Adams operations and \(\lambda\)-operations in \(\beta\)-rings
Let \(P_ n\) denote the set of conjugacy classes [H] of subgroups of the symmetric group \(S_ n\) on \(\{1,...,n\},\) R a commutative ring and \(R^ R\) the ring of all functions \(R\to R\) with pointwise addition and multiplication. The authors call \((R,\beta)\) a \(\beta\)-ring iff \(\beta:\cup_{n\geq 1}P_ n\to R^ R:[H]\to \beta_ H\) satisfies \(\beta_ F\beta_ H=\beta_{F\oplus H},\quad \beta_ F{\mathbb{O}}\beta_ H=\beta_{F[H]},\quad \beta_{S_ 1}=1,\) where \(F\oplus H\) and F[H] denote (for \(F\leq S_ m\), \(H\leq S_ n)\) the natural permutation representations of \(F\times H\) and \(H\wr F\) of degree \(m+n\) and mn. In particular the Burnside ring A(G) of a finite group, or, more generally, of a compact Lie group, is a \(\beta\)-ring. The paper under review is devoted to operations \(\psi_ H\) and \(\lambda_ H\) on a \(\beta\)-ring. The \(\psi_ H\) are such that the transition matrix expressing the \(\beta_ K\) in terms of the \(\psi_ H\) is the table of marks (which is also called the supercharacter table) of \(S_ n\), while the transition matrix from the \(\psi_ H\) to the \(\lambda_ K\) arises from the table of marks of \(S_ n\) by multiplication by a diagonal matrix with 1's and -1's along the diagonal. The transition matrix from the \(\beta_ K\) to the \(\lambda_ H\) is self-inverse. An important role plays the poset \(P_ G\) of conjugacy classes of subgroups of a finite group G, its incidence algebra, and the following remark: If \(L_ G\) denotes the lattice of subgroups of G, then an element \(\alpha\) of its incidence algebra, which satisfies \(\forall g (\alpha(K,H)=\alpha(g^{-1}Kg,g^{-1}Hg)),\) gives rise to \({\check \alpha}\) in the incidence algebra of \(P_ G: {\check \alpha}([K],[H]):=\sum_{H'\in [H]}\alpha(K,H').\) The authors show, among many other things, that the supercharacter table of G is of the form \(M({\check \xi})\)\(M({\check \eta})\)\(M({\check \gamma})\), where \(M({\check \gamma})\) is diagonal and \(M({\check \xi})\) is a direct sum of triangular matrices, for certain \(\xi\), \(\eta\) and \(\gamma\) in the incidence algebra of \(L_ G\).
scientific article; zbMATH DE number 3859373
scientific article