This project aims at finding extreme value theorems for a wide variety of algebraic statistics on permutation groups and general Coxeter groups. For example, two important statistics on random permutations are the number of inversions and the number of descents. These statistics can be generalized to Coxeter groups, where inversions and descents are defined with respect to a specific system of generators. In this context, we also want to investigate the Coxeter groups $B_n$ consisting of signed permutations and their subgroups $D_n \subseteq B_n$ consisting of all permutations with an even number of negative signs. Two further interesting statistics are $T(\pi) := \text{Inv}(\pi) + \text{Inv}(\pi^{-1})$ and $D(\pi) := \text{Des}(\pi) + \text{Des}(\pi^{-1})$ (where $\pi \in S_n, B_n$ or $D_n$).

The question of under which conditions these statistics satisfy a central limit theorem is well-understood. In contrast to that, the extremal type behavior of such statistics is rather unexplored yet. As the considered random permutation statistics only take finitely many values, we need to consider a suitably scaled triangular array of these statistics on groups of growing rank $n \rightarrow \infty$ and tackle the dependencies therein. In future work we will try to find analogous results for the combinatorial objects described above. For a start, we conjecture that the number of descents $\Des(\pi)$ on the symmetric group $S_n$ belongs to the domain attraction of the Gumbel distribution.