A canonical expansion of the product of two Stanley symmetric functions
نویسنده
چکیده
We study the problem of expanding the product of two Stanley symmetric functions Fw · Fu into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial Fw = limn→∞S1n×w, and study the behavior of the expansion of S1n×w · S1n×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. Résumé. Nous étudions le problème de l’développement du produit de deux fonctions symétriques de Stanley Fw ·Fu en fonctions symétriques de Stanley de façon naturelle. Notre méthode consiste à considerer une fonction symétrique de Stanley comme un polynôme du Schubert stabilisé Fw = limn→∞S1n×w, et à étudier le comportement de l’développement de S1n×W · S1n×u en polynômes de Schubert lorsque n augmente. Nous prouvons que cette développement se stabilise et donc nous obtenons une développement naturelle pour le produit de deux fonctions symétriques de Stanley. Dans le cas où l’une des permutations est Grassmannienne, nous avons une meilleure compréhension de cette stabilité.
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