Grant: Discovery Grants Program - Individual

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.