Matsumoto's theorem (group theory)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In algebra, Matsumoto's theorem, proved by Hideya Matsumoto (1964), gives conditions for two reduced words of a Coxeter group to represent the same element.

Statement[edit]

If two reduced words represent the same element of a Coxeter group, then Matsumoto's theorem states that the first word can be transformed into the second by repeatedly transforming

xyxy... to yxyx... (or vice versa)

where

xyxy... = yxyx...

is one of the defining relations of the Coxeter group.

Applications[edit]

Matsumoto's theorem implies that there is a natural map (not a group homomorphism) from a Coxeter group to the corresponding braid group, taking any element of the Coxeter group represented by some reduced word in the generators to the same word in the generators of the braid group.

References[edit]

Matsumoto, Hideya (1964), "Générateurs et relations des groupes de Weyl généralisés", C. R. Acad. Sci. Paris 258: 3419–3422, MR 0183818