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I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't really make a great deal of use of the concept so I'm not really clear on how it differs from a "normal" character, why we would want to use such a thing, etc.

To me, a "normal" character $\chi$ is the trace of a representation $\rho$; that is $\chi(g) := Tr(\rho(g))$, where $\rho: G \to GL(M)$. I am almost always working over $\mathbb{C}$. Now the notes say the following:

Characters of topological groups: Let $T_n \leq GL_n$ be the n-dimensional torus. If $\rho: GL_n \to GL(W)$ is a rational representation, then the character of $\rho$ is $\chi_W = \chi_\rho: (x_1,\ldots,x_n) \mapsto Tr(\rho(\operatorname{diag}(x_1,\ldots,x_n)))$.

They then go on to detail some basic properties such as $\chi_\rho$ being symmetric. I am hoping maybe someone with more familiarity than me might be able to help me decode these notes; I thought maybe I had written something down wrong but there are some other sources talking about characters from topological groups related to the torus (such as http://mathoverflow.net/questions/86089/two-definitions-of-character-of-topological-groups).

If the above does actually make sense - $T_n$ is isomorphic to the product of $n$ circles $S_1$, right? So then the only way I can see it embedding into $GL_n$ is as a diagonal matrix which we apply to some basis. This seems like some sort of extension of considering $\mathbb{C}^\times$ as a group of linear maps "multiplication by $\mathbb{C}$". However, why for example would we choose the torus? What benefit does it provide to use the torus (rather than e.g. $\mathbb{C}^n$)? Also, it seems like most of the other mentions of this topological group character seem to have maps going to the torus, not from it, so it's possible I'm not working with the normal definition.

I guess this is a slightly vague question so apologies for that, but what I'd really appreciate, if someone can make sense of the definition above, is why we would use this definition and what for; why is it better than the normal character definition I've stated? Any small introductory-level context you could give me about why people care about the character as defined in this manner would be greatly appreciated - no enormous detail needed, I probably won't actually use it for anything but rather I'm just curious about what sort of use this additional layer of structure for characters provides. Many thanks.

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Dear Spyam, In this context, torus means diagonal matrix. So if $GL_n$ is over $\mathbb C$, then $T_n$ is a product of $n$ copies of $\mathbb C^{\times}$, not a product of $n$ circles. Regards, –  Matt E May 28 '12 at 13:34
    
Ah, that makes sense, this was poorly explained (or rather, not explained at all) in the notes; thank you for the help. –  Spyam Jun 1 '12 at 16:14
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up vote 2 down vote accepted

The point is that the character of a representation of $GL_n$ is a priori a function of $n^2$ variables. But since we are looking at rational (in particular: continous) representations, it's enough to define it on a dense subset. It turns out that the set of diagonalizable matrices is dense, and since the trace is invariant by conjugation you can restrict to actual diagonal matrices.

In fact every rational representation of $GL_n$ is the tensor product of a polynomial representation by a representation of the form $g \mapsto \det(g)^{-k}$ for some integer $k$. Now characters of polynomial representations are, as you point out, symmetric polynomial in $n$ variables. In fact, characters of irreducibles representations are the so called Schur polynomials and gives a linear basis of the algebra of symmetric polynomials.

The point with all of this is that there are a lot of combinatorial results and methods in the area of symmetric polynomials, which you can gives a representation theoretic interpretation (the most notable example is the Littlewood-Richardson rule).

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Thanks Adrien, that helped a lot. –  Spyam Jun 1 '12 at 16:15
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