The combinatorial invariance conjecture of Kazhdan-Lusztig polynomials: hypercube decompositions of Bruhat intervals

Kazhdan-Lusztig polynomials are integer polynomials indexed by pairs of elements in Coxeter groups; they encode key information on the representation theory of Hecke algebras and singularities of Schubert varieties. The topic of this thesis is the combinatorial invariance conjecture, an intriguing problem formulated by Lusztig in the 1980s which suggests that the Kazhdan-Lusztig polynomials might be determined by combinatorial data. In the first part, we introduce the necessary preliminary notions to understand the conjecture. We then focus on recent advances obtained through the novel concept of hypercube decompositions of Bruhat intervals. This approach has emerged in the past few years, thanks to a machine learning model used to test the conjecture in the case of symmetric groups, and has already produced some partial results.