On Modular Homology in the Boolean Algebra

Jones, P.R; Siemons, I. J.

doi:10.1006/jabr.1996.0009

Cited by 10 publications

(3 citation statements)

References 29 publications

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“…The following generalizes the rank formula of Wilson in [17]. As in [9,10] one can use the exactness of the sequences also to determine the p-rank for the orbit inclusion matrix for any group whose order is not divisible by p.…”

Section: The Homology Of K-shellable Complexesmentioning

confidence: 96%

“…In [1,9,10] we have determined the modular homology of the simplex (or equivalently, of the Boolean algebra) and finite projective spaces (which may be considered as q-simplices) were dealt with in [11]. One consequence of [9] in particular is that modular homology is not homotopy invariant and hence does not coincide with the standard simplicial homology.…”

Section: Introductionmentioning

confidence: 98%

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On Modular Homology of Simplicial Complexes: Shellability

Mnukhin¹,

Siemons²

2001

Journal of Combinatorial Theory, Series A

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For a simplicial complex 2 and coefficient domain F let F2 be the F-module with basis 2. We investigate the inclusion map given by : { [ _ 1 +_ 2 +_ 3 + } } } +_ k which assigns to every face { the sum of the co-dimension 1 faces contained in {. When the coefficient domain is a field of characteristic p>0 this map gives rise to homological sequences. We investigate this modular homology for shellable complexes. Academic Press

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Section: The Homology Of K-shellable Complexesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

On Modular Homology of Simplicial Complexes: Shellability

Mnukhin¹,

Siemons²

2001

Journal of Combinatorial Theory, Series A

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“…Such computations are quite involved and, with the kind permission of the author, we will record some of his findings to make these more widely available. For instance, it is shown that AΓL(1, 16) has reconstruction index 7 in the natural representation, while AGL (1,16), AGL (1,11), AGL(1, 23) and subgroups of index 2 in AΓL(1, 16) all have index 6. The projective linear groups P SL(2, 11), P GL(2, 11), P GL(2, 16), P GL (2,17), P SL (2,19) have index 7 in their natural representations, P SL(2, 17) has index 6 while P GL(2, 13) and P SL(2, 13) have index 5.…”

Section: The Main Boundsmentioning

confidence: 99%

On the reconstruction index of permutation groups: semiregular groups

Maynard

Siemons

2002

Aequ. math.

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On Modular Homology in the Boolean Algebra, II

1998

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Let R be an associative ring with identity and ⍀ an n-element set. For k F n consider the R-module M with k-element subsets of ⍀ as basis. The r-stepis the linear map defined on this basis throughwhere the ⌫ are the k y r -element subsets of ⌬.For m -r one obtains chainsof inclusion maps which have interesting homological properties if R has charac-Ž teristic p ) 0. V. B. Mnukhin and J. Siemons J. Combin. Theory 74, 1996 . 287᎐300; J. Algebra 179, 1995, 191᎐199 introduced the notion of p-homology to examine such sequences when r s 1 and here we continue this investigation when r is a power of p. We show that any section of M M not containing certain middle terms is p-exact and we determine the homology modules for such middle terms. Numerous infinite families of irreducible modules for the symmetric groups arise in this fashion. Among these the semi-simple inducti¨e systems discussed by A. Ž . Kleshchev J. Algebra 181, 1996, 584᎐592 appear and in the special case p s 5 we Ž obtain the Fibonacci representations of A. J. E. Ryba J. Algebra 170, 1994, . 678᎐686 . There are also applications to permutation groups of order co-prime to p, resulting in Euler᎐Poincare equations for the number of orbits on subsets of such groups. ᮊ

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On Modular Homology in the Boolean Algebra

Cited by 10 publications

References 29 publications

On Modular Homology of Simplicial Complexes: Shellability

On Modular Homology of Simplicial Complexes: Shellability

On the reconstruction index of permutation groups: semiregular groups

On Modular Homology in the Boolean Algebra, II

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