Length of a Weyl group element

In mathematics, the length of an element w in a Weyl group 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 of W. It follows immediately from the definition that l(w⁻¹) = l(w) and that l(ww'⁻¹) ≤ l(w) + l(w' ).

• Kac, Victor G. (1994). Infinite dimensional Lie algebras (3rd ed.). Cambridge: Cambridge University Press. ISBN 9780521466936. 

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