The Rees product and cubical complexes - Armstrong Atlantic

43 Pages·2010·0.47 MB·English

by  Tricia Muldoon Brown

Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
The Rees product and cubical complexes
Tricia Muldoon Brown
ArmstrongAtlanticStateUniversity
April 18, 2010
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Outline
Introduction
Results
Cubical Results
Rees multiple
Almost cubical posets
Questions
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Assumptions
P is a poset which is:
bounded below
graded, with rank function ρ.
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Rees product
Definition
For two graded posets P and Q with rank function ρ the Rees
product P (cid:63)Q, is the set of ordered pairs (p,q) in the Cartesian
product P ×Q with ρ(p) ≥ ρ(q). These pairs are partially ordered
by (p,q) ≤ (p(cid:48),q(cid:48)) if p ≤ p(cid:48), q ≤ q(cid:48), and
P Q
ρ(p(cid:48))−ρ(p) ≥ ρ(q(cid:48))−ρ(q).
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
C (cid:63) C
4 4
◦◦◦◦ ∗ ◦◦◦◦ = ◦...............................................................◦...................................................................................................................................◦◦........................................................................................................................................................................................................◦◦.............................................................................................................................................................................................................◦◦...............................................................................................................................................◦..........................................................................◦......
Figure: The Rees product of two chains
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Notation: Rees(P,Q) = ((P \{ˆ0})(cid:63)Q)∪{ˆ0,ˆ1}
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
C
2
◦◦..................................................................................................................................................................................................................................................................................◦◦..................................................................................................................................................................................................................................................................◦◦..........................................................................................................................................................................................................................................................................◦◦............................................................................................................................................................................................................................................................................................................◦◦..............
Figure: The face lattice of the square, C
2
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
C
2
◦◦..................................................................................................................................................................................................................................................................................◦◦..................................................................................................................................................................................................................................................................◦◦..........................................................................................................................................................................................................................................................................◦◦............................................................................................................................................................................................................................................................................................................◦◦..............
Figure: The face lattice of the square, C
2
Notation: Rees(P,Q) = ((P \{ˆ0})(cid:63)Q)∪{ˆ0,ˆ1}
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Rees(C ,C )
2 3
•.................................................................................................................................................................................................................................................••.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•••.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................••..............................................................................................................................................................................................................................................................................•................
Figure: Rees(C ,C )
2 3
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Observation
For P, a rank n poset, the Rees product Rees(P,C ) is isomorphic
n
to the Segre product ((P \{ˆ0})◦(C (cid:63)C ))∪{ˆ0,ˆ1}.
n n
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity
Introduction Results Cubicalresults Reesmultiple Almostcubicalposets Conclusion
Cohen-Macaulay
Theorem (Bjo¨rner–Welker)
The Rees product of two Cohen-Macaulay posets is
Cohen-Macaulay.
TriciaMuldoonBrown:TheReesproductandcubicalcomplexes ArmstrongAtlanticStateUniversity