We'll provide combinatorial descriptions of Coxeter groups of type A, B, and D, their length function, reflections and longest elements. We use this descriptions to show that the poset formed by the shortest path in the Bruhat graph from the identity to the longest element is formed by Boolean posets. This gives a combinatorial interpretation of the lowest-degree terms of the complete cd-index of a finite coxeter group W. The complete cd-index is a polynomial that encodes the descent set of the Bruhat graph with a "nice" labeling (called reflection order). This polynomial can be used to obtain the Kazhdan-Lusztig polynomials.
The talk will be understandable to first-year graduate students.