Search

A graphical regular representation (GRR) of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper we study cubic GRRs of PSL n (q) (n = 4 , 6 , 8), PSp n (q) (n = 6 , 8), P Ω n + (q) (n = 8 , 10 , 12) and P Ω n − (q) (n = 8 , 10 , 12), where q = 2 f with f ≥ 1. We prove that for each of these groups, with probability tending to 1 as q → ∞ , any element x of odd prime order dividing 2 e f − 1 but not 2 i − 1 for each 1 ≤ i < e f together with a random involution y gives rise to a cubic GRR, where e = n − 2 for P Ω n + (q) and e = n for other groups. Moreover, for sufficiently large q , there are elements x satisfying these conditions, and for each of them there exists an involution y such that { x , x − 1 , y } produces a cubic GRR. This result together with certain known results in the literature implies that except for PSL 2 (q) , PSL 3 (q) , PSU 3 (q) and a finite number of other cases, every finite non-abelian simple group contains an element x and an involution y such that { x , x − 1 , y } produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR. [ABSTRACT FROM AUTHOR]

In this paper, we prove certain multiplicity one theorems and define GL-twisted gamma factors for irreducible generic cuspidal representations of quasi-split classical groups G r = Sp 2 r , U 2 r , U 2 r + 1 , SO 2 r + 1 over finite fields of odd characteristic, using Rankin-Selberg method. As applications, we prove converse theorems for these groups, namely, GL n -twisted gamma factors, n = 1 , 2 , ... , r , will uniquely determine irreducible generic cuspidal representations of G r (F q). [ABSTRACT FROM AUTHOR]

Let R be a unital subring of a commutative ring S, which is a free R-module of rank m. In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups GL(n, S) and E(n, S) in G = GL(mn,R), and showed that they are equal and coincide with the set {g ∈ G:E(n, S)g ≤ GL(n, S)} = Aut(S=R) x GL(n, S). Moreover, for any proper ideal A of R, NG (E(n, S) E(mn,R,A)), = ... (NGL(mn,R/A)) (E(n,S/SA))). In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups EO(n, S), SO(n, S),O(n, S) and GO(n, S) in G are equal and coincide with the set {g ∈ G: EO(n, S)g ≤ GO(n, S)} = Aut(S/R) x GO(n, S). Similarly, the ones of subgroups Ep(n, S), Sp(n, S), and GSp(n, S) are equal and coincide with the set {g ∈ G:Ep(n, S)g ≤ GSp(n, S)} = Aut(S/R) x GSp(n, S). Moreover, for any proper ideal A of R, NG(EO(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (EO(n,S/SA))) and NG(Ep(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (Ep(n,S/SA))). When R = S, we obtain the known results of N. A. Vavilov and V. A. Petrov. [ABSTRACT FROM AUTHOR]

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then N X (G) is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic. [ABSTRACT FROM AUTHOR]

Abstract We show that the minimal base size b (G) of a finite primitive permutation group G of degree n is at most 2 (log ⁡ | G | / log ⁡ n) + 24. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups G of degrees n such that b (G) = ⌊ 2 (log ⁡ | G | / log ⁡ n) ⌉ − 2 and b (G) is unbounded. As a corollary we show that a primitive permutation group of degree n that does not contain the alternating group Alt (n) has a base of size at most max ⁡ { n , 25 }. [ABSTRACT FROM AUTHOR]

We show that the proportion of permutations g in Sn or An such that g has even order and g|g|/2 is an involution with support of cardinality at most ⌈nε⌉ is at least a constant multiple of ε. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension n in odd characteristic that have even order and power up to an involution with (-1)-eigenspace of dimension at most ⌈nε⌉ for a linear or unitary group, or 2⌈⌊n/2⌋ε⌉ for a symplectic or orthogonal group. [ABSTRACT FROM AUTHOR]

We show that inside the real quaternionic symplectic group $${\mathrm {Sp}}(p,q)$$ , every element is conjugate to its inverse. Similar result is proved for the real quaternionic orthogonal group $$\mathrm{O }^{*}(2n)$$ . We also show that no similar result is possible in the p-adic case. [ABSTRACT FROM AUTHOR]

Abstract: We prove uniqueness of Fourier–Jacobi models for general linear groups, unitary groups, symplectic groups and metaplectic groups, over an Archimedean local field. [Copyright &y& Elsevier]

Let G be a finite classical group of characteristic p. In this paper, we give an arithmetic criterion of the primes r ≠ p, for which the Steinberg character lies in the principal r-block of G. The arithmetic criterion is obtained from some combinatorial objects (the so-called partition and symbol). [ABSTRACT FROM AUTHOR]

A theorem is proved on the structure of the group of isometries of a Hermitian map b : V x V → W, where V and W are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given a Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection over a set of classical subgroups of GL(V ), giving rise to the first polynomial-time solution of this old problem. The approach yields new algorithmic tools for algebras with involution, which in turn have applications to other computational problems of interest. Implementations of the various algorithms in the Magma system demonstrate their practicability. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that does not vanish, where is the Dirichlet polynomial associated to the group G. [Copyright &y& Elsevier]

We give a classification of maximal subgroups of odd index in finite groups whose socle is isomorphic to one of the groups PSL( q) , PSU( q), or PSp( q) for n ≥ 13. [ABSTRACT FROM AUTHOR]

Abstract: Let be a quasisimple classical group in its natural representation over a finite vector space , and let . We construct the projection from to and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of . A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find the number of classes of conjugate π-Hall subgroups in all finite almost simple groups. We also complete the classification of π-Hall subgroups in finite simple groups and correct some mistakes from our previous paper. [Copyright &y& Elsevier]

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite simple orthogonal groups in odd dimensions are defined on cosets with respect to maximal parabolic subgroups. [ABSTRACT FROM AUTHOR]

The construction of BN-pairs of groups is an important topic in Building theory. Since every BN-pair of a group corresponds to a group of Weyl type, the properties of the group are obtained by discussing the group of Weyl type. Therefore, BN-pair is a useful tool in studying groups. Let R be a local ring. Using matrix method, BN-pairs of the general linear group, orthogonal group and symplectic group over local ring R are constructed, respectively. Furthermore, BN-pairs of a family of general linear groups over local ring R are also discussed, and these general linear groups and their corresponding BN-pairs satisfy a communicative diagram of groups. [ABSTRACT FROM AUTHOR]

Let V be a three-dimensional vector space over a finite field. We show that any irreducible subgroup of GL( V) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V. In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups. [ABSTRACT FROM AUTHOR]

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite symplectic groups are defined on cosets with respect to maximal parabolic subgroups. [ABSTRACT FROM AUTHOR]

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkin's sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7, 9]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras. [ABSTRACT FROM AUTHOR]

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. [ABSTRACT FROM AUTHOR]

Abstract: This paper starts the classification of the primitive permutation groups such that G contains a regular subgroup X. We determine all the triples with an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples with . In order to obtain all these triples, we also study the almost simple groups G with . The case is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657–673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups. [Copyright &y& Elsevier]

Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form. [ABSTRACT FROM AUTHOR]

Abstract: We provide upper and lower bounds for the number of completely reducible homomorphisms from a finite group Γ to general linear and unitary groups over arbitrary finite fields, and to orthogonal and symplectic groups over finite fields of odd characteristic. [Copyright &y& Elsevier]

We give formulae relating the value Xλ ( g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL 2 case [1]. [ABSTRACT FROM AUTHOR]

Abstract: Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where q is transcendental. This is a ring theoretic formulation of the well-known fact that the representation theory of is completely analogous to its classical counterpart. The subalgebras of cocommutative elements in the corresponding FRT-bialgebras (defined by Faddeev, Reshetikhin, and Takhtadzhyan) are explicitly determined, using a bialgebra embedding of the FRT-bialgebra into the tensor product of the quantized coordinate ring and the one-variable polynomial ring. A parallel analysis of the subalgebras of adjoint coinvariants is carried out as well, yielding similar results with similar proofs. The basic adjoint coinvariants are interpreted as quantum traces of representations of the corresponding quantized universal enveloping algebra. [Copyright &y& Elsevier]

For 2&les;m&les;l/2, let G be a simply connected Lie group with g0=so(2m,2l−2m) as Lie algebra, let g=k⊕p be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra k∩g0, and let U(g) be the universal enveloping algebra of g. This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g.The roots with respect to the usual compact Cartan subalgebra are all ±ei±ej with 1&les;i. In the usual positive system of roots, the simple root em−em+1 is noncompact and the other simple roots are compact. Let q=l⊕u be the parabolic subalgebra of g for which em−em+1 contributes to u and the other simple roots contribute to l, let L be the analytic subgroup of G with Lie algebra l∩g0, let LC=Intg(l), let 2δ(u) be the sum of the roots contributing to u, and let q¯=l⊕u¯ be the parabolic subalgebra opposite to q.The members of u∩p are nilpotent members of g. The group LC acts on u∩p with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on u∩p that carries a fully reducible representation of LC.For s∈Z, let λs=∑lower limit k=1, upper limit m (−l+s/2)ek. Then λs defines a one-dimensional (l,L) module Cλs. Extend this to a (q¯,L) module by having u¯ act by 0, and define N(λs+2δ(u))=U(g)⊗q¯Cλs+2δ(u). Let N′(λs+2δ(u)) be the unique irreducible quotient of N(λs+2δ(u)). The representations under study are πs=ΠS(N(λs+2δ(u))) and πs′=ΠS(N′(λs+2δ(u))), where S=dim(u∩k) and ΠS is the Sth derived Bernstein functor.For s>2l−2, it is known that πs=πs′ and that πs′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m&les;s&les;2l−2 that πs=πs′ and that πs′ is still unitary. The present paper shows that πs′ is unitary for 0&les;s&les;m−1 even though πs≠πs′, and it relates the K spectrum of the representations πs′ to the representation of LC on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in πs′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on πs′ does not make πs′ unitary for s<0 and that the K spectrum of πs′ in these cases is not related in the above way to the representation of LC on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra g0=so(2m,2l−2m+1), 2&les;m&les;l/2. [Copyright &y& Elsevier]

D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τλ of U(n) has nonnegative highest weight λ of depth &les;n/2. It says that the multiplicity in τλ&z.sfnc;SO(n) of an irreducible representation of SO(n) of highest weight ν is the sum over μ of the multiplicities of τλ in the U(n) tensor product τμ⊗τν, the allowable μ''s being all even nonnegative highest weights for U(n). Littlewood''s proof is character-theoretic. The present paper gives a geometric interpretation of this theorem involving the tensor products τμ⊗τν explicitly. The geometric interpretation has an application to the construction of small infinite-dimensional unitary representations of indefinite orthogonal groups and, for each of these representations, to the determination of its restriction to a maximal compact subgroup. The other Littlewood branching theorem is for restriction from U(2r) to the rank-r quaternion unitary group Sp(r). It concerns nonnegative highest weights for U(2r) of depth &les;r, and its statement is of the same general kind. The present paper finds an analogous geometric interpretation for this theorem also. [Copyright &y& Elsevier]

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group. [Copyright &y& Elsevier]

Let G be the classical group, and let Hom (Z m , G) denote the space of commuting m -tuples in G. First, we refine the formula for the Poincaré series of Hom (Z m , G) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom (Z m , G) on the parity of m and the rational hyperbolicity of Hom (Z m , G) for m ≥ 2. Next, we give a minimal generating set of the cohomology of Hom (Z m , G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom (Z m , G) with the best possible stable range. Baird proved that the cohomology of Hom (Z m , G) is identified with a certain ring of invariants of the Weyl group of G , and our approach is a direct calculation of this ring of invariants. [ABSTRACT FROM AUTHOR]

We continue the description of embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections which was begun in Part I [sb1]. We deal with the embeddings in orthogonal groups. [ABSTRACT FROM AUTHOR]

We are concerned with finite-dimensional classical groups over arbitrary commutative fields. In an orthogonal group a Siegel transvection, that is, an element centralizing l⊥ for some totally singular 2-dimensional subspace l, plays the same role as a transvection in the linear, symplectic or unitary groups. The Main Theorem of this paper describes the possible embeddings of classical groups in classical groups such that (Siegel) transvections act as (Siegel) transvections. [ABSTRACT FROM AUTHOR]