Young-derived sequences of \(S_ n\)-characters
For any sequence \(\psi = \{\psi_ n\}_{n \geq 0}\) of characters of the symmetric groups \(S_ n\), \(n = 0,1,2, \dots\), the author constructs the Young-derived sequence \[ y(\psi) = \{y_ n(\psi)\}_{n \geq 0} =\biggl\{\sum_{j=0}^ n \psi_ j \widehat{\otimes} \chi_{(n- j)}\biggr\}_{n \geq 0}, \] where \(\chi_{(k)}\) is the trivial \(S_ k\)- character and \(\widehat{\otimes}\) denotes the outer product expressed in terms of the Young rule. The aim of the paper is to analyze the relations between the sequences \(\psi\) and \(y(\psi)\). It turns out that the Young- derived sequences arise naturally in representation theory, combinatorics and the theory of algebras with polynomial identities. The author studies the asymptotic behaviour of the degrees of the characters \(\psi_ n\) and \(y_ n(\psi)\). Among other results he obtains an integration formula which extends some Macdonald type integrals and he evaluates some of these integrals. The author also gives a method which approximately determines the sequence \(\psi\) from the sequence \(y(\psi)\). Finally he applies this machinery to the cocharacters and codimensions of the ordinary and trace polynomial identities of matrix algebras.
scientific article; zbMATH DE number 638526