Timeline for Dimension of $\mathrm{Hom}_G(V, W)$ in terms of characteristic polynomial
Current License: CC BY-SA 4.0
9 events
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| Jan 27, 2022 at 7:41 | comment | added | Alphonse | @GeorgLehner : thanks, I forgot that irreps of an abelian group are 1-dimensional... (it should also be true for infinite abelian groups, see math.stackexchange.com/questions/354926, en.wikipedia.org/wiki/…) | |
| Jan 27, 2022 at 7:35 | vote | accept | Alphonse | ||
| Jan 26, 2022 at 23:31 | comment | added | Georg Lehner | So knowing the $a_i$ which appear in the characteristic polynomial $\rho_V(\phi)$ completely determines the decomposition into irreducibles. The formula for the dimension of $Hom_G$ now follows from Schurs Lemma. (And I hope the same reasoning goes through for the continuous case as well) | |
| Jan 26, 2022 at 23:30 | comment | added | Georg Lehner | If $V$ decomposes as a sum of irreducibles $V_i$, then the characteristic polynomial of $\rho_V(\phi)$ is the product of the characteristic polynomials $\rho_{V_i}(\phi)$. Since $G$ is abelian, all irreducible representations are 1-dimensional, therefore each irreducible $V_i$ gives you one eigenvalue $a_i$. Conversely, since $G$ is cyclic, the character of $V_i$ is completely defined by $a_i$ (the values are powers of it) hence $V_i$ is determined as well. | |
| Jan 26, 2022 at 15:30 | comment | added | Alphonse | Thank you very much. May I ask how exactly the eigenvalues 𝑎𝑖 give a decomposition of $V$ into irreps? The $a_i$'s might not be all distinct a priori. (Sorry if this is a silly question, I am not very familiar with representation theory...). This has probably to do with the group being (pro)-cyclic? | |
| Jan 26, 2022 at 8:41 | comment | added | Johannes Hahn | You are right. Then, we're in the first case again. | |
| Jan 25, 2022 at 19:26 | comment | added | Benjamin Steinberg | Doesn't the question demand V,W be semisimple? | |
| Jan 25, 2022 at 17:09 | history | edited | Johannes Hahn | CC BY-SA 4.0 |
Fixed typo
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| Jan 25, 2022 at 11:51 | history | answered | Johannes Hahn | CC BY-SA 4.0 |