The Orchard relation of a generic symmetric or antisymmetric function

We associate to certain symmetric or antisymmetric functions on the set ( E d+1 ) of (d+ 1)−subsets in a finite set E an equivalence relation on E and study some of its properties. 1 Definitions and main results We consider a finite set E and denote by (E d ) the set of subsets containing exactly d elements of E. In the sequel we move often freely from sets to sequences: we identify a subset {x1, . . . , xd} ∈ (E d ) with the finite sequence (x1, . . . , xd) where the order of the elements is for instance always increasing with respect to a fixed total order on E. A function φ : (E d ) −→ R is symmetric if φ(x1, . . . , xi, xi+1, . . . , xd) = φ(x1 . . . , xi−1, xi+1, xi, xi+1, . . . , xd) for 1 ≤ i < d and all {x1, . . . , xd} ∈ (E d ) . Similarly, such a function φ : (E d ) −→ R is antisymmetric if φ(x1, . . . , xi, xi+1, . . . , xd) = −φ(x1 . . . , xi−1, xi+1, xi, xi+2, . . . , xd) for 1 ≤ i < d and all x1, . . . , xd ∈ E. φ is generic if φ(x1, . . . , xd) 6= 0 for all subsets {x1, . . . , xd} ∈ (E d ) of d distinct elements in E. In the sequel of this paper all functions will be generic. We will mainly be concerned with sign properties of generic symmetric or antisymmetric functions: Given any symmetric generic function σ : (E d ) −→ R>0 and a symmetric or antisymmetric generic function φ : (E d ) −→ R, the two functions (x1, . . . , xd) 7−→ φ(x1, . . . , xd) and (x1, . . . , xd) 7−→ σ(x1, . . . , xd)φ(x1, . . . , xd)