Invariant subspaces and isotypic decomposition (reference request)

Question

If $G$ is a (say) compact group and $V=\bigoplus_{i\in I}V_i$ the isotypic (a.k.a. primary) decomposition of a $G$ -module, then

any $G$ -invariant subspace $W\subset V$ writes $W=\bigoplus_{i\in I}(W\cap V_i)$ .

While this isn’t hard to prove (similar to Hoffman-Kunze 1971, §7.5 for a single operator), it seems silly to redo it in a paper. Unfortunately, the only reference I’m familiar with omits the proof (Kirillov 1976, §8.3), and when using it (e.g. Lie groups VIII.3.1) Bourbaki refers to such an abstrusely worded version (Algebra VII.2.2) that unpacking it takes as much work as a direct proof.

Q: What is a good reference to quote for this? Bonus points if the case of non-algebraically closed fields is spelled out.

A better Bourbaki reference is Algèbre VIII, §4, Proposition 4 (unfortunately not yet translated, as far as I know). It works for semi-simple modules, no fields involved. — abx, Mar 29 '20 at 12:44
@abx Would you like to make your comment an answer? It is the one I wish I could accept, so far. — Francois Ziegler, Apr 02 '20 at 11:58

score 2 · Accepted Answer · answered Apr 02 '20 at 13:43

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As suggested by the OP, I am turning my comment into an answer: a better Bourbaki reference is Algèbre VIII (new edition), §4, Proposition 4 d) (unfortunately not yet translated, as far as I know). It works for semi-simple modules over an arbitrary ring.

answered Apr 02 '20 at 13:43

abx

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Seems perfect, thanks. The earlier edition has it as VIII, §3, Proposition 9 b): “Tout sous-module $\mathrm N$ de $\mathrm M$ est somme directe des $\mathrm N\cap\mathrm M_\lambda$ ”. – Francois Ziegler Apr 02 '20 at 21:30

score 1 · Answer 2 · edited Apr 01 '20 at 23:28

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For compact groups you can quote IV.2.7 in Naimark-Stern. There the $T$ -isotypical component is described as the image of an operator $E^T$ .

For general semisimple categories it may be better to give a short modern proof. Define the center of the category. It is a product of division rings. Each simple object gives an idempotent in the center. This idempotent gives compatible idempotents in $Hom(V,V)$ and $Hom(W,W)$ , which you can split off.

edited Apr 01 '20 at 23:28

Francois Ziegler

29,500

answered Mar 31 '20 at 06:47

Bugs Bunny

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Thank you. Naimark and Stern’s long section 2.7 (pp. 199-205) has much useful material on the primary decomposition of a given $V$ (what they call $S$ ) and projectors onto a $V_i$ (what they call $\smash{E^T}$ ), but they don’t seem to state let alone prove results on restriction to arbitrary invariant $W\subset V$ , do they? – Francois Ziegler Apr 01 '20 at 23:27
They don't, you are right. They prove that $E^T$ is the projection $V\rightarrow V_{i}$ , killing all other isotypical components. The operator does not depend on a representation. Since $V_i$ (or $W_i$ ) is the 1-eigenspace of $E^T$ on $V$ (or $W$ ), it is a basic Linear Algebra exercise to deduce the statement you need. – Bugs Bunny Apr 02 '20 at 06:11

Invariant subspaces and isotypic decomposition (reference request)

2 Answers2