arXiv subjects

Carles Marín

Publications and source records attributed to Carles Marín.

3 recordsLinked to original sources

arXiv2026

An Affine Semigroup from Orbifold Boundary Conditions: cut, phylogenetic and hierarchical models in the unit-weight sector, and weighted configurations beyond them

The equivalence classes of boundary conditions of a gauge theory on a two-dimensional orbifold are the fibres of a marginal map, indexed by an affine semigroup: one generator per alphabet label, graded by weight, embedded by its local data at the fixed points. This note identifies that semigroup. Without weights the configuration has a name and a literature, whose results about our cases are attributed here: over $\mathbb{Z}_2$ it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant --- the four-cycle for $T^2/\mathbb{Z}_2$, the wheel $W_4$ for $S^1/\mathbb{Z}_2\times S^1/\mathbb{Z}_2$ --- verified as an equality of configurations; over $\mathbb{Z}_m$ with equal cone orders, the group-based phylogenetic model on a claw tree; with unequal orders, a mixed-order variant we do not find in the literature; for higher products, the binary hierarchical model of a cross-polytope boundary complex. The product orbifold's ring is a row of a 2008 table --- codimension, degree, minimal generators, normality --- every invariant of which our machinery reproduced without knowing of it. What none of the three covers is the alphabet with weights, which arise from induction to higher-dimensional irreducibles of a non-abelian space group and from conjugate-pair recombination over real or quaternionic ground. That sector is adjacent to, but not identified with, the non-abelian direction Sturmfels and Sullivant raised in 2005, and is where our contributions sit: gluing trees for the weighted alphabets and the orthogonal and symplectic columns, and the group-based model on the tripod, a complete intersection exactly when the finite abelian group has order at most three. The first group beyond $\mathbb{Z}_3$ separates local from global: the $\mathbb{Z}_4$ tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.

math.CO↗

arXiv2026

Schur polynomials twisted by roots of unity and reciprocal pairs: exactly three factors, and where they vanish

Let $μ_t$ be the full set of $t$-th roots of unity. Adjoining $r$ free reciprocal pairs gives a two-parameter family of alphabets; we settle three parts of it. At $r=1$, for every $t\ge2$ and every $λ$ with at most $t+2$ parts, $s_λ(μ_t,z,z^{-1})$ is a signed product of exactly three factors over a fixed denominator, or zero, the arguments read off core and quotient. What it sees of $λ$ is a multiset of three integers and a sign, and exactly that: two partitions of any sizes share a nonzero value if and only if they agree on that datum. The proof is a Laplace expansion along the $t$ frozen rows with one cancellation lemma, and delivers the sign, of which Littlewood's is one factor. At $t=2$ and every $r$, $s_λ(1,-1,z_1^{\pm1},\dots,z_r^{\pm1})$ vanishes exactly when the beta set has constant parity or $λ$ is self-complementary of odd width; that direction is a corollary of complementation over an index family of Ayyer and Behrend, the converse an extremal argument in the degree filtration, modulo one rigidity theorem for Schur products. Equivalently: exactly those $V_λ$ restrict to $O(N,\mathbb{C})$ $\det$-stably. At odd $t$ and every $r$ it vanishes exactly when a residue class is absent, at no external cost. And for every $t$ and $r$, a reflection of the beta set's excess part with one increment hitting its centre forces vanishing. Three consequences of the first. A vanishing criterion: an empty residue class, or two distinguished classes concentric as intervals, the second only for even $t$. An extension of Ayyer-Kumari's independence criterion: on the reciprocal locus it acquires one further family, classified by core and quotient. And at $t=2$ a $(-1)$-enumeration of plane partitions in a box refined by a parameter that stays free. The factorization is isolated: it fails under each of four deformations of the alphabet, for one reason.

math.CO↗

arXiv2026

Schur polynomials twisted by roots of unity and reciprocal pairs: torsion filters, fusion quotients, and total unimodularity at odd order

Write $μ_t$ for the $t$-th roots of unity and $z^{\pm1}$ for $r$ free reciprocal pairs. We study $Φ_{t,r}(β)=s_λ(μ_t,z^{\pm1})$, $β=λ+δ$, and the question the companion paper left open after $r=1$: when does it vanish? We factor the evaluation into classical branching followed by a torsion filter, and the shape depends on the parity of $t$: the point lies in the orthogonal group with determinant $(-1)^{t+1}$. For odd $t$ it sits in the identity component: an ordinary restriction $SO_{2R'+1}\downarrow SO_{2m'+1}\times SO_{2r}$, the filter an odd orthogonal character at a principal element of order $h+1$. For even $t$ in the other: a twining, a virtual expansion, and a torsion element regular but not principal; there we prove the filter, with its sign. One description covers both: the filter is nonzero exactly when the shifted point is regular semisimple. Both are minimal-level fusion projections: the even of type $C$, the odd's tensor sector of type $B$. Affine folding accounts for $0,\pm1$; what it does not survives as conjectures. The highest surviving weight is the dominant vertex of the numerator's Newton polytope minus the denominator's --- the latter proved, the former conditional on a single-orbit property --- and the class there is conjecturally primitive, $\pm$ the generator of the rank-one quotient. For odd $t$ and one $Λ$ that numerator is a signed transversal count in $\{0,\pm1\}$ by the equal-rank character formula, leaving one division. We invert it in closed form, along an arithmetic progression; the quotient is $\pmε_t\det M$ for an explicit $0/{\pm}1$ matrix --- an interval matrix up to signs, hence totally unimodular, which settles (L1). The fibre count is a permanent, odd only when $1$, so at a dominant index a multi-hit fibre sums to zero. Two extremal statements remain. What is unproved is measured, in both parities.

math.CO↗