Une combinatoire du pléthysme. (Combinatorics of plethysm)
The plethysm \(\{\alpha\}\odot \{\beta\}\) of Schur polynomials \(\{\alpha \}\), \(\alpha\vdash m\), and \(\{\beta\}\), \(\beta\vdash n\), was introduced by D. E. Littlewood long ago. It corresponds to the representation \([\alpha]\odot [\beta]\) of the symmetric group \(S_{mn}\) induced by the irreducible representation \((\alpha;\beta)\) of the wreath product \(S_ m\wr S_ n\), which is naturally embedded in \(S_{mn}\) (see e.g. the reviewer's joint book with G. D. James for details and notation). Closely related to this construction is the so-called composition of cycle indicator polynomials of actions of G on X and H on Y, which yields the cycle indicator of \(G\wr H\) on \(X\times Y\), and which is used for the enumeration of orbits of \(G\wr H\) on \(Y^{X\times Y}\) by weight (cf. the book of James/Kerber). The problem of decomposing such compositions or plethysms of representations or Schur polynomials is difficult and yet unsolved. The various applications in mathematics and sciences suggest to examine the natural generalization of this method of substituting one polynomial into another to formal power series. This was recently begun by Nava and Rota. Bergeron further elaborates this consideration by considering the category S the elements of which are pairs \((A,\sigma)\), A a finite set, \(\sigma\) a permutation of A. The morphisms \(\phi: (A,\sigma)\to (B,\tau)\) are the bijections which satisfy \(\tau =\phi \circ \sigma \circ \phi^{- 1}\). An espèce is then a functor T from S to Ens, the category of finite sets. O. A. Nava continues his joint work with G.-C. Rota by constructing three large reduced incidence algebras which are the linear duals of incidence bialgebras arising from transversals of partitions, partitions compatible with permutations, and linear transversals of linear partitions, respectively. Morphisms are introduced and counted, and connection constants are obtained using Möbius inversion. Use is made of W. Schmitt's thesis (MIT 1986) and its theory of incidence coalgebras of families of posets. [This review also covers the paper, the bibliographical data of which are given in the following item.]
scientific article; zbMATH DE number 4049651