Border Strip Decompositions On Two-Dimensional Surfaces
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Transcript of Border Strip Decompositions On Two-Dimensional Surfaces
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Border Strip Decompositions OnTwo-Dimensional SurfacesWei-Chia Tsai ()[email protected] University of KaohsiungAdvisor: Sen-Peng Eu ()2012/06/26
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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
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1. Introduction
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- |S|=1 |S|=2 |S|=3
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-|S|=1, |S|=2, |S|=3
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2. Main results
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123456 112358
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0 1 2 3 4 56 213471118
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The b.s.d. on the surfaces
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--- 2, 6, 10, 14,
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Fixed points of b.s.d.on the surfaces
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3. Conclusion & Future work
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Thanks for your attention!!
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Symmetric functionThe symmetric function is a polynomial satisfyfor any permutations is symmetric function, too.The symmetric function is a vector space.Ex:---
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Basis of symmetric functionPower sum
Ex:
Schur polynomialEx:
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Theorem[Littlewood-Richardson,1934]For any and we have
summed over all partitions for which is a border strip of size-Ex:
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Ex:-
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Ex:
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For each surface(cylinder, Mobius band, torus, Klein bottle, projective plane) above, we obtain explicit formula b.s.d.
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TorusAll snakes in torus are cyclic.One part isrepeatedin the torus.There are cyclic snake.Each snake has boses.
Thm 3
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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.
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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.
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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.
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If there are xs snakes sizes 2m, then there are sankes sizes 4m. If n divides m then
Otherwise
Thm 4
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Ex: (Klein bottle)
Ex:
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Projective planeAll the snakes in projective plane are cyclic.There is no differenceincase mn.
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Projective planeThe snake pass the left and right side.
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|S|=18|S|=14|S|=10|S|=6|S|=2
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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.
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Ex: (projective plane)
Ex:|S|=2, |S|=6, |S|=10,|S|=12=|S|,|S|=6
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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
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Fixid points of mobius band2.The bijection of fixed point between and under action of
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Fixid points of mobius band3.The bijection of fixed point between and under action of
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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
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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