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Pablo S. Ocal

Publications and source records attributed to Pablo S. Ocal.

10 recordsLinked to original sources
arXiv2026

Structure and geometry of the tableaux algebra

We study the monoid algebra ${}_{n}\mathcal{T}_{m}$ of semistandard Young tableaux, which coincides with the Gelfand--Tsetlin semigroup ring $\mathcal{GT}_{n}$ when $m = n$. Among others, we show that this algebra is commutative, Noetherian, reduced, Koszul, and Cohen--Macaulay. We provide a complete classification of its maximal ideals and compute the topology of its maximal spectrum. Furthermore, we classify its irreducible modules and provide a faithful semisimple representation. We also establish that its associated variety coincides with a toric degeneration of certain partial flag varieties constructed by Gonciulea--Lakshmibai. As an application, we show that this algebra yields injective embeddings of $\mathfrak{sl}_n$-crystals, extending a result of Bossinger--Torres.

math.AC↗
arXiv2026

Locally recoverable algebro-geometric codes from projective bundles

A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that is, evaluation codes where each code word symbol can be reconstructed from several disjoint sets of other symbols. The simplest case, where the code's underlying variety is a plane, exhibits noteworthy properties: When $r = 1$, $2$, $3$, they are optimal; when $r \geq 4$, they are optimal with probability approaching $1$ as the alphabet size grows. Additionally, their information rate is close to the theoretical limit. In higher dimensions, our codes form a family of asymptotically good codes.

cs.IT↗
arXiv2025

Realizing compatible pairs of transfer systems by combinatorial $N_\infty$-operads

We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg--Hill relate via the correspondence between indexing systems and $N_{\infty}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\infty}$-operads.

math.AT↗
arXiv2024

A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.

q-bio.MN↗
arXiv2024

A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the twisted tensor product of bialgebras is a bialgebra, and when the twisted tensor product of Frobenius algebras is a Frobenius algebra. Our methods are purely diagrammatic, so these results hold for (braided) monoidal categories. As an application, we recover that some quantum complete intersections are Frobenius algebras, and we construct families of noncommutative symmetric Frobenius algebras. Along the way, we also characterize when twisted tensor products of separable algebras are separable, and we prove that twisted tensor products of special Frobenius algebras are special Frobenius.

math.RA↗
arXiv2023

Symmetric monoidal equivalences of topological quantum field theories in dimension two and Frobenius algebras

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.

math.QA↗
arXiv2023

A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry

We review several techniques that twist an algebra's multiplicative structure. We first consider twists by an automorphism, also known as Zhang twists, and we relate them to 2-cocycle twists of certain bialgebras. We then outline the classification and properties of twisted tensor products, and we examine twisted Segre products. Our exposition emphasizes clarity over generality, providing a wealth of interconnecting examples.

math.RA↗
arXiv2021

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov's homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

math.RA↗