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

TL;DR

This paper proves the existence of a specific linear map related to symmetric tensor invariants, with applications in geometry and combinatorics, confirming a previously conjectured mathematical object.

Contribution

The paper establishes the existence of the $det^{S^2}$ map for vector spaces, providing a concrete construction and demonstrating its applications.

Findings

Existence of the $det^{S^2}$ map confirmed.

Map vanishes when certain triples are equal.

Applications demonstrated in geometry and combinatorics.

Abstract

In this paper we show that for a vector space $V_d$ of dimension $d$ there exists a linear map $det^{S^2}:V_d^{\otimes d(2d-1)}\to k$ with the property that $det^{S^2}(\otimes_{1\leq i<j\leq 2d}(v_{i,j}))=0$ if there exists $1\leq x<y<z\leq 2d$ such that $v_{x,y}=v_{x,z}=v_{y,z}$. The existence of such a map was conjectured in [4]. We present two applications of the map $det^{S^2}$ to geometry and combinatorics.