A canonical expansion of the product of two Stanley symmetric functions
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We study the problem of expanding the product of two Stanley symmetric functions F[subscript w]⋅F[subscript u] into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert F[subscript w] = lim[subscript n →∞] S[subscript 1[superscipt n]x w], and study the behavior of the expansion of S[subscript 1[superscript n] x w]⋅S[subscript 1[superscript n] x u] into Schubert polynomials as n increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stability properties, providing a second proof of the main result.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Algebraic Combinatorics
Li, Nan. “A Canonical Expansion of the Product of Two Stanley Symmetric Functions.” Journal of Algebraic Combinatorics 39.4 (2014): 833–851.
Author's final manuscript