Representation Theory

As corollaries, we establish geometric criteria for finiteness of the dimension of $\Hom_G(\pi,\Ind_H^G \tau)$ (induction) and of $\Hom_H(\pi|_H,\tau)$ (restriction) by means of the real flag variety $G/P$, and criteria for uniform boundedness of these multiplicities by means of the complex flag variety.

Resumo: O conceito de "mônada" é vasto e amplo. Neste ensaio, será apresentado o conceito de "mônada" leibniziano ([1714] (2016)). Para Leibniz, a mônada é uma parte, bem como o universo inteiro. Porém, aqui, será apresentado um único ponto da sua monadologia, a saber, a mônada in potentia (em potência). Ou seja, de que modo estas substâncias simples se transformam em compostas. Nosso método é fenomenológico. E o objetivo é clarear um pouco mais sobre a formação destas substâncias.

Resumo: Leibniz escreveu uma série de manuscritos, dentre eles, a sua obra principal-a "Monadologia"-que é um estudo aprofundado do que ele descobriu e chamou de "mônada". Para tanto, este artigo tem o intuito de apresentar, em linhas gerais, a teoria das mônadas de Leibniz. Em um primeiro momento, apresentamos o conceito capital de "mônada"-"como um universo inteiro". E, em segundo momento, apresentamos os dois tipos de substâncias simples-as "finitas e irrefletidas"-e-as "infinitas e refletidas". Por fim, concluímos, de modo inicial, que a mônada leibniziana é um conceito puramente abstrato, universal e que pode ser representativo para as várias áreas do saber, incluindo, a filosofia.

Resumo: Este estudo trata, de modo aprofundado, da concepção de mônada em Leibniz. Concernente ao conceito de mônada em Leibniz, em um primeiro momento, abrimos este artigo/ensaio, apresentando a natureza incorpórea das substâncias simples individuais. Em um segundo momento, tratamos da representação in conceptus de uma mônada na outra. No terceiro momento, apresentamos uma das características fundamentais das mônadas; a sua capacidade de espelhamento (speculum). Num quarto momento, discutimos o aspecto de "Deus ser o conjunto infinito de substâncias simples. Em um quinto momento, tratamos do conceito de "metempsicose das substâncias". E, por fim, em um último momento, com o intuito de concluir este ensaio, trazemos a questão de como o intellectus age animando razão de uma forma substancial.

For any normal commutative Hopf subalgebra K = k G of a semisimple Hopf algebra we describe the ring inside kG obtained by the restriction of H-modules. If G = Z p this ring determines a fusion ring and we give a complete description for it. The case G = Z p n and some other applications are presented.

In this work, we construct fundamental domains for congruence subgroups of SL2 (Fq[t]) and PGL2 (Fq[t]). Our method uses Gekeler's description of the fundamental domains on the Bruhat-Tits tree X = Xq+1 in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.

We prove a classification-level result for compact connected Lie groups acting on C 3. Under three independently motivated assumptions-a determinant constraint, the presence of a cyclic 3-symmetry, and operational distinguishability of chirally conjugate states-the only compact connected Lie group admitting a faithful irreducible unitary representation on C 3 is SU(3). The operational distinguishability assumption is formulated via an Information Preservation Principle (IPP): states related by an antiunitary intertwiner should remain distinguishable within the physically admissible observable algebra. We argue that the IPP motivates the requirement that the representation be complex (V ̸ ∼ = V), eliminating SU(2) (whose 3-dimensional adjoint is real) and leaving SU(3) as the unique solution. The proof proceeds via the Peter-Weyl structure theorem, the Weyl dimension formula, and a center-quotient argument. A minimality corollary characterizes SU(3) as the minimal compact Lie group supporting a faithful irreducible unitary action on C 3 together with a cyclic permutation symmetry and a genuinely complex representation.

This study interrogates the algorithmically mediated construction of khawaja sira (transgender) identity within Pakistan’s rapidly evolving YouTube ecosystem. Moving beyond legacy broadcast stereotypes, it explores how digital platforms simultaneously amplify transphobic comedic tropes and facilitate emergent regimes of socio-political visibility. Anchored in Stuart Hall’s theory of representation and Robert Entman’s framing theory, the research employs qualitative content analysis to examine eight purposively selected high-engagement videos spanning entertainment and socio-political genres.

The analysis identifies a deep representational bifurcation. Entertainment-oriented content reproduces “Comedic Othering,” wherein trans bodies are rendered intelligible through exaggerated performance, linguistic ridicule, and spectacle-driven engagement, reinforcing normative heteropatriarchal ideologies while optimizing algorithmic visibility. In contrast, socio-political discourse enacts “Socio-Political Humanization” through constitutional framing, rights-based narratives, and intellectual discourse that repositions transgender subjects as political agents rather than objects of humor.

The findings suggest that Pakistani YouTube operates as a contradictory ideological apparatus—simultaneously reproducing structural violence and enabling counter-hegemonic articulation. The study underscores the urgency of situated digital governance frameworks and ethical content regulation to address algorithmic exploitation and representational harm in the Global South.

Keywords: Representation, Framing, Algorithmic Culture, Transgender Studies, Pakistani YouTube, Digital Inequality

Below is a summary of the principal discoveries of each investigation within the TCFQ programme, from Note R178 through R185, presented in the requested order:
R178. Establishment of the symbolic involutivity of the projected Grassmannian jet tower associated with idempotent operators. It achieves the identification of the first universal non‑linear correction ($\Psi_{i,j}$) and the exact computation of the first diagonal defects ($D_{ij}, D_{ijk}$), demonstrating that symbolic involutivity is independent of effective local realizability.
R179. Proof of universal Gram saturation for generic exponential models of any rank ($r \geq 2$), confirming the absence of linear or algebraic obstructions to quadratic closure. It also establishes by induction the universal formula for all diagonal defects ($D_\alpha$), thereby validating the structure of the jet tower.
R180. Identification of the exact symmetry of the constitutive operator ($\hat{H}\nu = \hat{H}{-\nu}$) and its scale invariance, which places the infrared scale $m_*$ as an external parameter. It localises the symmetry breaking specifically within the spectral coefficient $D_0(\nu)$, an essential step for mass generation in the theory.
R181. Localisation of the first carrier of asymmetry in the McMahon asymptotic shift ($c_\nu = \nu/2 - 1/4$), which arises from the phase of Bessel functions. It establishes the “even‑parity barrier”, proving mathematically that the odd structure required for $D_0(\nu)$ cannot originate directly from the operator.
R182. Determination that the Friedrichs extension constitutively selects the regular branch ($J_\nu$) of the spectral problem by excluding the singular solution because it requires infinite energy. It further proves the uniqueness of the Dirichlet condition as the only local self‑adjoint extension compatible with the observed Bessel‑zero spectrum.
R183. Exact numerical computation of the primary spectral object $D_0^{\text{exact}}(\nu)$ via a sum over Bessel zeros with rigorous error control. It discovers that this observable is strictly decreasing with respect to the parameter $\nu$, enabling precise evaluation of derived physical constants.
R184. Constitutional first‑principles reconstruction of the relation $D_0(\nu) = \frac{1}{2}\sum j_{n,\nu}^{-3}$ via the Taylor expansion of the spectral generating function $\Psi_\nu$. It explains that the polygamma structure central to the programme emerges naturally from the McMahon linearisation of the spectrum.
R185. Confirmation that the Spencer compatibility identities generate effective differential constraints (Scenario B), overcoming the possibility that they might be mere tautologies. It establishes the definitive structural bridge between the Grassmannian projector geometry and the spectral observable $D_0(\nu)$ as a closure condition of the system.

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described in . After that we describe some linear dynamical systems in K [[x]] with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to the Pascal Triangle.

In this note, we mainly study the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities. By using the generalized Hölder inequality for symmetric gauge functions, we obtain a more general version of a norm inequality for positive real numbers α, β, γ and r satisfying 1 α + 1 β = 1 γ and r ⩾ max{ 1 α , 1 β }. The obtained result generalizes the inequalities of

In this note, we mainly study the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities. By using the generalized Hölder inequality for symmetric gauge functions, we obtain a more general version of a norm inequality for positive real numbers α, β, γ and r satisfying 1 α + 1 β = 1 γ and r ⩾ max{ 1 α , 1 β }. The obtained result generalizes the inequalities of

La prueba de la democracia no es que el pueblo vote, sino que gobierne."-G. K. Chesterton La paradoja latinoamericana Hay preguntas que se vuelven más incómodas cuanto más tiempo permanecen sin respuesta. Una de ellas recorre América Latina, por no hablar aquí del mundo, desde hace décadas: ¿cómo es posible que una región que celebra elecciones de manera regular siga atrapada entre la corrupción, la fragilidad institucional, la desconfianza ciudadana y la inestabilidad política? Las urnas abundan. La democracia escasea. Sobre el papel, la región parece haber resuelto el problema democrático. Constituciones modernas, elecciones competitivas, alternancia en el poder, congresos representativos, tribunales constitucionales y sistemas multipartidistas conforman un paisaje institucional comparable al de muchas democracias consolidadas. No obstante, la realidad se obstina en desmentir la apariencia. Detrás de la arquitectura formal persisten Estados débiles, violentados por cualquier fenómeno de tráfico, narco u otros, sistemas judiciales vulnerables, corrupción estructural y una ciudadanía cada vez más desencantada y menos ciudadana. La distancia entre lo que las instituciones prometen y lo que efectivamente producen se ha convertido en una de las características centrales de la experiencia latinoamericana. La paradoja es tan evidente como persistente: las formas democráticas prosperan mientras su contenido se debilita y se pierde. Francis Fukuyama sostiene que toda democracia estable descansa sobre tres pilares fundamentales: un Estado competente, un imperio efectivo de la ley y mecanismos sólidos de rendición de cuentas. Cuando esos tres elementos convergen, la democracia deja de ser un simple procedimiento electoral y se convierte en un orden político legítimo y previsible. En gran parte de América Latina ocurre exactamente lo contrario. El Estado suele ser incapaz de aplicar las normas que proclama. La ley se ejerce de manera desigual. La rendición de cuentas se diluye entre clientelas, pactos de impunidad y redes de intereses que sobreviven a los cambios de gobierno. El problema no es la ausencia de democracia. El problema es su formalismo caricaturesco y vacío. Durante décadas se creyó que la celebración periódica de elecciones bastaría para consolidar instituciones republicanas. La experiencia demuestra lo contrario. Las elecciones son una condición necesaria de la democracia, pero están lejos de ser una condición suficiente. Una sociedad puede votar regularmente y seguir siendo políticamente precaria. Puede cambiar de gobernantes sin cambiar de prácticas. Puede celebrar elecciones impecables y continuar gobernada por mecanismos abstrusos e informales. La democracia puede existir como procedimiento mientras fracasa como proyecto de autogobierno, independientemente de que esta sea republicano o de otra índole. Ahí reside una de las grandes contrariedades latinoamericanas. Instituciones sin ciudadanía Las instituciones no flotan en el vacío. Douglass North explicó que las reglas formales solo funcionan cuando encuentran respaldo en hábitos, valores y expectativas compartidas. Cuando las normas escritas y las prácticas reales se

We construct radial dual lattice graphs for the Eisenstein, Hurwitz, and E 8 lattices using admissible hyperspherical inversion. The inversion induces exact bijections between outer zone vertices and rational inner zone representatives, and it gives transported-edge graph isomorphisms once the radialdual edge relation is defined. We verify the norm relation, involution, shell compression, and finiteshell adjacency identities using exact arithmetic. Composing the radial inversion with the Moxness E 8-to-H 4 folding matrix H 4fold gives a candidate golden linear-radial dual compressor Υ r for Cycle Clock Theory (CCT) workflows; on the E 8 root shell, the top 4 × 8 projection block Π of H 4fold maps the 240 roots into two 120-point layers (the regular 600-cell H 4 and its golden-ratio scaled copy H 4 Φ, with radius ratio Φ, equivalently squared-norm ratio Φ 2). For larger shells, this compressor is validated on finite domains and proposed as a proof target for full cycle-clock enumeration. The framework aligns naturally with the classical divisor-sum expression c 8 (n) = 240 σ 3 (n) for E 8 shell multiplicities and makes it computationally useful for folded inner zone enumeration: our benchmark reports 83 000-260 000× speedups over exact-arithmetic Jacobi theta polynomial expansion at moderate shell indices. Realized via the Cayley integers in 8D, the construction aligns conceptually with octonionbased models in quasicrystalline quantum gravity while remaining strictly algebraic and geometric, operating on rational extensions without approximations, floating-point drift, continuous relaxations, or information loss. The construction offers a practical exact-arithmetic method for shelling and scaling calculations, while full global injectivity of Υ r on L 8 , 8D-to-4D graph-isomorphism preservation across arbitrary shells, and end-to-end CCT simulation integration remain proof obligations for future work. The entire construction is formulated inside the real Clifford algebra Cl(8) as the enveloping associative algebra: A 2 , D 4 , and E 8 are realized as the grade-1 root subsystems of Cl(2), Cl(4), and Cl(8) respectively, and the classical Eisenstein, Hurwitz, and Cayley integer rings are recovered as the even subalgebras Cl + (d) acting on these generators. Under this enveloping algebra we adopt the uniform packing-radius root-length convention ⟨α, α⟩ = 2 (Euclidean length √ 2) across all three lattices, in agreement with the Bourbaki/Conway-Sloane/Viazovska normalization and with the maximally dense sphere-packing radius in each dimension; the canonical admissible inversion radius is therefore r = √ 2 uniformly, and the hyperspherical inversion ι r is Clifford-equivariant under Pin(d) ⊂ Cl(d).

This paper explores the enduring perception of Soltaniyeh’s magnificence in Western narratives, particularly in connection to its grand mausoleum (1302-12), which boasts the largest dome in Iran. Strategically positioned along the Silk Road, linking Tabriz to the central Iranian plateau, Soltaniyeh served as the Ilkhanate Empire’s summer capital for a brief period (1306–1335) before its decline due to the Timurid invasion and the Black Death.
My research highlights why the city was well-known in Europe, focusing on its diverse international community—primarily Italians—who resided in Soltaniyeh before its decline. It also examines how the city’s brief prominence persisted in Western accounts for centuries. Soltaniyeh features prominently in maps such as the Catalan Atlas (c. 1375) and Fra Mauro’s map (c. 1450), as well as in travelogues written by ambassadors, clergy, and merchants. These sources reveal the rapid erosion of historical knowledge about the city and its mausoleum, with both locals and visitors often perpetuating contradictory narratives.
The architectural significance of the monumental dome was also the key factor in renewed interest among archaeologists and architects from the late 19th century onward. These international efforts ultimately played a crucial role in revitalizing the site’s importance for Iranian institutions, eventually leading to Soltaniyeh’s designation as a UNESCO World Heritage Site.

This paper revisits a remark by Wittgenstein that the statement "That's him" contains the whole problem of representation. It argues that the difficulty does not lie in resemblance, perception, or internal representation, but in how something comes to be taken or treated as representing something else within a practice. By examining familiar cases-from portraits to memory images and mediated interaction-the paper suggests that representation is not secured by any intrinsic relation between image and object, but by the ways in which appearances are used and recognised. The result is a clarification of the limits of resemblance-based and representational accounts, and a reframing of representation in terms of shared practices rather than internal structures or perceptual equivalences.

This paper examines a basic constraint on the relation between representation and perception. It argues that experiencing a representation of X is not a way of perceiving X, and that this holds independently of the richness, realism, or interactivity of the representation. From this it follows that accounts of perception framed in purely representational terms cannot, by themselves, explain perceptual contact with the world. The aim is not to deny the importance of representation, but to clarify its limits and to distinguish it from perception as ordinarily understood.

If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig * to be in the wave front set of Ind G H τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.