Some directed strongly regular graphs constructed from linear groups

Regular graphs and discrete subgroups of projective linear groups

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G) . The minimal degree of a faithful representation of G by quasi-permutation matrices over the rationals and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G . In this paper p(G), q(G), c(G) and r(G) are calculated for the group G 2 ( q n ), q ≠ 3. AMS Classification: 20C15 Keywords: General linear group, Quasi-permutation. Introduction Let G be a finite linear group of degree n , that is, a finite group of automorphisms of an n -dimensional complex vector space. We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n , its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V , induce automorphisms of V forming a group isomorphic to G . The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x and so is a non-negative integer.

In this paper we classify distance-regular graphs‎, ‎including strongly regular graphs‎, ‎admitting a transitive action of the linear groups $L(3,2)$‎, ‎$L(3,3)$‎, ‎$L(3,4)$ and $L(3,5)$ for which the rank of the permutation representation is at most 15‎. ‎We give details about constructed graphs‎. ‎In addition‎, ‎we construct self-orthogonal codes from distance-regular graphs obtained in this paper‎.

The ranks, degrees, subdegrees, and double stabilizers of permutation representations of finite special linear and unitary groups on cosets of parabolic maximal subgroups are found.

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.

We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

Let g be a connected reductive linear algebraic group, and let G = gσ be the finite subgroup of fixed points, where σ is the generalized Frobenius endomorphism of g. Let x be a regular semisimple element of G and let w be a corresponding element of the Weyl group W. In this paper we give a formula for the number of right cosets of a parabolic subgroup of G left fixed by x, in terms of the corresponding action of w in W. In case G is untwisted, it turns out thta x fixes exactly as many cosets as does W in the corresponding permutation representation.

An Aschbacher–OʼNan–Scott theorem for countable linear groups

A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left, see Definition 2) such that ⁠. This is well known in the finite case. We prove this conjecture when Γ<GLn(F) is a linear group, where F is any field with char(F)≠2 and that p−char(Γ)≠2 (see Definition 2.2).

Method for separation of acid pollution gas

It is known that every polycyclic-by-finite group – even if it admits no affine structure – allows a polynomial structure of bounded degree. A major obstacle to a further development of the theory of these polynomial structures is that the group of the polynomial diffeomorphisms of \(\mathbb{R}^n\), in contrast to the group of affine motions, is no longer a finite dimensional Lie group. In this paper we construct a family of (finite dimensional) Lie groups, even linear algebraic groups, of polynomial diffeomorphisms, which we call weighted groups of polynomial diffeomorphisms. It turns out that every polycyclic-by-finite group admits a polynomial structure via these weighted groups; in the nilpotent (and other) case(s), we can sharpen, by specifying a nice set of weights, the existence results obtained in earlier work. We introduce unipotent polynomial structures of nilpotent groups and show how the existence of such polynomial structures is closely related to the existence of simply transitive actions of the corresponding Mal`cev completion. This, and other properties, provide a strong analogy with the situation of affine structures and simply transitive affine actions considered e.g. in the work of Fried, Goldman and Hirsch.

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

The importance of interactions between groups, linear codes and t-designs has been well recognized for decades. Linear codes that are invariant under groups acting on the set of code coordinates have found important applications for the construction of combinatorial t-designs. Examples of such codes are the Golay codes, the quadratic-residue codes, and the affine-invariant codes. Let \(q=5^m\). The projective general linear group PGL(2, q) acts as a 3-transitive permutation group on the set of points of the projective line. This paper is to present two infinite families of cyclic codes over GF\((5^m)\) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL(2, q), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters \([q+1,4,q-5]_q\), where \(q=5^m\), and \( m\ge 2\). A code from the second family has parameters \([q+1,q-3,4]_q\), \(q=5^m,~m\ge 2\). This paper also points out that the set of the support of all codewords of these two kinds of codes with any nonzero weight is invariant under \(\textrm{Stab}_{U_{q+1}}\), thus the corresponding incidence structure supports 3-design.KeywordsLinear codeCyclic codet-designProjective general linear groupAutomorphism group

We investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group G admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then G must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local eld of characteristic zero having cut-points in one asymptotic cone. We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of nitely generated lacunary hyperbolic groups to the setting of locally compact groups. We moreover answer a question of Olshanskii, Osin and Sapir about subgroups of lacunary hyperbolic groups.

Motivated by the physical concept of special geometry, two mathematical constructions are studied which relate real hypersurfaces to tube domains and complex Lagrangian cones, respectively. Methods are developed for the calssification of homogeneous Riemannian hypersurfaces and the classification of linear transitive reductive algebraic group actions on pseudo-Riemannian hypersurfaces. The theory is applied to the case of cubic hypersurfaces, which is the one most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special Kähler manifolds.