Asymptotics in cocharacters of simple algebras with involution. (Q713392)

The paper under review studies the \(*\)-polynomial identities of finite dimensional simple algebras with involution over a field \(F\) of characteristic 0. It is well known that up to a PI equivalence one may consider three types of algebras, namely \((F_k,t)\), the \(k\times k\) matrices over \(F\) with the transpose involution; \((F_k,s)\), the \(k\times k\) matrices with the symplectic involution for even \(k\); and the algebras \(D_k=F_k\oplus F_k\) with the exchange involution. As the field \(F\) is of characteristic 0 one may consider multilinear identities only. In the study of \(*\)-identities one uses the fact that the vector space of the multilinear elements of degree \(n\) is a module over the hyperoctahedral group \(B_n\) (a semidirect product of \(S_n\) and \(\mathbb Z_2\)). Thus one may study the \(S_n\)-cocharacter and also the \(B_n\)-cocharacter of an algebra with involution. If \((A,*)\) is an algebra with involution one considers the corresponding relatively free algebra with involution and its vector subspaces of multilinear elements of degree \(n\); these are \(B_n\)-modules in a natural way. The \(n\)-th colength of \(A\) is the sum of the multiplicities of the irreducible \(B_n\)-modules in their decomposition while the maximal multiplicity of \(A\) is the largest among the multiplicities. The author describes the asymptotics of the maximal multiplicities for the \(B_n\)-cocharacter for each one of the three types of \(*\)-algebras defined above. The asymptotics of the maximal colengths of these algebras are also computed. Analogous results are obtained in the case of the \(S_n\)-cocharacters.