Dear all,
Consider the $(n+1)\times (n+1)$ matrix $A$ with indeterminates $X_i, Y_i$, $0\leq i\leq n$ such that the $(i,j)$-th entry is given by $X_i^jY_i^{n-j}$. The $i$-th row is $(X_i^n,X_i^{n-1}Y_i,\dots,Y_i^n)$ that can be thought as given by the Veronese embedding $\mathbb{P}^1\to\mathbb{P}^n$. The determinant of $A$ is multi-homogeneous and hence defines a variety in $\prod_{i=0}^n\mathbb{P}^1$. It can be checked that $\det(A)$ has the following nice factorization
$$
\det(A)=\prod_{0\leq i<j\leq n}\det\left(\begin{smallmatrix}X_j & Y_j\\X_i & Y_i\end{smallmatrix}\right)
$$
which gives the decomposition of the corresponding variety. Actually, setting $Y_i=1$, $0\leq i\leq n$, the matrix $A$ simply becomes the Vandermonde matrix and the above factorization becomes $\det(A)=\prod_{0\leq i<j\leq n}(X_j-X_i)$, as is well known.
We may also consider its higher dimensional generalization: for $n,d\geq 1$, let $N={n+d\choose d}$. Consider the $N\times N$ matrix $A$ whose $i$-th row is given by the $n$-uple Veronese embedding $\mathbb{P}^d\to\mathbb{P}^N$ that sents $(X_{i,0},\dots,X_{i,d})$ to the $N$-tuple consisting of all monomials in $X_{i,0},\dots,X_{i,d}$ with total degree $n$.
Question: for $d>1$, does $\det(A)$ admit similar factorization? Or is it simply irreducible?
Setting $X_{i,0}=1$ for all $i$ gives the higher dimensional analogue of the Vandermonde matrix which arises naturally in, for example, the problem of multivariate interpolation. For the multivariate case, I have never seen a simple explicit form of $\det(A)$ as the univariate one, though. So I guess it must be complicated. And my understanding in this case is very limited. I tried to calculate the determinant in a naive way (by mimicing the calculation in the 1-dimension case) but the amount of calculation blows up quickly. But maybe there are structures that I didn't recognize which help simplifying the calculation. So I am also looking for any suggestion for better calculation methods (grobner basis, tools from algebraic combinatorics, representation theory, etc).
Thanks a lot!
Best Answer
Looking at the same issue i found this paper : D’Andrea, Carlos, and Luis Tabera. "Tropicalization and irreducibility of generalized Vandermonde determinants." Proceedings of the American Mathematical Society 137.11 (2009): 3647-3656. It's a bit above my algebra skills for now but maybe it could help.