To compute minors of Jacobian of symmetric polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:00:48Z http://mathoverflow.net/feeds/question/99527 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99527/to-compute-minors-of-jacobian-of-symmetric-polynomials To compute minors of Jacobian of symmetric polynomials Neeraj 2012-06-13T22:57:28Z 2013-03-14T15:55:40Z <p>For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$ one has Jacobian, expressed by the $(n \times n)$-determinants: $$J(f_1,\dots,f_n):=|\frac{\partial}{\partial x_i}(f_j)|_{1 \leq i,j,\leq n}$$ And, one has for elementary symmetric polynomials $e_i$ of degree $i$, $$J(e_1,e_2,\dots,e_n)=\prod_{1 \leq i,j,\leq n}(x_i-x_j)=\triangle$$ also for complete symmetric polynomials and power sum symmetric polynomials, one has a very nice formula for Jacobian, see the following link:</p> <p><a href="http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&amp;s1=1955517&amp;loc=fromreflist" rel="nofollow">http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&amp;s1=1955517&amp;loc=fromreflist</a></p> <p><strong>Question</strong>: Let $n \geq 4$. Is the similar results also known for minors of Jacobian of symmetric polynomials (complete symmetric polynomials). </p> <p>Remark: For power sum and elementary symmetric polynomials, one can derive easily, I am looking for complete symmetric polynomials.</p> http://mathoverflow.net/questions/99527/to-compute-minors-of-jacobian-of-symmetric-polynomials/124482#124482 Answer by Duchamp Gérard H. E. for To compute minors of Jacobian of symmetric polynomials Duchamp Gérard H. E. 2013-03-14T04:06:54Z 2013-03-14T15:55:40Z <p>The Jacobian of a family of complete functions, or of power sums, is given in the article: A.Lascoux, P.Pragacz, {\it Jacobians of symmetric polynomials}, Annals of Comb. 6(2002) 169-172. An interesting fact is that Schur functions of "multiple" of alphabets occur in the case of complete functions.</p>