6533b86cfe1ef96bd12c8ab0

RESEARCH PRODUCT

Central polynomials and matrix invariants

subject

description

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 $$\begin{gathered} F^{\lambda ,\mu } = \sum\limits_{\sigma ,\tau \in S_n 2} {\left( {\operatorname{sgn} \sigma \tau } \right)x_\sigma (1) \cdot \cdot \cdot x_\sigma (\lambda _1 )^{y_\tau } (1)^{ \cdot \cdot \cdot } y_\tau (\mu _1 )^{x\sigma } (\lambda _1 + 1)} \hfill \\ \cdot \cdot \cdot x_\sigma (\lambda _1 + \lambda _2 )^{y_\tau } (\mu _1 ^{ + 1} )^{ \cdot \cdot \cdot y_\tau } (\mu _1 + \mu _2 ) \hfill \\ \cdot \cdot \cdot x_\sigma (\lambda _1 + \cdot \cdot \cdot + \lambda _{\mu - 1} ^{ + 1} ) \hfill \\ \cdot \cdot \cdot x_\sigma (n^2 )^{y_\tau } (\mu _1 ^{ + \cdot \cdot \cdot + \mu } \mu _{\mu - 1} ^{ + 1} )^{ \cdot \cdot \cdot y_\tau (n^2 )} \hfill \\ \end{gathered} $$ 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).