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Cesar Galindo

Publications and source records attributed to Cesar Galindo.

6 recordsLinked to original sources
arXiv2026

Unitary Yang--Baxter Operators: Towards a Classification

There is a well-known circle of conjectures relating unitary solutions of the Yang--Baxter equation, unitary braided fusion categories, topological quantum computation, and link invariants. Progress is limited by the lack of a classification of unitary Yang--Baxter operators. We propose a conjectural classification with three generating sources: monomial solutions, group-type solutions arising from Yetter--Drinfeld modules, and solutions from twisted group-algebra towers. We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources. We provide evidence for our classification conjecture by means of computational searches for unitary Yang--Baxter operators among Clifford groups and a polynomial-in-Pauli ansatz. This search yields only a few solutions that do not appear to be equivalent to monomial or group type solutions. Direct calculations identify their link invariants with specializations of the HOMFLYPT and BMW/Kauffman invariants; the latter is a squared Jones specialization at a primitive sixth root, and the corresponding qutrit operator strictly localizes the semisimple quotient of the BMW algebra specializations $\mathcal{C}_n(q^3,e^{\pi\ii/6})$ associated with the braided fusion category $SO(4)_2$. The quaternionic Family III specialization realizes a two-eigenvalue class and gives tensor-power realization of the semisimple Hecke algebra tower associated with \(SU(3)_3\). Significantly, this answers a recent existence question of Lechner, and verifies an open case of the Rowell-Wang localization conjecture from 2010.

math.QA↗
arXiv2026

Group Structure from Subgroup and Cyclic Subgroup Counts

For a finite group \(G\), let \(\sub(G)\) be the number of subgroups of \(G\), let \(\cyc(G)\) be the number of cyclic subgroups, and let \(\pi(G)\) be the number of distinct prime divisors of \(|G|\). We study the normalized counts \(\lambda(G)=\sub(G)/2^{\pi(G)}\) and \(\eta(G)=\cyc(G)/2^{\pi(G)}\). We prove that \(\eta(G)<5/4\) or \(\lambda(G)<3/2\) implies that \(G\) is cyclic of squarefree order. The inequalities \(\eta(G)<2\) and \(\lambda(G)<5/2\) each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding \(ZM\)-presentation. We determine all values with \(1<\eta(G)<2\) and \(1<\lambda(G)<5/2\), and prove that \(\lambda(G)<59/8\) or \(\eta(G)<4\) implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.

math.GR↗
arXiv2021

On the convergence of the orthogonal spectral sequence

We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence.

math.AG↗
arXiv2016

Fermionic Modular Categories and the 16-fold Way

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towards a classification of the low-rank cases.

math.QA↗
arXiv2007

Normal Hopf subalgebras in cocycle deformations of finite groups

Let $G$ be a finite group and let $π: G \to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^σ$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $σ$. The (normal) Hopf subalgebra $k^{G'} \subseteq k^G$ corresponds to a Hopf subalgebra $L' \subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$.

math.QA↗