To compute minors of Jacobian of symmetric polynomials

2012-06-13T22:57:28Z

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:

http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=1955517&loc=fromreflist

Question: Let $n \geq 4$. Is the similar results also known for minors of Jacobian of symmetric polynomials (complete symmetric polynomials).

Remark: For power sum and elementary symmetric polynomials, one can derive easily, I am looking for complete symmetric polynomials.

Answer by Duchamp Gérard H. E. for To compute minors of Jacobian of symmetric polynomials

2013-03-14T04:06:54Z

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.

To compute minors of Jacobian of symmetric polynomials - MathOverflow

To compute minors of Jacobian of symmetric polynomials

Answer by Duchamp Gérard H. E. for To compute minors of Jacobian of symmetric polynomials