Configuration spaces of points, symmetric groups and polynomials of several variables

Abstract

Denoting by $C_n(X)$ the configuration space of $n$ distinct points in $X$, with $X$ being either Euclidean $3$-space $\mathbb{E}^3$ or hyperbolic $3$-space $\mathbb{H}^3$ or $\mathbb{C}P^1$ , by $\mathscr{P}_{k,d}$ the vector space of homogeneous complex polynomials in the variables $z_0, \ldots, z_k$ of degree $d$, and by $\mathrm{Obs}^n_d$ the set of all $d$-subsets of $\{1,\ldots,n\}$, the symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^3)$ by permuting the $n$ points and also acts in a natural way on $\mathrm{Obs}^n_d$. With $n = k+d$, the space $\mathscr{P}_{k,d}$ has dimension $\binom{n}{d}$, which is also the number of elements in $\mathrm{Obs}^n_d$. It is thus natural to ask the following question. Is there a family of continuous maps $f_I: C_n(X) \to \mathbb{P}\mathscr{P}_{k,d}$, for $I \in \mathrm{Obs}^n_d$ (here $\mathbb{P}$ is complex projectivization), which satisfies $f_I(\sigma.\mathbf{x}) = f_{\sigma.I}(\mathbf{x})$, for all $\sigma \in \Sigma_n$ and all $\mathbf{x} \in C_n(X)$, and such that, for each $\mathbf{x} \in C_n(X)$, the polynomials $f_I(\mathbf{x})$, for $I\in \mathrm{Obs}^n_d$, each defined up to a scalar factor, are linearly independent over $\mathbb{C}$? We provide two closely related smooth candidates for such maps for each of the two cases, Euclidean and hyperbolic, which would be solutions to the above problem provided a linear independence conjecture holds. Our maps are natural extensions of the Atiyah-Sutcliffe maps. Moreover, we get two constructions of actual solutions of the above problem for $X = \mathbb{C}P^1$, as we prove linear independence for these last two constructions. These last two constructions are classical in character, and can be viewed as higher dimensional versions of Lagrange polynomial interpolation. They appear to be new.