Timeline for How to write down explictly the isomorphism of two finite dimensional representation of compact groups?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2013 at 12:41 | comment | added | Bruce Westbury | Look up "Koszul complex" | |
| Jan 29, 2013 at 12:38 | comment | added | Claudio Gorodski | As a general rule, if $V=\oplus_i V_i$, $W=\oplus_j W_j$ are decompositions into $G$-irreducible components, then $\mathrm{Hom}(V,W)=\oplus_{i,j}\mathrm{Hom}(V_i,W_j) =\oplus_{i,j}(V_i)^*\otimes W_j$ and for the $G$-equivariant homomorphisms we have ${\mathrm{Hom}}_G(V,W)=\oplus_{i,j}((V_i)^*\otimes W_j)^G$ where Schur s lemma tells us that $((V_i)^*\otimes W_j)^G$ is zero$ if $V_i$, $W_j$ are not equivalent and one-dimensional otherwise. So the problem is equivalent to finding fixed vectors of a representation. | |
| Jan 29, 2013 at 12:10 | answer | added | ACL | timeline score: 1 | |
| Jan 29, 2013 at 11:25 | comment | added | Allen Knutson | You could ask your general question for trivial representations, which is about showing two vector spaces with the same dimension are isomorphic. Of course we usually do that by picking bases of each. The $G$-analogue is to break into irreps, first. Now we want to isomorph two irreps $V,W$ of the same type. An irrep is linearly spanned by the $G$-orbit of the high weight vector, so once we find high weight vectors in $V,W$ (unique up to scale) and correspond them, $G$-equivariance tells us how to correspond their orbits, and linearity does the rest. | |
| Jan 29, 2013 at 11:20 | comment | added | S. Carnahan♦ | Perhaps you mean $U(V)$. | |
| Jan 29, 2013 at 10:03 | answer | added | Johannes Ebert | timeline score: 3 | |
| Jan 29, 2013 at 9:33 | comment | added | Johannes Ebert | ""isomorphic as repn of GL(V)" is equivalent to isomorphic as repn of SU(V) (when chosen a metric) since GL(V) and SU(V) generate the same subalgebra in End(V), by density theorem." This is not true: you can twist any rep of $GL(V)$ by a power of the determinant; without changing the restriction of the representation to $SL(V)$. | |
| Jan 29, 2013 at 8:25 | history | asked | user22381 | CC BY-SA 3.0 |