Noncommutative Schur functions, Fomin-Greene monoids and LR coefficients

The study of Schur functions and their generalizations is a very active area of research in algebraic combinatorics. I will discuss two analogues of Schur functions: one quasisymmetric and the other noncommutative. The quasisymmetric Schurs are defined combinatorially via composition tableaux whereas noncommutative Schur functions are defined implicitly as Hopf duals to quasisymmetric Schurs. In the talk, I will give a direct combinatorial definition of noncommutative Schur functions using the theory of Fomin-Greene monoids. I will subsequently discuss the computation of the structure constants - called noncommutative LR coefficients - that arise upon multiplying noncommutative Schur functions. Both of the aforementioned works rely crucially on Lascoux-Schützenberger's notion of frank words and crystals operators on tableaux. Time permitting, I will discuss a polytopal description for certain noncommutative LR coefficients and relate it to Pak-Vallejo's LR cone. This is joint work with Edward Richmond.