Involution codimensions and trace codimensions of matrices are asymptotically equal (Q677445)

Let \((M_p(F),*)\) be the \(p\times p\) matrix algebra with transpose or symplectic involution over a field \(F\) of characteristic 0. Important invariants of the \(*\)-polynomial identities of \((M_p(F),*)\) are the sequences of ordinary and trace \(*\)-codimensions. The main result of the paper under review is that these sequences are asymptotically equal. This is an analogue of a result of one of the authors [\textit{A. Regev}, Isr. J. Math. 47, 246-250 (1984; Zbl 0537.16014)] for codimensions with and without trace for the ordinary polynomial identities of matrices. The proof is based on representation theory of the wreath product \(B_n\) of \(\mathbb{Z}/2\mathbb{Z}\) with the symmetric group \(S_n\). In particular, the authors calculate the exact asymptotics of the trace \(*\)-codimensions. As a by-product they also obtain some results on \(B_n\)-cocharacters which are of independent interest.