permutahedra and the saneblidze-umble diagonal
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Permutahedra and the Saneblidze-Umble Diagonal. By Stephen Weaver Directed by Dr. Ron Umble. Computational Geometry the study of algorithms to solve problems in geometry. Permutahedra. - PowerPoint PPT PresentationTRANSCRIPT
Permutahedra and the Permutahedra and the Saneblidze-Umble DiagonalSaneblidze-Umble Diagonal
By Stephen WeaverBy Stephen WeaverDirected by Dr. Ron UmbleDirected by Dr. Ron Umble
Computational Geometry the study of algorithms to solve problems in geometry
Permutahedra● Geometric shapes based on permutations of
the partitions of a set {1,2} {3}
● Each permutation corresponds to a face of some permutahedron
● We use bar notation for convenience {3} {1,2} = 3|12
● The single partition {1,2,3,…,n} corresponds to the top dimensional face
Permutahedra● The boundary of a face consists of adding a
single bar in every possible way
121|2 2|1
● Two faces are adjacent if their boundaries intersect
12|3 1|231|2|3
2|1|3 1|3|2
Permutahedra - Examples
1
P1P2
121|2 2|1
P3
123
12|3
1|23 2|13
23|1
3|12
13|2
2|1|31|2|3
2|3|1
3|2|13|1|2
1|3|2
P4
Permutahedra - Examples
1
P1P2
121|2 2|1
P3
123
12|3
1|23 2|13
23|1
3|12
13|2
2|1|31|2|3
2|3|1
3|2|13|1|2
1|3|2
P4
Permutahedra - Examples
1
P1P2
121|2 2|1
P3
123
12|3
1|23 2|13
23|1
3|12
13|2
2|1|31|2|3
2|3|1
3|2|13|1|2
1|3|2
P4
Permutahedra - Examples
1
P1P2
121|2 2|1
P3
123
12|3
1|23 2|13
23|1
3|12
13|2
2|1|31|2|3
2|3|1
3|2|13|1|2
1|3|2
P4
12|34123|4
12|3
|4
DiagonalGiven a set S,Diagonal of S S { (x,x) | x є S }
S
S
Diagonal on P2 P2
1|2 1|2 2|1 1|2
1|2 2|1 2|1 2|2
1|2 12
12 2|1
Step Matrix Example
2 31 9
84 7
5 6
Reading a Step Matrix
14|2|3 4|123
41 2 3
Step Matrix Example
2 31 9
84 7
5 6
Reading a Step Matrix
14|2|3
41 2 3
Step Matrix Example
2 31 9
84 7
5 6
Reading a Step Matrix
14|2|3 4|123
41 2 3
Transforming a Step Matrix
231
45 2
31
45
231
4 52 3
1
4 5
1 2 3 123
12
3
12 3
12 3
1 23
1 23
123
S-U Diagonal on P3
1|2|3123 1|23 13|2 12|32|13
12|323|1
2|1323|1 13|23|12
1|233|12
1233|2|1
Calculating the S-U Diagonal● Skip step matrix stage – Use permutations and
strings● One – to – one correspondence between
permutations and step matrices
Calculating the S-U Diagonal● Skip step matrix stage – Use permutations and
strings● One – to – one correspondence between
permutations and step matrices
4132
41
23
A=4
B=4
Calculating the S-U Diagonal● Skip step matrix stage – Use permutations and
strings● One – to – one correspondence between
permutations and step matrices
4132
41
23
A=41
B=4|1
Calculating the S-U Diagonal● Skip step matrix stage – Use permutations and
strings● One – to – one correspondence between
permutations and step matrices
4132
41
23
A=41|3
B=4|13
Calculating the S-U Diagonal● Skip step matrix stage – Use permutations and
strings● One – to – one correspondence between
permutations and step matrices
4132
41
23
A=41|32
B=4|13|214|23 4|13|2
S-U Diagonal● Acts multiplicatively w.r.t. the bar operation
(12|34) = (12) | (34)
= (1|212 + 122|1) | (3|434 + 344|3)
= (1|2|3|412|34) + (1|2|3412|4|3) +
(12|3|42|1|34) + (12|342|1|4|3)
Generic Diagonal2|134
a|b|c abc
a|bc ac|b
ab|c b|ac
b|ac bc|a
ac|b c|ab
abc c|b|a
ab|c bc|a
a|bc c|ab
Three Element Diagonal
abc
(2|134) = (2)| (134) = 2|1|3|4 2|134
2|1|34 2|14|3
2|13|4 2|3|14
2|3|14 2|34|1
2|14|3 2|4|13
2|134 2|4|3|1
2|13|4 2|34|1
2|1|34 2|4|13
Associativity● (ab)c = a(bc)● m( m(a,b) , c ) = m( a , m(b,c) )● m(m x id)(a,b,c) = m(id x m)(a,b,c)● S-U Diagonal takes one input and produces two
outputs – “comultiplication”● ( id) (X) = (id ) (X) ?
Is “coassociative?”
Not Coassociative - Example
(x1) (123) + (1x) (123) =
2|1|32|1323|1 + 1|2|32|1323|1+1|3|213|23|12 + 1|2|313|23|12+12|32|133|2|1 + 12|32|132|3|1+12|32|1|323|1 + 12|32|3|123|1+1|2313|23|2|1 + 1|2313|23|2|1+1|231|3|23|12 + 1|233|1|23|12
(mod 2)
This measures the error from being coassociative.
Homotopy Coassociativity● Let Vi be the Z2-vector space whose basis is the i
dimensional faces of Pn
● Let : Vi Vi-1 be the boundary operation● Let H : Vi (V* V* V*) i+1 such that
H+H=(id) + (id)● H acts multiplicatively with respect to bar H(13|24) = H(13) | H(24)
Homotopy Function● H(1) = 0● H(12) = 0● H(123) = 12|3x2|13x23|1 + 1|23x13|2x3|12 ● H(1234) = ?
Calculating H(1234)● H = (id) + (id) + H● H(1234) = [(id) + (id) + H](1234)● X = H(1234) є (V* V* V*)4
● (X) = [(id) + (id) + H](1234)
(V* V* V*)4
(V* V* V*)3
Calculating H(1234)● H = (id) + (id) + H● H(1234) = [(id) + (id) + H](1234)● X = H(1234) є (V* V* V*)4
● (X) = [(id) + (id) + H](1234)
120,960 x 73,729
Calculating H(1234)● H = (id) + (id) + H● H(1234) = [(id) + (id) + H](1234)● X = H(1234) є (V* V* V*)4
● (X) = [(id) + (id) + H](1234)(V* V* V*)4
(V* V* V*)3
One Solution for H(1234)H(1234) = 12|34x24|13x4|2|3|1 + 12|34x24|13x4|3|2|1 + 123|4x3|24|1x34|2|1 + 123|4x3|2|14x34|2|1 + 123|4x3|2|14x3|24|1 + 124|3x4|2|13x4|23|1 + 12|34x24|1|3x4|2|13 + 12|34x24|3|1x4|23|1 + 12|34x2|14|3x24|3|1 + 12|34x2|14|3x2|4|13 + 12|34x2|14|3x4|23|1 + 12|34x2|4|13x4|23|1 + 13|24x34|1|2x4|3|12 + 13|24x3|14|2x34|2|1 + 13|24x3|14|2x3|4|12 + 14|23x4|13|2x4|3|12 + 1|234x14|3|2x4|13|2 + 1|234x14|3|2x4|3|12 + 1|234x4|13|2x4|3|12 + 23|14x3|24|1x34|2|1 + 2|134x24|3|1x4|23|1 + 12|34x2|1|4|3x24|13 + 12|34x2|4|1|3x24|13 + 13|24x3|1|4|2x34|12 + 13|24x3|4|1|2x34|12 + 12|3|4x23|14x34|2|1 + 12|3|4x23|14x3|24|1 + 12|3|4x2|134x24|3|1 + 12|3|4x2|134x4|23|1 + 12|4|3x24|13x4|23|1 + 13|2|4x3|124x34|2|1 + 1|23|4x134|2x4|3|12 + 1|23|4x13|24x34|1|2 + 1|23|4x13|24x3|14|2 + 1|23|4x13|24x4|3|12 + 1|23|4x3|124x34|2|1 + 1|24|3x14|23x4|13|2 + 1|24|3x14|23x4|3|12 + 1|2|34x124|3x4|23|1 + 1|2|34x124|3x4|2|13 + 1|2|34x14|23x4|13|2 + 1|2|34x14|23x4|3|12 + 1|2|34x24|13x4|23|1 + 1|3|24x134|2x4|3|12 + 2|13|4x23|14x34|2|1 + 2|13|4x23|14x3|24|1 + 2|14|3x24|13x4|23|1 + 12|3|4x2|13|4x234|1 + 12|3|4x2|13|4x23|14 + 12|3|4x2|1|34x24|13 + 12|3|4x2|3|14x234|1 + 12|4|3x2|14|3x24|13 + 13|2|4x3|1|24x34|12 + 13|4|2x3|14|2x34|12 + 1|23|4x13|2|4x34|12 + 1|23|4x13|2|4x3|124 + 1|23|4x1|3|24x134|2 + 1|23|4x3|14|2x34|12 + 1|23|4x3|1|24x34|12 + 1|24|3x14|2|3x4|123 + 1|2|34x14|2|3x4|123 + 1|2|34x1|24|3x14|23 + 1|2|34x1|24|3x4|123 + 1|2|34x2|14|3x24|13 + 1|3|24x3|14|2x34|12 + 2|13|4x2|3|14x234|1
Future WorkFuture Work
Finding an H with minimal number of termsFinding an H with minimal number of terms Modifying / Parallelizing the row reduction Modifying / Parallelizing the row reduction
algorithm to calculate H for n > 4algorithm to calculate H for n > 4
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