Degenerate Representations Research Articles

Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains

Let Ω=G/K be a bounded symmetric domain of non-compact type. In this paper the image of the Poisson transform on the degenerate principal series representations of G attached to the Shilov boundary of Ω is considered. We characterize the images in terms of the third-order Hua operators Uν and Wν. When Ω is the exceptional domain of type V, we give the explicit formulas for the operators Uν and Wν.

Explicit n-particle harmonic oscillator states in products of SU(1,1) representations

In this paper, we present a simple algorithm for the explicit construction of [Formula: see text]-particle harmonic oscillator states simultaneously belonging to irreducible representations of [Formula: see text] (or [Formula: see text]) and [Formula: see text]. For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with [Formula: see text] and [Formula: see text] are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the [Formula: see text] structure of Bell states.

InfoCTM: A Mutual Information Maximization Perspective of Cross-Lingual Topic Modeling

Cross-lingual topic models have been prevalent for cross-lingual text analysis by revealing aligned latent topics. However, most existing methods suffer from producing repetitive topics that hinder further analysis and performance decline caused by low-coverage dictionaries. In this paper, we propose the Cross-lingual Topic Modeling with Mutual Information (InfoCTM). Instead of the direct alignment in previous work, we propose a topic alignment with mutual information method. This works as a regularization to properly align topics and prevent degenerate topic representations of words, which mitigates the repetitive topic issue. To address the low-coverage dictionary issue, we further propose a cross-lingual vocabulary linking method that finds more linked cross-lingual words for topic alignment beyond the translations of a given dictionary. Extensive experiments on English, Chinese, and Japanese datasets demonstrate that our method outperforms state-of-the-art baselines, producing more coherent, diverse, and well-aligned topics and showing better transferability for cross-lingual classification tasks.

Picture-in-Picture Strategy-Based Complex Graph Neural Network for Remaining Useful Life Prediction of Rotating Machinery

Graph neural networks are increasingly explored in the field of Prognostics and Health Management (PHM) due to their excellent performance when dealing with non-Euclidean data. However, current graph neural networks are mostly based on real domain modeling. In addition, existing graph construction methods rely on the prior positional relationship of multiple sensors. In view of the above, this paper proposes complex graph neural network based on the picture-in-picture strategy (CGNN-PIP) to realize the remaining useful life (RUL) prediction of rotating machinery under multi-channel signals. Specifically, the classical graph convolution operation is upgraded to generalized complex graph convolution, and complex graph neural network is further constructed to extract deep degenerate feature representations. Meanwhile, the picture-in-picture strategy is designed to guide graph construction, which takes the single-path graph as a node of the new graph to build a deeper-level graph. We verified the effectiveness and superiority of the proposed method through two case studies on different run-to-failure datasets. The results show that the proposed CGCN-PIP can reasonably construct the topology map of the complex domain data, and extract the temporal and structural information reflecting the equipment degradation. The comparison with state-of-the-art methods also proves that CGCN-PIP has advantages in terms of prediction accuracy and training consumption.

Degenerate Representation and Racism in Heart of Darkness

This paper has analyzed Joseph Conrad’s Heart of Darkness from Orientalist perspective. The term “Orientalism” has been a critical tool after Edward Said described how understanding of Europe about the non-Europeans became a weapon of maintaining power over the latter. This research has applied analytical and interpretivist approaches in discussion. The objective is to pinpoint why and how colonial rulers in the novel stereotypically produce representations of the Orient (here African is included) and create a false illusion that the West is “self” and the East is “other”. The selected text is analyzed in light of Edward Said’ description of the stereotypes of Oriental land and race as strange, inferior and wild. The search shows that Conrad presents Africans as uncivilized people, living in backward condition. He derogatorily describes them in their physical appearance and social behavior. Congo land is presented as wild, uncommon and mysterious. Europeans in the novel express their obsession that they are intelligent, skilled and much superior to the native and try to justify the colonial rule as an effort to civilize the African race and develop the place.

Local multiplet formation around a single vacancy in graphene: An effective Anderson model analysis based on the block-Lanczos DMRG method

To better understand the electronic structure of a single vacancy in graphene, we study the ground state property of an effective Anderson model, consisting of three dangling $sp^2$ orbitals of the surrounding carbon atoms around the vacancy and the $\pi$ orbitals of carbon atoms that form the honeycomb lattice with a single vacancy. Employing the block-Lanczos density-matrix renormalization group method, we show that there are two phases in the relevant parameter space, i.e., a nonmagnetic phase in the weak coupling region and a free magnetic moment phase in the realistic parameter region. The systematic analysis finds that, in the free magnetic moment phase, local multiplets of the doubly degenerate irreducible representation of the $\mathcal{C}_3$ point group with spin 1 become dominant in the ground state, and approximately half of this local spin 1 is screened by electrons in the surrounding $\pi$ orbitals, indicating the emergence of the residual spin-1/2 free magnetic moment. Furthermore, we find that the emergence of the free magnetic moment is robust against carrier doping, which is in sharp contrast to the case of graphene with an adatom, thus explaining the qualitative difference observed experimentally in these two classes of systems. We also find the enhancement of the spin correlation function between $\pi$ electrons around the vacancy and those in the conduction band away from the vacancy in the undoped case, as compared to that in the doped case, while the spin correlation function between the $\sigma$ electrons in the $sp^2$ dangling orbitals around the vacancy and the $\pi$ electrons in the conduction band remains large in both undoped and doped cases. Our calculations thus support qualitatively the previous experiment that suggests the emergence of free magnetic moment with two distinct origins.

Degenerate operators in JT and Liouville (super)gravity

We derive explicit expressions for a specific subclass of Jackiw-Teitelboim (JT) gravity bilocal correlators, corresponding to degenerate Virasoro representations. On the disk, these degenerate correlators are structurally simple, and they allow us to shed light on the 1/C Schwarzian bilocal perturbation series. In particular, we prove that the series is asymptotic for generic weight h ∉ −ℕ/2. Inspired by its minimal string ancestor, we propose an expression for higher genus corrections to the degenerate correlators. We discuss the extension to the mathcal{N} = 1 super JT model. On the disk, we similarly derive properties of the 1/C super-Schwarzian perturbation series, which we independently develop as well. As a byproduct, it is shown that JT supergravity saturates the chaos bound λL = 2π/β at first order in 1/C. We develop the fixed-length amplitudes of Liouville supergravity at the level of the disk partition function, the bulk one-point function and the boundary two-point functions. In particular we compute the minimal superstring fixed length boundary two-point functions, which limit to the super JT degenerate correlators. We give some comments on higher topology at the end.

Conical spaces, modular invariance and cp,1 holography

We propose a non-unitary example of holography for the family of two-dimensional logarithmic conformal field theories with negative central charge c = cp,1 = −6p + 13 − 6p−1. We argue that at large p, these models have a semiclassical gravity-like description which contains, besides the global AdS3 spacetime, a tower of solitonic solutions describing conical excess angles. Evidence comes from the fact that the central charge and the natural modular invariant partition function of such a theory coincide with those of the cp,1 model. These theories have an extended chiral W-algebra whose currents have large spin of order |c|, and which in the bulk are realized as spinning conical solutions. As a by-product we also find a direct link between geometric actions for exceptional Virasoro coadjoint orbits, which describe fluctuations around the conical spaces, and Felder’s free field construction of degenerate representations.

The degenerate principal series representations of exceptional groups of type E7 over p-adic fields

In this paper, we study the degenerate principal series of a split, simply-connected, simple p-adic group of type $E_7$. We determine the points of reducibility and the maximal semi-simple subrepresentation at each point.

The degenerate principal series representations of exceptional groups of type E6 over p-adic fields

In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type E6. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations.

Linear surface gravity waves on current for a general inertial viewer

Marine measurement instrumentation, such as free-floating wave buoys, drones, and autonomous unmanned vehicles, often propagates in different directions and velocities relative to the fluid and waves. Convention assumes that these different instrumentations provide Galilean invariant descriptions of the wave field. Herein, it is shown that Galilean invariance exists for the water wave problem only in a restricted sense. The impact of this loss of invariance is investigated using a new formulation of the water wave problem, which is generalized for both current and an arbitrary inertial viewer. In the still water limit, the boundary value problem is shown to be non-invariant under Galilean transformations. This impacts the dispersion relation and interpretation of measurements. It also explains the appearance of wave modes on current, which have no analogy on still water. These modes do not appear in a still water formulation because it is a degenerate representation exhibiting a loss of Galilean symmetries. The approach provides a more complete solution of the wave–current boundary value problem by making a clear distinction between current and viewer velocity effects. Numerical examples that demonstrate the importance of the results on calculating wave characteristics are given.

PBW Degenerate Schubert Varieties: Cartan Components and Counterexamples

In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type A but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for \(\mathfrak {sl}_{6}\). The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.

The compact picture of symmetry-breaking operators for rank-one orthogonal and unitary groups

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method describes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calculate these relations explicitly and use them to classify intertwining operators. We further show that in these cases automatic continuity holds, i.e. every intertwiner between the Harish-Chandra modules extends to an intertwiner between the Casselman--Wallach completions, verifying a conjecture by Kobayashi. Altogether, this establishes the compact picture of the recently studied symmetry breaking operators for orthogonal groups by Kobayashi--Speh, gives new proofs of their main results and extends them to unitary groups.

Rectangular W algebras and superalgebras and their representations

We study W-algebras obtained by quantum Hamiltonian reduction of $sl(Mn)$ associated to the $sl(2)$ embedding of rectangular type. The algebra can be realized as the asymptotic symmetry of higher spin gravity with $M \times M$ matrix valued fields. In our previous work, we examined the basic properties of the W-algebra and claimed that the algebra can be realized as the symmetry of Grassmannian-like coset even with finite central charge based on a proposal of holography. In this paper, we extend the analysis in the following ways. First, we compute the operator product expansions among low spin generators removing the restriction of $n = 2$. Second, we investigate the degenerate representations in several ways, and see the relations to the coset spectrum and the conical defect geometry of the higher spin gravity. For these analyses, we mainly set $M=n=2$. Finally, we extend the previous analysis by introducing $\mathcal{N}=2$ supersymmetry.

Stochastic interface dynamics in the Hele-Shaw cell.

A one-parametric stochastic regularized dynamics of the interface in the Hele-Shaw cell is introduced. The short-distance regularization suggested by the aggregation model stabilizes the growth by preventing the formation of cusps at the interface and makes the interface dynamics chaotic. The introduced stochastic growth process generates universal complex patterns with the well-developed fjords of oil separating the fingers of water. In a long time asymptotic, by coupling a conformal field theory to the stochastic growth process, we introduce a set of observables (the martingales), whose expectation values are constant in time. The martingales are closely connected to degenerate representations of the Virasoro algebra and can be written in terms of conformal correlation functions. A direct link between Laplacian growth and conformal Liouville field theory with the central charge c≥25 is proposed.

Extending representations of Banach algebras to their biduals

We show that a representation of a Banach algebra A on a Banach space X can be extended to a canonical representation of $$A^{**}$$ on X if and only if certain orbit maps $$A\rightarrow X$$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if A has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in $$C^*$$-algebras and in abstract harmonic analysis. For example, a $$C^*$$-algebra A has an isometric representation on an $$L^p$$-space, for $$p\in [1,\infty ){\setminus }\{2\}$$, if and only if A is commutative. Moreover, the $$L^p$$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $$L^p$$-spaces.

Dimension of the space of intertwining operators from degenerate principal series representations

Let $X$ be a homogeneous space of a real reductive Lie group $G$. It was proved by T. Kobayashi and T. Oshima that the regular representation $C^{\infty}(X)$ contains each irreducible representation of $G$ at most finitely many times if a minimal parabolic subgroup $P$ of $G$ has an open orbit in $X$, or equivalently, if the number of $P$-orbits on $X$ is finite. In contrast to the minimal parabolic case, for a general parabolic subgroup $Q$ of $G$, we find a new example that the regular representation $C^{\infty}(X)$ contains degenerate principal series representations induced from $Q$ with infinite multiplicity even when the number of $Q$-orbits on $X$ is finite.

Fourier supports of K-finite Bessel integrals on classical tube domains

Let [Formula: see text] be the symmetric tube domain associated with the Jordan algebra [Formula: see text], [Formula: see text], [Formula: see text], or [Formula: see text], and [Formula: see text] be its Shilov boundary. Also, let [Formula: see text] be a degenerate principal series representation of [Formula: see text]. Then we investigate the Bessel integrals assigned to functions in general [Formula: see text]-types of [Formula: see text]. We give individual upper bounds of their supports, when [Formula: see text] is reducible. We also use the upper bounds to give a partition for the set of all [Formula: see text]-types in [Formula: see text], that turns out to explain the [Formula: see text]-module structure of [Formula: see text]. Thus, our results concretely realize a relationship observed by Kashiwara and Vergne [[Formula: see text]-types and singular spectrum, in Noncommutative Harmonic analysis, Lecture Notes in Mathematics, Vol. 728 (Springer, 1979), pp. 177–200] between the Fourier supports and the asymptotic [Formula: see text]-supports assigned to [Formula: see text]-submodules in [Formula: see text].

Analytic Continuation of Equivariant Distributions

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

Symmetry breaking operators for the restriction of representations of indefinite orthogonal groups $O(p,q)$

For the pair $(G, G') =(O(p+1, q+1), O(p,q+1))$, we construct and give a complete classification of intertwining operators (symmetry breaking operators) between most degenerate spherical principal series representations of $G$ and those of the subgroup $G'$, extending the work initiated by Kobayashi and Speh [Mem. Amer. Math. Soc. 2015] in the real rank one case where $q=0$. Functional identities and residue formulæ of the regular symmetry breaking operators are also provided explicitly. The results contribute to Program C of branching problems suggested by the first author [Progr. Math. 2015].