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Bargain hunting in a Coxeter group

Joel Brewster Lewis, Bridget Eileen Tenner

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 (à 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.

Bargain hunting in a Coxeter group

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 be the distance between the integers transposed by in the combinatorial representation of the group (à 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.
Paper Structure (13 sections, 6 theorems, 46 equations)

This paper contains 13 sections, 6 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Who are the groups?
  4. What are the statistics?
  5. Main theorems
  6. First main theorem: unbranched George groups
  7. Second main theorem: finite type
  8. Further remarks and open questions
  9. Different combinatorial realizations in affine types
  10. Conjectures in affine types $\bm{B}$ and $\bm{D}$
  11. An aesthetically pleasing construction
  12. Ordering the statistics
  13. Depth in affine types

Key Result

Theorem 3.1

For any unbranched George group $W$ and any element $w$ in $W$, we have

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • ...and 10 more