The University of Michigan Combinatorics Seminar
I will discuss an approach to the theory of symmetric functions (and generalizations) in which the Grothendieck ring of the symmetric group S_n is replaced by a certain subring of the group algebra of S_n.
This may be seen as part of a general trend in mathematics that consists of passing to the non-commutative world with the hope of simplifying things that in the commutative world might be obscured by all sorts of coefficient counting.
This approach has its roots in the works of many people including Solomon, Garsia, Reutenauer, Thibon et al, and Blessenohl et al, and involves various objects such as descent algebras, non-commutative symmetric functions, and quasi-symmetric functions.
I will adopt a Hopf algebraic point of view that allows us to unify these constructions and which suggests further generalizations. I plan to discuss one of these in some detail: the existence of a new product among symmetric functions which interpolates between the familiar "internal" and "external" products.
This will be preceded by a VIGRE pre-talk, 3:15-3:45pm, in the same room.