Hecke Groups Research Articles

Certain Types of the Orbits of Real Quadratic Fields by Hecke Groups

Certain Types of the Orbits of Real Quadratic Fields by Hecke Groups

Some Normal Subgroups of Extended Generalized Hecke Groups

Generalized Hecke group H p,∞ (λ) is generated by X(z) = −(z − λ p ) −1 and Y (z) = −(z + λ) −1 where λ p = 2 cos πp , p ≥ 2 integer and λ ≥ 2 . Extended generalized Hecke group H p,∞ (λ) is obtained by adding the reection R(z) = 1/z to the generators of generalized Hecke group H p,∞ (λ). In this paper, we study the commutator subgroups of ex- tended generalized Hecke groups H p,∞ (λ). Also, we determine the power subgroups of generalized Hecke groups H p,∞ (λ) and extended generalized Hecke groups H p,∞ (λ) .

Hecke Groups, Dessins d'Enfants, and the Archimedean Solids

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.

On the parametrized modular group

Generalizing the modular and Hecke groups, we consider the subgroup Π of SL(2, ℤ[ξ]) generated by the parabolic element \(\left( {_{01}^{1\xi } } \right)\) and the elliptic element \(\left( {_{10}^{0 - 1} } \right)\), where ℤ[ξ] is the ring of polynomials in the variable ξ. For ζ ∈ ℂ and W ∈ Π, we denote by W (ζ) the matrix in SL(2, ℂ) obtained when evaluating the parameter ξ at ζ. We enumerate the elements of Π and study the relators, defined as those W ∈ ℂ for which there exists ζ ∈ ℂ with W(ζ) = ±I. Then, for W ∈ Π, we investigate the sets of ζ for which W(ζ) is not loxodromic; their union is the singular set S(Π) ⊂ ℂ. The closure of the singular set for the two-parabolic group, which is isomorphic to a free subgroup Π of index 4, has been studied extensively.

Hecke-symmetry and rational period functions

In this paper, we continue our work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper.

Power subgroups of the extended Hecke groups

Power subgroups of the extended Hecke groups

The commutator subgroup and the index formula of the Hecke group H 5

Abstract Let A be an ideal of ℤ[2cos(π/5)] and let H(A) be the principal congruence subgroup of level A of the Hecke group H 5. The present article gives explicit formulas for [H 5 : H(A)]. As a byproduct, we prove that the commutator subgroup of H 5 is not congruence. Consequently, the fact that the commutator subgroup of the modular group is congruence cannot be carried over to the Hecke groups.

A Hecke correspondence theorem for automorphic integrals with infinite log-polynomial sum period functions

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

On Subsets of Q(√<span style=" text-decoration:overline;">m</span>) Q under the Action of Hecke Groups H(λ<sub>q</sub>)

is the disjoint union of for all , where is the set of all roots of primitive second degree equations , with reduced discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the group of linear-fractional transformations on . In particular, we investigate the action of on for finding different orbits.

Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups

Abstract.In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in anyHecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic ℤ[λ]-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic ℤ[λ]-binary quadratic forms.

Some properties of the generalized Fibonacci and Lucas sequences related to the extended Hecke groups

In this paper, we define a sequence, which is a generalized version of the Lucas sequence, similar to the generalized Fibonacci sequence given in Koruoglu and Sahin in Turk. J. Math. 2009, doi:10.3906/mat-0902-33. Also, we give some connections between the generalized Fibonacci sequence and the generalized Lucas sequence, and we find polynomial representations of the generalized Fibonacci and the generalized Lucas sequences, related to the extended Hecke groups given in Koruoglu and Sahin in Turk. J. Math. 2009, doi:10.3906/mat-0902-33.

Gröbner-Shirshov bases for extended modular, extended Hecke, and Picard groups

In this paper, Grobner-Shirshov bases (noncommutative) for extended modular, extended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of elements and hence solving the word problem in these groups is proposed.

A Method for Determining the Mod-$2^k$ Behaviour of Recursive Sequences, with Applications to Subgroup Counting

We present a method to obtain congruences modulo powers of $2$ for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß--Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of $2$.

The minimal polynomial of 2 cos(π/q) and Dickson polynomials

The number λq=2cos(π/q),q∈N,q⩾3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Möbius inversion for this minimal polynomial by means of the Dickson polynomials and the Möbius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments.

Hecke graphs, Ramanujan graphs and generalized duality transformations for lattice spin systems

I discuss two related subjects: (1) Hecke surfaces and k-regular graphs, (2) duality transformations for lattice spin models. Each of them is related to deep mathematical and physical theories, and at first glance, they have nothing in common. However, it became evident in recent years that there exist deep internal relations between these two problems. Especially interesting (and mysterious) is the role of Hecke groups in this context. I consider the following relevant example: Hecke graphs and Ramanujan graphs.

Bicuspid F -structures and Hecke groups

Bicuspid <i>F</i> -structures and Hecke groups

Commutator subgroups of the power subgroups of some Hecke groups

Let q≥3 be a prime and let H(λq) be the Hecke group associated to q. Let m be a positive integer and Hm(λq) be the mth power subgroup of H(λq). In this work, we study the commutator subgroups of the power subgroups Hm(λq) of H(λq). Then, we give the derived series for all triangle groups of the form (0;2,q,n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.

A remarkable family of left-ordered groups: Central extensions of Hecke groups

We provide an infinite family of left-ordered groups that have a positive cone that is generated by two elements as a semigroup. This family corresponds to that of certain central extensions of Hecke groups, and includes the Klein bottle group and the braid group B 3 . Using the classical convex extension (flipping) procedure, on each of the groups of this family we define an ordering sharing many properties with Dehornoy's. Several related questions and problems are addressed.

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function.

Hecke surfaces and duality transformations in lattice spin systems

We discuss two related subjects: (1) Hecke surfaces and K-regular graphs and (2) duality transformations for generalized Potts models. Each is related to deep mathematical and physical theories; at a first glance, they have nothing in common. But it has recently become more evident that there are deep internal relations between these two problems. The role of Hecke groups is especially interesting and mysterious in this context. We consider a few examples. The exposition is basically descriptive.