most recent 30 from http://mathoverflow.net 2013-05-24T23:43:08Z http://mathoverflow.net/feeds/question/108253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108253/schurs-di-lemma-finite-and-lie-groups-different Hauke Reddmann 2012-09-27T15:00:05Z 2012-09-27T15:00:05Z

<p>For a finite group it's nothing special if two one-dimensional irreps pop up in a product, e.g. for $C_{3v}$ symmetry, $E\bigotimes{E}=A_1\bigoplus{A_2}\bigoplus{E}$ or in dimensions, $2*2=1+1+2$. Schur's Lemma merely forces that only one $A_1$ can be in the product.<br> Now I never saw a Lie group irrep with dimension $1$ which is <em>not</em> the trivial irrep. Have I just not looked around far enough? :-) (Maybe e.g. beyond semisimple ones?)<br> (Background: Playing around with tensors for 3-nodes and crossings, I found a six-dimensional irrep which gives an invariant for tangled graphs, the Clebsch-Gordan series being $6*6=1+1+6+8+8+12$. The second $1$ is kind of an "antisymmetric one", like $A_2$. In every respect (I could check), this invariant behaves like the invariants coming from Lie groups.) </p>