Combinatorial bases for modules of coinvariants.

摘要

The module of coinvariants is the quotient of the ring of polynomials and the ideal generated by positive degree G-invariants. We construct combinatorial bases for various modules of coinvariants. A well known basis of monomials for the coinvariants of the symmetric group, referred to as the Artin basis is incorporated into many of our results.;In particular, we find bases for the space of coinvariants involving symmetric functions when the group G is the Weyl group of type Dn, Bn, F4, and E6. In addition, we construct a basic set of invariants for Weyl group of type E 6, which we express in terms of well know invariants of the Weyl group of type D5. Two combinatorial bases for the coinvariants of the wreath product of the symmetric group with the cyclic group are found. The first basis involves symmetric functions, and is a generalization of our result on the coinvariants of the Weyl group of type Bn. The second is basis of monomials. An alternate proof of Steinberg's result which gives a basis of monomials for the coinvariants when the group under consideration is the general linear group over a finite field is provided.