## Can one give a “nice” expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.

Let $(x_{ij})$ be an $n \times n$ matrix, and define $p(T_1,\ldots,T_n) = \det (x_{ij} - \delta_{ij}T_i)$. Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $(x_{ij})$ counted with algebraic multiplicity. The problem is to find a "reasonable" general formula for $$\prod_{\sigma \in \mathfrak{S}_n} p(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(n)}).$$

For example, when $n=1$ this expression vanishes identically. When $n=2$ one gets $x_{12}x_{21}\Delta$, where $\Delta = (\lambda_1-\lambda_2)^2$ is the discriminant of the characteristic polynomial. A conceptual way of seeing that this is the right answer for $n=2$ is that the formula must vanish on all lower or upper triangular matrices, and when the eigenvalues coincide all factors of the product vanish.

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 Some computations plus handwaving seem to tell me that this expression, seen as a homogeneous polynomial in the entries of the matrix, is irreducible when $n=3$. I could post the argument here, but I'd prefer it if you tell me first whether the problem is still open and this negative result of any interest (none of these is clear to me, given that the question is 3 months old...). – darij grinberg Dec 10 2011 at 5:51