## Coxeter-Like Complexes (2001)

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Venue: | Discrete Math. Theor. Comput. Sci |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Babson01coxeter-likecomplexes,

author = {Eric Babson and Victor Reiner},

title = {Coxeter-Like Complexes},

journal = {Discrete Math. Theor. Comput. Sci},

year = {2001},

volume = {6},

pages = {223--252}

}

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### Abstract

Motivated by the Coxeter complex associated to a Coxeter system (W, S), we introduce a simplicial regular cell complex #(G,S) with a G-action associated to any pair (G, S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion.

### Citations

884 |
Enumerative combinatorics
- Stanley
- 1997
(Show Context)
Citation Context ...plex representation of S n on its homology H . (# T , C). For this purpose, we will make use of standard terminology about the symmetric group and its complex representations, such as can be found in =-=[25, 32]-=-. In what follows, all simplicial chain groups and homology groups are taken with C coe#cients, unless explicitly stated otherwise. One useful feature of the setting (G, S) = (S n , S T ) is that Prop... |

869 | Algebraic Topology - Spanier - 1966 |

464 | Reflection Groups and Coxeter Groups,” Cambridge - Humphreys - 1990 |

438 | Atlas of Finite Groups - Conway, Curtis, et al. - 1985 |

238 |
The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions
- Sagan
- 1991
(Show Context)
Citation Context ...plex representation of S n on its homology H . (# T , C). For this purpose, we will make use of standard terminology about the symmetric group and its complex representations, such as can be found in =-=[25, 32]-=-. In what follows, all simplicial chain groups and homology groups are taken with C coe#cients, unless explicitly stated otherwise. One useful feature of the setting (G, S) = (S n , S T ) is that Prop... |

215 |
Finite unitary reflection groups
- Shephard, Todd
(Show Context)
Citation Context ...ace with positive definite Hermitian bilinear form) and generated by unitary reflections, that is, elements of finite order which fix some hyperplane. Such groups were classified by Shephard and Todd =-=[28]-=-, and contain many interesting examples. There is one infinite family of such groups G(de, e, r), consisting of the r r matrices with one non-zero entry in each row and column for which all COXETER-LI... |

156 |
Topological methods, in: Handbook of Combinatorics
- Björner
- 1995
(Show Context)
Citation Context ... section gives the basic construction, and explores some of its general properties. Good references for some of the terminology and facts regarding posets, simplicial complexes and cell complexes are =-=[4]-=- and [5]. Let G be a (finitely generated) group, and S a finite generating set for G which is minimal with respect to inclusion. Given any subset J # S, COXETER-LIKE COMPLEXES 3 let G J denote the sub... |

127 | Complex reflection groups, braid groups, Hecke algebras - Broué, Malle, et al. - 1998 |

115 |
Topological methods
- Björner
- 1995
(Show Context)
Citation Context ... section gives the basic construction, and explores some of its general properties. Good references for some of the terminology and facts regarding posets, simplicial complexes and cell complexes are =-=[3]-=- and [4]. Let G be a (finitely generated) group, and S a finite generating set for G which is minimal with respect to inclusion. Given any subset J ⊆ S, let GJ denote the subgroup 〈J〉 generated by J i... |

71 | Some aspects of groups acting on finite posets - Stanley - 1982 |

60 |
The colored Tverberg’s problem and complexes of injective functions
- Zivaljevic, Vrecica
- 1992
(Show Context)
Citation Context ...icity sequence m r by setting m i = 1 for each leaf vertex i, and m v = r. Then one can easily check that # T ,mr is isomorphic to the (n - 1) (n + r - 1) chessboard complex # n-1,n+r-1 considered in =-=[1, 9, 15, 17, 26, 34, 35]-=-, whose faces correspond to placements of non-attacking rooks on an (n - 1) (n - 1 + r) chessboard. In particular, when T is an n-vertex star, # T = # T,(1,1,...,1) = # T,m1 # = # n-1,n . It was noted... |

53 |
regular CW complexes and Bruhat order
- Björner, Posets
- 1984
(Show Context)
Citation Context ... gives the basic construction, and explores some of its general properties. Good references for some of the terminology and facts regarding posets, simplicial complexes and cell complexes are [4] and =-=[5]-=-. Let G be a (finitely generated) group, and S a finite generating set for G which is minimal with respect to inclusion. Given any subset J # S, COXETER-LIKE COMPLEXES 3 let G J denote the subgroup #J... |

45 |
On discrete Morse functions and combinatorial decompositions
- Chari
- 2000
(Show Context)
Citation Context ...2 [m] at the rank Dm := {m 1 , m 1 +m 2 , . . . , m 1 +m 2 + +m n-1 }. The Coxeter complex is shellable, a property which is automatically inherited by all of its type-selected subcomplexes (see e.g. =-=[4, 11]-=-). Hence in this case # T,m is homotopy equivalent to a wedge of (n - 2)- spheres, which is (n-3)-connected, in agreement with Conjecture 4.12. 26 ERIC BABSON AND VICTOR REINER The homology is also we... |

38 | Complexes of groups and orbihedra, Group theory from a geometrical viewpoint - Haefliger - 1990 |

35 |
Chessboard complexes and matching complexes
- Björner, Lovasz, et al.
- 1994
(Show Context)
Citation Context ...on 4.2 below), and that it is isomorphic to the 34 chessboard complex, first considered by Garst [17] in the context of coset complexes of groups, and later by Bjorner, Lovasz, Vrecica and Zivaljevic =-=[9]-=- and many other authors (see Example 4.5 below). In [9, p. 30] it was also pointed out that it is a 2-torus. In Section 4, we discuss the case where W = S n in more detail. Example 3.2. The previous e... |

28 |
f -vectors and h-vectors of simplicial posets
- Stanley
- 1991
(Show Context)
Citation Context ...whose elements are the cosets {gG J : g # G, J # S} with ordering by reverse inclusion, i.e. gG Js# G J # if gG J # g # G J # . Proposition 2.1. P (G, S) is a simplicial poset in the sense of Stanley =-=[31]-=-, that is, every lower interval in P (G, S) is isomorphic to a Boolean algebra. Proof. It su#ces to show that if gGK # g # GK # then g # GK # = gGK # and K # K # , since then the interval below gGK in... |

27 | On the Betti numbers of chessboard complexes
- Friedman, Hanlon
- 1998
(Show Context)
Citation Context ...icity sequence m r by setting m i = 1 for each leaf vertex i, and m v = r. Then one can easily check that # T ,mr is isomorphic to the (n - 1) (n + r - 1) chessboard complex # n-1,n+r-1 considered in =-=[1, 9, 15, 17, 26, 34, 35]-=-, whose faces correspond to placements of non-attacking rooks on an (n - 1) (n - 1 + r) chessboard. In particular, when T is an n-vertex star, # T = # T,(1,1,...,1) = # T,m1 # = # n-1,n . It was noted... |

26 |
Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings
- Björner
- 1984
(Show Context)
Citation Context ...1 , and may be identified with the simplicial decomposition of the unit sphere in V by the reflecting hyperplanes for the reflections in W . There is an extensive literature on Coxeter complexes; see =-=[7] for some -=-references. 2 Some authors might apply the term "Euclidean reflection group" to the case where W is possibly infinite but generated by a#ne reflections. For this reason, one should perhaps c... |

18 |
Group actions on Stanley-Reisner rings and invariants of permutation groups
- Garsia, Stanton
- 1984
(Show Context)
Citation Context ...g the atom gG S-{s} the color s. We have the following immediate consequence. Corollary 2.2. There is a unique (up to isomorphism) balanced regular cell complex of Boolean type [5] or Boolean complex =-=[16]-=- having P (G, S) as its poset of faces. # We denote this regular cell complex having face poset P (G, S) by #(G, S); it will be our main object of study. The regular nature of the face poset P (G, S) ... |

18 |
Shellability of chessboard complexes
- Ziegler
- 1994
(Show Context)
Citation Context ...icity sequence m r by setting m i = 1 for each leaf vertex i, and m v = r. Then one can easily check that # T ,mr is isomorphic to the (n - 1) (n + r - 1) chessboard complex # n-1,n+r-1 considered in =-=[1, 9, 15, 17, 26, 34, 35]-=-, whose faces correspond to placements of non-attacking rooks on an (n - 1) (n - 1 + r) chessboard. In particular, when T is an n-vertex star, # T = # T,(1,1,...,1) = # T,m1 # = # n-1,n . It was noted... |

17 |
Non-positively curved triangles of groups, in: Group Theory from a Geometrical Viewpoint
- Stallings
- 1991
(Show Context)
Citation Context ...codimension 1 faces of some fixed maximal face of #, and # # = #(G, S). We also note 1 that #(G, S) is a very special case of what has been called a (developable) simplex of groups (see [18, 2.4] and =-=[30]-=-). Although #(G, S) has simplicial cells, it need not be a simplicial complex; see Example 3.4 below. However, there is a simple criterion for this to occur. Given any Boolean complex # with vertex se... |

16 |
Some Cohen-Macaulay complexes and group actions
- Garst
- 1979
(Show Context)
Citation Context ...equivalently, if and only if # is a simplicial complex. # In the case of # = #(G, S), there is a natural alternative description of # which ties it in with Tits coset complexes, as studied in [6] and =-=[17]-=-. Let C(G, S) = {gG S-s : g # G, s # S}. 1 Thanks to Mike Davis for pointing this out. COXETER-LIKE COMPLEXES 5 denote the covering of the set G by the cosets of maximal (proper) parabolic subgroups. ... |

12 |
Regular complex polytopes, Second edition
- Coxeter
- 1991
(Show Context)
Citation Context ... of Euclidean reflection groups (i.e. extending the action of a Euclidean reflection group acting on R n to C n ), and . the Shephard groups introduced by Shephard [27] and studied further by Coxeter =-=[12]-=-, which are the automorphism groups of regular complex polytopes. For Shephard groups and their distinguished generating sets S, the complex #(G, S) has many di#erent descriptions, including some whic... |

6 |
Decompositions and connectivity of matching and chessboard complexes
- Athanasiadis
(Show Context)
Citation Context ... a group, S is a finite minimal generating set and s # J # S then there is a short exact sequence of complexes of C[G]-modules 0 # C . (#(G, S) J-s ) # C . (#(G, S) J ) # (C . (#(G S-s , S - s) J-s ))=-=[1]-=- # G GS-s # 0. Here C . [1] denotes the chain complex C . with degree shift by 1, i.e. C i [1] = C i-1 , and # G H denotes induction of a representation from a subgroup H to G. Proof. The injective ma... |

6 | The coset poset and probabilistic zeta function of a finite group
- Brown
(Show Context)
Citation Context ...ties of #(G, S) proven above. In the case where G is a group with BN-pair having associated Coxeter system (W, S), this #(G, B, S) is the usual Tits building. Remark 2.8. We should mention that Brown =-=[8]-=- recently studied a (di#erent) topological space built from proper cosets of a group ordered by inclusion. We are not aware of a direct link with his work. 6 ERIC BABSON AND VICTOR REINER 2.2. Pseudom... |

6 | Free resolutions of simplicial posets - Duval - 1997 |

6 | Reduced decompositions of permutations in terms of star transpositions, generalized catalan numbers and k-ary
- Pak
(Show Context)
Citation Context ...trees T on [n], less is known, although the case where T is the star graph (so that # T is the chessboard complex # n-1,n as in Example 4.5) was considered in [14, 5], and studied more extensively in =-=[23]-=-. 4.2. Deletion-contraction and flossing. For the remainder of the paper, we examine the topology of # T , and particularly the complex representation of S n on its homology H . (# T , C). For this pu... |

6 |
Regular complex polytopes
- Shephard
- 1952
(Show Context)
Citation Context ...ch have them . the complexifications of Euclidean reflection groups (i.e. extending the action of a Euclidean reflection group acting on R n to C n ), and . the Shephard groups introduced by Shephard =-=[27]-=- and studied further by Coxeter [12], which are the automorphism groups of regular complex polytopes. For Shephard groups and their distinguished generating sets S, the complex #(G, S) has many di#ere... |

3 |
Some Cohen-Macaulay complexes arising in group theory, Commutative algebra and combinatorics (Kyoto
- Björner
- 1985
(Show Context)
Citation Context ...ces, or equivalently, if and only if # is a simplicial complex. # In the case of # = #(G, S), there is a natural alternative description of # which ties it in with Tits coset complexes, as studied in =-=[6]-=- and [17]. Let C(G, S) = {gG S-s : g # G, s # S}. 1 Thanks to Mike Davis for pointing this out. COXETER-LIKE COMPLEXES 5 denote the covering of the set G by the cosets of maximal (proper) parabolic su... |

3 |
On inversions and cycles in permutations
- Edelman
- 1987
(Show Context)
Citation Context ...h have been answered. For other spanning trees T on [n], less is known, although the case where T is the star graph (so that # T is the chessboard complex # n-1,n as in Example 4.5) was considered in =-=[14, 5]-=-, and studied more extensively in [23]. 4.2. Deletion-contraction and flossing. For the remainder of the paper, we examine the topology of # T , and particularly the complex representation of S n on i... |

3 |
personal communication
- Hersh
(Show Context)
Citation Context ...rem 5.3. However, see the discussion of chessboard complexes in Example 4.15 below as an illustration of the looseness of this conjectural connectivity bound in general. Some recent ideas of P. Hersh =-=[19]-=- regarding a notion of weak orderingson (n - #(T ))-faces of # T ,m may lead to a stronger assertion than Conjecture 4.12, namely that the (n - #(T ))-skeleton is shellable. Similar results were prove... |

3 |
fiber complexes for Shephard groups
- Orlik, Milnor
- 1990
(Show Context)
Citation Context ... For Shephard groups and their distinguished generating sets S, the complex #(G, S) has many di#erent descriptions, including some which make no reference to the choice of the generators S- see Orlik =-=[21]-=-. In this situation, #(G, S) turns out to be a simplicial complex which is homotopy equivalent to a wedge of spheres of dimension |S|-1, and the homology representation H |S|-1 (#(G, S), Z) has many b... |

3 |
Quotients of Coxeter complexes and P -partitions
- Reiner
- 1992
(Show Context)
Citation Context ...12-13], where it was incorrectly asserted that H\P (G, S) is the poset of all double cosets {HgG J : g # G, J # S} ordered by reverse inclusion. Fortunately, this has no e#ect on the later results of =-=[24]-=-, as they proceed from the (correct) assumption that the faces of H\#(G,S) having color set S - J are in bijection with double cosets of the form HgG J inside G. The slight subtlety here is that whene... |

2 |
A presentation for the unipotent group over
- Biss
- 1998
(Show Context)
Citation Context ...he map # : s i ## -1 extends to the homomorphism G # # Zs(a ij ) n i,j=1 ## (-1) # n-1 i=1 a i,i+1 Therefore #(G, S) is an orientable pseudomanifold by Proposition 2.9. COXETER-LIKE COMPLEXES 15 Biss =-=[3]-=- has shown that all relations among the s i are generated by the following Coxeter-like relations (3.1) s 2 i = 1 (s i s i+1 ) 4 = 1 (s i s j ) 2 = 1 for |i - j| > 1, along with the extra relations (s... |

2 | Homology of matching and chessboard complexes, extended abstract
- Shareshian, Wachs
- 2001
(Show Context)
Citation Context |

1 | A.Shepler, The sign representation for a Shephard group
- Orlik, Reiner
(Show Context)
Citation Context ... to be a simplicial complex which is homotopy equivalent to a wedge of spheres of dimension |S|-1, and the homology representation H |S|-1 (#(G, S), Z) has many beautiful guises, which are studied in =-=[22]-=-. Remark 3.3. Motivated by the Coxeter and Shephard cases, along with Corollary 2.16 and Proposition 2.17, one might naively hope that H |S|-1 (#(G, S), Z) carries some canonical representation of G, ... |

1 |
A presentation for the unipotent group over F2
- Biss
- 1998
(Show Context)
Citation Context ...t the map ɛ : si ↦→ −1 extends to the homomorphism G ɛ → Z × (aij) n i,j=1 ↦→ (−1)� n−1 i=1 ai,i+1 Therefore ∆(G, S) is an orientable pseudomanifold by Proposition 2.9.sCOXETER-LIKE COMPLEXES 15 Biss =-=[3]-=- has shown that all relations among the si are generated by the following Coxeter-like relations (3.1) s 2 i = 1 (sisi+1) 4 = 1 (sisj) 2 = 1 for |i − j| > 1, along with the extra relations (sisi+1si+2... |