Multiple Free Space Research Articles

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=Un,SOn. Extending these results to groups of other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie(K)⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G=KC.We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K-spaces.An essential tool in our study is the dual Poisson-Lie group K⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K⁎, and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie(K)⁎→K⁎.

Local normal forms for multiplicity free U(n) actions on coadjoint orbits

Actions of $U(n)$ on $U(n+1)$ coadjoint orbits via embeddings of $U(n)$ into $U(n+1)$ are an important family of examples of multiplicity free spaces. They are related to Gelfand-Zeitlin completely integrable systems and multiplicity free branching rules in representation theory. This paper computes the Hamiltonian local normal forms of all such actions, at arbitrary points, in arbitrary $U(n+1)$ coadjoint orbits. The results are described using combinatorics of interlacing patterns; gadgets that describe the associated Kirwan polytopes.

Free Space Optical Communication in Long-Reach Unidirectional Ring-Architecture Fiber Network

In the work, a ring-constructed wavelength-division-multiplexing (WDM) access system for delivering multiple free space optical (FSO) wireless signals is investigated. To reach the FSO transmission through single-mode fiber (SMF) connection, the optical wireless unit (OWU) and remote OWU are designed in the WDM network. The downstream and upstream WDM FSO channels could be transmitted in unidirectional propagation with the same wavelength. Hence, the presented ring-based WDM network not only can transmit the FSO signal, but also can circumvent the Rayleigh backscattering (RB) interferometric beat noise. In addition, according to the experimental results, the SMF and FSO transmission lengths can reach 50 km and 500 to 1000 m at the forward error correction (FEC) level without using optical amplification and dispersion compensation, respectively, when four WDM wavelengths are exploited experimentally for demonstration.

Analysis using multiple free space optic channel with amplification to mitigate haze attenuation

Weather severity has unfavorable effect on FSO transmission performance. The impact could result in poor quality of transmission and communication failure. This paper presents the analysis using multiple free space optic channels with amplification to mitigate haze attenuation. The visibility is progressively reduced due to the attenuation effects, causing the bit error rate to increase. Three designs of FSO system are proposed in this research as they are simulated and analysed thoroughly using Optisys software. The achieved result shows improved performance for Design 3, as the minimum BER achieved is 10<sup>-127</sup> when the power received is -26 dBm, and maximum Q-factor of 23.6 at the aperture diameter of 40 cm. It is proved to improve the system efficiently in low visibility condition compared to Design 1 and Design 2. It clearly has shown the superior capabilities of Design 3 to intercept and mitigate the atmospheric attenuation in haze condition as the power loss during transmission also reduced significantly.

A combinatorial smoothness criterion for spherical varieties

We suggest a combinatorial criterion for the smoothness of an arbitrary spherical variety using the classification of multiplicity-free spaces, generalizing an earlier result of Camus for spherical varieties of type A.

Invariant differential operators on a class of multiplicity-free spaces

If $(G,V)$ is a multiplity free space with a one dimensional quotient we give generators and relations for the non-commutative algebra $D(V)^{G'}$ of invariant differential operators under the semi-simple part $G'$ of the reductive group $G$. More precisely we show that $D(V)^{G'}$ is the quotient of a Smith algebra by a completely described two-sided ideal.

Visible Actions on Reducible Multiplicity-Free Spaces

The notion of (strongly) visible actions was introduced by T. Kobayashi [10, 11] for the biholomorphic action on a complex manifold with (possibly) infinitely many orbits. A holomorphic action of a Lie group G on a complex manifold D is strongly visible with a slice S if D = G ⋅ S, and there exists an anti-holomorphic and orbit preserving diffeomorphism σ on D such that σ|S = idS. Generalizing our previous result [22] about strongly visible actions on irreducible multiplicity-free spaces à la Kac, we treat reducible multiplicity-free spaces classified by Benson–Ratcliff and Leahy and prove that any multiplicity-free space naturally admits a strongly visible action. Further, we give an explicit description of S and σ. Our result gives an evidence to Kobayashi's conjecture [12, Conjecture 3.2] in the case of linear actions, asserting that we can take S to have the same dimension with the rank of V .

Algebras of invariant differential operators on a class of multiplicity free spaces

Let G be a connected reductive algebraic group and let G ′ = [ G , G ] be its derived subgroup. Let ( G , V ) be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra D ( V ) G ′ of G ′ -invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to U ( sl 2 ) . Our result generalizes the case of the Weil representation, where the associative algebra generated by Q ( x ) and Q ( ∂ ) ( Q being a non-degenerate quadratic form on V) is a quotient of U ( sl 2 ) . Other structure results are obtained when ( G , V ) is a regular prehomogeneous vector space of commutative parabolic type. To cite this article: H. Rubenthaler, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Visible Actions on Irreducible Multiplicity-Free Spaces

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with a sliceS if D′ ≔ G · S is open in D and there exists an antiholomorphic and orbit-preserving diffeomorphism σ of D′ such that σ |S = id S. In this article, we study linear, strongly visible actions. We prove that irreducible multiplicity-free space V of a connected compact Lie group is strongly visible. Furthermore, we find an explicit description of S and σ according to Kac's classification. Our result gives an evidence to Kobayashi's conjecture [10, Conjecture 3.2] in the case of irreducible multiplicity-free spaces, asserting that we can take S to have the same dimension as the rank of V.

Combinatorics and invariant differential operators on multiplicity free spaces

We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.

Construction of commuting difference operators for multiplicity free spaces

The analysis of invariant differential operators on certain multiplicity free spaces led recently to the introduction of a family of symmetric polynomials that is more general than Jack polynomials (see [KS], but also [OO1], [OO2]). They are called interpolation Jack polynomials, shifted Jack polynomials, or Capelli polynomials. Apart from being inhomogeneous, they are distinguished from classical Jack polynomials by their very simple definition in terms of vanishing conditions. One of the most important and non-obvious properties of Capelli polynomials is that they are eigenfunctions of certain explicitly given difference (as opposed to differential) operators (see [KS]). This readily implies that their top homogeneous term is in fact a (classical) Jack polynomial. Other consequences include a binomial theorem, a Pieri formula, and much more. It is well-known that Jack polynomials are tied to root systems of type A and that they have natural analogues for other root systems (see e.g., [He]). Therefore, it is a natural problem as to whether this holds for the Capelli polynomials as well. Okounkov [Ok] proposed such an analogue for root systems of type BC and proved that these share some of the nice properties of Capelli polynomials. But, unfortunately, Okounkov’s polynomials do not satisfy difference equations. Also their representation-theoretic significance is not clear. In this paper we go back to the origin and let ourselves be guided by the theory of multiplicity free actions. It is known (see section 7 for details) that these actions give rise to combinatorial structures consisting of four data (Γ,Σ,W, l). Here Γ is a lattice, Σ ⊂ Γ a basis, W ⊆ AutΓ a finite reflection group, and l ∈ Γ some element. These data alone suffice to formulate the definition of a (generalized) Capelli polynomial but, in that generality, neither existence nor uniqueness will hold, let alone any other good property.

Multiplicity-free spaces

Multiplicity-free spaces