Filters in the partition lattice

Richard Ehrenborg; Dustin Hedmark

Filters in the partition lattice

Mathematics

Research output: Contribution to conference › Paper › peer-review

Abstract

Given a filter Λ in the poset of compositions of n, we form the filter Π_Δ^∗in the partition lattice. We determine all the reduced homology groups of the order complex of Π_Δ^∗as S_n-1-modules in terms of the reduced homology groups of the simplicial complex A and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank-Hanlon-Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a + d,⋯,a + (a - 1) • d, extending work of Browdy.

Original language	English
State	Published - 2006
Event	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017 - London, United Kingdom Duration: Jul 9 2017 → Jul 13 2017

Conference

Conference	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017
Country/Territory	United Kingdom
City	London
Period	7/9/17 → 7/13/17

Bibliographical note

Publisher Copyright:
© 29th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.

Funding

The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram and Michelle Wachs for essential references. The authors thank Margaret Readdy for her comments on an earlier draft. Both authors were partially supported by National Security Agency grant H98230-13-1-0280. The first author wishes to thank the Princeton University Mathematics Department where this work began.

Funders	Funder number
National Security Agency	H98230-13-1-0280

Keywords

Composition lattice
Equivariant Quillen's Fiber Lemma
Frobenius complex
Partition lattice
Specht module

ASJC Scopus subject areas

Algebra and Number Theory

Cite this

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abstract = "Given a filter Λ in the poset of compositions of n, we form the filter ΠΔ∗in the partition lattice. We determine all the reduced homology groups of the order complex of ΠΔ∗as Sn-1-modules in terms of the reduced homology groups of the simplicial complex A and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank-Hanlon-Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a + d,⋯,a + (a - 1) • d, extending work of Browdy.",

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AU - Ehrenborg, Richard

AU - Hedmark, Dustin

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N2 - Given a filter Λ in the poset of compositions of n, we form the filter ΠΔ∗in the partition lattice. We determine all the reduced homology groups of the order complex of ΠΔ∗as Sn-1-modules in terms of the reduced homology groups of the simplicial complex A and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank-Hanlon-Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a + d,⋯,a + (a - 1) • d, extending work of Browdy.

AB - Given a filter Λ in the poset of compositions of n, we form the filter ΠΔ∗in the partition lattice. We determine all the reduced homology groups of the order complex of ΠΔ∗as Sn-1-modules in terms of the reduced homology groups of the simplicial complex A and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank-Hanlon-Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a + d,⋯,a + (a - 1) • d, extending work of Browdy.

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KW - Equivariant Quillen's Fiber Lemma

KW - Frobenius complex

KW - Partition lattice

KW - Specht module

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Filters in the partition lattice

Abstract

Conference

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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