arXiv subjects

Minh-Tâm Quang Trinh

Publications and source records attributed to Minh-Tâm Quang Trinh.

11 recordsLinked to original sources

arXiv2026

Algebraic loops, braids, and tropicalization

Let $Y$ be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in $Y^{\mathrm{an}}$ associated with formal loops $\mathrm{Spec}\, \mathbb{C}(\!(t)\!)\to Y$. When $Y$ is very affine, every formal loop determines an integral point of $\mathrm{Trop}(Y)$. For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when $Y$ is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type $A$, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.

math.AG↗

arXiv2026

Free Immersions in Critical Dimension from Lie Groups

We construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo constructs a map that is symmetric with respect to a representation of the Lie group formed by the $(m - 1)$-torus; we replace this group with $\mathrm{SO}(3)$. In our examples, the osculating determinant is constant.

math.DG↗

arXiv2026

Level-Rank Dualities for Finite Reductive Groups

This is an extended abstract of our work "Level-Rank Dualities from $Φ$-Cuspidal Pairs..." We present evidence for a family of surprising coincidences within the representation theory of a finite reductive group $G$: more precisely, dualities between blocks of cyclotomic Hecke algebras attached by Broué-Malle to $Φ$-cuspidal pairs of $G$, where the Hecke parameters are specialized not to the order of the underlying finite field, but to roots of unity. For the groups $G = \mathrm{GL}_n(\mathbf{F}_q)$, these coincidences can be expressed very concretely in terms of the combinatorics of partitions, and the whole story recovers an avatar of the level-rank duality studied by Frenkel, Uglov, Chuang-Miyachi, and others.

math.RT↗

arXiv2026

Haiman's Conjecture and Springer's Representations

For any connected complex reductive group $G$ and element $z$ of its Weyl group $W$, we use work of Lusztig and Abreu-Nigro to compute the graded $W$-character of the intersection cohomology of any closed Lusztig variety for $z$ over the regular semisimple locus of $G$. We relate the resulting formula to unipotent Lusztig varieties, giving a new geometric model for unicellular LLT polynomials. We then consider Laurent polynomials $α_{ψ, G}^z$ indexed by irreducible characters $ψ$, encoding how our formula decomposes into ungraded characters arising from the Springer theory of $G$. From evidence in low rank, we conjecture that if $ψ$ is inflated from type $A$ in a particular way, then the nonzero coefficients of $α_{ψ, G}^z$ are positive and unimodal. This offers an answer to a 1993 question of Haiman about generalizing a conjecture he posed for symmetric groups. We also prove that the matrix formed by the $α_{ψ, G}^z$ is partially triangular, and that their positivity and unimodality properties are stable under inclusions of Levi subgroups.

math.RT↗

arXiv2026

Partial Resolutions and Noncrossing Combinatorics

For any finite reductive group, we compute the central elements in its Hecke algebra that arise from partial Springer resolutions via the Harish-Chandra transform. Of the two kinds of partial resolution, the larger is the more interesting case. We deduce formulas for associated Hecke traces, generalizing work of Wan-Wang beyond type $A$, and Deodhar-like decompositions of braid varieties that map to partial Springer resolutions. From the latter, we construct noncrossing sets that interpolate between rational Catalan and parking objects, generalizing our work with Galashin-Lam. In parallel, we establish new formulas for arbitrary $a$-degrees of the HOMFLYPT invariants of positive braid closures, from which we construct noncrossing sets for rational Kirkman numbers.

math.RT↗

arXiv2025

The Hilb-vs-Quot Conjecture

Let $R$ be the complete local ring of a complex plane curve germ and $S$ its normalization. We propose a "Hilb-vs-Quot" conjecture relating the virtual weight polynomials of the Hilbert schemes of $R$ to those of the Quot schemes that parametrize $R$-submodules of $S$. By relating the Quot side to a type of compactified Picard scheme, we show that our conjecture generalizes a conjecture of Cherednik's, and that it would relate the perverse filtration on the cohomology of the Picard side to a more elementary filtration. Next, we propose a Quot version of the Oblomkov-Rasmussen-Shende conjecture, relating parabolic refinements of our Quot schemes to Khovanov-Rozansky link homology. It becomes equivalent to the original version under (refined) Hilb-vs-Quot, but can also be strengthened to incorporate polynomial actions and $y$-ification. For germs $y^n = x^d$, where $n$ is either coprime to or divides $d$, we prove the Quot version of ORS through combinatorics. When $n = 3$ and $3 \nmid d$, we deduce Hilb-vs-Quot by an asymptotic argument, and hence, establish the original ORS conjecture for these germs.

math.AG↗

arXiv2025

Level-Rank Dualities from $Φ$-Cuspidal Pairs and Affine Springer Fibers

We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Broué-Malle, which conjecturally describe the endomorphisms of Lusztig induction modules, and a generalization of Harish-Chandra theory due to Broué-Malle-Michel. For any generic finite reductive group $\mathbb{G}$ and integers $e, m > 0$, we conjecture that: (1) the intersection of a $Φ_e$-Harish-Chandra series and a $Φ_m$-Harish-Chandra series is parametrized by a union of blocks of the Hecke algebra of the $Φ_e$-cuspidal pair at an $m$th root of unity, and similarly for the Hecke algebra of the $Φ_m$-cuspidal pair at an $e$th root of unity; (2) these parametrizations match the blocks on the two sides; (3) when two blocks match, the bijection between them lifts to a derived equivalence between associated blocks of rational DAHAs. Surprisingly, these structures also appear in bimodules formed from the cohomology of affine Springer fibers studied by Oblomkov-Yun. When $\mathbb{G} = \mathbb{GL}_n$ and $e, m$ are coprime, we show that (1)-(3) hold, and that (3) recovers the level-rank dualities conjectured by Chuang-Miyachi and later proved through the work of several other people. Finally, we verify for many cases where $\mathbb{G}$ is exceptional that Broué-Malle's parameters are numerically compatible with our conjectures.

math.RT↗

arXiv2024

Simple Braids Tend toward Positive Entropy

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as $n \to \infty$, the proportion of simple braids on $n$ strands that have positive topological entropy tends toward $100\%$. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple $3$-strand braids that may be of independent interest.

math.GT↗

arXiv2022

Unipotent Elements and Twisting in Link Homology

Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$ restricts to a map from $U_+U_- \cap gB_+$ into $\mathcal{U} \cap gB_+$, for any $g$. We conjecture that the restricted map forms half of a homotopy equivalence between these varieties, and thus, induces a weight-preserving isomorphism between their compactly-supported cohomologies. Noting that the map is equivariant with respect to certain actions of $B_+ \cap gB_+g^{-1}$, we prove for type $A$ that an equivariant analogue of this isomorphism exists. Curiously, this follows from a certain duality in Khovanov-Rozansky homology, a tool from knot theory.

math.RT↗

arXiv2022

Rational Noncrossing Coxeter-Catalan Combinatorics

We solve two open problems in Coxeter-Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type-uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter-Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.

math.CO↗

arXiv2021

From the Hecke Category to the Unipotent Locus

Let $W$ be the Weyl group of a split semisimple group $G$. Its Hecke category $\mathsf{H}_W$ can be built from pure perverse sheaves on the double flag variety of $G$. By developing a formalism of generalized realization functors, we construct a monoidal trace from $\mathsf{H}_W$ to a category of bigraded modules over a certain graded ring: namely, the endomorphisms of the equivariant Springer sheaf over the unipotent locus of $G$. We prove that: (1) On objects attached to positive braids $β$, the output is the weight-graded, equivariant Borel-Moore homology of a generalized Steinberg scheme $\mathcal{Z}(β)$. (2) Our functor contains, as a summand, one used by Webster-Williamson to construct the Khovanov-Rozansky link invariant. In particular, the Khovanov-Rozansky homology of the link closure of $β$ is fully encoded in the Springer theory of $\mathcal{Z}(β)$. Decategorifying, we get a trace on the Iwahori-Hecke algebra, valued in graded virtual characters of $W$. We give a formula for it in terms of a pairing on characters of $W$ called Lusztig's exotic Fourier transform, which generalizes it to all finite Coxeter groups. Using this formula, we establish properties of the trace like rationality, symmetry, and compatibility with parabolic induction. We also show that on periodic braids, the trace produces the characters of explicit virtual modules over Cherendik's rational double affine Hecke algebra. For $W = S_n$, we recover an identity of Gorsky-Oblomkov-Rasmussen-Shende relating these modules to the HOMFLY polynomials of torus knots.

math.RT↗