Commutative and non-commutative structures in algebraic combinatorics

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Classical results relating the action of subgroups of matricies on the ring of polynomials have been used to develop a classical theory of invariants and coinvariants.  Inspired by these results, we considered [1] the permutation action on the space of non-commutative polynomials and have begun to develop a theory of non-commutative invariants and coinvariants.  From this connection between commutative and non-commutative algebras we propose to investigate using the theory of languages a relationship of non-commutative series and classes of commutative generating functions which are studied in enumerative combinatorics.Arising from a connection with earlier research we are proposing to develop in a group research project a theory of affine Schubert calculus.  Classical Schubert calculus is the outcome of Hilbert's fifteenth problem and is a branch of enumerative algebraic geometry concerned with counting subspaces with certain intersection conditions.  This extension to the classical theory will have implications for mathematicains, physicsts and computer scientists.