| Résume||The PBW filtration and degeneration for a semi-simple complex Lie algebra provides a fruitful bridge from Lie theory to commutative algebra. We will be focussing on $sl_n$ and recalling the degenerate flag variety, a flat degeneration of the classical flag variety. I'll introduce monomial bases for the PBW degenerate simple $sl_n$-modules and describe the associated flat toric degeneration of the flag variety (this is joint work with E. Feigin and P. Littelmann).
In order to set this in context with previously known toric degenerations (due to Caldero, Littelmann, Alexeev-Brion et al), I'll introduce ''birational sequences'', which are in some sense factorizations of a maximal unipotent subgroup of $SL_n$. This provides a framework of toric degenerations that covers for all types the construction of string polytopes, Lusztig polytopes and many more (the latter is joint work with X. Fang and P. Littelmann).|