Minimal modularity lifting for nonregular symplectic representations

We associate one or two posets (which we call "semistandard posets") to any given irreducible representation of a rank two semisimple Lie algebra over ${\Bbb C}$. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these "semistandard lattices") can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the "solitary" and "edge-minimal" properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.

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Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

International Journal of Mathematics ◽

10.1142/s0129167x18500854 ◽

2019 ◽

Vol 30 (03) ◽

pp. 1850085

Author(s):

Artur de Araujo

Keyword(s):

Reductive Group ◽

Kähler Manifold ◽

Kahler Manifold ◽

Quiver Bundles ◽

Symplectic Representations

We review the theory of quiver bundles over a Kähler manifold, and then introduce the concept of generalized quiver bundles for an arbitrary reductive group [Formula: see text]. We first study the case when [Formula: see text] or [Formula: see text], interpreting them as orthogonal (respectively symplectic) bundle representations of the symmetric quivers introduced by Derksen–Weyman. We also study supermixed quivers, which simultaneously involve both orthogonal and symplectic symmetries. In particular, we completely characterize the polystable forms of such representations. Finally, we discuss Hitchin–Kobayashi correspondences for these objects.

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Symplectic Representations of Inertia Groups

Journal of Algebra ◽

10.1006/jabr.2000.8638 ◽

2001 ◽

Vol 238 (1) ◽

pp. 400-410 ◽

Cited By ~ 1

Author(s):

A Silverberg ◽

Yu.G Zarhin

Keyword(s):

Symplectic Representations

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Adapted hyperbolic polygons and symplectic representations for group actions on Riemann surfaces

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2012.06.030 ◽

2013 ◽

Vol 217 (3) ◽

pp. 409-426 ◽

Cited By ~ 4

Author(s):

Antonio Behn ◽

Rubí E. Rodríguez ◽

Anita M. Rojas

Keyword(s):

Riemann Surfaces ◽

Group Actions ◽

Symplectic Representations

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SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

International Journal of Modern Physics A ◽

10.1142/s0217751x13500139 ◽

2013 ◽

Vol 28 (05n06) ◽

pp. 1350013 ◽

Cited By ~ 6

Author(s):

R. G. G. AMORIM ◽

M. C. B. FERNANDES ◽

F. C. KHANNA ◽

A. E. SANTANA ◽

J. D. M. VIANNA

Keyword(s):

Quantum Mechanics ◽

Group Theory ◽

Phase Space ◽

Wigner Function ◽

Physical Theory ◽

Star Product ◽

Unitary Representations ◽

Theory Approach ◽

Galilei Group ◽

Symplectic Representations

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

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