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Calcul de Schubert affine et formules de Pieri

Abstract : Pieri's formulas are a gateway to understanding the algebra structure of the (affine) Grassmannian or even that of Flag varieties. Several are already established in a few particular types and cases. However, this problem remains open for most affine cases, especially to find Pieri formulas in $H (\mathcal{G}r_G)$ in types $B$, $C$ and $D$. In this thesis, even if some results are generalized for non-twisted affine Weyl groups, we mainly explore types A and C. In the flag variete of affine type A, we find a formula for multiplying, in the cohomology algebra of a flag variety, one element of the base $\xi^w$ by another (special) element that will be called ''crochet''. This result is shown using the Pieri formula given by Lam et al in \cite{insertion}. In the affine Type $C$, we propose a conjecture for a Pieri formula in Cohomology, showing that it is valid in degree $1$ and "almost" all cases of degree $2$. It is also checked, by testing many examples using the computer. In Homology, the Pieri formula in type C \cite{lam2010schubert}, is re-demonstrated, using a new simplified strategy. This new approach could eventually be used to establish formulas of exceptional types. In the finite dimensional flag varieties, we find an upper bound for the littlewood-Richardson's coefficients and generalize it, in all types, to particular classes that will be called ''small Schubert classes''.
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https://hal.univ-lorraine.fr/tel-03203728
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Dimitry Kfoury. Calcul de Schubert affine et formules de Pieri. Mathématiques [math]. Université de Lorraine, 2020. Français. ⟨NNT : 2020LORR0215⟩. ⟨tel-03203728⟩

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