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| Format: | Preprint |
| Veröffentlicht: |
2004
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| Online-Zugang: | https://arxiv.org/abs/math/0401404 |
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| _version_ | 1866909038063648768 |
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| author | Reading, Nathan |
| author_facet | Reading, Nathan |
| contents | We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let η_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of η_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0401404 |
| institution | arXiv |
| publishDate | 2004 |
| record_format | arxiv |
| spellingShingle | Lattice congruences of the weak order Reading, Nathan Combinatorics 20F55, 06B10 (Primary) 52C35 (Secondary) We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let η_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of η_K constitute the smallest lattice congruence with 1\equiv s for every s\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order. |
| title | Lattice congruences of the weak order |
| topic | Combinatorics 20F55, 06B10 (Primary) 52C35 (Secondary) |
| url | https://arxiv.org/abs/math/0401404 |