#### Event Title

Hopf Algebra of k-Schur Functions and its Self-Duality

#### Advisor(s)

Dr. Aaron Lauve; Loyola University

#### Discipline

Mathematics

#### Start Date

21-4-2021 10:45 AM

#### End Date

21-4-2021 11:05 AM

#### Abstract

The k-Schur functions can be defined using several different methods, including using k-tableaux, weak tableaux, and strong tableaux (Lam et. al. 2013). They can be generated by a function derived from the weak Pieri rule for Schur functions, by summing over k-bounded partitions. These k-Schur functions also reside in a subring of the symmetric functions. The dual k-Schur functions can be generated using a weak Pieri rule on affine Grassmannian polynomials within the affine symmetric group, summing over cyclically decreasing reduced words. They are equivalent to Schur functions under certain conditions (k is greater than lambda’s absolute

value and t = 0).

We intend to connect the self-duality of the Hopf algebra of k-Schur functions to unsolved problems in the fields of mathematics, including representation theory. The self-duality of the kSchur functions opens all its definitions to be linked together, allowing for a wider range of possibilities when solving for open questions in the fields, including finding a k-MurnaghanNakayama rule, coefficients on the coproducts of symmetric functions, a model of k-Schur functions, and more (Lam et. al. 2013).

Hopf Algebra of k-Schur Functions and its Self-Duality

The k-Schur functions can be defined using several different methods, including using k-tableaux, weak tableaux, and strong tableaux (Lam et. al. 2013). They can be generated by a function derived from the weak Pieri rule for Schur functions, by summing over k-bounded partitions. These k-Schur functions also reside in a subring of the symmetric functions. The dual k-Schur functions can be generated using a weak Pieri rule on affine Grassmannian polynomials within the affine symmetric group, summing over cyclically decreasing reduced words. They are equivalent to Schur functions under certain conditions (k is greater than lambda’s absolute

value and t = 0).

We intend to connect the self-duality of the Hopf algebra of k-Schur functions to unsolved problems in the fields of mathematics, including representation theory. The self-duality of the kSchur functions opens all its definitions to be linked together, allowing for a wider range of possibilities when solving for open questions in the fields, including finding a k-MurnaghanNakayama rule, coefficients on the coproducts of symmetric functions, a model of k-Schur functions, and more (Lam et. al. 2013).