On generating the ring of matrix semi-invariants

الأصل · EN

For a field F, let R(n, m) be the ring of invariant polynomials for the action of SL(n, F) × SL(n, F) on tuples of matrices -- (A, C)(n, F) × SL(n, F) sends (B₁,, Bₘ)∈ M(n, F)⊕ ᵐ to (AB₁C⁻¹,, ABₘC⁻¹). In this paper we call R(n, m) the ring of matrix semi-invariants. Let β(R(n, m)) be the smallest D s.t. matrix semi-invariants of degree ≤ D generate R(n, m). Guided by the Procesi-Razmyslov-Formanek approach of proving a strong degree bound for generating matrix invariants, we exhibit several interesting structural results for the ring of matrix semi-invariants R(n, m) over fields of characteristic 0. Using these results, we prove that β(R(n, m))=Ω(n³/²), and β(R(2, m))≤ 4.