose vertex set is
$\unicode[STIX]{x1D70C}(G)$
, and there is an edge between two distinct primes
$p$
and
$q$
if and only if
$pq$
divides
$\unicode[STIX]{x1D712}(1)$
for some
$\unicode[STIX]{x1D712}\in \text{Irr}(G)$
. We prove that
$\unicode[STIX]{x1D6E5}(G)$
is
$k$
-regular for some natural number
$k$
if and only if
$\overline{\unicode[STIX]{x1D6E5}}(G)$
is a regular bipartite graph.
We consider the relationship between structural information of a finite group
$G$
and
$\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$
, the set of all irreducible projective character degrees of
$G$
with factor set
$\unicode[STIX]{x1D6FC}$
. We show that for nontrivial
$\unicode[STIX]{x1D6FC}$
, if all numbers in
$\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$
are prime powers, then
$G$
is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.
Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).
We study the germs at the origin of
$G$
-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group
$G$
is either
$\text{SL}_{2}(\mathbb{C})$
or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either
$G=\text{SL}_{n}(\mathbb{C})$
for some
$n\geqslant 3$
, or the depth is greater than 1, then certain natural inclusions of germs are strict.
For a group G, a weak Cayley table isomorphism is a bijection f : G → G such that f(g1g2) is conjugate to f(g1)f(g2) for all g1, g2 ∈ G. The set of all weak Cayley table isomorphisms forms a group (G) that is the group of symmetries of the weak Cayley table of G. We determine (G) for each of the 17 wallpaper groups G, and for some other crystallographic groups.
Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group
$\text{Sp}_{4}$
over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup
$I_{n}$
of
$\text{GL}_{n}(F)$
, where
$F$
is a non-archimedean local field. We establish a bijection between the set of irreducible representations of
$I_{n}$
and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.
Let
$G$
be a finite solvable group and let
$p$
be a prime. In this note, we prove that
$p$
does not divide
$\unicode[STIX]{x1D711}(1)$
for every irreducible monomial
$p$
-Brauer character
$\unicode[STIX]{x1D711}$
of
$G$
if and only if
$G$
has a normal Sylow
$p$
-subgroup.
We prove that a finite coprime linear group
$G$
in characteristic
$p\geq \frac{1}{2}(|G|-1)$
has a regular orbit. This bound on
$p$
is best possible. We also give an application to blocks with abelian defect groups.
Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.
When
$G$
is a finite solvable group, we prove that
$|G|$
can be bounded by a function in the number of irreducible characters with values in fields where
$\mathbb{Q}$
is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.
Let
$UY_{n}(q)$
be a Sylow
$p$
-subgroup of an untwisted Chevalley group
$Y_{n}(q)$
of rank
$n$
defined over
$\mathbb{F}_{q}$
where
$q$
is a power of a prime
$p$
. We partition the set
$\text{Irr}(UY_{n}(q))$
of irreducible characters of
$UY_{n}(q)$
into families indexed by antichains of positive roots of the root system of type
$Y_{n}$
. We focus our attention on the families of characters of
$UY_{n}(q)$
which are indexed by antichains of length
$1$
. Then for each positive root
$\unicode[STIX]{x1D6FC}$
we establish a one-to-one correspondence between the minimal degree members of the family indexed by
$\unicode[STIX]{x1D6FC}$
and the linear characters of a certain subquotient
$\overline{T}_{\unicode[STIX]{x1D6FC}}$
of
$UY_{n}(q)$
. For
$Y_{n}=A_{n}$
our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of
$\text{Irr}(UE_{i}(q))$
,
$6\leqslant i\leqslant 8$
, and
$\text{Irr}(UF_{4}(q))$
.
In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact,
$2^{10}$
to
$2^{12}$
points provide satisfactory convergence. Even for
$g=2$
, the classical approach using moment statistics requires about
$2^{30}$
sample points to obtain such information.
In 1981, Thompson proved that, if
$n\geqslant 1$
is any integer and
$G$
is any finite subgroup of
$\text{GL}_{n}(\mathbb{C})$
, then
$G$
has a semi-invariant of degree at most
$4n^{2}$
. He conjectured that, in fact, there is a universal constant
$C$
such that for any
$n\in \mathbb{N}$
and any finite subgroup
$G<\text{GL}_{n}(\mathbb{C})$
,
$G$
has a semi-invariant of degree at most
$Cn$
. This conjecture would imply that the
${\it\alpha}$
-invariant
${\it\alpha}_{G}(\mathbb{P}^{n-1})$
, as introduced by Tian in 1987, is at most
$C$
. We prove Thompson’s conjecture in this paper.
Let
$G$
be a finite group and
$\mathsf{cd}(G)$
denote the set of complex irreducible character degrees of
$G$
. We prove that if
$G$
is a finite group and
$H$
is an almost simple group whose socle is a sporadic simple group
$H_{0}$
and such that
$\mathsf{cd}(G)=\mathsf{cd}(H)$
, then
$G^{\prime }\cong H_{0}$
and there exists an abelian subgroup
$A$
of
$G$
such that
$G/A$
is isomorphic to
$H$
. In view of Huppert’s conjecture, we also provide some examples to show that
$G$
is not necessarily a direct product of
$A$
and
$H$
, so that we cannot extend the conjecture to almost simple groups.