Hi, I also strongly support both left and right actions, with the syntax
Left action: s1(s2(x)) == (s1*s2)(x) Right action: (x^s1)^s2 == x^(s1*s2) Currently the left (functional) notation is implemented in Sage with the right order of application: s1(s2(x)) == (s2*s1)(x) is True. This needs to be either accepted or deprecated in preference to a new right operator. Magma implements only the right action with ^ notation, and GAP also (only?) implements the same. Attention: Unless I'm mistaken, Magma does something bad with operator precedence in order to have x^s1^s2 evaluate to (x^s1)^s2 rather than x^(s1^s2) -- the latter is defined but not equal. The significant change is that a left acting symmetric group will have its multiplication reversed. Implementing the right action involves some thought, since it must be a method on the domain class. Either the domain must be a designated class of G-sets, which knows about its acting group G, or for each object with a "natural" permutation action, one needs to implement or overload ^ on a case-by-case basis (e.g. integers, free modules, polynomial rings and other free objects [action on the generators], etc.). Please correct me if there is another way to drop through and pick up an action operation on the right-hand argument. Defining the (left or right) action by * would probably be a nightmare with the coercion model, since it is handled as a symmetric operator. Some thought needs to be given to the extension from a group action to a group algebra action. The ZZ-module structure of an abelian group with left G-action would give an argument to admitting * as a possible (left) notation for this operation, despite the obvious headaches. The ^ notation would give a compatible extension of the natural ZZ-module structure acting on the right on a multiplicatively represented abelian group. As an example, in number theory, when G is a Galois group Gal(L/K), it is typical to left K[G] act on the left on L by *, and ZZ[G] act on the right on multiplicative group L^*. Although G is not a permutation group in this example, we want to set up the framework for G-sets to admit these natural actions. The left action is problematic, since the coercion model is susceptible to building L[G] and carry out * with the trivial G-action on L. To avoid this, L needs to be identified as an object in the category of K-modules (with G-action) rather than a K-algebra. Until someone comes up with a well-developed and coherent model for treating the operator * in general G-actions, I would avoid any implementations using this operator with permutation actions. --David -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/groups/opt_out.
