Borel and shifted category O
We prove a precise relation between simple modules in the Borel category O and the shifted category O for a symmetrizable Kac-Moody Lie algebra.
math.RT↗
arXiv subjects
David Hernandez
Publications and source records attributed to David Hernandez.
Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models
In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster categorification and quantum integrable models. We will also give new examples and conjectures.
math.RT↗
Linking Young Stellar Object Morphology to Evolutionary Stages with Self-Organizing Maps
Studies in the past few decades have investigated young stellar object evolution based on their spectral energy distribution (SED). The SED is heavily influenced not only by evolutionary stage, but also the morphology of the young star. This work is part of the NEMESIS project which is aiming to revisit star formation with the aid of machine learning techniques and provides the framework for this work. In a first effort towards a novel spectro-morphological classification we analyzed young stellar object morphologies and linked them to the currently used observational classes. Thereby we aim to lay the foundation for a spectro-morphological classification, and apply the insights learned in this study in a future, revisited classification scheme. We obtained archival high-resolution survey images from VISTA for approximately 10,000 literature young stellar object candidates towards the Orion star formation complex (OSFC). Utilizing a Self-Organizing map (SOM) algorithm, an unsupervised machine learning method, we created a grid of morphological prototypes from near- and mid-infrared images. Furthermore, we determined which prototypes are most representative of the different observational classes, derived from the infrared spectral index, via Bayesian inference. We present our grids of morphological prototypes of young stellar objects in the near-infrared, which were created purely from observational data. They are thus non-dependent on theoretical models. In addition, we show maps that indicate the probability for a prototype belonging to any of the observational classes. We find that SOMs created from near-infrared images are a useful tool, with limitations, to identify characteristic morphologies of young stellar objects in different evolutionary stages. This first step lays the foundation for a spectro-morphological classification of young stellar objects to be developed in the future.
astro-ph.SR↗
Bounded confidence dynamics generates opinion cascades on growing scale-free networks
We study the pairwise bounded confidence model on scale-free networks where new agents regularly arrive over time. The probability that arriving agents form links to preexisting ones depends on both agent degree and opinion proximity. In parameter value ranges where both factors impact the link choice, a new phenomenon is observed. Minor clusters continuously form on the periphery of the opinion space and remain stable for a time, before suddenly merging with a major cluster in the network. We label these processes as "opinion cascades", and analyse their origin and behavior. They are triggered by the arrival of agents acting as "bridges" between the previously disconnected minor and major clusters. Lastly, we propose theoretical approximations to describe the varying shapes and merging behavior of opinion cascades under different conditions.
physics.soc-ph↗
Extremal monomial property of q-characters and polynomiality of the X-series
The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar "extremal weight property" for every weight obtained by applying an element of W to the highest weight. In this paper we conjecture an analogous "extremal monomial property" of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in arXiv:math/9911112. Here we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each w in W turns out to be equivalent to polynomiality of the "X-series" corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all w in W. Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated to quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of arXiv:1308.3444.
math.QA↗
Exploring Young Stellar Variability with Gaia DR3 light curves
Context: Photometric variability is a defining characteristic of young stellar objects (YSO), which can be traced back to a range of physical processes taking place at different stages of young stars' formation and early evolution. Gaia's third Data Release (GDR3) has provided an unprecedented dataset of photometric time series, including 79375 light curves for YSO candidates. With its all-sky coverage, Gaia provides a unique opportunity for large-scale studies of YSO variability. Aims: Our goal was to characterise the GDR3 sample of YSO variables further and verify the recurrence of YSO variability modes due to accretion, extinction, rotation modulation, etc. By adapting the Q&M methodology for Gaia's sparse and long-term light curves, we seek to bridge the gap between low and high-cadence surveys' insights on YSO variability. Methods: We piloted the application of the asymmetry (M) and periodicity (Q) variability metrics to characterise YSO variability with Gaia light curves. Through refined sample selection, we identified sources with appropriate sampling for the Q&M methodology. We used the Generalised Lomb Scargle periodogram and structure functions to infer variability timescales. Results: We successfully derived Q&M indices for ~23000 sources in the GDR3 YSO sample. These variables were then classified into eight variability morphological classes. We linked morphological classes with physical mechanisms by using H$\alpha$ as a proxy of accretion and $\alpha_\mathrm{IR}$-indices to gauge circumstellar material's presence. Conclusions: We demonstrate that the Q&M metrics can be successfully applied to Gaia's sparse time-series. We applied them to distinguish between the several variability modes. While our results are generally consistent with previous high-cadence, short-term studies, we find that GDR3's long timespan yields an enhanced variety of variability mechanisms.
astro-ph.SR↗
SED Modeling of Young Stellar Objects in the Orion Star Formation Complex
We performed a large-scale spectral energy distribution (SED) fitting analysis for young stellar objects (YSOs) in the Orion star formation complex (OSFC) to derive key physical parameters; temperature, luminosity, mass, and age, using SED models. Our goal is to establish a statistically robust characterization of the stellar population and its evolutionary state across the entire complex. We utilize a set of new radiative transfer model SEDs that span a variety of geometries and parameter spaces. These SEDs are fitted to multi-wavelength photometric data from optical to submillimeter wavelengths. We conducted SED fitting on a sample of 15,396 sources. Among these, 5,062 have at least a reliable AllWISE W3 (12~$\mu$m) detection at longer wavelengths, and 63 have sub-millimeter detections in APEX/SABOCA at 350~$\mu$m or APEX/LABOCA at 870~$\mu$m. The resulting physical parameters are cross-referenced with stellar evolutionary tracks to ensure consistency with theoretical predictions. Sources placed on the Hertzsprung-Russell diagram show distinct evolutionary sequences. The results are provided with varying levels of completeness and reliability, depending on the available data for each source. The catalog includes quality indicators such as the flux code, which represents the longest detected wavelength for each source, as well as Prob_W3 and Prob_W4 values that quantify the reliability of the AllWISE W3 and W4 detections. All results, including SED fitting outcomes, uncertainty estimates, and source metadata, are publicly available in a comprehensive CDS table. This dataset provides a statistically significant view of the evolutionary processes within the OSFC. The publicly accessible dataset offers a valuable resource for future studies on star and planet formation.
astro-ph.SR↗
Jordan-H\"older property for shifted quantum affine algebras
We prove that finite length representations of shifted quantum affine algebras in category $\mathcal{O}^{\mathrm{sh}}$ are stable by fusion product. This implies that in the topological Grothendieck ring $K_0(\mathcal{O}^{\mathrm{sh}})$ the Grothendieck group of finite length representations forms a non-topological subring. We also conjecture this subring is isomorphic to the cluster algebra discovered in arXiv:2401.04616. In the course of our proofs, we establish that any simple representation in category $\mathcal{O}^{\mathrm{sh}}$ descends to a truncation, for certain truncation parameters as conjectured in arXiv:2010.06996 in terms of Langlands dual $q$-characters.
math.QA↗
Symmetries of Grothendieck rings in representation theory
This is a written version of the invited lecture at the 9th European Congress of Mathematics in July 2024 in Sevilla. We review certain new symmetries of Grothendieck rings that have emerged in representation theory.
math.RT↗
The new Herschel/PACS Point Source Catalogue
Herschel operated as an observatory, therefore it did not cover the whole sky, but still observed ~8% of it. The first version of an overall Herschel/PACS Point Source Catalogue was released in 2017. The data are still unique and are very important for research, especially because no new far-infrared mission is foreseen for at least the next decade. In the framework of the NEMESIS project, we revisited all the photometric observations obtained by the PACS instrument on-board of the Herschel space observatory. We aimed to build the most complete and most accurate Herschel/PACS catalogue to date. Our primary goal was to increase the number of real sources, and decrease the number of spurious sources identified on a strongly variable background. Our goal was to build a blind catalogue, meaning that source extraction is conducted without relying on prior detections at various wavelengths, allowing us to detect sources never catalogued before. We define a hybrid strategy that includes classical and ML source identification and characterisation methods, providing catalogues at much higher completeness levels than before. Quality assessment also involves ML techniques. Our source extraction methodology facilitates a systematic and impartial comparison of sensitivity levels across various Herschel fields, a task that was typically beyond the scope of individual programs. We created a high-reliability and a rejected source catalogue for each PACS passband, i.e., at 70, 100, and 160 {\mu}m. With the high-reliability catalogue, we managed to significantly increase the completeness in all bands. At the same time, while the number of high-reliability detections decreased, the number of sources matching with existing catalogues increased, suggesting that the purity is also higher than before. The photometric accuracy of our pipeline is ~1% based on comparison with the standard star models.
astro-ph.IM↗
Monoidal Jantzen filtrations
We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal abelian categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum loop algebras, we prove the associativity and we establish that the resulting quantization coincides with the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in a geometric manner. Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring. As a second main example, we establish an analogous result for a monoidal category of finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated with a Weyl group element. We obtain various applications, in particular on the homological structure of representations.
math.RT↗
Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case
We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories $\mathcal{O}_\mathbb{Z}$ of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called $QQ$-system. We conjecture that all cluster monomials are classes of simple objects of $\mathcal{O}_\mathbb{Z}$. In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply-connected algebraic group. This explains the similarity between $QQ$-system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.
math.QA↗
Extended Baxter relations and QQ-systems for quantum affine algebras
Generalized Baxter's TQ-relations and the QQ-system are systems of algebraic relations in the category O of representations of the Borel subalgebra of the quantum affine algebra U_q(g^), which we established in our earlier works arXiv:1308.3444 and arXiv:1606.05301. In the present paper, we conjecture a family of analogous relations labeled by elements of the Weyl group W of g, so that the original relations correspond to the identity element. These relations are closely connected to the W-symmetry of q-characters established in arXiv:2211.09779. We prove these relations for all w in W if g has rank two, and we prove the extended TQ-relations if w is a simple reflection. We also generalize our results and conjectures to the shifted quantum affine algebras.
math.QA↗
Isomorphisms among quantum Grothendieck rings and cluster algebras
We establish a cluster theoretical interpretation of the isomorphisms of [F.-H.-O.-O., J. Reine Angew. Math., 2022] among quantum Grothendieck rings of representations of quantum loop algebras. Consequently, we obtain a quantization of the monoidal categorification theorem of [Kashiwara-Kim-Oh-Park, arXiv:2103.10067]. We establish applications of these new ingredients. First we solve long-standing problems for any non-simply-laced quantum loop algebras: the positivity of $(q,t)$-characters of all simple modules, and the analog of Kazhdan-Lusztig conjecture for all reachable modules (in the cluster monoidal categorification). We also establish the conjectural quantum $T$-systems for the $(q,t)$-characters of Kirillov-Reshetikhin modules. Eventually, we show that our isomorphisms arise from explicit birational transformations of variables, which we call substitution formulas. This reveals new non-trivial relations among $(q, t)$-characters of simple modules.
math.RT↗
VISIONS: The VISTA Star Formation Atlas -- I. Survey overview
VISIONS is an ESO public survey of five nearby (d < 500 pc) star-forming molecular cloud complexes that are canonically associated with the constellations of Chamaeleon, Corona Australis, Lupus, Ophiuchus, and Orion. The survey was carried out with VISTA, using VIRCAM, and collected data in the near-infrared passbands J, H, and Ks. With a total on-sky exposure time of 49.4 h VISIONS covers an area of 650 deg$^2$, and it was designed to build an infrared legacy archive similar to that of 2MASS. Taking place between April 2017 and March 2022, the observations yielded approximately 1.15 million images, which comprise 19 TB of raw data. The observations are grouped into three different subsurveys: The wide subsurvey comprises shallow, large-scale observations and has visited the star-forming complexes six times over the course of its execution. The deep subsurvey of dedicated high-sensitivity observations has collected data on the areas with the largest amounts of dust extinction. The control subsurvey includes observations of areas of low-to-negligible dust extinction. Using this strategy, the VISIONS survey offers multi-epoch position measurements, is able to access deeply embedded objects, and provides a baseline for statistical comparisons and sample completeness. In particular, VISIONS is designed to measure the proper motions of point sources with a precision of 1 mas/yr or better, when complemented with data from VHS. Hence, VISIONS can provide proper motions for sources inaccessible to Gaia. VISIONS will enable addressing a range of topics, including the 3D distribution and motion of embedded stars and the nearby interstellar medium, the identification and characterization of young stellar objects, the formation and evolution of embedded stellar clusters and their initial mass function, as well as the characteristics of interstellar dust and the reddening law.
astro-ph.GA↗
VISIONS: The VISTA Star Formation Atlas -- II. The data processing pipeline
The VISIONS public survey provides large-scale, multiepoch imaging of five nearby star-forming regions at subarcsecond resolution in the near-infrared. All data collected within the program and provided by the European Southern Observatory (ESO) science archive are processed with a custom end-to-end pipeline infrastructure to provide science-ready images and source catalogs. The data reduction environment has been specifically developed for the purpose of mitigating several shortcomings of the bona fide data products processed with software provided by the Cambridge Astronomical Survey Unit (CASU), such as spatially variable astrometric and photometric biases of up to 100 mas and 0.1 mag, respectively. At the same time, the resolution of coadded images is up to 20% higher compared to the same products from the CASU processing environment. Most pipeline modules are written in Python and make extensive use of C extension libraries for numeric computations, thereby simultaneously providing accessibility, robustness, and high performance. The astrometric calibration is performed relative to the Gaia reference frame, and fluxes are calibrated with respect to the source magnitudes provided in the Two Micron All Sky Survey (2MASS). For bright sources, absolute astrometric errors are typically on the order of 10 to 15 mas and fluxes are determined with subpercent precision. Moreover, the calibration with respect to 2MASS photometry is largely free of color terms. The pipeline produces data that are compliant with the ESO Phase 3 regulations and furthermore provides curated source catalogs that are structured similarly to those provided by the 2MASS survey.
astro-ph.IM↗
Weyl group symmetry of q-characters
We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of W-invariants of Y is precisely the ring of q-characters, which is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of U_q(g^). This resolves an old puzzle in the theory of q-characters. We also identify the screening operators, which were previously used to describe the ring of q-characters, as the subleading terms of simple reflections from W in a certain limit. Our results have already found applications to the study of the category O of representations of the Borel subalgebra of U_q(g^) in arXiv:2312.13256 and to the categorification of cluster algebras in arXiv:2401.04616.
math.QA↗
Isomorphisms among quantum Grothendieck rings and propagation of positivity
Let ($\mathfrak{g},\mathsf{g})$ be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with $\mathsf{g}$ being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories $\mathscr{C}_{\mathfrak{g}}$ and $\mathscr{C}_{\mathsf{g}}$ of finite-dimensional representations over the quantum loop algebras of $\mathfrak{g}$ and $\mathsf{g}$ respectively. As a consequence, we solve long-standing problems : the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced $\mathfrak{g}$. In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [H., Adv. Math., 2004]) for simple modules in remarkable monoidal subcategories of $\mathscr{C}_{\mathfrak{g}}$ for any non-simply-laced $\mathfrak{g}$, and for any simple finite-dimensional modules in $\mathscr{C}_{\mathfrak{g}}$ for $\mathfrak{g}$ of type $\mathrm{B}_n$. In the course of the proof we obtain and combine several new ingredients. In particular we establish a quantum analog of $T$-systems, and also we generalize the isomorphisms of [H.-Leclerc, J. Reine Angew. Math., 2015] and [H.-O., Adv. Math., 2019] to all $\mathfrak{g}$ in a unified way, that is isomorphisms between subalgebras of the quantum group of $\mathsf{g}$ and subalgebras of the quantum Grothendieck ring of $\mathscr{C}_\mathfrak{g}$.
math.RT↗