Matsumoto's theorem (group theory)

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

Statement

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

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.

gollark: - which is why I think all government workers should be randomly selected, similarly to jury duty

gollark: - which is why I think anyone in government who makes a mistake of any kind should be immediately fired

gollark: - I support an efficient and adaptable government- which is why I think we should replace all civil servants with small swarms of bees in balloons

gollark: - I support the right to free speech!- In order to preserve freedom of speech and ensure disagreeing views can be heard, I will ban anyone who agrees with me from this website and promote anyone who disagrees.

gollark: Hmm, maybe I should have a list of political positions, but half of them are true (EDIT: i.e. really mine) and half of them are bizarre metaironical things.

References

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

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