[Paper Review] Kraśkiewicz-Pragacz modules and Ringel duality

Masaki Watanabe|arXiv (Cornell University)|Apr 17, 2015

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes that the highest weight category of Kraśkiewicz-Pragacz (KP) modules over the upper-triangular Lie algebra 𝔟 is self-Ringel dual, revealing a symmetry in Ext groups between KP modules. It further shows that the Ringel duality functor commutes with a modified tensor product operation on these modules, providing a deep structural duality in the representation theory of Schubert polynomials.

ABSTRACT

Kraśkiewicz and Pragacz introduced representations of the upper-triangular Lie algebras whose characters are Schubert polynomials. In a previous work the author studied the structure of Kraśkiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual: this leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on b-modules is compatible with Ringel duality functor.

Motivation & Objective

• To construct a highest weight category C_n whose standard objects are Kraśkiewicz-Pragacz (KP) modules.
• To prove that this category C_n is self-Ringel dual, establishing a contravariant equivalence on the subcategory of modules with KP filtrations.
• To investigate the compatibility between the Ringel duality functor and a modified tensor product operation on 𝔟-modules.
• To reveal symmetry in Ext groups between KP modules via the self-duality of C_n.
• To provide a representation-theoretic framework for Schubert positivity using highest weight category theory.

Proposed method

• Construct a highest weight category C_n using the results from Watanabe (2012) on KP modules and highest weight categories.
• Define the Ringel dual category C_n^∨ and prove it is equivalent to C_n via a contravariant equivalence on the standardly filtered subcategory.
• Show that under this duality, the standard KP module S_w maps to S_{w₀ww₀}, where w₀ is the longest element in S_n.
• Introduce a modified tensor product operation on C_n that is closed under the category and compatible with Ringel duality.
• Use the theory of tilting modules and exact functors to establish isomorphisms between Ext groups via the Ringel duality functor.
• Leverage the fact that the Ringel duality functor F restricts to a contravariant equivalence between the standardly filtered subcategories of C_n and its dual.

Experimental results

Research questions

• RQ1 Is the highest weight category of KP modules self-Ringel dual?
• RQ2 How does the Ringel duality functor act on the standard modules S_w in the category C_n?
• RQ3 Does the tensor product operation on 𝔟-modules commute with the Ringel duality functor when appropriately modified?
• RQ4 What symmetry does self-Ringel duality induce on Ext groups between KP modules?
• RQ5 Can the representation-theoretic structure of KP modules be used to prove Schubert positivity results?

Key findings

• The highest weight category C_n whose standard modules are KP modules is self-Ringel dual, meaning C_n^∨ ≅ C_n as highest weight categories.
• Under the self-Ringel duality, the standard module S_w is mapped to S_{w₀ww₀}, where w₀ is the longest permutation in S_n.
• The Ringel duality induces a contravariant equivalence between the subcategory of modules with KP filtrations in C_n and itself.
• The Ext groups between KP modules satisfy the symmetry Ext^i(M,N) ≅ Ext^i(FN,FM), where F is the Ringel duality functor.
• The modified tensor product operation on C_n commutes with the Ringel duality functor, meaning the duality functor preserves the tensor structure up to isomorphism.
• The self-duality implies a hidden duality in the cohomological structure of KP modules, reflecting the symmetry of the Weyl group action on Schubert polynomials.