-
Border Strip Decompositions OnTwo-Dimensional SurfacesWei-Chia
Tsai ()[email protected] University of KaohsiungAdvisor:
Sen-Peng Eu ()2012/06/26
-
Outline1. Introduction Background Border strips Border strip
decompositions Two-dimensional surfaces2. Main results The b.s.d.
on the surfaces Fixed points of b.s.d. on the surfaces3. Conclusion
& Future work
-
1. Introduction
-
-
-
-
- -
-
-
-
-
-
-
-
-
-
-
-
--
- |S|=1 |S|=2 |S|=3
-
-
-|S|=1, |S|=2, |S|=3
1234511235
-
-
-
------
-
-
-
-
-
-
-
-
-
-
-
2. Main results
-
-
-AA
-
-
-
-
-
-
-
--
-
-
-
-
-
-
123456 112358
-
-
-
0 1 2 3 4 56 213471118
-
The b.s.d. on the surfaces
-
---
-
-
-
-
----
-
--- 2, 6, 10, 14,
-
-
----
-
-
---
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Fixed points of b.s.d.on the surfaces
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3. Conclusion & Future work
-
-
-
-
-
-
-
-
-
-
- 2
-
Thanks for your attention!!
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Symmetric functionThe symmetric function is a polynomial
satisfyfor any permutations is symmetric function, too.The
symmetric function is a vector space.Ex:---
--
-
Basis of symmetric functionPower sum
Ex:
Schur polynomialEx:
-
Theorem[Littlewood-Richardson,1934]For any and we have
summed over all partitions for which is a border strip of
size-Ex:
-
Ex:-
-
Ex:
-
For each surface(cylinder, Mobius band, torus, Klein bottle,
projective plane) above, we obtain explicit formula b.s.d.
-
TorusAll snakes in torus are cyclic.One part isrepeatedin the
torus.There are cyclic snake.Each snake has boses.
Thm 3
-
Klein bottleAll the snakes are cyclic.One part isrepeatedin
theSo we can reduce intoWe can know the property of snake in
theEvery snake has the same number of boxes in a row.
-
Klein bottleThe snake pass the left and right side.m is oddone
snake has two boxes in a rowand other sankes have four boxes in a
row. m is evenall snakes has four boxes in a row.
-
Klein bottleThe snake do not pass the left and right side. n-m
is oddone snake has two boxes in a rowand other sankes has four
boxes in a row. n-m is evenall snakes have four boxes in a row.
-
If there are xs snakes sizes 2m, then there are sankes sizes 4m.
If n divides m then
Otherwise
Thm 4
-
Ex: (Klein bottle)
Ex:
-
Projective planeAll the snakes in projective plane are
cyclic.There is no differenceincase mn.
-
Projective planeThe snake pass the left and right side.
-
|S|=18|S|=14|S|=10|S|=6|S|=2
-
Projective planeThe snake do not pass the left and right side.
n-m is oddone snake has 2m boxesand other snakes have 4m boxes. n-m
is evenall snakes have 4m boxes.
-
Ex: (projective plane)
Ex:|S|=2, |S|=6, |S|=10,|S|=12=|S|,|S|=6
-
Fixid points of mobius band1.we transpose mobius band.
2.The bijection of fixed point between and under action of
3.The bijection of fixed point between and under action of
x123456789112358132134
-
Fixid points of mobius band2.The bijection of fixed point
between and under action of
-
Fixid points of mobius band3.The bijection of fixed point
between and under action of
-
The bijection of fixed point between cylinder and mobius band
under action has 3 fixed points.
The bijection of fixed point betweenand under action of
Mobius band has fixed points.Fixid points of mobius band
x3456789101123581321345589
bsd
*Littlewood-Richardson(http://www.emis.de/journals/SLC/wpapers/s66vortrag/schilling_lecture3.pdf
)SPSborder strip
border strip border stripbox..skew shapeStanleyskew shape
Stanley*Klein bottleProjective plane Klein bottle projective
planemobius bandshow a_k count the number of orbit whose
stabilizer-order divides
k*Littlewood-Richardson(http://www.emis.de/journals/SLC/wpapers/s66vortrag/schilling_lecture3.pdf
)*We can know the property of snake in the
