In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group.
In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders,
J. Algebra163 (1994), 3, 861–879].
We also show that, for a non-finite and non-cofinite biclosed set 𝐵 in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed 𝐵-twisted length is infinite.
This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases.
Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.
A. Björner and F. Brenti,
Combinatorics of Coxeter Groups,
Grad. Texts in Math. 231,
Springer, New York, 2005.
A. Björner, P. H. Edelman and G. M. Ziegler,
Hyperplane arrangements with a lattice of regions,
Discrete Comput. Geom. 5 (1990), no. 3, 263–288.
M. J. Dyer,
Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders,
Kazhdan–Lusztig Theory and Related Topics (Chicago 1989),
Contemp. Math. 139,
American Mathematical Society, Providence (1992), 141–165.
M. J. Dyer,
Bruhat intervals, polyhedral cones and Kazhdan–Lusztig–Stanley polynomials,
Math. Z. 215 (1994), no. 2, 223–236.
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