User talk:LittleDan

Welcome to Groupprops! We hope you will contribute much and well. You will probably want to read the help pages. Again, welcome and have fun! Vipul 20:21, 6 November 2008 (UTC)

Hi Dan,

Thanks for your contributions related to the pages on group actions. You seem to have got the knack for editing and organizing the pages despite there being very few formal guidelines for it. I'll be working on improving the pages you've started at some later point in time. Let me know if you have suggestions on the other pages. Vipul 15:03, 8 November 2008 (UTC)

Hi Dan,

I'll think about your questions and get back to you soon. Similar issues have cropped up in other areas where there are different possible notions of equivalence, and I haven't yet decided how these should be dealt with. Vipul 13:42, 9 November 2008 (UTC)

Hi Dan,

My personal inclination for notions of equivalence are the strong notion of equivalence, though there are many properties that are invariant upto the weaker notion of equivalence. The real issue here is whether we are thinking of the group and the set on which it acts as living in a possibly entangled world, or whether they live in separate worlds. When they live in separate worlds, the weak notion of similarity is good, but when their worlds are entangled, the stronger notion is necessary.

I think the word equivalent is used for the strong similarity. I don't think there is an agreed-upon word for the weak similarity -- you could just call it similar group actions (as you did); some people talk of quasi-equivalent or pseudo-equivalent. As you pointed out, there is absolutely nothing to be lost my thinking of equivalent actions as the same thing, but in many situations, we do lose something by thinking of similar actions as the same thing.

Similar issues come up in other related things. For instance, when studying the set of homomorphisms between two groups $$G$$ and $$H$$, do we view this set upto equivalence by composition with automorphisms of $$G$$? What about equivalence by composition with automorphisms of $$H$$? How about a weaker notion of equivalence only by composition with inner automorphisms of $$H$$? The last notion is useful and coincides with the strong similarity/equivalence of group actions, when the group action is viewed as a homomorphism to the symmetric group. That's because two actions of $$G$$ on $$S$$ are similar in the strong sense if there is a bijection of $$S$$ taking one to the other -- which is equivalent to an inner automorphism of $$\operatorname{Sym}(S)$$ conjugating the homomorphism for one representation to the homomorphism for the other.

In linear representations, where we study actions on vector spaces instead of sets, there are again two broad notions of equivalence: equivalence arising from inner automorphisms in the linear group to which the map is going (the analogue of the strong similarity) and equivalences arising from a combination of inner automorphisms in the linear group to which the map is going, and automorphisms of the group from which the map is going. The default notion of equivalence is the first one. The second one gives rise to a notion of pseudo-equivalent representations. (I've adopted this convention in the wiki as well: see the page on linear representation and the section about equivalence).

Note also that if the automorphism of $$G$$ by which we're changing things is an inner automorphism of $$G$$, it can be pushed forward to an inner automorphism of the target group -- in other words, the real problem arises when we start using outer automorphisms of the group, because these may not correspond to an inner automorphism on the other side. (by the way, this is also related to the fact that any inner automorphism is a pushforwardable automorphism and in particular an extensible automorphism -- whether the converse is true is an open question that I've raised and has not been settled). Vipul 21:15, 9 November 2008 (UTC)

Hi Dan,

Yes, you're right. Any reasonable "group action property" should be invariant under the notion of similarity of group/action pairs: an isomorphism between the groups and a bijection between the sets such that everything corresponds. When studying a relation between multiple actions of the same group, the issue of what kind of equivalence we're taking starts mattering. Vipul 00:40, 10 November 2008 (UTC)