We give bijective proofs for Jacobi{Trudi-type and Giambelli-type identities for symplectic and orthogonal characters. These proofs base on interpreting King and El-Sharkaway's symplectic tableaux, Proctor's odd and intermediate symplectic tableaux, Proctor's and King and Welsh's orthogonal tableaux, and Sundaram's odd orthogonal tableaux in terms of certain families of nonintersecting lattice ...

We consider 2n× 2n symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by a...

In this paper, we study a multi-symplectic scheme for three dimensional Maxwell’s equations in a simple medium. This is a system of PDEs with multi-symplectic structures. We prove that this multi-symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence. Furthermore, we extend the discussion to several dispersion properties of the m...

Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homotopical obstruction is described which detects when an action is Hamiltonian. This new entity, the AA-invariant, allows many results of symplectic geometry to be generalized to manifo...

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes’ basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the Symplectic Thom Conjecture: a symplectic surface in a symplectic four-manifold is genus...

In [23] a nice looking formula is conjectured for a deformed product of functions on a symplectic manifold in case it concerns a hermitian symmetric space of non-compact type. We derive such a formula for simply connected symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional...

2 The symplectic double of a cluster X -variety 9 2.1 Cluster X and A-varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Construction of the symplectic double . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 A decomposition of mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 The motivic dilogarithm structure on the sym...

Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The structure-preservation region provides a practical criterion for choosing step-size in symplectic computation. Examples are given to justify the investigation.

In this paper, we give a symplectic proof for Seiberg-Witten blow-up formula of four dimensional symplectic manifolds, especially we interpret a strange phenomenon that the genera of embedding J-holomorphic curves will decrease when we symplectically blow-up the four dimensional symplectic manifold.

We prove Gray–Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a oneform is replaced by the condition that its kernel hyperplane distribution have constant class in the sense of ...