What is known about this plethysm? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:28:55Z http://mathoverflow.net/feeds/question/23391 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm What is known about this plethysm? David Speyer 2010-05-04T00:33:09Z 2012-03-28T20:23:49Z <p>Let $S^{\lambda}$ be a Schur functor. Is there a known <strong>positive</strong> rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps?</p> <hr> <p>In response to Vladimir's request for clarification, the ideal answer would be a finite set whose cardinality is the multiplicity of $S^{\mu}(\mathbb{C}^n)$ in $S^{\lambda}(\bigwedge^2 \mathbb{C}^2)$. As an example, the paper <a href="http://www.ams.org/mathscinet-getitem?mr=1331743" rel="nofollow">Splitting the square of a Schur function into its symmetric and anti-symmetric parts</a> gives such a rule for $\bigwedge^2(S^{\lambda}(\mathbb{C}^n))$.</p> <p>Formulas involving evaluations of symmetric group characters, or involving alternating sums over stable rim hooks, are not good because they are not positive.</p> <p>And, yes, it is easy to relate the answers for $\bigwedge^2 \mathbb{C}^n$ and $\mathrm{Sym}^2(\mathbb{C}^n)$, so feel free to answer with whichever is more convenient.</p> http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm/23412#23412 Answer by Vladimir Dotsenko for What is known about this plethysm? Vladimir Dotsenko 2010-05-04T10:52:16Z 2010-05-04T10:52:16Z <p>What kind of a formula will you find satisfactory? Formulas for the plethysm $s_\lambda\circ h_n$ where coefficients are expressed in terms of $S_n$-characters and generalized Kostka numbers are in Macdonald's book (see pp.138-140), so putting $n=2$ and applying the standard involution will give you some result for $e_2$ as well (which is your question, I presume)...</p> http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm/23491#23491 Answer by Bruce Westbury for What is known about this plethysm? Bruce Westbury 2010-05-04T21:42:58Z 2010-05-05T08:49:06Z <p>If I remember this correctly the cases $\mathrm{Sym}^k(\bigwedge^2 \mathbb{C}^n)$ and $\mathrm{Sym}^k(\mathrm{Sym}^2(\mathbb{C}^n))$ are known; and hence $\bigwedge^k(\bigwedge^2 \mathbb{C}^n)$ and $\bigwedge^k(\mathrm{Sym}^2(\mathbb{C}^n))$. I will look up the references tomorrow (if this is of interest).</p> <p><strong>Edit</strong> The result has now been stated. I learnt this from R.P.Stanley "Enumerative Combinatorics" Vol 2, Appendix 2. Specifically, A2.9 Example (page 449) which refers to (7.202) on page 503. This gives as the original reference (11.9;4) of the 1950 edition of:</p> <p>Littlewood, Dudley E. "The theory of group characters and matrix representations of groups."</p> <p>P.S. In the Notes at the end of 7.24 (bottom of page 404 in CUP 1999 edition) it discusses the origin and the etymology of "plethysm". It says:</p> <p>Plethysm was introduced in<br> MR0010594 (6,41c) Littlewood, D. E. Invariant theory, tensors and group characters.<br> Philos. Trans. Roy. Soc. London. Ser. A. 239, (1944). 305--365</p> <p>The term "plethysm" was suggested to Littlewood by M. L. Clark after the Greek word <em>plethysmos</em> $\pi\lambda\eta\theta\nu\sigma\mu o\zeta$ for "multiplication". </p> <p>(the Greek is an approximation)</p> http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm/23510#23510 Answer by Gjergji Zaimi for What is known about this plethysm? Gjergji Zaimi 2010-05-04T23:26:52Z 2010-05-04T23:26:52Z <p>From Weyman's book "Cohomology of Vector bundles and Syzygies" Chapter 2 gives the following decompositions: $$\mathrm{Sym}^m \left(\bigwedge^2 E\right)=\bigoplus_{\lambda \in A_m}S^{\lambda}E$$ $$\bigwedge^m \left(\bigwedge^2E\right)=\bigoplus_{\lambda \in B_m}S^{\lambda}E$$ where $A_m$ is the set of all $\lambda$ with $|\lambda|=2m$ such that all parts $\lambda_i$ are even. $B_m$ is the set of all partitions $\lambda$ of $2m$ so that when you write it in hook notation $\lambda=(a_1,\dots,a_r|b_1,\dots,b_r)$ you have $a_i=b_i+1$ for all $i$. Also, maybe <a href="http://www.mathnet.or.kr/mathnet/kms_tex/980064.pdf" rel="nofollow">this</a> article has some useful references.</p> http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm/23557#23557 Answer by mingming for What is known about this plethysm? mingming 2010-05-05T08:12:32Z 2010-05-05T08:12:32Z <p>Did you check the book by Procesi?</p> http://mathoverflow.net/questions/23391/what-is-known-about-this-plethysm/92500#92500 Answer by Maxim Leyenson for What is known about this plethysm? Maxim Leyenson 2012-03-28T20:17:17Z 2012-03-28T20:23:49Z <p>You may also use <a href="http://sagemath.org/" rel="nofollow">SAGE</a> , (for example, the <a href="https://sagenb.kaist.ac.kr/" rel="nofollow">Sage online notebook</a> )</p> <p>Example: </p> <p>The Riemann curvature tensor $R$ lives in the space $Sym^2(\Lambda^2 V)$ (after identifying $V$ with $V^{\vee}$)</p> <p>Decomposing it in Sage:</p> <p>$s = SFASchur(QQ)$ </p> <p>(let s be the Schur functor) </p> <p>$s([2])(s([1,1]))$</p> <p>(compute plethysm $Sym^2 \Lambda^2$)</p> <blockquote> <p>s[1, 1, 1, 1] + s[2, 2] </p> </blockquote> <p>-- i.e., $\Lambda^4 V + S_{[2,2]}$, as it should be </p> <p>\$ s([3])(s([1,1]))</p> <blockquote> <p>s[1, 1, 1, 1, 1, 1] + s[2, 2, 1, 1] + s[3, 3]</p> </blockquote> <p>-- though i understand that the explicit formula is better :)</p>