Existence of the detS2$det^{S^2}$ map

Abstract

In this paper we show that for a vector space 𝑉 𝑑 of dimension 𝑑 there exists a linear map𝑣 𝑥,𝑧 = 𝑣 𝑦,𝑧 . The existence of such a map was conjectured by Staic [The exterior graded Swiss-Cheese Operad Λ 𝑆 2 (𝑉) (with an appendix by Ana Lorena Gherman and Mihai D. Staic), to appear in Comm. Algebra.]. We present applications of the map 𝑑𝑒𝑡 𝑆 2 to geometry and combinatorics. M S C 2 0 2 0 15A15 (primary), 05C70 (secondary) 1 𝑉 𝑑 [2𝑑]) = 1. This conjecture is equivalent with the existence and uniqueness (up to a scalar) of a non-trivial linear map 𝑑𝑒𝑡 𝑆 2 ∶ 𝑉 ⊗𝑑(2𝑑−1