Some hybrid symplectic group phenomena

Symplectic Group Lattices

Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pn +1)/2 realized over Q(√ p), where = (−1)(p−1)/2. We study the integral lattices related to these representations for the case pn ≡ 1 mod 4. (The case pn ≡ 3 mod 4 has been considered in a previous paper.) We show that the class of invariant lattices co...

The Hybrid Phenomena Theory

The Hybrid Phenomena Theory (HPT) is a framework for formalizing how dynamic state space models of physical systems are built from first principles. The HPT descends from the Qualitative Process Theory (QPT), [Forb us, 1984], from which it inherits basic concepts like views, phenomena and influences. However, the HPT redefines some of these concepts in a more strict manner in order to represent...

Some Remarks on Symplectic Automorphisms

Compactification of the Symplectic Group via Generalized Symplectic Isomorphisms

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic zero. We have a left (G×G)-action on G defined as (g1, g2) ·x := g1xg −1 2 . A (G×G)-equivariant embedding G ↪→ X is said to be regular (cf. [BDP], [Br, §1.4]) if the following conditions are satisfied: (i) X is smooth and the complement X \G is a normal crossing divisor D1 ∪ · · · ∪Dn. (ii) Ea...

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Given a Lagrangian sphere in a symplectic 4-manifold (M,ω) with b+ = 1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension κ of (M,ω) is −∞, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is...