Abstract
In their 1987 paper Kraskiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraskiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely Sw Sd(Ki) and Sw Vd(Ki), corresponding to Pieri and dual Pieri rules for Schubert polynomials.
Highlights
Schubert polynomials are one of the main subjects in algebraic combinatorics
One of the tools for studying Schubert polynomials is the modules introduced by Kraskiewicz and Pragacz
In [Wat15a] the author showed that the tensor product of two KP modules always has a filtration by KP modules and gave a representation theoretic proof for this positivity
Summary
Schubert polynomials are one of the main subjects in algebraic combinatorics. One of the tools for studying Schubert polynomials is the modules introduced by Kraskiewicz and Pragacz. The proof there does not give explicit construction for the KP filtrations, it may provide a new viewpoint for the notorious SchubertLR problem, i.e. finding a combinatorial positive rule for the coefficient in the expansion of products of Schubert polynomials into a sum of Schubert polynomials. There are some cases where the expansions of products of Schubert polynomials are explicitly known Examples of such cases are the Pieri and the dual Pieri rules for Schubert polynomials conjectured in [BB93] and proved in [Win98] ( appearing with different formulations in [Las82] and [Sot96]). They are the cases where one of the Schubert polynomials is a complete symmetric function hd(x1, .
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