Isomorphism problem of Coxeter groups

{{unsolved|mathematics|Given two Coxeter groups $\backslash Gamma\_1$ and $\backslash Gamma\_2$, decide whether $W(\backslash Gamma\_1)\backslash simeq\{W(\backslash Gamma\_2)\}$.}}

It is an unresolved problem in the mathematical field of group theory to determine whether or not two Coxeter groups (specified by their Coxeter diagrams) are isomorphic as abstract groups. Equivalently, the problem asks to determine, for a given Coxeter group $W$, the possible subsets $S$ of $W$ that are Coxeter generating sets for $W$ (that is, for which $(W\; ,\; S)$ is a Coxeter system).

A slight generalization of the problem can be made by asking to find to all isomorphisms from one group onto the other.{{cite arXiv

|last = Mühlherr

|first = Bernhard

|date = 2005-06-28

|title = The isomorphism problem for Coxeter groups

|arxiv = math.GR/0506572

}}

In 2022, Yuri Santos Rego and Petra Schwer introduced a new framework to deal with the problem (a finite dimensional, locally finite, ranked simplicial complex to capture isomorphisms between finite rank Coxeter systems) and asked more

related open questions motivated by it.{{cite journal

|last1 = Santos Rego

|first1 = Yuri

|last2 = Schwer

|first2 = Petra

|journal = Journal of Algebra

|volume = 656

|date = 2024-10-15

|title = The galaxy of Coxeter groups

|pages = 406–445

|arxiv = 2211.17038

}}