Action groups

HarF70a Harary, F. Enumeration under group action Unsolved problems in graphical enumeration IV. J. Combinatorial Theory 8 (1970) 1-11 9 (1970) 221. 

B. Smith, Handbook of Ocular Pharmacology, Publication Sciences Group, Action, MA, 1974. 

Question 4.6 (Hitchin). Gonsider a hnite group action on a K3 surface which preserves a hyper-Kahler structure. (Such actions were classified by Mukai [58].) It naturally induces the action on the Hilbert scheme of points on the K3 surface. Its fixed point component is a compact hyper-Kahler manifold as in 4.2. Is the component a new hyper-Kahler manifold The known compact irreducible hyper-Kahler manifolds are equivalent to the Hilbert scheme of points on a /F3 surface, or the higher order Kummar variety (denoted by Kr in [6]) modulo deformation and birational modification, (cf. [57, p.l68j)... 

When df is a Kahler manifold, the stable and the unstable manifolds can be expressed purely in terms of the group action. Notice that an T-action on X extends uniquely to a holomorphic T -action on X. 

Exercise 6.10 Show that the invariant integral on SU(2) given in Equation 6.1 is invariant under the group action. 

See Boothby [Bo, II.6] or Bamberg and Sternberg [BaS, p. 237] for a proof of the inverse function theorem on R". The corresponding theorem for manifolds follows by restricting to coordinate neighborhoods of m and f(m). We will use the following theorem about group actions on differentiable manifolds. 

Theorem B.3 Suppose M is a differentiable manifold, G is a compact Lie group and G, M, o) is a group action. Suppose further that... 

It was also felt that group performance was affected in subtle ways. This may be evidence that some sort of group action was going on all the time."... 

Let G be a group functor, X a set functor. An action of G on X is a natural map G x X - X such that the individual maps G(R) x X(R)- X(R) are group actions. These will come up later for general X, but the only case of interest now is X(R)= V R, where V is a fixed k-module. If the action of G(R) here is also R-linear, we say we have a linear representation of G on V. The functor GLV(R) = Aut (F R) is a group functor a linear representation of G on V clearly assigns an automorphism to each g and is thus the same thing as a homomorphism G - GLK. If V is a finitely generated free module, then in any fixed basis automorphisms correspond to invertible matrices, and linear representations are maps to GL . 

Write down the commutative diagrams saying that G x X - X is a group action. For representable G and X, write down the corresponding algebra diagrams. 

A. Kerber, Applied Finite Group Actions, Springer, Berlin, 1998 A. Kerber, Applied Finite Group Actions, 2nd edition, Springer, Berlin, 1999. 

For such an easy, familiar example, Schur s lemma doesn t gain us any new or particularly enlightening information. But for the case of rotations in three-dimensional space, it certainly will. In this situation, the invariant subspaces that are fixed by the group action will not be one-dimensional. Yet the lemma tells us that every element of such a subspace will be an eigenfunction of our spherical Laplace operator, A, and will always have the same eigenvalue. So we need to only compute the action of A on one function in each space to determine the eigenvalue. Furthermore, the dimensions of these invariant subspaces (which we can calculate) will have an actual physical interpretation, as we will see later. 

Substantial mathematical difficulties arise because the group action p is only strongly continuous. Moreover the group SE 2) is non-compact due to its translational component. We suppress these technicalities in the following. 

B. Fiedler, B. Sandstede, A. Scheel, and C. Wulff. Bifurcation from relative equilibria of noncompact group actions Skew products, meanders, and drifts. Doc. Math., J. DMV, 1 479-505, 1996. 

B. Fiedler and D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Ration. Mech. Anal., 145(2) 129-159, 1998. 

B is the subsheaf of An consisting of the invariants under the group action, i.e., the kernel of the couple... 

The existence of a is clear in order to see that a has the above described property with respect to the group action of H it suffices to look to the functor and to check the compatibility pointset theoretically, which is immediately clea ). Vie state some simple properties ... 

Remarki In 2.3.4- we mean, of course, isomorphisms as schemes because we don t have a group action on X itself. 

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