arXiv subjects

Matheus Brito

Publications and source records attributed to Matheus Brito.

16 recordsLinked to original sources

arXiv2026

Monoidal categorification from alternating snakes

In a recent paper, the authors introduced the notion of an alternating snake and a corresponding family of finite dimensional modules for the quantum affine algebra associated to $A_n$. We prove that under some restrictions, an alternating snake defines a canonical monoidal category. We prove that this category has finitely many prime objects. As a consequence we prove that the Grothendieck ring is isomorphic to the Grothendieck ring of the category $\mathscr C_\xi$ for a suitable height function. In particular it follows that the special family of alternating snakes provides a monoidal categorification of a cluster algebra of type $A_N$ for a suitable value of $N$.

math.QA↗

arXiv2025

Imaginary modules arising from tensor products of snake modules

Motivated by the limitations of cluster algebra techniques in detecting imaginary modules, we build on the representation-theoretic framework developed by the first author and Chari to extend the construction of such modules beyond previously known cases, which arise from the tensor product of a higher-order Kirillov--Reshetikhin module and its dual. Our first main result gives an explicit description of the socle of tensor products of two snake modules, assuming the corresponding snakes form a covering pair of ladders. By considering a higher-order generalization of the covering relation, we describe a sequence of inclusions of highest-$\ell$-weight submodules of such tensor products. We conjecture all the quotients of subsequent modules in this chain of inclusions are simple and imaginary, except for the socle itself, which might be real. We prove the first such quotient is indeed simple and, assuming an extra mild condition, we also prove it is imaginary, thus giving rise to new classes of imaginary modules within the category of finite-dimensional representations of quantum loop algebras in type A.

math.RT↗

arXiv2025

On dominant $\ell$--weights and maps between Weyl modules for quantum affine $A_n$

We determine the set of dominant $\ell$--weights in the Weyl (or standard) modules for quantum affine $A_n$. We then prove that the space of homomorphisms between standard modules is at most one-dimensional and give a necessary and sufficient condition for equality to hold. We also describe the socle of the standard module and prove that the socle is simple for large $n$. Finally, we give applications of our results to mixed Weyl modules, calculating extensions in the category and identify new families of tensor subcategories of finite dimensional representations.

math.QA↗

arXiv2024

Alternating snake modules and a determinantal formula

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $\mu$ for which the non-zero Kazhdan-Lusztig coefficients $c_{\mu, \nu}$ are $\pm 1$.

math.RT↗

arXiv2024

Totally ordered pseudo q-factorization graphs and prime factorization

In an earlier publication, the last two authors showed that a finite-dimensional module for a quantum affine algebra of type $A$ whose $q$-factorization graph is totally ordered is prime. In this paper, we continue the investigation of the role of totally ordered pseudo $q$-factorization graphs in the study of the monoidal structure of the underlying abelian category. We introduce the notions of modules with (prime) snake support and of maximal totally ordered subgraphs decompositions. Our main result shows that modules with snake support have unique such decomposition and that it determines the corresponding prime factorization. Along the way, we also prove that prime snake modules (for type $A$) can be characterized as the modules for which every pseudo $q$-factorization graph is totally ordered.

math.RT↗

arXiv2024

On the Quantum Metaplectic Howe Duality

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of Drinfeld-Jimbo quantum groups $(\mathcal U_{q^2}(\mathfrak{sl}_2),\mathcal U_{q}(\mathfrak{sl}_2))$ realized in a suitable algebra of $q$-differential operators acting on the space of symplectic polynomial spinors. We obtain $q$-analogues for the symplectic Dirac operator, the Fischer decomposition, the expression for the symplectic polynomial monogenics and for the projection operators onto the monogenics. We also discuss $q$-analogues of generalized symmetries of the $q$-symplectic Dirac operator raising the homogeneous polynomial degree.

math.QA↗

arXiv2024

Reality determining subgraphs and strongly real modules

The concept of pseudo q-factorization graphs was recently introduced by the last two authors as a combinatorial language which is suited for capturing certain properties of Drinfeld polynomials. Using certain known representation theoretic facts about tensor products of Kirillov Reshetikhin modules and qcharacters, combined with special topological/combinatorial properties of the underlying q-factorization graphs, the last two authors showed that, for algebras of type A, modules associated to totally ordered graphs are prime, while those associated to trees are real. In this paper, we extend the latter result. We introduce the notions of strongly real modules and that of trees of modules satisfying certain properties. In particular, we can consider snake trees, i.e., trees formed from snake modules. Among other results, we show that a certain class of such generalized trees, which properly contains the snake trees, give rise to strongly real modules.

math.RT↗

arXiv2022

Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$

We study the family of irreducible modules for quantum affine $\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m$ with $m\le n$. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category $\mathscr C^-$. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov--Reshetikhin module with its dual always contains an imaginary module in its Jordan--Holder series and give an explicit formula for its Drinfeld polynomial. Together with the results of \cite{HL13a} this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of \cite{LM18}. Finally, we use our methods to give a family of imaginary modules in type $D_4$ which do not arise from an embedding of $A_r$ with $r\le 3$ in $D_4$.

math.QA↗

arXiv2022

Quantum affine algebras, graded limits, and flags

In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations of these algebras. This connection arises via the notion of the graded and classical limit of finite-dimensional representations of quantum affine algebras. We explain how this study has led to interesting connections with Macdonald polynomials and discuss a BGG-type reciprocity result. We also discuss the role of Demazure modules in this theory and several recent results on the presentation, structure, and combinatorics of Demazure modules.

math.RT↗

arXiv2020

BrazilDAM: A Benchmark dataset for Tailings Dam Detection

In this work we present BrazilDAM, a novel public dataset based on Sentinel-2 and Landsat-8 satellite images covering all tailings dams cataloged by the Brazilian National Mining Agency (ANM). The dataset was built using georeferenced images from 769 dams, recorded between 2016 and 2019. The time series were processed in order to produce cloud free images. The dams contain mining waste from different ore categories and have highly varying shapes, areas and volumes, making BrazilDAM particularly interesting and challenging to be used in machine learning benchmarks. The original catalog contains, besides the dam coordinates, information about: the main ore, constructive method, risk category, and associated potential damage. To evaluate BrazilDAM's predictive potential we performed classification essays using state-of-the-art deep Convolutional Neural Network (CNNs). In the experiments, we achieved an average classification accuracy of 94.11% in tailing dam binary classification task. In addition, others four setups of experiments were made using the complementary information from the original catalog, exhaustively exploiting the capacity of the proposed dataset.

cs.CV↗

arXiv2019

Local Weyl modules and fusion products for the current superalgebra $\mathfrak{sl}(1|2)[t]$

We study a class of modules, called Chari-Venkatesh modules, for the current superalgebra $\mathfrak{sl}(1|2)[t]$. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules.

math.RT↗

arXiv2019

Tensor products and $q$-characters of HL-modules and monoidal categorifications

We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.

math.RT↗

arXiv2017

Resolutions and a Weyl Character formula for prime representations of quantum affine sl_{n+1}

In this paper we study the family of prime irreducible representations of quantum affine $\lie{sl}_{n+1}$ which arise from the work of D. Hernandez and B. Leclerc. These representations can also be described as follows: the highest weight is a product of distinct fundamental weights with parameters determined by requiring that the representation be minimal by parts. We show that such representations admit a BGG-type resolution where the role of the Verma module is played by the local Weyl module. This leads to a closed formula (the Weyl character formula) for the character of the irreducible representation as an alternating sum of characters of local Weyl modules. In the language of cluster algebras our Weyl character formula describes an arbitrary cluster variable in terms of the generators $x_1,\cdots,x_n,x_1',\cdots, x_n'$ of an appropriate cluster algebra. Our results also exhibit the character of a prime level two Demazure module as an alternating linear combination of level one Demazure modules.

math.QA↗

arXiv2015

Graded limits of simple tensor product of Kirillov-Reshetikhin modules for $U_q(\tilde{\mathfrak {sl}}_{n+1})$

We study the graded limits of simple $U_q(\tilde{\mathfrak{sl}}_{n+1})$-modules which are isomorphic to tensor products of Kirillov-Reshetikhin modules associated to a fix fundamental weight. We prove that every such module admits a graded limit which is isomorphic to the fusion product of the graded limits of its tensor factors. Moreover, using recent results of Naoi, we exhibit a set of defining relations for these graded limits.

math.QA↗

arXiv2015

Demazure modules of level two and prime representations of quantum affine $\mathfrak{sl}_{n+1}$

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to $\mathfrak{sl}_{n+1}[t]$--stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. More generally, we prove that any level two Demazure module is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

math.QA↗

arXiv2014

Representations of quantum affine algebras of type $B_N$

We study finite-dimensional representations of quantum affine algebras of type $B_N$. We show that a module is tame if and only if it is thin. In other words, the Cartan currents are diagonalizable if and only if all joint generalized eigenspaces have dimension one. We classify all such modules and describe their $q$-characters. In some cases, the $q$-characters are described by super standard Young tableaux of type $(2N|1)$.

math.QA↗