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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:


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.

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1 Answer 1

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.

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Dear Duchamp, Infact I have given the same reference. Jacobian is given, but not for the minors. It seems difficult to me, to get anything out of that paper to formulate for the minors. If you know any case, explicitely. I would appreciate that. –  Neeraj Mar 16 '13 at 20:06

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