Advances in the combinatorics of shifted tableaux

2018-08-07In this work we explore shifted combinatorics, making new constructions and proving results about existing ones. Chapters 1 and 2 are expository in nature and discuss partitions, Young tableaux and symmetric functions in the contexts of regular and shifted combinatorics. In Chapter 3, we consider the type B quasisymmetric Schur functions defined by Jing and Li in 2015. We prove their conjecture that these functions have a positive, integral and unitriangular expansion into peak functions, and refine their combinatorial model to give explicit expansions in monomial, fundamental and peak bases. We also show that these functions are not quasisymmetric Schur, Young quasisymmetric Schur or dual immaculate positive, and do not have a positive multiplication rule. In Chapter 4, we introduce a shifted analogue of the ribbon tableaux defined by James and Kerber [16]. For any positive integer k, we give a bijection between the k-ribbon fillings of a shifted shape and regular fillings of a k/2-tuple of shapes called its k-quotient. We define the corresponding generating functions, prove that they are symmetric, Schur positive and Schur Q-positive, and introduce a Schur Q-positive q-refinement. In Chapter 5, we introduce crystal operators on semistandard shifted tableaux, giving a new proof that Schur P-functions are Schur positive. We define a queer crystal operator that constructs a connected queer crystal graph on semistandard shifted tableaux of a given shape,providing a new proof that products of Schur P-functions are Schur P-positive