Nontriviality of det S_r for r ≥ 4

Establish whether the determinant-like linear functional det_{S_r}: ⊗_{1≤i_1<⋯<i_r≤r d} V_d → k, characterized by the vanishing condition that det_{S_r}(⊗ v_{i_1,…,i_r}) = 0 whenever there exist indices 1 ≤ x_1 < ⋯ < x_{r+1} ≤ r d with v_{x_1,…,x_r} = v_{x_1,…,x_{r−1},x_{r+1}} = ⋯ = v_{x_2,…,x_{r+1}}, is nontrivial (i.e., not identically zero) for every integer d ≥ 1 when r ≥ 4.

Background

The paper introduces determinant-like maps det S_r that generalize the classical determinant and relate to r-equilibrium problems and hypergraph invariants. For r = 2 and r = 3, such maps are known to be nontrivial, but the situation for higher r is unresolved.

Determining nontriviality for r ≥ 4 is central to understanding whether these maps provide meaningful invariants across all dimensions d and to extending their applications in combinatorics and mechanics.