What is Gelfand-Tsetlin basis for an irreducible representation of sl(n)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:39:35Z http://mathoverflow.net/feeds/question/68644 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68644/what-is-gelfand-tsetlin-basis-for-an-irreducible-representation-of-sln What is Gelfand-Tsetlin basis for an irreducible representation of sl(n)? Melania 2011-06-23T19:02:31Z 2011-06-24T05:32:59Z <p>Let us consider, for example, the standard irreducible $\mathfrak{sl}_3$-module </p> <p>$\Gamma_{1,1}$ with the highest weight $(1,1),$ $\dim \Gamma_{1,1}=8.$ </p> <p>The set of all weight of $\dim \Gamma_{1,1}$ is $(0,0),(1,1),(2,-1),(1,-2),(-2,1),(-1,2),(-1,-1).$</p> <p><strong>Question.</strong> What is the Gelfand-Tsetlin basis for $\Gamma_{1,1}$?</p> <p>As I understood from literature (Zhelobenko ) there is a combinatorial structure $\Lambda$ depended of $(1,1)$ such that a basis of $\dim \Gamma_{1,1}$ can be labeled via the $\Lambda$ but I cant do it. Anybody can help?</p> http://mathoverflow.net/questions/68644/what-is-gelfand-tsetlin-basis-for-an-irreducible-representation-of-sln/68694#68694 Answer by Sasha for What is Gelfand-Tsetlin basis for an irreducible representation of sl(n)? Sasha 2011-06-24T05:03:08Z 2011-06-24T05:03:08Z <p>Let $V$ denote the standard representation of $sl_3$. Then your $\Gamma_{1,1}$ is $V\otimes V^*/k$. To obtain a GT-basis we have to choose a flag of subspaces $0 \subset V_1 \subset V_2 \subset V_3 = V$ ($\dim V_i = i$) and restrict to $sl(V_i) \subset sl(V)$. First, let us take $U = V_2$. Then $V = U \oplus k$ (as $sl(U)$-module), hence $\Gamma = sl(U) + U + U^* + k$. Now if $e_1,e_2,e_3$ is a basis of $V$ such that <code>$V_i = &lt;e_j&gt;_{j \le i}$</code> and $e^i$ is the dual basis of $V^*$ then <code>$$sl(U) = &lt; e_1\otimes e^2, e_2\otimes e^1, e_1\otimes e^1 - e_2\otimes e^2&gt;.$$</code> Further, <code>$$U = &lt;e_1\otimes e^3, e_2\otimes e^3&gt;,\ U^* = &lt;e_3\otimes e^1, e_3\otimes e^2&gt;,\ k = &lt; e_1\otimes e^1 + e_2\otimes e^2 - 2e_3\otimes e^3 &gt;.$$</code> So, collecting all these vectors you get the GT-basis.</p> http://mathoverflow.net/questions/68644/what-is-gelfand-tsetlin-basis-for-an-irreducible-representation-of-sln/68697#68697 Answer by Victor Protsak for What is Gelfand-Tsetlin basis for an irreducible representation of sl(n)? Victor Protsak 2011-06-24T05:23:50Z 2011-06-24T05:32:59Z <p>It's more common to talk about the GT basis of $\mathfrak{gl}_n$-modules. In the case of the adjoint representation, the first row of the GT scheme is $(1,0,\ldots,0,-1)$ and each subsequent row satisfies the interlacing condition, which implies that the left (resp, right) edge consists of a string of $1$s (resp $-1$s), followed by a string of $0$s (possibly empty), except that the bottom entry can be $1$ (resp $-1$) if all entries on the left (resp right) are $1$s (resp $-1$), and all other entries are zeros. It follows that the whole scheme can be reconstructed from the positions of the lowest $1$ along the left edge of the triangle and the lowest $-1$ along the right edge. They may occupy any of the $n$ possible rows/positions each, except that both cannot occur in the $n$th row, corresponding to the impossibility of the bottom entry being simultaneously $1$ and $-1.$</p> <p>In your special case $n=3$ the diagrams will look like this:</p> <p><code>$$\begin{array}{rrrrr} 1 &amp; &amp; 0 &amp; &amp; -1\\ &amp; 1 &amp; &amp; 0 &amp; \\ &amp; &amp; 1 &amp; &amp; \end{array} \quad$$</code></p> <p>(this is the highest weight; there are 7 more).</p>