It is an unresolved problem in the mathematical field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

to determine whether or not two Coxeter groups (specified by their

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

s) are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

abstract group In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...

s. Equivalently, the problem asks to determine, for a given Coxeter group

W

, the possible subsets

S

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. 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 In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

to capture isomorphisms between finite rank Coxeter systems) and asked more related open questions motivated by it.

References

{{Reflist Unsolved problems in mathematics Coxeter groups