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Group Representations on the Homology of Products of Posets

Sheila Sundaram, Volkmar Welker
1996 Journal of combinatorial theory. Series A  
In this note we give a description of the representation of the wreath product S n [G] of the symmetric group S n and a finite group G on the homology of the product of n copies of a partially ordered  ...  set (poset for short) P on which G acts as a group of automorphisms.  ...  The action of these two groups induces an action of the wreath product S n [G] on the poset-products.  ... 
doi:10.1006/jcta.1996.0013 fatcat:yvlv3v3vk5df3fxxyqaymkj2gy

Homology representations of unitary reflection groups [article]

Justin Koonin
2013 arXiv   pre-print
The emphasis in this paper is on the representation theory of unitary reflection groups. The main tool is the theory of poset extensions due to Segev and Webb ([16]).  ...  The new results place the well-known representations of unitary reflection groups on the top homology of the lattice of intersections of hyperplanes into a natural family, parameterised by eigenvalue.  ...  The central theme of this paper is the study of homological properties of various posets of eigenspaces associated with unitary reflection groups, and associated homology representations of these groups  ... 
arXiv:1303.5155v2 fatcat:xjtqe2w52nfghhudwut6hnhkfq

A new proof of a theorem of Solomon-Tits

C. W. Curtis, G. I. Lehrer
1982 Proceedings of the American Mathematical Society  
The group G acts on A by conjugation, and on the rational homology ii*(A) of A. In this note we shall prove  ...  Let A be the combinatorial building of a finite group of Lie type G. A new proof is given of the theorem of Solomon-Tits on the G-module structure of the rational homology H»(A) of A.  ...  Let G be a finite group acting on the vertices of X, preserving the order relation, and hence the simplicial structure.  ... 
doi:10.1090/s0002-9939-1982-0652431-1 fatcat:yxax4buevzd5xhndnyo27ucjsi

A New Proof of a Theorem of Solomon-Tits

C. W. Curtis, G. I. Lehrer
1982 Proceedings of the American Mathematical Society  
The group G acts on A by conjugation, and on the rational homology ii*(A) of A. In this note we shall prove  ...  Let A be the combinatorial building of a finite group of Lie type G. A new proof is given of the theorem of Solomon-Tits on the G-module structure of the rational homology H»(A) of A.  ...  Let G be a finite group acting on the vertices of X, preserving the order relation, and hence the simplicial structure.  ... 
doi:10.2307/2044270 fatcat:v3rsxzg2jvdrvmw2r7xozjnjwu

The symmetric group action on rank-selected posets of injective words [article]

Christos A. Athanasiadis
2016 arXiv   pre-print
The symmetric group S_n acts naturally on the poset of injective words over the alphabet {1, 2,...,n}. The induced representation on the homology of this poset has been computed by Reiner and Webb.  ...  We generalize their result by computing the representation of S_n on the homology of all rank-selected subposets, in the sense of Stanley.  ...  The induced representation on the homology of this poset has been computed by Reiner and Webb.  ... 
arXiv:1606.03829v2 fatcat:pxyyi5fj55cr5dqqfyvcdbh27u

Page 77 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews  
Here S,(G) is the wreath product of the symmetric group S, and a finite group G, which acts as a group of (order) automorphisms on the poset P.  ...  Hoo (3-AB; Edmonton, AB) 97a:05215 OSE25 06A09 Sundaram, Sheila (1-MIAM; Coral Gables, FL); Welker, Volkmar (D-ESSN-EM; Essen) Group representations on the homology of products of posets. J. Combin.  ... 

A q-analogue and a symmetric function analogue of a result by Carlitz, Scoville and Vaughan [article]

Yifei Li
2018 arXiv   pre-print
Here the elementary symmetric function e_i is the Frobenius characteristic of the representation of S_i on the top homology of the subset lattice B_i, whereas our identity involves the representation of  ...  S_n×S_n on the Segre product of B_n with itself.  ...  The author would like to thank Washington University in St. Louis Department of Mathematics and Statistics for their support.  ... 
arXiv:1811.06180v1 fatcat:hf2gpbx7vbbhllonchfsoi2icu

Poset topology: Tools and applications [chapter]

Michelle Wachs
2007 Geometric Combinatorics  
In order to transfer the representation of the symmetric group on homology of the kequal partition poset to representations of the symmetric group on the homology of the complement M A n,k := R n − ∪A  ...  These bases are used to obtain further results on the representations of the symmetric group on homology of the posets and to relate these representations to representations on homology of certain interesting  ...  Let NCG n be the poset of disconnected graphs on node set [n] ordered by inclusion of edge sets.  ... 
doi:10.1090/pcms/013/09 fatcat:ipixgnk2bjcpxnqqysvqoxyvni

A generating function approach to new representation stability phenomena in orbit configuration spaces [article]

Christin Bibby, Nir Gadish
2020 arXiv   pre-print
We apply this point of view to the homology and combinatorics of orbit configuration spaces: using the notion of twisted commutative algebras, which essentially categorify exponential generating functions  ...  This idea allows for a factorization of the orbit configuration space "generating function" into an infinite product, whose terms are surprisingly easy to understand.  ...  A central tool in the study of poset homology, as well as the cohomology of subspace arrangements, is the Whitney homology: we shall think of this as the collection of poset homology groups H •−2 (P ≤p  ... 
arXiv:1911.02125v2 fatcat:ful563csozezfjxwdoxu2b3y2a

Poset Topology: Tools and Applications [article]

Michelle L. Wachs
2006 arXiv   pre-print
Some of the topics covered are: subspace arrangements, graph complexes, group actions on poset homology, shellability, recursive techniques, and fiber theorems.  ...  These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology.  ...  In order to transfer the representation of the symmetric group on homology of the kequal partition poset to representations of the symmetric group on the homology of the complement M A n,k := R n − ∪A  ... 
arXiv:math/0602226v2 fatcat:wck4csepvbg6pe2iu7k2me5nt4

Page 3179 of Mathematical Reviews Vol. , Issue 2000e [page]

2000 Mathematical Reviews  
We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d.  ...  Summary: “We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the ho- mology of posets of partitions with restricted block size.  ... 

The combinatorics and the homology of the poset of subgroups of p-power index

Michael Weidner, Volkmar Welker
1993 Journal of Pure and Applied Algebra  
We describe the representation of G on the homology groups of the order complex of S' (G) and show that this representation can be realized by matrices with entries in the set { + 1, -1, 0).  ...  Welker, The combinatorics and the homology of the poset of subgroups of p-power index, Journal of Pure and Applied Algebra 90 (1993) 253-274.  ...  Acknowledgment The second author would like to thank the Mittag-Leffler Institute for its hospitality. Most of his work on the paper has been done there. We also would like to thank A.  ... 
doi:10.1016/0022-4049(93)90044-t fatcat:dkafoj32gbajnaol2zajcxh7ay

Page 2334 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
Carlsson’s theorem on free actions on products of spheres and to show that for any such module M there is a finite nonnegative complex of projective kG-modules with zero homology M and all other homology  ...  The author gives a very interesting algebraic characterisation of the Chern classes of a complex representation of a finite group G.  ... 

The Symmetric Group Action on Rank-selected Posets of Injective Words

Christos A. Athanasiadis
2016 Order  
The induced representation on the homology of this poset has been computed by Reiner and Webb.  ...  The symmetric group S n acts naturally on the poset of injective words over the alphabet {1, 2, . . . , n}.  ...  homology group of (the order complex of) PS := P S { 0, 1}.  ... 
doi:10.1007/s11083-016-9417-9 fatcat:7b2ohwzr2jejnaacbmvdsizpdu

Page 7121 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
In the case of group algebras k[G] (G a discrete group), the au- thor relates this definition of cyclic homology and the usual cyclic homology of k[G] as an associative algebra.  ...  group B, on n —1 generators acts transitively on the set of complete exceptional sequences.  ... 
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