Ha–Hai–Nghia Delta-Operator Basis Conjecture for GL_n-invariants

Establish, for all integers n ≥ 1 and m ≥ 1 over a finite field F_q, that the set B_m(n) = { δ_{n−s}(f) : f ∈ _s^m, 0 ≤ s ≤ min(m,n) } forms a basis of the GL_n(F_q)-invariant subspace of Q = F_q[x_1, …, x_n]/(x_1^{q^m}, …, x_n^{q^m}), where δ_{s;m} is the determinantal delta operator acting on suitable Dickson algebra subspaces _s^m as defined by Ha–Hai–Nghia.

Background

Ha–Hai–Nghia introduced a determinantal “delta operator” acting on structured subspaces of the Dickson algebra and proposed a candidate basis B_m(n) for invariants of truncated polynomial rings under GL_n(F_q). They verified this basis for ranks n ≤ 3. The present paper proves the rank-4 case under an explicit matching hypothesis (H_match), thereby conditionally extending their program.

Although the paper’s main theorem confirms the conjecture in rank four assuming (H_match), the abstract explicitly refers to the broader Ha–Hai–Nghia conjecture, which concerns the general validity of this delta-operator basis construction beyond low ranks.