John Campbell, Department of Pure Mathematics, University of Waterloo
“The Immaculate Basis and the Shin Basis of NSym”
The algebra Sym of symmetric functions may be defined as the free commutative Q-algebra with one generator in each degree, and the algebra NSym of non-commutative symmetric functions may be defined as the free Q-algebra with one generator in each degree. In this talk, I will describe some basic properties of NSym and Sym, and I will define the Schur basis of Sym, which is a remarkable basis that has many applications in algebraic combinatorics and representation theory. Finding a basis of NSym which somehow “resembles” the Schur basis is a very important problem in algebraic combinatorics. In this talk I will explore this problem by defining the immaculate basis and the shin basis of NSym, and explaining why these bases are regarded as natural analogues of the Schur basis.