We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters. We also classify finite primitive permutation groups in which every real element has a fixed point.
Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup $P_n$. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra $\mathcal{H}(\mathfrak{S}_n, P_n, 1_{P_n})$.
A permutation group $G\le\operatorname{Sym}(Ω)$ is said to be $2$-closed if no group $H$ such that $G<H\le\operatorname{Sym}(Ω)$ has the same orbits on $Ω\timesΩ$ as $G$. A simple and efficient inductive criterion for the $2$-closedness is established for abelian permutation groups with cyclic transitive constituents.
We extend the notion of a {$p$-permutation equivalence} between two $p$-blocks $A$ and $B$ of finite groups $G$ and $H$, from the definition in [Boltje-Xu 2008] to a virtual $p$-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of $A$ and $B$ are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that $p$-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of $p$-permutation equivalences between $A$ and $B$ is fini…
School of Mathematics and Statistics, Guangxi Normal University（广西师范大学数学与统计学院）；Department of Mathematics, Shanghai University（上海大学数学系）；Newtouch Center for Mathematics of Shanghai University（上海大学新拓数学中心）
Let $G$ be a finite simple group of Lie type, and let $π_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of $π_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails. Let St be the Steinberg representation of $G$. We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $St\otimes St$, with the same exceptions as in the previous statement.
Semi-direct products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general idea is applied to the finite subgroups of the special unitary group of $(2\times 2)$-matrices. Amusing diagrams are developed that describe the unit quaternions, the binary tetrahedral, octahedral, and icosahedral group as well as the dicyclic groups. In all cases, the quotients …
We study the solvable groups $G$ that have an irreducible character $χ\in \Irr(G)$ such that $χ\barχ$ has at most two non-principal irreducible constituents.
We investigate the question which Q-valued characters and characters of Q-representations of finite groups are Z-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large.
In this paper we classify all maximal subgroups M of a nearly simple primitive rank 3 group G of type L=Omega_{2m+1}(3), m > 3; acting on an L-orbit E of non-singular points of the natural module for L such that 1_P^G <=1_M^G where P is a stabilizer of a point in E. This result has an application to the study of minimal genera of algebraic curves which admit group actions.
We investigate finite non-Abelian simple groups $G$ for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless $G$ is of Lie type in defining characteristic.
Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic…
We prove that a finite group is rational if and only if it has a set of permutation characters which separate conjugacy classes. It follows from this that a finite group is rational if and only if it has a representation as a permutation group in which any two elements fixing the same number of letters are conjugate.
We characterise acylindrical hyperbolicity of a group in terms of properties of an action of the group on a set (without any extra structure). In particular, this applies to the action of the group on itself by left multiplication, as well as the action on a (full measure subset of the) Furstenberg-Poisson boundary.
The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups $H$ of finite quasiprimitive permutation groups $G$. If the socle of $G$, denoted by $\mbox{soc}(G)$, is nonabelian, then $H$ lies in $\mbox{soc}(G)$. An explicit description is given for the possibilities of $H$ under the condition that $H$ does not contain a nontrivial normal subgroup of $\mbox{soc}(G)$.
Let $G$ be a finite primitive permutation group and let $κ(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $κ(G) \geqslant 1$. In this paper we classify the groups $G$ with $κ(G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $κ(G)=2$, including a complete classification for almost simple groups.
We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $μ$ a dominant weight and $W$ the Weyl group of ${\mathbf G}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $μ$ of ${\mathbf G}$ the sum $s_μ$ of distinct weights $w(μ)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit cha…
Let $G$ be a finite solvable group and $χ, ψ\in \Irr(G)$ be complex characters of $G$. Let $α$ be an irreducible constituent of the product $χψ$. We show that the derived length of $\Ker(α)/\Ker(χψ)$ is bounded by a linear function on the number of distinct irreducible constituents of $χψ$
Let G be a finite solvable group and $χ\in \Irr(G)$ be a faithful character. We show that the derived length of G is bounded by a linear function of the number of distinct irreducible constituents of $χ\barχ$. We also discuss other properties of the decomposition of $χ\barχ$ into its irreducible constituents.
We study the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ over an algebraically closed field and $D$ is a divisor on $X$ left stable by $G$ then we show the irreducible constituents of the natural representation of $G$ on the Riemann-Roch space $L(D)=L_X(D)$ are of dimension $\leq d$, where $d$ is the size of the smallest $G$-orbit acting on $X$. We give an example to show that this is, in general, sharp (i.e., that dimension $d$ irreducible constituents can occur). Connections…
Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note we state explicit formulae for the primitive idempotents of K\otimes pp_k(G), where pp_k(G) is the ring of p-permutation kG-modules.
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.
For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(Λ^{++})$ of strictly dominant weights $Λ^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(Λ^{++})$, there is a very peculiar subset $\wp(Λ^{++})$ for which we always have $$ dim\wp(Λ^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ Λ^+ $, the elements of $\wp(Λ^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(Λ^{++})$ and $\wp(ρ)$ where $ρ$ is the Weyl vector of $G_N$. The concept of…
