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.
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Notes
This is the same as the depth in the root system of W of the positive root orthogonal to the reflecting hyperplane of t.
In fact, in principle, one could place both mirrors between consecutive pairs of integers, so that there are no fixed points in the action of the group on \(\mathbb {Z}\). However, this clashes with the extremely natural convention to have \(S^B_n\) act on \(\pm [n]\) (with 0 fixed) rather than a string of 2n consecutive integers like \(\{-n + 1, \ldots , -1, 0, 1, \ldots , n\}\).
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Acknowledgements
The authors are grateful for the careful reading and helpful advice of the anonymous referees. On behalf of all authors, the corresponding author states that there are no conflicts of interest.
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Communicated by Jang Soo Kim.
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Research of Joel Brewster Lewis was partially supported by a grant from the Simons Foundation (634530). Research of Bridget Eileen Tenner was partially supported by NSF under Grant DMS-2054436
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Lewis, J.B., Tenner, B.E. Bargain Hunting in a Coxeter Group. Ann. Comb. (2023). https://doi.org/10.1007/s00026-023-00670-2
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DOI: https://doi.org/10.1007/s00026-023-00670-2