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
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
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
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
) 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
{{group-theory-stub