On minimal lengths of expressions of Coxeter group elements as products of reflections

Dyer, Matthew

doi:10.1090/S0002-9939-01-05876-2

On minimal lengths of expressions of Coxeter group elements as products of reflections
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Proc. Amer. Math. Soc. 129 (2001), 2591-2595 Request permission

Abstract:

It is shown that the absolute length $l’(w)$ of a Coxeter group element $w$ (i.e. the minimal length of an expression of $w$ as a product of reflections) is equal to the minimal number of simple reflections that must be deleted from a fixed reduced expression of $w$ so that the resulting product is equal to $e$, the identity element. Also, $l’(w)$ is the minimal length of a path in the (directed) Bruhat graph from the identity element $e$ to $w$, and $l’(w)$ is determined by the polynomial $R_{e,w}$ of Kazhdan and Lusztig.

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