# Parameterizing chains in generalized fixed point free involutions

## Description

The W-sets of elements of weak order posets are useful in the study of the co-homology of the closures of spherical subgroups in generalized flag varieties (Can et al., 2016). This motivated the study of the W-sets of involutions, fixed-point-free involutions, and charged involutions in previous research (Can et al., 2016; Can and Joyce, 2013; Can et al., 2018). Generalized involutions, or μ-involutions, have similar geometric interpretations; however, parametrization of their W-sets remained asan open question. Ourmain resultpresents a theorem that parametrizes the W-set of generalized fixed-point-free involutions. The achievements of this thesis are threefold. Firstly, we provide a new prooffor the parametrization of the W-sets of fixed-point-free involutions by Can et al. (2016). The proof from Can et al. (2016) is formulated as a corollary of the rule for the W-sets of all involutions, whereas our proof is self-contained. Secondly, using computational experiments, we generate the W-sets for generalized fixed-point-free involutions n≤16. Then, based on theW-sets that we the generated, we proposea conjecture for parametrizing W-sets of generalized fixed-point-free involutions. Thirdly, building on the arguments we used to prove the rule for fixed-point-free involutions, we prove a complete characterization of the W-set for the generalized fixed-point-free involutions.