Outer plethysms and \(\lambda\)-rings
Let \(R(S_ n)\) be the Grothendieck group formed from isomorphism classes [M] of \(S_ n\)-modules M, where \(S_ n\) is the symmetric group, and let \(R=\{R(S_ n)\}\) be the direct sum of the \(R(S_ n)\) formed into a graded ring by outer tensor product of representations. It is known that R is a free \(\lambda\)-Hopf ring as it is isomorphic to the Hopf ring of symmetric functions in countably many variables. In this paper the \(\lambda\)-structure on R is defined directly by using the outer plethysm \(\sigma^ k\); thus if M is an \(S_ n\)-module, one forms \(M^{\otimes k}\), the outer tensor product representation of the k-fold direct product \(S_ n^{\times k}\) and extends this to get a representation of the wreath product \(S_ n wr S_ k\) \((=S_ k[S_ n])\); \(\sigma^ k([M])\) is the isomorphism class of the induced \(S_{kn}\)-module, \(Ind^{S_{kn}}_{S_ k[S_ n]}(M^{\otimes k})\). If one forms the formal power series in \(t: \sigma_ t([M])=1+\sigma^ 1([M])t+\sigma^ 2([M])t^ 2+...\), etc., then \(\lambda^ k([M])\) is given by \(\lambda_ t([M])=1/(\sigma_{-t}([M]))\). By using the kth Newton polynomial \(Q_ k\) in the elementary symmetric functions, the Adams operations \[ \psi^ k: R(S_ n)\to R(S_{kn}),\quad \psi^ k(x)=Q_ k(\lambda^ 1(x),...,\lambda^ k(x))=(-1)^{k-1}Q_ k(\sigma^ 1(x),...,\sigma^ k(x)) \] are defined. This makes R a \(\psi\)-Hopf ring. Further, R is a special-\(\psi\)-Hopf ring and a special free \(\lambda\)-Hopf ring on a single generator. These facts follow as R is \(\lambda\)-isomorphic, via the character map \(\chi\) : \(R\to C\), to the ring \(C=\{C(S_ n)\}\), which is a priori verified to have these same properties, where \(C(S_ n)\) is the group of integer valued class functions on \(S_ n\). The calculation of the character values of the outer plethysms \(\sigma^ k([M])\) is required to verify this \(\lambda\)-isomorphism.
scientific article; zbMATH DE number 3983459