Let a linear finite-dimensional isotypic (i.e., decomposable into a direct sum of isomorphic irreducible subrepresentations) representation of a compact Lie group $G$ in a finite-dimensional space $V$ is given. How to describe all linear transformations of the space $V$, that commute with all elements of the image of this representation? This is probably known. I have my own assumption about this, but I want to know a specific link to the formulation and proof of a way to describe all such transformations
$\begingroup$
$\endgroup$
6
-
2$\begingroup$ I assume you're working with complex representations. The space of transformations of $V^{\oplus n}$, which is $G$-isomorphic to $V \otimes_{\mathbb C} \mathbb C^n$ (where $\mathbb C^n$ is the $n$-dimensional isotrivial representation of $G$), that commute with $G$ are parameterised by $\operatorname{End}(\mathbb C^n)$, acting purely on the second tensor factor. Proof: for each pair $(i, j)$, the map $V \xrightarrow{\operatorname{in}_i} V^{\oplus n} \to V^{\oplus n} \xrightarrow{\operatorname{out}_j} V$ is $G$-equivariant, hence a scalar. Now assemble the scalars into a matrix. $\endgroup$– LSpiceCommented Sep 23, 2023 at 17:45
-
$\begingroup$ No, I work with real representations! I supposed tensor product too. $\endgroup$– Vladimir47Commented Sep 23, 2023 at 19:21
-
1$\begingroup$ With real representations you get $\text{End}(V^n) \cong M_n(\text{End}(V))$ where $\text{End}(V)$ could be $\mathbb{R}, \mathbb{C}$, or $\mathbb{H}$. $\endgroup$– Qiaochu YuanCommented Sep 23, 2023 at 20:37
-
$\begingroup$ Thank you for your answers. But I want to have some reference. And what is End(V)? By V you mean a simple G-module (in my question V is a isotypic one)? And I want to have a concrete answer for G=SU(2) (real representations). $\endgroup$– Vladimir47Commented Sep 24, 2023 at 7:48
-
$\begingroup$ Re, I misread your notation, so we can call the simple $W$, and then $V = W^{\oplus n}$. The space $\operatorname{End}(W)$ (I write $\operatorname{End}_G(W)$) consists of linear endomorphisms of $W$ that commute with the action of $G$. These are multiplications by (complex) scalars for $W$ complex. A real $W$ carries a $G$-equivariant structure of a module over $\mathbb R$, $\mathbb C$, or $\mathbb H$, and then the operators are again scalars from the appropriate division algebra. That seems pretty concrete! $\endgroup$– LSpiceCommented Sep 24, 2023 at 17:43
|
Show 1 more comment