Identities of symmetric and skew-symmetric matrices in characteristicp

Abstract

It is well known how the Kostant-Rowen Theorem extends the validity of the famous Amitsur-Levitzki identity to skew-symmetric matrices. Here we give a general method, based on a graph theoretic approach, for deriving extensions of known permanental-type identities to skew-symmetric and symmetric matrices over a commutative ring of prime characteristic. Our main result has a typical Kostant-Rowen flavour: IfM≥p[n+1/2] then $$C_M (X,Y) = \sum\limits_{\alpha ,\beta \in Sym(M)} {x_{\alpha (1)} y_{\beta (1)} x_{\alpha (2)} y_{\beta (2)} } ...x_{\alpha (M)} y_{\beta (M)} = 0$$ is an identity onM n − (Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1Ω=0 (provided that $$P > \sqrt {[n + 1/2)} $$ ). Otherwise, the stronger conditionM≥pn implies thatC M(X,Y)=0 is an identity on the full matrix ringM n(Ω).