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Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.


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TL;DR: In this paper, the weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights were studied and compared to those of the non-weyl modules.
Abstract: (1983). The weyl modules and the irreducible representations of the symplectic group with the fundamental highest weights. Communications in Algebra: Vol. 11, No. 12, pp. 1309-1342.

76 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a universal closed 2-form with integral periods can be obtained from a complex projective space, where the manifold is a manifold of the form 2*=i dft A dgt, where gi are real valued functions on the manifold.
Abstract: The existence of universal connections was shown by Narasimhan and Ramanan [5], and Kostant [3] showed that any integral closed 2-form is the curvature form of a connection on some circle bundle. These results can be combined to show the existence of a universal closed 2-form with integral periods. In this paper we will use the symplectic structure of a complex projective space to give an elementary proof of this result the precise statement is given in Theorem A. The result of Kostant is in fact a corollary of the existence of a universal closed 2-form, as is indicated below. Another immediate corollary of Theorem A is the result of Gromov [3] that closed symplectic manifolds can be symplecticalΓy immersed in CP, for large enough n see Theorem B. First we indicate why the proof which we are going to give here is a simple and natural generalization of an elementary fact about exact 2-forms. Consider the standard symplectic form Ω = Σιl=i dXidyt on R . Any exact 2-form on a manifold M can be induced from Ω by a mapping to R for some n, since any exact 2-form on M can be written in the form 2*=i dft A dgt, where /*, gi are real valued functions on M. CP has a symplectic structure Ωo which is locally given by Ωo = 2?=i dxi A dyt. Furthermore, CP n is the 2π-skeleton of an Eilenberg-MacLane space of type K(Z, 2). It is thus natural to expect that any closed 2-form with integral periods can be induced from Ωo by a map to CP, because there is some map to CP, for large n, which pulls back Ωo to within an exact 2-form of the given closed 2-form. The only complication that is met in CP to adjusting the map to account for the exact 2-form is that, unlike in R, the symplectic charts on CP have finite radius, so the fi9 g/s utilized would have to be bounded. The proof we give of Theorem A depends only on estimating the bounds on fi9 gt as n becomes large. A closed &-form on a manifold M will be said to be integral if its de Rham cohomology class is in the image of the canonical coefficient map H(M Z) -*H(M;R). Complex projective space CP has a Kahlerian structure, and we will denote its Kahler form by flj. The 2-form Ω% can be chosen to represent a generator in the image of H\\CP Z) -> H\\CP R), and we can assume that /*(βJ0 = Ωl where i is the standard inclusion of CP in CP,

75 citations

Journal ArticleDOI
TL;DR: In this article, the Pfafi lattice is shown to be integrable for a splitting involving the a−ne symplectic algebra, and the tau-functions for the skew-Borel decomposition for the wave vectors skew-orthogonal polynomials are given.
Abstract: Consider a semi-inflnite skew-symmetric moment matrix, m1 evolving according to the vector flelds @m=@tk = ⁄ k m + m⁄ >k ; where ⁄ is the shift matrix. Then The skew-Borel decomposition m1 := Q i1 JQ >i1 leads to the so-called Pfafi Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the a‐ne symplectic algebra. The tau-functions for the system are shown to be pfa‐ans and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).

75 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202221
202113
20208
201910
201818