arXiv subjects
Publications and source records attributed to Alexandre Zalesski.
This is an expository paper aimed to outline the current situation with problems related with the occurrence of eigenvalue $1$ of elements in linear groups and group representations.
We determine the irreducible representations of alternating and symmetric groups and their universal central extensions that contain a non-scalar element with all but one eigenvalues of multiplicity 1. The ground field is algebraically closed of arbitrary characteristic.
The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type.
We determine the irreducible representations of simple Lie algebras with maximum weight multiplicity 2.
Extending earlier results of the authors on minimal polynomials of $p$-elements of finite groups of Lie type in cross-characteristic representations, this paper focuses on the case where Sylow $p$-subgroups are cyclic and $p$ is distinct from the representation field characteristic.
A linear group is called unisingular if every element of it has eigenvalue 1. A certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements. A more special, but an important question is on the existence of such subgroups in the symplectic groups of particular degree. We answer this question for almost all degrees $2n<250$, specifically, the question remains open only 7 values of $n$. Additionally, the paper contains results of general nature on the structure of unisingular irreducible linear groups.
We study the irreducible representations of simple algebraic groups in which some non-central semisimple element has at most one eigenvalue of multiplicity greater than 1. We bound the multiplicity of this eigenvalue in terms of the rank of the group.
A finite order element $g$ of a group $G$ is called rational if $g$ is conjugate to $g^i$ for every integer $i$ coprime to the order $g$. We determine all triples $(G,g,ϕ)$, where $G$ is a simple algebraic group of type $A_n,B_n$ or $C_n$ over an algebraically closed field of characteristic $p\geq 0$, $g\in G$ is a rational odd order semisimple element and $ϕ$ is an irreducible representation of $G$ such that $ϕ(g)$ has eigenvalue 1.
Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $ϕ$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $α(s)\neβ(s)$ for all distinct $T$-roots $α$ and $β$ of $G$. Our main result states that if all but one of the eigenvalues of $ϕ(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regular and all non-zero weights of $ϕ$ are of multiplicity 1.
The irreducible representations $ϕ_n^1$ and $ϕ_n^2$ of the symplectic group $G_n=Sp_{2n}(P)$ over an algebraically closednfield $P$ of characteristic $p>2$ with highest weights $ω_{n-1}+\frac{p-3}{2}ω_n$ and $\frac{p-1}{2}ω_n$, respectively, are investigated. It is proved that the dimension of $ϕ_n^i$ ($i=1,2$) is equal to $(p^n+(-1)^i )/2$, all weight multiplicities of these representations are equal to $1$, their restrictions to the group $G_k$ naturally embedded into $G_n$ are completely reducible with irreducible constituents $ϕ_k^1$ and $ϕ_k^2$, and their restrictions to $Sp_{2n}(p)$ can be obtained as the result of the reduction modulo $p$ of certain complex irreducible representations of the group $Sp_{2n}(p)$.
Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $ϕ$ of $H$ such that $ϕ(G)$ is irreducible and $ϕ(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $ϕ( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.
We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible representation of G in some GL(V) for which there exists a non-regular non-central semisimple element s in G such that f(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at most one eigenvalue is of multiplicity greater than 1. This yields a kind of characterization of the natural representation (up to their Frobenius twists) of classical algebraic groups in terms of the behavior of semisimple elements.
Let G be a subgroup of GL(V), where V is a finite dimensional vector space over a finite field of characteristic p >0. If det(g-1) = 0 for all g \in G then we call G a fixed-point subgroup of GL(V). Motivated in parallel by questions in arithmetic and linear group theory, we classify all irreducible fixed-point subgroups of Sp_8(2) and give new infinite series of irreducible fixed-point subgroups of symplectic groups Sp_m(2) for various m arising from certain representations of groups of Lie type.
Let $G$ be a simple algebraic group in defining characteristic $p>0$, and let $V$ be an irreducible $G$-module which is the tensor product of exactly two non-trivial modules. We obtain a criterion for $V$ to have the zero weight. In addition, we provide a uniform criterion for an irreducible representation of a simple Lie algebra over the complex numbers to have a multiple of a prescribed fundamental weight.
Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,.., n\},$ respectively. In this paper we are interested in the second largest eigenvalue $λ_{2}(Γ)$ of the Cayley graph $Γ=Cay(G,H)$ over $G=S_{n}$ or $A_{n}$ for certain connecting sets $H.$ Let $1<k\leq n$ and denote the set of all $k$-cycles in $S_{n}$ by $C(n,k).$ For $H=C(n,n)$ we prove that $λ_{2}(Γ)=(n-2)!$ (when $n$ is even) and $λ_{2}(Γ)=2(n-3)!$ (when $n$ is odd). Further, for $H=C(n,n-1)$ we have $λ_{2}( Γ)=3(n-3)(n-5)!$ (when $n$ is even) and $λ_{2}(Γ)=2(n-2)(n-5) !$ (when $n$ is odd). The case $H=C(n,3)$ has been considered in X. Huang and Q. Huang, The second largest eigenvalue of some Cayley graphs on alternating groups, J. Algebraic Combinatorics} 50(2019), $99-111$. Let $1\leq r<k<n$ and let $C(n,k;r) \subseteq C(n,k)$ be set of all $k$-cycles in $S_{n}$ which move all the points in the set $\{1,2,..., r\}.$ That is to say, $g=(i_{1},i_{2}... i_{k})(i_{k+1})\dots(i_{n})\in C(n,k;r)$ if and only if $\{1,2,..., r\}\subset \{i_{1},i_{2},..., i_{k}\}.$ Our main result concerns $λ_{2}( Γ)$, where $Γ=Cay(G,H)$ with $H=C(n,k;r)$ with $1\leq r<k<n$ when $G=S_{n}$ if $k$ is even and $G=A_{n}$ if $k$ is odd. Here we observe that $$λ_{2}( Γ)\geq (k-2)! {n-r \choose k-r} \frac{1}{n-r} \big((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big).$$ We show that this bound is sharp in the special case $k=r+1$ , giving $λ_{2}(Γ)=r!(n-r-1)$. The cases with $H=C(n,3;1)$ and $H=C(n,3;2)$ were considered earlier in the same paper of X. Huang and Q. Huang.
We determine the irreducible 2-modular representations of the symplectic group $G=Sp_{2n}(2)$ whose restriction to every abelian subgroup has a trivial constituent. A similar result is obtained for maximal tori of $G$. There is significant information on the existence of eigenvalue 1 of elements of $G$ in a given irreducible representation of $G$.
The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group $G$ of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group $G$ in terms of its Lie rank and defining characteristic. When $G$ is specified as $G(q)$ and $q$ is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of $G$.
Let $Γ$ be a finite graph and let $A(Γ)$ be its adjacency matrix. Then $Γ$ is {\it singular} if $A(Γ)$ is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we investigate this problem for Cayley graphs ${\rm Cay}(G,H)$ when $G$ is a finite group and when the connecting set $H$ is a union of conjugacy classes of $G.$ In this situation the singularity problem reduces to finding an irreducible character $χ$ of $G$ for which $\sum_{h\in H}\,χ(h)=0.$ At this stage we focus on the case when $H$ is a single conjugacy class $h^G$ of $G.$ Here the above equality is equivalent to $χ(h)=0$. Much is known in this situation, with essential information coming from the block theory of representations of finite groups. An element $h\in G$ is called vanishing if $χ(h)=0$ for some irreducible character $χ$ of $G.$ We study vanishing elements mainly in finite simple groups and in alternating groups in particular. We suggest some approaches for constructing singular Cayley graphs.