Filters in the partition lattice

Richard Ehrenborg; Dustin Hedmark

doi:10.1007/s10801-017-0780-4

Filters in the partition lattice

Richard Ehrenborg, Dustin Hedmark

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Given a filter Δ in the poset of compositions of n, we form the filter in the partition lattice. We determine all the reduced homology groups of the order complex of as S_n _- ₁-modules in terms of the reduced homology groups of the simplicial complex Δ 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 with 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
Pages (from-to)	403-439
Number of pages	37
Journal	Journal of Algebraic Combinatorics
Volume	47
Issue number	3
DOIs	https://doi.org/10.1007/s10801-017-0780-4
State	Published - May 1 2018

Bibliographical note

Funding Information:
Acknowledgements The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began.

Funding Information:
The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant?H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began.

Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

Keywords

Homology
Partition
Representation theory
Symmetric group

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

Access to Document

10.1007/s10801-017-0780-4

Cite this

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abstract = "Given a filter Δ in the poset of compositions of n, we form the filter in the partition lattice. We determine all the reduced homology groups of the order complex of as Sn - 1-modules in terms of the reduced homology groups of the simplicial complex Δ 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 with 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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note = "Funding Information: Acknowledgements The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began. Funding Information: The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant?H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began. Publisher Copyright: {\textcopyright} 2017, Springer Science+Business Media, LLC.",

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N2 - Given a filter Δ in the poset of compositions of n, we form the filter in the partition lattice. We determine all the reduced homology groups of the order complex of as Sn - 1-modules in terms of the reduced homology groups of the simplicial complex Δ 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 with 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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Filters in the partition lattice

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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