Hurwitz Problem Research Articles

Solution of the Hurwitz problem with a length-2 partition

Solution of the Hurwitz problem with a length-2 partition

Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.

The Hurwitz existence problem for surface branched covers

To a branched cover f:Σ ˜→Σ between closed surfaces one can associate a combinatorial datum given by the topological types of Σ ˜ and Σ, the degree d of f, the number n of branching points of f, and the n partitions of d given by the local degrees of f at the preimages of the branching points. This datum must satisfy the Riemann-Hurwitz condition plus some extra ones if either Σ or both Σ and Σ ˜ are non-orientable. A very old question posed by Hurwitz [14] in 1891 asks whether a combinatorial datum satisfying these necessary conditions is actually realizable (namely, associated to some existing f) or not (in which case it is called exceptional). Or, more generally, to count the number of realizations of the datum up to a natural equivalence relation. Many partial answers have been given to the Hurwitz problem over the time, but a complete solution is still missing. In this short course we will report on ancient and recent results and techniques employed to attack the question.

Translation surfaces and periods of meromorphic differentials

Let S $S$ be an oriented surface of genus g $g$ and n $n$ punctures. The periods of any meromorphic differential on S $S$ , with respect to a choice of complex structure, determine a representation χ : Γ g , n → C $\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where Γ g , n $\Gamma _{g,n}$ is the first homology group of S $S$ . We characterise the representations that thus arise, that is, lie in the image of the period map Per : Ω M g , n → Hom ( Γ g , n , C ) $\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$ . This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S $S$ with the prescribed holonomy χ $\chi$ . Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

Counting surface branched covers

Abstract To a branched cover f between orientable surfaces one can associate a certain branch datum, that encodes the combinatorics of the cover. This satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum there exists a branched cover f such that . One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.

Almost-regular Dessins d'Enfant on a torus and sphere

The Hurwitz problem asks which ramification data are realizable, i.e., appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. A similar description holds in the case of genus 0 ramification data.

On the existence of covers of $\mathbb P_1$ associated to certain permutations

In this short note we prove the impossibility of realizing finite topological covers of the Riemann sphere minus three points, associated to certain explicit combinatorial (permutation) data. This comes from a question of M. Zieve and falls in the framework of the so-called ‘‘Hurwitz problem’’, asking for a ‘‘simple’’ description of the combinatorial data which can be so realized.

Elementary solution of an infinite sequence of instances of the Hurwitz problem

We prove that there exists no branched cover from the torus to the sphere with degree 3h and 3 branching points in the target with local degrees (3, … , 3), (3, … , 3), (4, 2, 3, … , 3) at their preimages. The result was already established by Izmestiev, Kusner, Rote, Springborn, and Sullivan, using geometric techniques, and by Corvaja and Zannier with a more algebraic approach, whereas our proof is topological and completely elementary: besides the definitions, it only uses the fact that on the torus a simple closed curve can only be trivial (in homology, or equivalently bounding a disc, or equivalently separating) or non-trivial.

On Orthogonal Tensors and Best Rank-One Approximation Ratio

As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented.

A Generalization of the Doubling Construction for Sums of Squares Identities

The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple $[r,s,n]$ a series of new ones $[r+\rho(2^{m-1}),2^ms,2^mn]$ for all positive integer $m$, where $\rho$ is the Hurwitz-Radon function.

ℤ₂ⁿ-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms

We introduce a series of Z 2 n \mathbb {Z}_2^n -graded quasialgebras P n ( m ) \mathbb {P}_n(m) which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The constructed series of algebras and their minor perturbations are applied to contribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvinsky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.

Diagonal solutions to the 2-Toda hierarchy

Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize diagonal solutions and use this to prove that a special family of formal power series, called content-type series, are solutions to the 2-Toda hierarchy. As examples, we prove that various generating series for permutation factorization problems (including the double Hurwitz problem), correlators for the Schur measure on partitions and the formal character expansion of the HCIZ integral are all solutions to the 2-Toda hierarchy.

New solutions to the Hurwitz problem on square identities

The Hurwitz problem of composition of quadratic forms, or of “sum of squares identity” is tackled with the help of a particular class of ( Z / 2 Z ) n -graded non-associative algebras generalizing the octonions. This method provides explicit formulas for the classical Hurwitz–Radon identities and leads to new solutions in a neighborhood of the Hurwitz–Radon identities.

Branched covers of the sphere and the prime-degree conjecture

To a branched cover $${\widetilde{\Sigma} \to \Sigma}$$ between closed, connected, and orientable surfaces, one associates a branch datum, which consists of Σ and $${\widetilde{\Sigma}}$$ , the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann–Hurwitz formula. A candidate surface cover is an abstract branch datum, a priori not coming from a branched cover, but satisfying the Riemann– Hurwitz formula. The old Hurwitz problem asks which candidate surface covers are realizable by branched covers. It is now known that all candidate covers are realizable when Σ has positive genus, but not all are when Σ is the 2-sphere. However, a long-standing conjecture asserts that candidate covers with prime degree are realizable. To a candidate surface cover, one can associate one $${\widetilde {X} \dashrightarrow X}$$ between 2-orbifolds, and in Pascali and Petronio (Trans Am Math Soc 361:5885–5920, 2009), we have completely analyzed the candidate surface covers such that either X is bad, spherical, or Euclidean, or both X and $${\widetilde{X}}$$ are rigid hyperbolic orbifolds, thus also providing strong supporting evidence for the prime-degree conjecture. In this paper, using a variety of different techniques, we continue this analysis, carrying it out completely for the case where X is hyperbolic and rigid and $${\widetilde{X}}$$ has a 2-dimensional Teichmüller space. We find many more realizable and non-realizable candidate covers, providing more support for the prime-degree conjecture.

PERIODICITY THEOREM FOR STRUCTURE FRACTALS IN QUATERNIONIC FORMULATION

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 27. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 24-dimensional "bipetals" for p = 9 and 27-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.

Towards an intersection theory on Hurwitz spaces

Moduli spaces of algebraic curves are closely related to Hurwitz spaces, that is, spaces of meromorphic functions on curves. All of these spaces naturally arise in numerous problems of algebraic geometry and mathematical physics, especially in connection with string theory and Gromov-Witten invariants. In particular, the classical Hurwitz problem of enumerating the topologically distinct ramified coverings of the sphere with prescribed ramification type reduces to the study of the geometry and topology of these spaces. The cohomology rings of such spaces are complicated even in the simple case of rational curves and functions. However, the cohomology classes most important for applications (namely, the classes Poincaré dual to the strata of functions with given singularities) can be expressed in terms of relatively simple “basic” classes (which are, in a sense, tautological). The aim of the present paper is to identify these basic classes, to describe relations between them, and to find expressions for the strata in terms of them. Our approach is based on Thom's theory of universal polynomials of singularities, which has been extended to the case of multisingularities by the first author. Although the general Hurwitz problem still remains open, our approach enables one to achieve significant progress towards its solution and an understanding of the geometry and topology of Hurwitz spaces.

On realizability of branched coverings of the sphere

We introduce a simple geometric method of determining which abstract branch data can be realized by branched coverings of the two-dimensional sphere S 2 (the Hurwitz problem). Using this method we show the realizability of some classes of data.

Generalized Hurwitz maps of the type S × V → W

The notion of J 3-triple ( J 3 = Jos e ́ , Julian, Jakub ) is introduced in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements.

Generalized Hurwitz maps of the type S × V → W, anti-involutions, and quantum braided Clifford algebras

The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz m

Hurwitz problem, harmonic morphisms and generalized Hadamard matrices

We use real representations of Clifford algebras associated with Hurwitz factorizations in order to generate harmonic morphisms on pseudo-Euclidean spaces and generalized Hadamard matrices.