Bargain hunting in a Coxeter group

TL;DR

This paper extends the concept of the depth statistic from symmetric groups to all classical Weyl groups by defining a new cost function based on transposition distances, providing an intrinsic formula for computation.

Contribution

It generalizes the depth statistic to classical Weyl groups using a new transposition-based cost function and derives a simple intrinsic formula for calculating it.

Findings

Cost of elements can be computed directly from the elements.

The generalized cost function applies to classical finite and affine Weyl groups.

Provides a unified approach to measure element complexity in Coxeter groups.

Abstract

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost function on transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection $t$ be the distance between the integers transposed by $t$ in the combinatorial representation of the group (\`a la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula.