The Homology Representations of the Symmetric Group on Cohen-Macaulay Subposets of the Partition Lattice

Sundaram, Sheila

doi:10.1006/aima.1994.1030

Cited by 82 publications

(140 citation statements)

References 4 publications

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“…From these facts the assertion then follows by Proposition 2.9 and Theorem 3.2. The recursive formula in the preceding proposition for the characteristics π (k ) follows from the equivariant acyclicity of Whitney homology established in [Su,Lemma 1.1]. For k = 2 this gives π .…”

Section: Another Example Of An Orbit Arrangement Is Thementioning

confidence: 87%

“…The first assertion follows from the "equivariant acyclicity" of the Whitney homology established in [Su,Lemma 1.1]. This means that the virtual G-module…”

Section: Note That the Dimension Of The Space H I (|K|)mentioning

confidence: 90%

“…In the sequel we will need much of the notation of [Su,Section 1]. For background on symmetric functions and plethysm, we refer to [Macd].…”

Section: Note That the Dimension Of The Space H I (|K|)mentioning

confidence: 99%

“…Now use [Theorem 4.8 (iii)] and the defining equation in [C-H-R] (see also [Su,Theorem 5.1]) for the top homology of the lattice Π (k ) . The Frobenius characteristic of the Whitney homology W H +1−t (Π (k ) ) may thus be rewritten as the degree (k )-term in the plethysm…”

Section: Then These Numbers Reappear As Character Values Of the Cohommentioning

confidence: 99%

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Group actions on arrangements of linear subspaces and applications to configuration spaces

Sundaram

Welker²

1997

Trans. Amer. Math. Soc.

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Abstract. For an arrangement of linear subspaces in R n that is invariant under a finite subgroup of the general linear group Gln(R) we develop a formula for the G-module structure of the cohomology of the complement M A . Our formula specializes to the well known Goresky-MacPherson theorem in case G = 1, but for G = 1 the formula shows that the G-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space M A /G. Our motivating examples are arrangements in C n that are invariant under the action of Sn by permuting coordinates. A particular case is the "k-equal" arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases M A and M A /Sn are spaces of ordered and unordered point configurations in C n many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in R d and provide explicit results for the "k-equal" and the "k-divisible" cases.

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Section: Another Example Of An Orbit Arrangement Is Thementioning

confidence: 87%

“…The first assertion follows from the "equivariant acyclicity" of the Whitney homology established in [Su,Lemma 1.1]. This means that the virtual G-module…”

Section: Note That the Dimension Of The Space H I (|K|)mentioning

confidence: 90%

“…In the sequel we will need much of the notation of [Su,Section 1]. For background on symmetric functions and plethysm, we refer to [Macd].…”

Section: Note That the Dimension Of The Space H I (|K|)mentioning

confidence: 99%

Section: Then These Numbers Reappear As Character Values Of the Cohommentioning

confidence: 99%

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Group actions on arrangements of linear subspaces and applications to configuration spaces

Sundaram

Welker²

1997

Trans. Amer. Math. Soc.

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“…The homology of the partition lattice n+1 = L n,1 has been extensively studied in order to find representations of the symmetric group; see [23] and [28]. In the same spirit, Wachs has studied the signed partition lattice L n,2 ; see [29].…”

Section: Discussionmentioning

confidence: 99%

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

Ehrenborg¹,

Readdy²

1999

Discrete Comput Geom

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Abstract. We study complex hyperplane arrangements whose intersection lattices, known as the Dowling lattices, are a natural generalization of the partition lattice. We give a combinatorial description of the Dowling lattice via enriched partitions to obtain an explicit EL-labeling and then find a recursion for the flag h-vector in terms of weighted derivations. When the hyperplane arrangements are real they correspond to the braid arrangements A n and B n . By applying a result due to Billera and the authors, we obtain a recursive formula for the cd-index of the lattice of regions of the braid arrangements A n and B n .

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Convex Polytopes and Enumeration

Simion

1997

Advances in Applied Mathematics

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This is an expository paper on connections between enumerative combinatorics and convex polytopes. It aims to give an essentially self-contained overview of five specific instances when enumerative combinatorics and convex polytopes arise jointly in problems whose initial formulation lies in only one of these two subjects. These examples constitute only a sample of such instances occurring in the work of several authors. On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, Andre, and simsun permutations, q-Catalan and q-Schrodeŕn umbers. From the subject of polytopes, the examples involve the Ehrhart polynomial, the permutohedron, the associahedron, polytopes arising as intersections of cubes and simplices with half-spaces, and the cd-index of a polytope. ᮊ 1997 Academic Press

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The Homology Representations of the Symmetric Group on Cohen-Macaulay Subposets of the Partition Lattice

Cited by 82 publications

References 4 publications

Group actions on arrangements of linear subspaces and applications to configuration spaces

Group actions on arrangements of linear subspaces and applications to configuration spaces

On Flag Vectors, the Dowling Lattice, and Braid Arrangements

Convex Polytopes and Enumeration

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