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 ...
as
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
, the possible subsets
of
that are Coxeter generating sets for
(that is, for which
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