arXiv subjects

Davide Dal Martello

Publications and source records attributed to Davide Dal Martello.

4 recordsLinked to original sources

arXiv2026

Okamoto's symmetry on the representation space of the sixth Painlevé equation

The sixth Painlevé equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a Teichmüller $\mathcal{X}$-coordinatization for the Fuchsian monodromy group, we give Okamoto's symmetry $w_2$ of PVI a monodromic realization in the language of cluster $\mathcal{X}$-mutations. The explicit mutation formula is encoded in dual geometric terms of colored equilateral triangulations and star-shaped fat graphs. Moreover, this realization has a known additive analogue through the middle convolution for Fuchsian systems, and dual formulations for both the Birkhoff representation and its Stokes data exist. We give this quadruple of maps, each one realizing $w_2$, a unified diagrammatic description in purely convolutional terms.

math-ph↗

arXiv2026

Cluster topography

Using the LP algebraic toolkit, Conway's original topograph is rethought of as a cluster construction, paving the way for a wider topography based on mutation-type local rules. As a remarkable application of such cluster-driven upgrade, both the process of analytic continuation for Painlevé VI and the reduction algorithm for quadratic forms are endowed with the Laurent phenomenon. En passant, the rattlesnake is defined so to complete the bijection between snake graphs and rationals to the whole of $\mathbb{Q}$.

math.CO↗

arXiv2025

Generic Hecke algebras in the infinite

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point theorem. These new presentations behave à la Coxeter, encoding many of the group's properties at a glance, and their signature feature -- named the $x$-relation -- is fully understood in terms of configuration spaces. Crucially, the presentations further achieve the braid theorem, allowing to deform into the generic Hecke algebra. In particular, we revisit the affine GDAHA family in deformation terms of the most general class of Steinberg crystallographic complex reflection groups.

math.GR↗

arXiv2024

Generalized double affine Hecke algebras, their representations, and higher Teichmüller theory

Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of $2$-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the $\tilde D_4$-type GDAHA to representations of the $\tilde E_6$-type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type $\tilde D_4$ and $\tilde E_6$ into matrix algebras over quantum cluster $\mathcal{X}$-varieties, thus linking to the theory of higher Teichmüller spaces. For $\tilde E_6$, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.

math.QA↗