Wikipedia:Reference desk/Archives/Mathematics/2012 September 13

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 13

Irreducible nontrivial finite dimensional unitary representations of noncompact connected Lie groups operating on a complex vector space

Well, are there any such animals at all for any such group?

Please note the key words: noncompact, connected, irreducible, nontrivial, unitary, and complex.

I don't require the representation to be complex. There are subtleties in terminology here best illustrated by the induced Lie Algebra representation. Such a representation is complex if it is complex linear, (in which case we get a representation of the complexification of the algebra for free). Thus we can easily have a real representation on a complex space, see last paragraph.

In particular (if yes), are there any such representations for the (identity component of) the Lorentz group?

If yes, are there any n-dimensional ones where n > 1. (The determinant is always a 1-dimentional representation, in the case of the full Lorentz group on would get a nontrivial 1-dimensional representation, but not a very interesting one.)

I can think of a representation of R under addition taking t in R to a rotation matrix: t-> {(cos t, sin t),(-sin t, cos t)} in SO(2). This looks irreducible if operating on R x R. But if we allow it to operate on C x C, then it isn't irreducible because the matrix can then be diagonalized over C.

I have seen answers to closely related to this in the literature. They range from a plain "No" to "Such representations are very difficult to achieve". I suspect the anwer is "No, with the exception of a few nearly trivial examples". YohanN7 (talk) 10:59, 13 September 2012 (UTC)

I think that know the answer now.

• There are no finite dimensional irreducible representations of a noncompact, connected simple Lie group.
• The semisimple SL(2,C)×SU(2) is noncompact, but has many f d irreps by letting SL(2,C) act trivially.
• The connected component of the Lorentz group, SO(3,1) ,while being semi-simple and not simple, has no f d irreps. This was originally proved by Wigner. YohanN7 (talk) 14:07, 20 September 2012 (UTC)

Hmmmmmm. The algebra so(3;1) is simple (and semisimple of course). Thus the last point follows from the first. YohanN7 (talk) 14:28, 20 September 2012 (UTC)

plotting relations

I have a collection of items. I have a "weight" between each pair of items. For example, A-B is 10, A-C is 7, B-C is 19, and so on for around 1,000 items. I want to plot them in 2 dimensions, placing each item such that the mean difference between the plotted distance between any two points and the actual distance in 2D between those two points is minimized. Is there a "best" way to do this? — Preceding unsigned comment added by 128.23.113.249 (talk) 12:41, 13 September 2012 (UTC)

This amounts to plotting a "network visualization". In math, we just call them graphs, but that is confusing to outsiders. So other people use network, but that term is also overloaded. Anyway, I haven't used this program, but Graphvis [1] is a free, open source tool that should do what you want. You can use your weights between the nodes to determine spring constants as a start, though that won't generate a solution that truly minimizes the |given-plotted| error in distances. You could also try just setting the edge lengths equal to the weights. An exact solution might not be possible in 2D space, but it's still worth a try. SemanticMantis (talk) 14:53, 13 September 2012 (UTC)

See also Network_visualization#Layout_methods, and the list of software products at the bottom of that page. SemanticMantis (talk) 14:55, 13 September 2012 (UTC)

What you are trying to do is known in the trade as multidimensional scaling. It's an old concept and a number of methods have been developed, but none of them work perfectly, especially in cases where the "distances" are badly inconsistent. If you have access to a statistical analysis package, there is a good chance it will contain an MDS option. Looie496 (talk) 21:07, 13 September 2012 (UTC)

Looie, your answer seems much more relevant to the OP's problem than my shoddy attempt. Thanks! To partially redeem myself, I'll add that R_(programming_language) has MDS packages, and is free/open source. SemanticMantis (talk) 21:35, 13 September 2012 (UTC)

The link multidimensional scaling#Details is very interesting. At one point it says For some particularly chosen cost functions, minimizers can be stated analytically in terms of matrix eigendecompositions. (1) What objective function can this be done for? (2) And, for the quadratic function given in that link, I get interlocking nonlinear first order conditions that look like they would be impossible to solve in polynomial time--is this problem NP-hard? Duoduoduo (talk) 19:12, 15 September 2012 (UTC)