Algebraic combinatorics around a problem in enumerative geometry
Résumé
The research presented in this habilitation thesis falls within the field of algebraic combinatorics, and consists of several original contributions. It was initially motivated by an enumerative geometry problem: namely, the explicit computation of the number of intersection points $a_w$ between the permutahedral variety and a Schubert subvariety $X_w$ within the flag variety. The first part introduces the original problem and describes classical cohomological techniques that allow for the derivation of two distinct formulas for $a_w$. The first formula, based on Klyachko’s work, establishes the strict positivity of the numbers $a_w$ and highlights several of their structural properties. The second formula, obtained through substantial algebraic and combinatorial developments, provides an explicit combinatorial interpretation of these numbers. The second part gathers various contributions that emerged during the resolution of the initial problem. In particular, $q$-analogues of the quantities appearing in the first formula, named remixed Eulerian numbers, are introduced and studied. This rich family encompasses many classical combinatorial sequences, and the combinatorial and algebraic aspects of this $q$-deformation are studied in depth. Additionally, in connection with the second formula, a variant of the theory of $P$-partitions is developed, along with a theory of bilateral parking functions which sheds light on certain combinatorial aspects of the formula. The third and final part focuses on quasisymmetric polynomials, an important family playing a central role in the derivation of the second formula for $a_w$. A new operator-based approach is introduced, simplifying and extending classical methods for studying these polynomials. This construction is directly inspired by the theory of symmetric polynomials, with forest polynomials playing a role analogous to that of Schubert polynomials.
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