Timeline for Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?
Current License: CC BY-SA 3.0
7 events
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| Oct 28, 2012 at 17:39 | comment | added | Geoff Robinson | Well, if we extend the scalars to the complex field, the idempotent elements of the character ring are just the characteristic functions of the conjugacy classes, and these can be calculate by knowing how the basis elements multiply. On the other hand, the characteristic function of the class of $x$ is expressible as $\sum_{\chi}\frac{\chi(x^{-1}) \chi}{|C{G}(x)|}.$ | |
| Oct 28, 2012 at 16:49 | comment | added | Alexander Chervov | Thank you very nice comment ! Actually indeed I was assuming ring+basis, but your comment is very valuable still. So it seems to me it should be obvious that from ring+basis I can get char-table, but somehow I do not see it :( Would you be so kind to comment on this... | |
| Oct 28, 2012 at 13:45 | comment | added | Geoff Robinson |
I was assuming you were starting from the natural basis for the character ring. Without that, I think you may need the duality operation, given by complex conjugation in general. You then recognise the irreducible characters because you know the identity, and you can recognise the irreducible characters, given the integral character ring, because these are the elements $\theta$ for which $\theta(1) >0$ and the identity shows up with multiplicity $1$ in $\theta \overline{\theta}.$
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| Oct 28, 2012 at 12:44 | comment | added | Alexander Chervov | This is one direction: from c-table->ring, and from ring to c-table is it also possible ? | |
| Oct 28, 2012 at 12:30 | comment | added | Geoff Robinson |
Well, yes, because once you know the character table, you know how to decompose $\chi_{i}\chi_{j}$ for irreducible characters $\chi_{i}$ and $\chi_{j}$.
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| Oct 28, 2012 at 12:08 | comment | added | Alexander Chervov | Is "it" also equivalent to the knowledge of (ring+basis) of irreps ? | |
| Oct 28, 2012 at 11:17 | history | answered | Geoff Robinson | CC BY-SA 3.0 |