[Paper Review] Affine Fomin-Kirillov algebra

Seung Jin Lee|arXiv (Cornell University)|Jun 8, 2015

Advanced Combinatorial Mathematics16 references3 citations

TL;DR

This paper introduces the affine Fomin-Kirillov algebra to study affine Schubert calculus in type A, constructing Murnaghan-Nakayama and Dunkl elements that act commutatively on the homology of the affine flag variety via Bruhat actions. It establishes compatibility with cap and Pieri operators, proves the commutative subalgebra generated by these elements surjects onto the cohomology, and derives Murnaghan-Nakayama rules and a new formula for $k$-Schur functions using $k$-strong-ribbon tableaux.

ABSTRACT

We construct the affine version of the Fomin-Kirillov algebra, which we call affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements as elements in the affine FK algebra. We show that they give commutative operators acting on the homology of the affine flag variety via Bruhat actions. We show that the Murnaghan-Nakayama element as a Bruhat action is compatible with the cap operators defined by the author using the Kumar and Kostant's work \cite{Lee14}, and with the Pieri operators defined by Berg, Saliola and Serrano using strong strips \cite{BSS14}. This shows that the commutative subalgebra generated by those elements surgects onto the cohomology of affine flag variety. As a byproduct, we obtain Murnaghan-Nakayama rules for the affine flag variety, and for affine Stanley symmetric functions. We also define $k$-strong-ribbon tableaux from Murnaghan-Nakayama elements to provide the new formula of $k$-Schur functions.

Motivation & Objective

• To develop an affine version of the Fomin-Kirillov algebra for studying affine Schubert calculus in type A.
• To define and analyze Murnaghan-Nakayama and Dunkl elements within the affine FK algebra.
• To establish that these elements generate a commutative subalgebra acting via Bruhat actions on the homology of the affine flag variety.
• To show compatibility of these operators with existing constructions: cap operators (Kumar and Kostant) and Pieri operators (Berg, Saliola, Serrano).
• To derive Murnaghan-Nakayama rules for the affine flag variety and affine Stanley symmetric functions, and to define $k$-strong-ribbon tableaux for a new formula of $k$-Schur functions.

Proposed method

• Construct the affine Fomin-Kirillov algebra as a generalization of the finite Fomin-Kirillov algebra.
• Introduce Murnaghan-Nakayama elements and Dunkl elements as specific generators in the affine FK algebra.
• Define Bruhat actions of these elements on the homology of the affine flag variety, showing they commute.
• Establish compatibility of the Murnaghan-Nakayama element's Bruhat action with cap operators from Lee (2014) and with Pieri operators from Berg, Saliola, and Serrano (2014).
• Use the commutative action to prove that the subalgebra generated by these elements surjects onto the cohomology of the affine flag variety.
• Define $k$-strong-ribbon tableaux as combinatorial objects arising from the Murnaghan-Nakayama elements to express $k$-Schur functions.

Experimental results

Research questions

• RQ1 How can the Fomin-Kirillov algebra be extended to the affine setting to model affine Schubert calculus in type A?
• RQ2 Do the Murnaghan-Nakayama and Dunkl elements in the affine FK algebra act as commuting operators on the homology of the affine flag variety via Bruhat actions?
• RQ3 Is the action of the Murnaghan-Nakayama element compatible with the cap operators defined by Kumar and Kostant?
• RQ4 Is the action compatible with the Pieri operators defined via strong strips by Berg, Saliola, and Serrano?
• RQ5 Can the commutative subalgebra generated by these elements realize the full cohomology of the affine flag variety, and what combinatorial formulas arise from this construction?

Key findings

• The Murnaghan-Nakayama element acts as a commutative operator on the homology of the affine flag variety through Bruhat actions.
• The action of the Murnaghan-Nakayama element is compatible with the cap operators from Kumar and Kostant's work.
• The action is also compatible with the Pieri operators defined via strong strips by Berg, Saliola, and Serrano.
• The commutative subalgebra generated by the Murnaghan-Nakayama and Dunkl elements surjects onto the cohomology of the affine flag variety.
• The construction yields Murnaghan-Nakayama rules for the affine flag variety and for affine Stanley symmetric functions.
• The paper defines $k$-strong-ribbon tableaux from the Murnaghan-Nakayama elements, providing a new combinatorial formula for $k$-Schur functions.