Asymptotics in the cocharacter sequence of \(A\otimes B\).
By the already classical theorem of Regev the codimension sequence \(c_n(A)\) of a PI-algebra \(A\) grows exponentially. As a consequence, the tensor product \(A\otimes B\) of two PI-algebras \(A\) and \(B\) is a PI-algebra again. The more recent theorem of Giambruno and Zaicev states that, over a field \(F\) of characteristic 0, the limit \(e(A)=\lim_{n\to\infty}\root n\of{c_n(A)}\) always exists and is an integer called the exponent of \(A\). In the paper under review the author obtains upper and lower bounds for \(e(A\otimes B)\) in terms of \(e(A)\) and \(e(B)\), namely \(e(A\otimes B)=e(A)\cdot e(B)\) or \(e(A\otimes B)=e(A)\cdot e(B)-1\). Both bounds can occur. For a fixed algebra \(A\), the equality \(e(A\otimes B)=e(A)\cdot e(B)\) holds for all \(B\) if and only if there exists a verbally prime algebra \(A'\) which satisfies all polynomial identities of \(A\) and \(e(A')=e(A)\). The minimum value of \(\{e(A\otimes B)\mid e(B)=b\}\) occurs when \(B\) is the \(b\times b\) upper triangular matrix algebra over \(F\). The second problem considered in the paper deals with the cocharacter sequence \(\chi_n(A\otimes B)=\sum_{\lambda\vdash n}m_\lambda(A\otimes B)\chi_\lambda\). It is known that the partitions \(\lambda\) participating with nonzero multiplicities \(m_\lambda\) are in a hook. The author studies the shape of the hook in terms of the corresponding hooks for \(A\) and \(B\). He obtains upper and lower bounds for the arm and the leg of this hook.
scientific article; zbMATH DE number 5118776