Eulerian Polynomial Identities on Matrix Rings Original Research Article

We prove that [formula] is a polynomial identity on Mn(Ω) over any commutative ring Ω with 1; here Γ is an Eulerian directed graph with k vertices and N edges, N ≥ 2kn, and Π(Γ) is the set of covering directed paths of Γ (viewed as permutations with respect to an arbitrary but fixed ordering of the edges of Γ). The standard and double Capelli identities can be obtained from extremely simple Eulerian graphs.