In
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 ...
, Matsumoto's theorem, proved by , 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
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
, 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
*
Theorems in group theory
Braid groups
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