Filters in the partition lattice

Richard Ehrenborg; Dustin Hedmark

Filters in the partition lattice

Mathematics

Producción científica: Paper › revisión exhaustiva

Resumen

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.

Idioma original	English
Estado	Published - 2006
Evento	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017 - London, United Kingdom Duración: jul 9 2017 → jul 13 2017

Conference

Conference	29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017
País/Territorio	United Kingdom
Ciudad	London
Período	7/9/17 → 7/13/17

Nota bibliográfica

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

Financiación

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.

Financiadores	Número del financiador
National Security Agency	H98230-13-1-0280

ASJC Scopus subject areas

Algebra and Number Theory

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title = "Filters in the partition lattice",

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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T1 - Filters in the partition lattice

AU - Ehrenborg, Richard

AU - Hedmark, Dustin

PY - 2006

Y1 - 2006

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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T2 - 29th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2017

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Filters in the partition lattice

Resumen

Conference

Nota bibliográfica

Financiación

ASJC Scopus subject areas

Otros archivos y enlaces

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