Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.