Schur's Di-Lemma: finite and Lie groups different? - MathOverflow 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 Schur's 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>