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Group Representations on the Homology of Products of Posets

1996
*
Journal of combinatorial theory. Series A
*

In this note we give a description

doi:10.1006/jcta.1996.0013
fatcat:yvlv3v3vk5df3fxxyqaymkj2gy
*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*. ...##
###
Homology representations of unitary reflection groups
[article]

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*...

##
###
A new proof of a theorem of Solomon-Tits

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. ...

##
###
A New Proof of a Theorem of Solomon-Tits

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. ...

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

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. ...

##
###
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]

2018
*
arXiv
*
pre-print

Here

arXiv:1811.06180v1
fatcat:hf2gpbx7vbbhllonchfsoi2icu
*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. ...##
###
Poset topology: Tools and applications
[chapter]

2007
*
Geometric Combinatorics
*

In order to transfer

doi:10.1090/pcms/013/09
fatcat:ipixgnk2bjcpxnqqysvqoxyvni
*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. ...##
###
A generating function approach to new representation stability phenomena in orbit configuration spaces
[article]

2020
*
arXiv
*
pre-print

We apply this point

arXiv:1911.02125v2
fatcat:ful563csozezfjxwdoxu2b3y2a
*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 ...##
###
Poset Topology: Tools and Applications
[article]

2006
*
arXiv
*
pre-print

Some

arXiv:math/0602226v2
fatcat:wck4csepvbg6pe2iu7k2me5nt4
*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 ...##
###
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

1993
*
Journal of Pure and Applied Algebra
*

We describe

doi:10.1016/0022-4049(93)90044-t
fatcat:dkafoj32gbajnaol2zajcxh7ay
*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. ...##
###
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

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}. ...

##
###
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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