scholarly journals The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
B. Ntatin
Keyword(s):  
Parameter Space ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Flag Manifold ◽  
Open Orbit ◽  
Duality Relations ◽  
Geometric Objects ◽  
Complex Semisimple ◽  
Universal Domain ◽  
Lower Dimensional

The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.

scholarly journals Cycle Intersection for SOp,q-Flag Domains

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Faten Abu Shoga
Keyword(s):  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Schubert Varieties ◽  
Real Form ◽  
Precise Description ◽  
Closed Orbits ◽  
Complex Submanifolds ◽  
Complex Semisimple ◽  
Real Forms ◽  
Flag Domains

A real form G0 of a complex semisimple Lie group G has only finitely many orbits in any given compact G-homogeneous projective algebraic manifold Z=G/Q. A maximal compact subgroup K0 of G0 has special orbits C which are complex submanifolds in the open orbits of G0. These special orbits C are characterized as the closed orbits in Z of the complexification K of K0. These are referred to as cycles. The cycles intersect Schubert varieties S transversely at finitely many points. Describing these points and their multiplicities was carried out for all real forms of SLn,ℂ by Brecan (Brecan, 2014) and (Brecan, 2017) and for the other real forms by Abu-Shoga (Abu-Shoga, 2017) and Huckleberry (Abu-Shoga and Huckleberry). In the present paper, we deal with the real form SOp,q acting on the SO (2n, C)-manifold of maximal isotropic full flags. We give a precise description of the relevant Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number. The results in the case of G/Q for all real forms will be given by Abu-Shoga and Huckleberry.

scholarly journals Measure of Similarity between GMMs by Embedding of the Parameter Space That Preserves KL Divergence

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 957
Author(s):  
Branislav Popović ◽  
Lenka Cepova ◽  
Robert Cep ◽  
Marko Janev ◽  
Lidija Krstanović
Keyword(s):  
Computational Complexity ◽  
Parameter Space ◽  
Recognition Accuracy ◽  
Gaussian Mixture Models ◽  
Gaussian Mixture ◽  
Dimensional Manifold ◽  
Dimensional Parameter ◽  
Trade Off ◽  
Measure Of Similarity ◽  
Lower Dimensional

In this work, we deliver a novel measure of similarity between Gaussian mixture models (GMMs) by neighborhood preserving embedding (NPE) of the parameter space, that projects components of GMMs, which by our assumption lie close to lower dimensional manifold. By doing so, we obtain a transformation from the original high-dimensional parameter space, into a much lower-dimensional resulting parameter space. Therefore, resolving the distance between two GMMs is reduced to (taking the account of the corresponding weights) calculating the distance between sets of lower-dimensional Euclidean vectors. Much better trade-off between the recognition accuracy and the computational complexity is achieved in comparison to measures utilizing distances between Gaussian components evaluated in the original parameter space. The proposed measure is much more efficient in machine learning tasks that operate on large data sets, as in such tasks, the required number of overall Gaussian components is always large. Artificial, as well as real-world experiments are conducted, showing much better trade-off between recognition accuracy and computational complexity of the proposed measure, in comparison to all baseline measures of similarity between GMMs tested in this paper.

scholarly journals Yang–Mills connections on compact complex tori

2015 ◽  
Vol 07 (02) ◽  
pp. 293-307
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Structure Group ◽  
Kähler Structure ◽  
Yang Mills ◽  
Complex Torus ◽  
Complex Tori

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

scholarly journals An Application of Spherical Geometry to Hyperkähler Slices

2020 ◽  
pp. 1-30
Author(s):  
Peter Crooks ◽  
Maarten van Pruijssen
Keyword(s):  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Spherical Geometry ◽  
Spherical Subgroup ◽  
Cotangent Bundles ◽  
Complex Semisimple ◽  
Reductive Subgroup ◽  
Necessary And Sufficient

Abstract This work is concerned with Bielawski’s hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice with the data of a complex semisimple Lie group  $G$ , a reductive subgroup $H\subseteq G$ , and a Slodowy slice $S\subseteq \mathfrak{g}:=\text{Lie}(G)$ , defining it to be the hyperkähler quotient of $T^{\ast }(G/H)\times (G\times S)$ by a maximal compact subgroup of  $G$ . This hyperkähler slice is empty in some of the most elementary cases (e.g., when $S$ is regular and $(G,H)=(\text{SL}_{n+1},\text{GL}_{n})$ , $n\geqslant 3$ ), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$ -regularity of $(G,H)$ . This $\mathfrak{a}$ -regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$ . We also provide a classification of the $\mathfrak{a}$ -regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces.

scholarly journals The SI and SIR epidemics on general networks

2018 ◽  
Vol 37 (2) ◽  
pp. 229-234
Author(s):  
David Aldous
Keyword(s):  
Recent Result ◽  
Parameter Space ◽  
Epidemic Model ◽  
Small Proportion ◽  
High Probability ◽  
Bond Percolation ◽  
Sir Epidemic ◽  
Sir Epidemic Models ◽  
Lower Dimensional ◽  
General Networks

THE SI AND SIR EPIDEMICS ON GENERAL NETWORKSIntuitively one expects that for any plausible parametric epide mic model, there wil l be some region in parameter-space where the epidemicaffects with high probability only a small proportion of a largepopulation, another region where it affects with high probability a nonnegligible proportion, with a lower-dimensional “critical” interface. This dichotomy is certainly true in well-studied specific models, but we know o fno very general results. A recent result stated for a bond percolation modelcan be restated as giving weak conditions under which the dichotomy holdsfor an SI epidemic model on arbitrary finite networks. This result suggestsa conjecture for more complex and more realistic SIR epidemic models, and the purpose of this article is to record the conjecture.

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

scholarly journals Equivariant holomorphic maps into the Siegel disc and the metaplectic representation

1997 ◽  
Vol 62 (2) ◽  
pp. 160-174 ◽  
Author(s):  
Jean-Louis Clerc
Keyword(s):  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Holomorphic Maps ◽  
Harmonic Polynomials ◽  
Metaplectic Representation ◽  
Siegel Disc

AbstractWe restrict the metaplectic representation to subgroupsGof the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup ofG.

Generic Identication of Binary-Valued Hidden Markov Processes

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Alexander Schönhuth
Keyword(s):  
Stochastic Process ◽  
Markov Process ◽  
General Solution ◽  
Parameter Space ◽  
Markov Processes ◽  
Hidden Markov ◽  
Probability Distributions ◽  
Algebraic Statistics ◽  
Hidden Markov Processes ◽  
Lower Dimensional

The generic identication problem is to decide whether a stochastic process (Xt) is ahidden Markov process and if yes to infer its parameters for all but a subset of parametrizationsthat form a lower-dimensional subvariety in parameter space. Partial answers so far availabledepend on extra assumptions on the processes, which are usually centered around stationarity.Here we present a general solution for binary-valued hidden Markov processes. Our approach isrooted in algebraic statistics hence it is geometric in nature. We nd that the algebraic varietiesassociated with the probability distributions of binary-valued hidden Markov processes are zerosets of determinantal equations which draws a connection to well-studied objects from algebra. Asa consequence, our solution allows for algorithmic implementation based on elementary (linear)algebraic routines.

The relativistic SL (6, C ) strong vertex and kinematic form factors

1968 ◽  
Vol 305 (1480) ◽  
pp. 27-53
Keyword(s):  
Form Factor ◽  
Closed Form ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Form Factors ◽  
Large Momentum ◽  
Group Representations ◽  
Technical Problems ◽  
Infinite Dimensional ◽  
Dimensional Group

The SL (6, C ) invariant baryon-baryon-meson vertex, involving three infinite multiplets, is expanded in terms of representations of the maximal compact subgroup SU (6) p for particles of arbitrary momentum. The terms in the expansion which involve the lowest-dimensional SU (6) multiplets are evaluated in closed form. The assumption of SL (6, C ) invariance leads to a breaking of SU (6) by spurions in a well-defined way. Each type of spurion coupling is accompanied by a function of momentum which vanishes rapidly for large momentum transfer and can be interpreted as a kinematic form factor. All calculations are performed with the ‘generalized tensor’ formalism, and a number of mathematical results are obtained which greatly simplify the technical problems of dealing with infinite-dimensional group representations.

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