Timeline for What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
Current License: CC BY-SA 4.0
14 events
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| Dec 7, 2020 at 7:57 | history | edited | YCor | CC BY-SA 4.0 |
moved question to main text
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| Dec 7, 2020 at 4:47 | comment | added | Abdelmalek Abdesselam | The reason presentations stop after the PW Thm is these things are complicated. For $SU(2)$ or $SO(3)$ see, e.g., en.wikipedia.org/wiki/Wigner_D-matrix For higher dimension $SO(n)$, you might find formulas in the series of books springer.com/gp/book/9780792314660 | |
| Dec 7, 2020 at 4:47 | comment | added | Yemon Choi | I agree that it is not enough just to know the highest weight parametrization in order to obtain a basis, but as @LSpice and I have tried to point out, as soon as you have an explicit basis $v_1, \dots, v_n$ for $H_\pi$ then you can write down the corresponding matrix coefficient functions on $G$, namely $\langle \pi( \cdot) v_j, v_i \rangle$. Do you agree that this is enough? If so, then we can reduce your original question to the problem of computing an explicit ONB for each irrep, which I acknowledge is not trivial | |
| Dec 7, 2020 at 4:17 | comment | added | Andrew NC | It's somehow still not clear to me how to do that explicitly. Take for example the orthogonal group. I know how to parametrize the irreducible reps via the highest weight, but it's not clear to me what matrix coefficients are associated to each potential highest weight. I guess that the point is that the way that a basis of $H_{\pi}$ yields a basis of $A_{\pi}$ goes through knowing how $\pi(g)$ for any $g\in G$ acts on that basis... | |
| Dec 7, 2020 at 4:12 | comment | added | Yemon Choi | Schur orthogonality relations show that if you have an o.n. basis for each $H_\pi$ then this will give you a family of pairwise orthogonal coefficient functions in $A_\pi$, which can then be rescaled. | |
| Dec 7, 2020 at 4:12 | comment | added | LSpice | $\DeclareMathOperator\SO{SO}$ Re, it's just $K$ compact subgroup. As the article mentions, the motivating case is $K = \SO(2, \mathbb R)$ inside $G = \SO(3, \mathbb R)$, Your case is $K = 1$. \\ As to @YemonChoi's question, the point is there's obvious ONB for $A_\pi$: matrix coefficients of $v^* \otimes w$ where $v$ and $w$ range over ONB of $\pi$. What format answer do you want? | |
| Dec 7, 2020 at 4:11 | comment | added | Yemon Choi | In the case where G=SU(2) the irreps have an explicit description as symmetric tensor powers of the "tautological" 2-dim rep $V$. Picking an o.n. basis of the 2-dim space $V$ you can work out what norm to put on $\mathrm{Sym}^k({\mathbb C}^2)$ that makes it a unitary representation of $SU(2)$, and then this gives you an explicit o.n. basis for that irrep. Would this kind of construction be what you are after? | |
| Dec 7, 2020 at 4:08 | comment | added | Andrew NC | I mean giving an orthonormal basis for each $A_{\pi}$. | |
| Dec 7, 2020 at 4:05 | comment | added | Yemon Choi | When you say "compute the matrix coefficients" of a given irrep $\pi$, what do you mean? Do you mean some explicit parametrization with respect to a given o.n. basis of $H_\pi$? Or some kind of extrinsinc condition of the form "$f\in A_\pi$ if and only if some operator annihilates $f$"? | |
| Dec 7, 2020 at 3:41 | history | edited | Andrew NC | CC BY-SA 4.0 |
deleted 4 characters in body
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| Dec 7, 2020 at 3:41 | comment | added | Andrew NC | I'm confused about how that fits in -- it appears to me that zonal spherical functions are about L^2(G/K) where K is maximal compact. I am interested in the G is compact case. How should I be thinking about this? | |
| Dec 7, 2020 at 0:01 | history | edited | LSpice | CC BY-SA 4.0 |
\oplus -> \bigoplus
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| Dec 7, 2020 at 0:01 | comment | added | LSpice | A generalised version of this is called a zonal spherical function. | |
| Dec 6, 2020 at 23:41 | history | asked | Andrew NC | CC BY-SA 4.0 |