Kombinatorische Strukturen in Polynomringen

Michael Clausen

Séminaire lotharingien de combinatoire, Tome 14 (1986)

Résumé

The ring of polynomials over Z in the indetrminates X_i,j, i,j ∈ N, has a Z-basis of so-called standard bideterminants. Bideterminants are power products of minors of the matrix (X_i,j). This basis was perhaps first given by MEAD, but was probably not unknown to TURNBALL and HODGE. It became really widely known by the articles of ROTA and coworkkers. Numerous research programs around this basis were proposed. In the meantime, some of them have been taken up. Given the diversity of applications, it is the more surprising that the whole theory - from the point of view of techniques of proofs - is in principle built on two methods:

• LapIace-Entwicklungen;
• Capelli-Operatoren.

These methods have already been discussd extensively. Nevertheless, we revisit these methpds here, because the combinatorial and group-theoretic background became more and more apparent and caught more and more attention during the past few years. The following version is available: