arXiv subjects

Higinio Serrano

Publications and source records attributed to Higinio Serrano.

6 recordsLinked to original sources

arXiv2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ \nu_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ \nu_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

math.CO↗

arXiv2026

Many-point tropical relaxation and the Monge--Amp\`ere equation

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Amp\`ere equation. Let $\Omega\subset\mathbb R^2$ be a bounded open convex domain, fix $K\Subset\Omega$, and let $F_N=G_{P_N}(0_\Omega)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing a universally generic $N$-point set $P_N\subset K$. Set $u_N=N^{-1/2}F_N$ and $\mu_N=N^{-1}\sum_{p\in P_N}\delta_p$. For every compact $L\Subset\Omega$ we prove $\left|\int\varphi\,d(\mathrm{MA}(u_N)-\mu_N)\right|\le C(\Omega,K,L)N^{-1/2}(\|\varphi\|_\infty+\|\nabla\varphi\|_\infty)$ for $\varphi\in C_c^1(\Omega)$ supported in $L$. If $\mu_N\rightharpoonup\mu$, where $\mu$ is a probability measure supported in $K$, then $u_N$ converges uniformly on $\overline\Omega$ to the unique continuous concave zero-boundary Aleksandrov solution of $\mathrm{MA}(F)=\mu$, and $\mathrm{MA}(u_N)\rightharpoonup\mu$ vaguely in $\Omega$. No regularity or strict convexity of $\partial\Omega$ is assumed. For bounded rational convex polygons, strong genericity suffices. If $P\subset K$ is strongly generic with $|P|=N$ and $F_P=G_P(0_\Omega)$, its tropical curve has exactly $N$ bounded cells; the duals of the uncut marked carriers form a spanning tree; every compact internal edge has weight one; and $\mathrm{MA}(F_P)(\Omega^\circ)=N-1+\tfrac12D_{\mathrm{term}}(F_P)$, with $D_{\mathrm{term}}(F_P)=O_{\Omega,K}(\sqrt N)$. For strongly generic sequences satisfying the same empirical-measure hypothesis, the normalized curvature measures converge weakly on the closed polygon. We also obtain almost-sure limits for i.i.d. samples from absolutely continuous laws supported in $K$, affine covariance of the continuum solution, and, for source sequences covered by the polygonal theorem, a configuration-dependent Abelian-sandpile diagonal.

math.AP↗

arXiv2025

Magnetic Equivariant Graded Brauer Group

Given a magnetic finite group, we consider the similarity classes of magnetic equivariant central simple graded algebras over the complex numbers. We call this set the magnetic equivariant graded Brauer group and its structure as an abelian group is explicitly determined. Following Karoubi, we argue that the elements of this graded Brauer group parametrize the twistings of the magnetic equivariant K-theory of a point.

math.KT↗

arXiv2025

Magnetic Equivariant K-theory

We present the fundamental properties of the K-theory groups of complex vector bundles endowed with actions of magnetic groups. In this work we show that the magnetic equivariant K-theory groups define an equivariant cohomology theory, we determine its coefficients, we show Bott's, Thom's and the degree shift isomorphism, we present the Atiyah-Hirzeburh spectral sequence, and we explicitly calculate two magnetic equivariant K-theory groups in order to showcase its applicability. These magnetic equivariant K-theory groups are relevant in condensed matter physics since they provide topological invariants of gapped Hamiltonians in magnetic crystals.

math.KT↗

arXiv2024

Rational magnetic equivariant K-theory

We introduce the magnetic equivariant K-theory groups as the K-theory groups associated to magnetic groups and their respective magnetic equivariant complex bundles. We restrict the magnetic group to its subgroup of elements that act complex linearly, and we show that this restriction induces a rational isomorphism with the conjugation invariant part of the complex equivariant K-theory of the restricted group. This isomorphism allows to calculate the torsion free part of the magnetic equivariant K-theory groups reducing it to known calculations in complex equivariant K-theory

math.KT↗

arXiv2024

Spin Chern number in altermagnets

This work explores the topological properties of altermagnets, a novel class of collinear magnetic materials. We employ equivariant K-theory of magnetic groups and Hamiltonian models to formulate a robust $C^z_4 \mathbb{T}$ topological invariant to classify 2D and 3D altermagnetic systems. Our findings demonstrate that the spin Chern number serves as a robust topological index, corresponding to the half-quantized Chern number of the divided Brillouin zone. This indicator enables the prediction of a topologically protected 2D altermagnetic insulators and 3D Weyl altermagnetic semimetals, highlighting the relationship between altermagnetism and topological phases. Furthermore, our results provide a pathway to the exploration of topological applications in $d$-wave altermagnetic materials.

cond-mat.mtrl-sci↗