Depth in classical Coexter groups

TL;DR

This paper introduces algorithms to compute the depth statistic in classical Coxeter groups, characterizes elements with equal depth and length, and identifies boolean elements where reflection length, depth, and length coincide.

Contribution

It provides new algorithms for calculating the depth statistic and characterizes specific classes of elements in classical Coxeter groups based on their depth, length, and reflection length.

Findings

Algorithms yield simple formulas for depth calculation.

Elements with depth equal to length are exactly the short-braid-avoiding elements.

Boolean elements are characterized by the equality of reflection length, depth, and length.

Abstract

The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both the depth and the length. These are the boolean elements.