Explicit description of isomorphism when decomposing into irreps - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:14:15Z http://mathoverflow.net/feeds/question/120873 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120873/explicit-description-of-isomorphism-when-decomposing-into-irreps Explicit description of isomorphism when decomposing into irreps Garfield 2013-02-05T16:30:32Z 2013-02-05T16:30:32Z <p>I had a question which is slightly more general than <a href="http://mathoverflow.net/questions/97203/question-about-decomposition-of-exterior-product" rel="nofollow">this one on mathoverflow</a>: I am looking for an explicit description of the isomorphism <code>$\mathbb S_\nu(V\otimes W) \cong \bigoplus C_{\lambda\mu\nu} \mathbb S_\lambda V\otimes \mathbb S_\mu W$</code> from Exercise 6.11 in Fulton &amp; Harris. <a href="http://mathoverflow.net/questions/97203/question-about-decomposition-of-exterior-product" rel="nofollow">In said question</a>, the author asks whether <code>$${\textstyle\bigwedge}^p(V\otimes W) \cong \bigoplus\nolimits_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} \mathbb{S}_\lambda V \otimes \mathbb{S}_{\bar\lambda}W$$</code></p> <p>is given by</p> <p><code>$$(v_1\otimes w_1)\wedge\ldots\wedge(v_p\otimes w_p) \longmapsto \sum\nolimits_{\substack{\lambda\vdash p\\\\\ell(\lambda)\le n\\\\\lambda_1\le m }} c_\lambda(v_1\otimes\ldots\otimes v_p) \otimes c_{\bar\lambda}(w_1\otimes\ldots\otimes w_p).$$</code></p> <p>The answer is affirmative and in the comments it is said that this is due to Schur-Weyl duality. I frankly don't understand that argument: I do not see why the map is well-defined. One would have to show that simultaneous permutation of the <code>$v_i$</code> and the <code>$w_i$</code> with some <code>$\pi\in S_p$</code> introduces a factor of <code>$\mathrm{sgn}(\pi)$</code> on the right, and this is not at all clear to me. Can someone explain?</p>