Geodesic
A canonical expansion of the product of two Stanley symmetric functions
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 833-851.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We study the problem of expanding the product of two Stanley symmetric functions $F_w\cdot F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim_{n\to\infty}\mathfrak{S}_{1^n\times w}$, and study the behavior of the expansion of $\mathfrak{S}_{1^n\times w}\cdot\mathfrak {S}_{1^n\times u}$ 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.
Classification : 05E05, 05C05, 05C35
Keywords: Schubert polynomials, Stanley symmetric functions, maximal transition tree 
@article{JAC_2014__39_4_a6,
     author = {Li, Nan},
     title = {A canonical expansion of the product of two {Stanley} symmetric functions},
     journal = {Journal of Algebraic Combinatorics},
     pages = {833--851},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2014__39_4_a6/}
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Li, Nan. A canonical expansion of the product of two Stanley symmetric functions. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 833-851. http://geodesic.mathdoc.fr/item/JAC_2014__39_4_a6/