Schur-Weyl duality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:10:25Z http://mathoverflow.net/feeds/question/76497 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76497/schur-weyl-duality Schur-Weyl duality Ravi 2011-09-27T11:40:10Z 2011-09-27T18:32:07Z <p>I have a question about the interpretation of multiplicities and dimensions using Schur-Weyl duality.</p> <p>$V$ is an n-dimensional complex vector space. Then $V$ $\otimes$ $V$ $\otimes$ $V$ decomposes as:</p> <p>$V$ $\otimes$ $V$ $\otimes$ $V$ = $Sym^3$ $V$ $\oplus$ $\wedge^3$ $V$ $\oplus$ $S_{(2,1)}V$ $\oplus$ $S_{(2,1)}V$</p> <p>where $S_{(2,1)}V$ is a Schur module for the partition (2,1). (Fulton-Harris, Chapter 6).</p> <p>Then Schur-Weyl duality says that the multiplicity of $S_{(2,1)}V$ in this decomposition (it is 2) should be the dimension of the irrep of $S_3$ labelled by the partition (2,1) - which is correct.</p> <p>My question is about the other half of this duality: the dimension of $S_{(2,1)}V$ in this decomposition (it is 8) should correspond to some sort of multiplicity for the irrep of $S_3$ labelled by the partition (1,2) - but I am unable to see exactly what...</p> <p>Any help is most welcome.</p> http://mathoverflow.net/questions/76497/schur-weyl-duality/76500#76500 Answer by David Speyer for Schur-Weyl duality David Speyer 2011-09-27T12:04:39Z 2011-09-27T12:41:00Z <p>The dimension of $S_{21}(V)$ is $(n+1)n(n-1)/3$; it is only $8$ if $n=3$. The other half of Schur Weyl duality says that $S_{21}(V)$ is a $GL_n$ (not $S_3$) irrep; namely, the one which is indexed by the partition $(2,1)$. Similarly, $\mathrm{Sym}^3(V)$ and $\bigwedge^3 V$ have dimensions $(n+2)(n+1)n/6$ and $n(n-1)(n-2)/6$ and are $GL_n$, not $S_3$, irreps.</p> http://mathoverflow.net/questions/76497/schur-weyl-duality/76540#76540 Answer by David Hill for Schur-Weyl duality David Hill 2011-09-27T18:00:54Z 2011-09-27T18:00:54Z <p>Let $V$ be the vector representation of $GL_n(\mathbb{C})$, and let $d\leq n$. </p> <p>You want to see the multiplicities of a given irreducible $S_d$ module in $V^{\otimes d}$ in terms of the dimension of an associated irreducible $GL_n(\mathbb{C})$-module. To do this, observe that $S_d$ acts on $V^{\otimes d}$ by permuting the tensor factors and this action commutes with the diagonal action of $GL_n(\mathbb{C})$ on $V^{\otimes d}$. Therefore, for a partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\geq 0)$ (equivalently, a polynomial weight for $GL_n(\mathbb{C})$) we can consider the space $W_\lambda$ of highest weight vectors in $V^{\otimes d}$ of highest weight $\lambda$. Since the action of $S_d$ commutes with the action of $GL_n(\mathbb{C})$ (and, in particular, the action of the standard maximal torus, and standard Borel), $W_\lambda$ is an $S_d$-module. By character considerations, one can show that this space is the irreducible Specht module, $W_\lambda\cong S^{\lambda}$. </p> <p>More generally, if $X\in GL_n(\mathbb{C})$, then $X\cdot W_\lambda$ is another $S_d$-module since, again, the action of $S_d$ commutes with the action of $GL_n(\mathbb{C})$. Moreover, this module is obviously isomorphic to $S^\lambda$. This means that there are $\dim L(\lambda)$ copies of $S^\lambda$ in $V^{\otimes d}$, where $L(\lambda)$ is the irreducible $GL_n(\mathbb{C})$-module of highest weight $\lambda$. </p> <p>This proves half of the statement that, as a $(GL_n(\mathbb{C}),S_d)$-bimodule, $V^{\otimes d}\cong \bigoplus_{\lambda\vdash d}L(\lambda)\otimes S^\lambda.$ From your question, I assume you understand the other half. </p> http://mathoverflow.net/questions/76497/schur-weyl-duality/76542#76542 Answer by Andy B for Schur-Weyl duality Andy B 2011-09-27T18:32:07Z 2011-09-27T18:32:07Z <p>The multiplicity of a Specht module $Specht(\lambda)$ (Fulton-Harris Ch.4) in $V^{\otimes n}$ is the number semi-standard Young tableaux of shape $\lambda$ and entries in ${1,2,\dots,\dim V}$. In your example (where I've guessed you're assuming $\dim V = 3$) the multiplicity of the Spetch module $Specht((2,1))$ is the number of SSYT of this shape with entries in ${1,2,3}$. This is computed using the hook-content formula, and is 8 (or just write down the 8 tableuax).</p> <p>It is instructive to find a basis of the Schur module of shape $\lambda$ indexed by SSYT of shape $\lambda$. See Fulton's book on Young tableuax for this, or the book Constructive invariant theory by Sturmfels. </p>