Twisted Weak Orders of Coxeter Groups
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In this paper, we initiate the study of the twisted weak order associated to a twisted Bruhat order for a Coxeter group and explore the relationship between the lattice property of such orders and the infinite reduced words. We show that for a 2 closure biclosed set B in Φ+, the B-twisted weak order is a non-complete meet semilattice if B is the inversion set of an infinite reduced word and that the converse also holds in the case of affine Weyl groups.
KeywordsCoxeter groups Twisted weak order Closure operator Lattice
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The author acknowledges the support from Guangdong Natural Science Foundation Project 2018A030313581. Some results of the paper are based on part of the author’s thesis. The author wishes to thank his advisor Matthew Dyer, who was aware of the twisted weak order while studying the twisted Bruhat order, for his guidance. The author thanks the anonymous referees for their time and useful comments for improving the paper.
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