Group PGL Research Articles

Analytic Langlands correspondence for PGL_2 on {mathbb {P}}^1 with parabolic structures over local fields

We continue to develop the analytic Langlands program for curves over local fields initiated in our earlier papers, following a suggestion of Langlands and a work of Teschner. Namely, we study the Hecke operators which we introduced in those papers in the case of a projective line with parabolic structures at finitely many points for the group PGL_2. We establish most of our conjectures in this case.

On codes in the projective linear group PGL(2,q)

In this paper, we resolve a conjecture of Green and Liebeck (2019) [3] on codes in PGL(2,q). To be specific, we show that: if D is a dihedral subgroup of order 2(q+1) in G=PGL(2,q), and A={g∈G:gq+1=1,g2≠1}, then λG=A⋅D, where λ=q or q−1 according as q is even or odd.

Hamilton cycles in primitive vertex-transitive graphs of order a product of two primes – the case PSL(2, q^2) acting on cosets of PGL(2, q)

A step forward is made in a long standing Lovasz problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes arising from the group action of the projective special linear group PSL (2, q 2 ) on cosets of its subgroup isomorphic to the projective general linear group PGL (2, q ) contains a Hamilton cycle.

96120 : The degree of the linear orbit of a cubic surface

The projective linear group PGL(C,4) acts on cubic surfaces, considered as points of PC19. We compute the degree of the 15-dimensional projective variety defined by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry.

Self-dual, self-Petrie-dual and Möbius regular maps on linear fractional groups

Regular maps on linear fractional groups PSL(2, q) and PGL(2, q) have been studied for many years and the theory is well-developed, including generating sets for the associated groups. This paper studies the properties of self-duality, self-Petrie-duality and Mobius regularity in this context, providing necessary and sufficient conditions for each case. We also address the special case for regular maps of type (5, 5). The final section includes an enumeration of the PSL(2, q) maps for q ≤ 81 and a list of all the PSL(2, q) maps which have any of these special properties for q ≤ 49.

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

Let $\pi$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $\pi$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant sequence of measures is tested against an Eisenstein series.

Symmetry of Structures That Can Be Approximated by Chains of Regular Tetrahedra

The noncrystallographic symmetries of chains of regular tetrahedra are determined by mapping the system of algebraic geometry and topology designs to the structural level. It has been shown that the basic structural unit of such a chain is a tetrablock: a seven-vertex linear aggregation over faces of four regular tetrahedra, which is implemented in linear (right- and left-handed) and planar versions. The symmetry groups of linear and planar tetrablocks are isomorphic, respectively, to the projective special linear group PSL(2, 7) of order 168 and the projective general linear group PGL(2, 7) of order 336. A class of structures formed by an assembly of tetrablocks having no common tetrahedra is introduced. Examples of tetrablock assembly over common face, leading to a Boerdijk–Coxeter helix, an α helix, and a helix used as one of collagen models are presented.

New lower bounds for permutation arrays using contraction

A permutation array A is a set of permutations on a finite set $$\Omega $$ , say of size n. Given distinct permutations $$\pi , \sigma \in \Omega $$ , we let $$hd(\pi , \sigma ) = |\{ x\in \Omega : \pi (x) \ne \sigma (x) \}|$$ , called the Hamming distance between $$\pi $$ and $$\sigma $$ . Now let $$hd(A) =$$ min $$\{ hd(\pi , \sigma ): \pi , \sigma \in A \}$$ . For positive integers n and d with $$d\le n$$ we let M(n, d) be the maximum number of permutations in any array A satisfying $$hd(A) \ge d$$ . There is an extensive literature on the function M(n, d), motivated in part by suggested applications to error correcting codes for message transmission over power lines. A basic fact is that if a permutation group G is sharply k-transitive on a set of size $$n\ge k$$ , then $$M(n,n-k+1) = |G|$$ . Motivated by this we consider the permutation groups AGL(1, q) and PGL(2, q) acting sharply 2-transitively on GF(q) and sharply 3-transitively on $$GF(q)\cup \{\infty \}$$ respectively. Applying a contraction operation to these groups, we obtain the following new lower bounds for prime powers q satisfying $$q\equiv 1$$ (mod 3). These results resolve a case left open in a previous paper (Bereg et al. in Des Codes Cryptogr 86(5):1095–1111, 2018), where it was shown that $$M(q-1, q-3) \ge q^{2} - q$$ and $$M(q,q-3) \ge q^{3} - q$$ for all prime powers q such that $$q\not \equiv 1$$ (mod 3). We also obtain lower bounds for M(n, d) for a finite number of exceptional pairs n, d, by applying this contraction operation to the sharply 4 and 5-transitive Mathieu groups.

The covering radius of [formula omitted

In this note, with a purely geometric approach, the covering radius of the group PGL(3,q) is determined. Also, a new proof establishing the covering radii of PGL(2,q) and AGL(1,q) is provided.

On the Dickson–Guralnick–Zieve curve

The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field Fq arises naturally from the classical Dickson invariant of the projective linear group PGL(3,Fq). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q3−q2 and genus 12q(q−1)(q3−2q−2)+1. In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q−1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over Fq3, the DGZ curve is optimal with respect the number of its Fq3-rational points.

Differential and Integral Projective Invariants for the Groups of Diffeomorphisms

In this paper we study the differential and integral invariants for the action of the projective group PGL(n + 1) on the group of diffeomorphisms Diff(ℝP n ) by conjugations. Cases n = 1 and n = 2 are considered. For n = 1 the algebra of differential invariants is found and the criterion of the local equivalence of two diffeomorphisms is obtained. Also several integral invariants for n = 1 and n = 2 are calculated, the analogy with Calaby integral invariant for the symplectic groups is established.

Maximum scattered [formula omitted]-linear sets of [formula omitted

There are two known families of maximum scattered Fq-linear sets in PG(1,qt): the linear sets of pseudoregulus type and for t≥4 the scattered linear sets found by Lunardon and Polverino. For t=4 we show that these are the only maximum scattered Fq-linear sets and we describe the orbits of these linear sets under the groups PGL(2,q4) and PΓL(2,q4).

The isolated quantum group of the GL2 family

Quantum analogs of GL(2) are parameterized by PGL(2)-orbits in the commuting variety of the projective linear group PGL(2). The quantum group corresponding to the orbit constituting the second connected component of the commuting variety has previously received little attention. This paper investigates ring-theoretical properties of its coordinate ring.

An algorithm for the construction of substitution box for block ciphers based on projective general linear group

The aim of this work is to synthesize 8*8 substitution boxes (S-boxes) for block ciphers. The confusion creating potential of an S-box depends on its construction technique. In the first step, we have applied the algebraic action of the projective general linear group PGL(2,GF(28)) on Galois field GF(28). In step 2 we have used the permutations of the symmetric group S256 to construct new kind of S-boxes. To explain the proposed extension scheme, we have given an example and constructed one new S-box. The strength of the extended S-box is computed, and an insight is given to calculate the confusion-creating potency. To analyze the security of the S-box some popular algebraic and statistical attacks are performed as well. The proposed S-box has been analyzed by bit independent criterion, linear approximation probability test, non-linearity test, strict avalanche criterion, differential approximation probability test, and majority logic criterion. A comparison of the proposed S-box with existing S-boxes shows that the analyses of the extended S-box are comparatively better.

Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, 𝔽q) and PGL(2, 𝔽q)

We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, 𝔽q) and PGL(2, 𝔽q). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings.

On the unrecognizability by prime graph for the almost simple group PGL(2,9)

On the unrecognizability by prime graph for the almost simple group PGL(2,9)

Pentavalent symmetric graphs of order [formula omitted

A graph is symmetric (or arc-transitive) if its automorphism group is transitive on the arc set of the graph. Let r<q<p be three distinct primes. In this paper, we give a complete classification of connected pentavalent symmetric graphs of order 2pqr with r≥3. It is shown that a connected pentavalent symmetric graph Γ of order 2pqr is isomorphic to one of 20 sporadic coset graphs associated with some simple groups, a coset graph on the group PSL(2,p) or PGL(2,p) with p≥29 and Aut(Γ)≅PSL(2,p) or PGL(2,p) respectively, or a 1-regular graph given in Feng and Li (2011). The connected pentavalent symmetric graphs of order 4pq have been classified before.

C-groups of [formula omitted] and [formula omitted

We classify the C-group representations of the groups PSL(2,q) and PGL(2,q). We obtain that the C-rank of PSL(2,q) and PGL(2,q) is 3 except when q∈{7,9,11,19,31} for PSL(2,q) and when q=5 for PGL(2,q), in which case it is 4. We provide all representations of rank four.

Morphisms between Cremona groups, and characterization of rational varieties

AbstractWe classify all (abstract) homomorphisms from the group$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$to the group${\sf Bir}(M)$of birational transformations of a complex projective variety$M$, provided that$r\geq \dim _\mathbf{C}(M)$. As a byproduct, we show that: (i)${\sf Bir}(\mathbb{P}^n_\mathbf{C})$is isomorphic, as an abstract group, to${\sf Bir}(\mathbb{P}^m_\mathbf{C})$if and only if$n=m$; and (ii)$M$is rational if and only if${\sf PGL}_{\dim (M)+1}(\mathbf{C})$embeds as a subgroup of${\sf Bir}(M)$.

$N$-step energy of maps and the fixed-point property of random groups

We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT(0) spaces. We use Gromov’s criterion for the fixed-point property in terms of the growth of n -step energy of equivariant maps from a finitely generated group into a CAT(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL( m,\mathbb{Q}_r ), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with m bounded from above.