Exceptional Groups Research Articles

Dirac cohomology, branching laws and Wallach modules

The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) [11]. One of their results says that the Dirac cohomology of π completely determines π|K, where π is any irreducible unitarizable highest weight (g,K) module. This paper aims to develop this idea for the exceptional Lie groups E6(−14) and E7(−25): we recover the K-spectrum of the Wallach modules from their Dirac cohomology.

A Cubic Pure Half-Spinor Approach to High Performance Photon-Counting Computerized X-ray Tomography – Transcendence of the Brauer Mirror Symmetry Functor –

The goal of this paper is to present a spectral theory of advanced photon-counting computerized X-ray tomography, which is a promising microelectronical semiconductor wafer X-ray technique on the verge of becoming clinically feasible and has the potential to dramatically alter the clinical application of computer-aided medical diagnosis in the upcoming decades. In analogy to the deep fact that any real division algebra has dimension 1, 2, 4 or 8, the isogeneous third Galois cohomological, mirror symmetrical approach to high spatial resolution spectral photon-counting computerized tomography establishes that the exceptional Lie group Spin(8,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spin}(8,\\mathbb {R})$$\\end{document} is the only one of the spin group spectrum {Spin(n,R)∣n≥1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\ extrm{Spin}(n,\\mathbb {R})\\mid n \\ge 1\\}$$\\end{document} to admit a triality automorphism. It is the exceptional phenomenon of triality which is able to produce by the spin representation theory of the exceptional Lie group of octonions O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {O}$$\\end{document} the spectral shaped digital signals of spectral photon-counting computerized X-ray tomography. Similarly to spin echo magnetic resonance tomography, the mathematical approach to spectral high performance computerized tomography is based on the isogeneously invariant, projectively weighted, symplectic coadjoint orbit picture Lie(N)∗/CoAd(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {Lie}(\\mathcal {N})^*/\ extrm{CoAd}(\\mathcal {N})$$\\end{document} of the real 3-dimensional Heisenberg unipotent Lie group N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} of type GL(3,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{GL}(3,\\mathbb {R})$$\\end{document}, and its solvable diamond toric Lie group extension.

Gold-catalysed amine synthesis by reductive hydroamination of alkynes with nitroarenes.

Amines are the most pivotal class of organic motifs in pharmaceutical compounds. Here we provide a blueprint for a general synthesis of amines by catalyst differentiation enabled by triple Au-H/Au+/Au-H relay catalysis. The parent catalyst is differentiated into a set of catalytically active species to enable triple cascade catalysis, where each catalytic species is specifically tuned for one catalytic cycle. This strategy enables the synthesis of biorelevant amine motifs by reductive hydroamination of alkynes with nitroarenes. Using this triple cascade approach, we have achieved exceptional functional group tolerance, enabling the use of bulk chemical feedstocks as coupling partners for the amination of both simple and complex alkynes (>100 examples), including those derived from pharmaceuticals, peptides and natural products (>30 examples). The isolation and full crystallographic characterization of gold hydride and hydride-bridged gold complexes has garnered insights into the catalyst differentiation process of fundamental organometallic gold hydride complexes.

Affine Grassmannians for G2

We study affine Grassmannians for the exceptional group of type G2. This group can be given as automorphisms of octonion algebras (or para-octonion algebras). By using this automorphism group, we consider all maximal parahoric subgroups in G2, and give a description of affine Grassmannians for G2 as functors classifying suitable orders in a fixed space.

Early language experience and modality affect parietal cortex activation in different hemispheres: Insights from hearing bimodal bilinguals

The goal of this study was to investigate the impact of the age of acquisition (AoA) on functional brain representations of sign language in two exceptional groups of hearing bimodal bilinguals: native signers (simultaneous bilinguals since early childhood) and late signers (proficient sequential bilinguals, who learnt a sign language after puberty). We asked whether effects of AoA would be present across languages – signed and audiovisual spoken – and thus observed only in late signers as they acquired each language at different life stages, and whether effects of AoA would be present during sign language processing across groups. Moreover, we aimed to carefully control participants’ level of sign language proficiency by implementing a battery of language tests developed for the purpose of the project, which confirmed that participants had high competences of sign language.Between-group analyses revealed a hypothesized modulatory effect of AoA in the right inferior parietal lobule (IPL) in native signers, compared to late signers. With respect to within-group differences across languages we observed greater involvement of the left IPL in response to sign language in comparison to spoken language in both native and late signers, indicating language modality effects. Overall, our results suggest that the neural underpinnings of language are molded by the linguistic characteristics of the language as well as by when in life the language is learnt.

𝑝-adic limit of the Eisenstein series on the exceptional group of type 𝐸_{7,3}

In this paper, we show that the p p -adic limit of a family of Eisenstein series on the exceptional domain where the exceptional group of type E 7 , 3 E_{7,3} acts is an ordinary modular form for a congruence subgroup.

The Vinyl Group: Small but Mighty – Transition Metal Catalyzed and Non‐Catalyzed Vinylation Reactions

AbstractTransition metal catalysis allows for the efficient and selective introduction of vinyl moieties onto organic molecules and offers a versatile approach to synthesizing complex organic molecules. Nonetheless, few transition‐metal free vinylation reactions have also been reported, exhibiting exceptional functional group tolerance and circumventing selectivity issues. This review provides an overview of carbon, nitrogen and oxygen vinylation, presenting innovative strategies and key advancements in the field. Hence, it will serve as a valuable resource for organic chemists who are interested in the synthesis of vinyl‐containing compounds. By understanding the diverse strategies involved in vinylation, readers will gain insights into harnessing this powerful synthetic methodology for the efficient construction of carbon‐carbon and carbon‐heteroatom bonds.

Counting conjugacy classes of elements of finite order in exceptional Lie groups

Counting conjugacy classes of elements of finite order in exceptional Lie groups

Enforcing group fairness in privacy-preserving Federated Learning

With the expansive application of Federated Learning (FL) in critical domains such as healthcare and financial services, ensuring group fairness in FL has become imperative. The intrinsic distributed architecture of FL, while enhancing training efficiency, introduces significant challenges in maintaining group fairness due to client data privacy concerns and heterogeneous data distribution. For this reason, we present a suite of innovative strategies to navigate these challenges. First, we construct an approximate global dataset to achieve global group fairness. It was accomplished by collecting statistical information from clients and generating an approximate global dataset that circumvents the direct sharing of raw data. Moreover, we present the Local Sensitive Hashing-Categorical Attribute Data Completion (LSH-CADC) method, which improves the precision and efficiency of the generation of the approximate global dataset. In addition, an adaptive fairness-weighted aggregation (AFA) algorithm was developed to enhance the model’s group fairness. By integrating these methodologies within the classical privacy-preserving FL method HybridAlpha, we get a new framework, enforcing group fairness in privacy-preserving Federated Learning (GFL). GFL not only ensures privacy but also solidifies group fairness. Through extensive experimentation in various cross-dataset/domain non-IID scenarios, GFL demonstrated exceptional performance and group fairness metrics, surpassing existing benchmarks and highlighting its significant potential to promote group fairness.

C(sp3)-H (N-Phenyltetrazole)thiolation as an Enabling Tool for Molecular Diversification.

Methods enabling the broad diversification of C(sp3)-H bonds from a common intermediate are especially valuable in chemical synthesis. Herein, we report a site-selective (N-phenyltetrazole)thiolation of aliphatic and (hetero)benzylic C(sp3)-H bonds using a commercially available disulfide to access N-phenyltetrazole thioethers. The thioether products are readily elaborated in diverse fragment couplings for C-C, C-O, or C-N construction. The C-H functionalization proceeds via a radical-chain pathway involving hydrogen atom transfer by the electron-poor N-phenyltetrazolethiyl radical. Hexafluoroisopropanol was found to be essential to reactions involving aliphatic C(sp3)-H thiolation, with computational analysis consistent with dual hydrogen bonding of the N-phenyltetrazolethiyl radical imparting increased radical electrophilicity to facilitate the hydrogen atom transfer. Substrate is limiting reagent in all cases, and the reaction displays an exceptional functional group tolerance well suited to applications in late-stage diversification.

C(sp3)−H (N‐Phenyltetrazole)thiolation as an Enabling Tool for Molecular Diversification

AbstractMethods enabling the broad diversification of C(sp3)−H bonds from a common intermediate are especially valuable in chemical synthesis. Herein, we report a site‐selective (N‐phenyltetrazole)thiolation of aliphatic and (hetero)benzylic C(sp3)−H bonds using a commercially available disulfide to access N‐phenyltetrazole thioethers. The thioether products are readily elaborated in diverse fragment couplings for C−C, C−O, or C−N construction. The C−H functionalization proceeds via a radical‐chain pathway involving hydrogen atom transfer by the electron‐poor N‐phenyltetrazolethiyl radical. Hexafluoroisopropanol was found to be essential to reactions involving aliphatic C(sp3)−H thiolation, with computational analysis consistent with dual hydrogen bonding of the N‐phenyltetrazolethiyl radical imparting increased radical electrophilicity to facilitate the hydrogen atom transfer. Substrate is limiting reagent in all cases, and the reaction displays an exceptional functional group tolerance well suited to applications in late‐stage diversification.

Maximal subgroups of exceptional groups and Quillen’s dimension

Maximal subgroups of exceptional groups and Quillen’s dimension

The maximal Tori of finite exceptional groups of Lie type

Let G(q) be a twisted or untwisted finite exceptional group of Lie type over GF ( q ) , either adjoint or simply connected, and let W be the Weyl group of G(q). This paper compiles an extensive data set on the maximal tori of G(q). A systematic labeling is given, via the corresponding conjugacy classes of W, together with a complete list of the structures of the maximal tori in terms of their torsion coefficient decomposition. Additionally the associated admissible graphs and the structure or shape of centralizers of elements in W are given as well as compatible computer files. Also various apparent inconsistencies in the current literature on simply connected groups are discussed and resolved.

Synthesis of Methyl N-phenylcarbamate Derivatives by Xphos Pd G2 Catalyzed Intermolecular Amidation Reaction

Abstract: The utilization of palladium catalysts in cross-coupling reactions has emerged as a highly promising method for the facile formation of aryl C-N bonds, operating under mild conditions. In this study, we present an efficient approach for the synthesis of methyl N-phenyl carbamate derivatives through the intermolecular amidation of aryl chlorides, catalyzed by Xphos Pd G2. The developed protocol has demonstrated remarkable efficacy, offering several advantages. Notably, the intermolecular amidation reaction exhibited good chemoselectivity, allowing for the precise targeting of desired C-N bond formations while maintaining the integrity of other functional groups. Additionally, this methodology showcases exceptional functional group compatibility, accommodating a diverse array of moieties, including sensitive groups that are traditionally challenging to handle. The Xphos Pd G2 catalyst has proven to be instrumental in orchestrating this transformation, exhibiting high catalytic activity and selectivity. Furthermore, this protocol stands out for its operational simplicity, making it a practical choice for synthetic chemists seeking a straightforward and reliable route to access methyl N-phenyl carbamate derivatives. Overall, this study not only expands the synthetic toolbox for C-N bond formations, but also underscores the significance of palladium-catalyzed methodologies in modern organic synthesis. The reported findings hold substantial promise for applications in medicinal chemistry and material science, where the facile construction of aryl C-N bonds is of paramount importance.

5d Conformal matter

Six-dimensional superconformal field theories (SCFTs) have an atomic classification in terms of elementary building blocks, conformal systems that generalize matter and can be fused together to form all known 6d SCFTs in terms of generalized 6d quivers. It is therefore natural to ask whether 5d SCFTs can be organized in a similar manner, as the outcome of fusions of certain elementary building blocks, which we call 5d conformal matter theories. In this project we begin exploring this idea and we give a systematic construction of 5d generalized “bifundamental” SCFTs, building from geometric engineering techniques in M-theory. In particular, we find several examples of (e6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_6 $$\\end{document}, e6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_6 $$\\end{document}), (e7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_7 $$\\end{document}, e7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_7 $$\\end{document}) and (e8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_8 $$\\end{document}, e8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_8 $$\\end{document}) 5d bifundamental SCFTs beyond the ones arising from (elementary) KK reductions of the 6d conformal matter theories. We show that these can be fused together giving rise to 5d SCFTs captured by 5d generalized linear quivers with exceptional gauge groups as nodes, and links given by 5d conformal matter. As a first application of these models we uncover a large class of novel 5d dualites, that generalize the well-known fiber/base dualities outside the toric realm.

Collineation groups of octonionic and split-octonionic planes

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley–Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type [Formula: see text] and [Formula: see text].

The resurgence of the G(2) group for the strong sector and the emergence of dark matter

G(2) is the smallest exceptional group and it is the simplest and viable gauge group to minimally extend the strong interaction sector: G(2) includes the group SU(3) of Quantum Chromodynamics (QCD) as a maximal subgroup and it is equipped with six additional gluons that can acquire mass via a Higgs mechanism driven by a new Higgs particle and constitute dark matter. In this article I want to describe how the exceptional G(2) group can be a physical gauge group, capable of extending the Standard Model (SM) of particles and including a versatile dark sector, which is compatible with experimental observations. In fact, due to its peculiar mathematical features, the group G(2) manifests some complex features, not properly considered in literature, which guarantee its correct use in physics, as its {3}⊕{3‾} decompositions w.r.t. SU(3) can acquire a complex structure. The resulting framework can be a solid Beyond Standard Model (BSM) solution for the dark matter (DM) problem, in the form of massive complex scalar glueballs, and it includes the proper color representations for quarks and leptons. Several quantum field theory features are discussed, like the G(2) coupling constant running and its spectrum before and after the phase transition, along with all the DM astrophysical realizations, in order to present the unexpected potential of this gauge group.

A specific natural killer cells phenotypic signature associated to long term elite control of HIV infection.

Elite controllers (ECs) are an exceptional group of people living with HIV (PLWH) that control HIV replication without therapy. Among the mechanisms involved in this ability, natural killer (NK)-cells have recently gained much attention. We performed an in-deep phenotypic analysis of NK-cells to search for surrogate markers associated with the long term spontaneous control of HIV. Forty-seven PLWH (22 long-term EC [PLWH-long-term elite controllers(LTECs)], 15 noncontrollers receiving antiretroviral treatment [ART] [PLWH-onART], and 10 noncontrollers cART-naïve [PLWH-offART]), and 20 uninfected controls were included. NK-cells homeostasis was analyzed by spectral flow cytometry using a panel of 15 different markers. Data were analyzed using FCSExpress and R software for unsupervised multidimensional analysis. Six different subsets of NK-cells were defined on the basis of CD16 and CD56 expression, and the multidimensional analysis revealed the existence of 68 different NK-cells clusters based on the expression levels of the 15 different markers. PLWH-offART presented the highest disturbance of NK-cells homeostasis and this was not completely restored by long-term ART. Interestingly, long term spontaneous control of HIV (PLWH-LTEC group) was associated with a specific profile of NK-cells homeostasis disturbance, characterized by an increase of CD16dimCD56dim subset when compared to uninfected controls (UC) group and also to offART and onART groups (p < 0.0001 for the global comparison), an increase of clusters C16 and C26 when compared to UC and onART groups (adjusted p-value < 0.05 for both comparisons), and a decrease of clusters C10 and C20 when compared to all the other groups (adjusted p-value < 0.05 for all comparisons). These findings may provide clues to elucidate markers of innate immunity with a relevant role in the long-term control of HIV.

Unifying N-Sulfinylamines with Alkyltrifluoroborates by Organophotoredox Catalysis: Access to Functionalized Alkylsulfinamides and High-Valent S(VI) Analogues.

We describe an organophotoredox-catalyzed sp3 C-S coupling of N-sulfinylamines with bench-stable alkyltrifluoroborates as a latent nucleophilic counterpart en route to alkylsulfinamides in high efficiency. In contrast to the two-electron reactivity of traditional organometallic reagents, this catalytic method reports the single-electron process of an organometallic reagent with N-sulfinylamines in C-S coupling. This mild and scalable protocol offers operational simplicity and exceptional functional group compatibility, including ketone, ester, amide, nitrile, and halides, that is vulnerable to organolithium or Grignard reagents. Additionally, the sulfinamides are conveniently converted to a variety of important S(VI) compounds, like sulfonamides, sulfonimidamides, and sulfonimidates, among others.

Universal inter-molecular radical transfer reactions on metal surfaces.

On-surface synthesis provides tools to prepare low-dimensional supramolecular structures. Traditionally, reactive radicals are a class of single-electron species, serving as exceptional electron-withdrawing groups. On metal surfaces, however, such species are affected by conduction band screening effects that may even quench their unpaired electron characteristics. As a result, radicals are expected to be less active, and reactions catalyzed by surface-stabilized radicals are rarely reported. Herein, we describe a class of inter-molecular radical transfer reactions on metal surfaces. With the assistance of aryl halide precursors, the coupling of terminal alkynes is steered from non-dehydrogenated to dehydrogenated products, resulting in alkynyl-Ag-alkynyl bonds. Dehalogenated molecules are fully passivated by detached hydrogen atoms. The reaction mechanism is unraveled by various surface-sensitive technologies and density functional theory calculations. Moreover, we reveal the universality of this mechanism on metal surfaces. Our studies enrich the on-surface synthesis toolbox and develop a pathway for producing low-dimensional organic materials.