Mike Zabrocki

THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a nat- ural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommuta- tive polynomials and conclude two analogues of Chevalleys theorem in the noncommutative setting.

A LIFT OF THE SCHUR AND HALL-LITTLEWOOD BASES TO NON-COMMUTATIVE SYMMETRIC FUNCTIONS

We introduce a new basis of the algebra of non-commutative sym- metric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

Indecomposable modules for the dual immaculate basis of quasi-symmetric functions

We construct indecomposable modules for the 0-Hecke algebra whose characteristics are the dual immaculate basis of the quasi-symmetric functions.

Stanley Symmetric Functions and Peterson Algebras

This purpose of this chapter is to introduce Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The presentation roughly follows three lectures I gave at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.

KRONECKER COEFFICIENTS VIA SYMMETRIC FUNCTIONS AND CONSTANT TERM IDENTITIES

This work lies across three areas of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link led to the calculation of some Kronecker coefficients by computing constant terms and conversely the computations of certain constant terms by computing Kronecker coefficients by symmetric function methods. This led to results as well as methods for solving numerical problems in each of these separate areas.

Expansion of k-Schur functions for maximal rectangles within the affine nilCoxeter algebra

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra.

Multiplicative structures of the immaculate basis of non-commutative symmetric functions

Abstract We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the basis of Schur functions for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood–Richardson rule and the Murnaghan–Nakayama rule. A surprising relation develops among non-commutative Littlewood–Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.

The Pieri Rule for Dual Immaculate Quasi-Symmetric Functions

The immaculate basis of the non-commutative symmetric functions was recently introduced by the first and third authors to lift certain structures in the symmetric functions to the dual Hopf algebras of the non-commutative and quasi-symmetric functions. It was shown that immaculate basis satisfies a positive, multiplicity free right Pieri rule. It was conjectured that the left Pieri rule may contain signs but that it would be multiplicity free. Similarly, it was also conjectured that the dual quasi-symmetric basis would also satisfy a signed multiplicity free Pieri rule. We prove these two conjectures here.