In mathematics, the length of an element ''w'' in a
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
''W'', denoted by ''l''(''w''), is the smallest number ''k'' so that ''w'' is a product of ''k'' reflections by simple roots. (So, the notion depends on the choice of a
positive Weyl chamber
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
.) In particular, a simple reflection has length one. The function ''l'' is then an integer-valued function of ''W''; it is a
length function
In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Definition
A length function ''L'' : ''G'' → R+ on a group ''G'' is a function satisfy ...
of ''W''. It follows immediately from the definition that ''l''(''w''
−1) = ''l''(''w'') and that ''l''(''ww''
'−1) ≤ ''l''(''w'') + ''l''(''w' '').
References
* {{cite book, last1=Kac, first1=Victor G., title=Infinite dimensional Lie algebras, date=1994, publisher=Cambridge University Press, location=Cambridge, isbn=9780521466936, edition=3rd
Lie groups