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 t ...
''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 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 satisfyi ...
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