ice models and classical groups - university of minnesota
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Ice Models and Classical Groups
Yulia Alexandr1 Patricia Commins2 Alexandra Embry3 Sylvia Frank4
Yutong Li5 Alexander Vetter6
1Wesleyan University
2Carleton College
5Indiana University
4Amherst College
5Haverford College
6Villanova University
July 30th, 2018
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1 GL(n) Case
2 SO(2n+1) CaseSundaram TableauxKoike-Terada TableauProctor Tableaux
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GL(n) Case: Tableaux
Let λ = (λ1, λ2, · · · , λn)
Our alphabet is (1, 2, · · · , n)
To form a Young tableau, fill thetableaux with elements of thealphabet such that:
1 Weakly increasing along rows2 Strictly increasing along
columns
λ1 x x · · · x x
λ2 x x · · · x...
...
λn x x
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GL(n) case: Tableaux Example
Let λ = (4, 2, 1).Our tableau will be of the shape: A possible filling:
1 1 2 3
2 3
3
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Gelfand-Tsetlin Patterns
GL(n) ↓ GL(n − 1)
Gelfand-Tsetlin pattern rules:
1 Rows weakly decreasing
2 Interleaving
1 1 1 3 3
2 2 2
3 3
5 3 23 3
3
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Gelfand-Tsetlin Patterns
Gelfand-Tsetlin pattern rules:
1 Rows weakly decreasing
2 Interleaving
1 1 15 3 2
3 33
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Gelfand-Tsetlin Patterns
Gelfand-Tsetlin pattern rules:
1 Rows weakly decreasing
2 Interleaving
1 1 1
2 2 2
5 3 23 3
3
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Gelfand-Tsetlin Patterns
Gelfand-Tsetlin pattern rules:
1 Rows weakly decreasing
2 Interleaving
1 1 1 3 3
2 2 2
3 3
5 3 23 3
3
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Strict Gelfand-Tsetlin Patterns
ρ = (n − 1, ..., 0)
5 3 23 3
3
+ ρ = (2, 1, 0) −→+ ρ = (1, 0) −→+ ρ = (0) −→
7 4 24 3
3
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Tokuyama’s Formula
∑T∈SGT (λ+ρ)
(1 + t)S(T )tL(T )zwt(T ) =∏i<j
(zi + tzj)sλ(z1, ...zn)
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GL(n) Shifted Tableaux
7 4 24 3
3
1 1 1 2 3 3 3
2 2 2 3
3 3
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GL(n) Ice Models: Boundary Conditions
3
2
1
7 6 5 4 3 2 1 0
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GL(n) Ice Models: Gelfand-Tsetlin Pattern 1
3
2
1
7 6 5 4 3 2 1 0
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GL(n) Ice Models: Gelfand-Tsetlin Pattern 2
3
2
1
7 6 5 4 3 2 1 0
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GL(n) Ice Models: Gelfand-Tsetlin Pattern 3
3
2
1
7 6 5 4 3 2 1 0
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GL(n) Ice Models: Gelfand-Tsetlin Pattern 4
3
2
1
7 6 5 4 3 2 1 0
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GL(n) Ice Models: Gelfand-Tsetlin Pattern 5
3
2
1
7 6 5 4 3 2 1 0
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Boltzmann Weights
j j j j j j
1 zi ti zi zi (ti + 1) 1
Figure: Boltzmann weights for Gamma Ice
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Introduction
tableauxGroup
Gelfand-Tsetlin-type
patterns
shiftedtableaux
strictGT-typepatterns
ice models
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Branching Rule for Sundaram Tableaux
ssoλ =∑µ⊆λ
ssp(µ)
Sp(2n) ↓ Sp(2n − 2)⊗ U(1)
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Sundaram Tableaux
Partition: λ = (λ1 ≥ λ2 ≥ . . . λn ≥ 1)Alphabet: {1 < 1 < · · · < n < n < 0}
1 Rows are weakly increasing.
2 Columns are strictly increasing, but 0s do not violate this condition.
3 No row contains multiple 0s.
4 In row i, all entries are greater than or equal to i.
1 1 0
2 0
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Sundaram Gelfand-Tsetlin-type patterns
Gelfand-Tsetlin-type pattern rules:
1 Rows weakly decreasing
2 Interleaving
3 Difference between top rows ≤ 1
4 Even rows cannot end in 0
1 1 0
2 0
3 2 03 1
2 12
1
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Sundaram Strict Gelfand-Tsetlin-Type Patterns
3 2 03 1
2 12
1
add ρ −→5 3 0
5 24 2
32
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Sundaram Shifted Tableaux
5 3 04 3
3 22
1
1 1 2 2 0
2 2 2
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Sundaram Ice Models: Boundary Conditions
2
1
1
2
0
3 2 1
Figure:
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Sundaram Ice Models: Modeling GT-Type Pattern
2
1
1
2
0
3 2 1
3 2 03 1
2 12
1
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Sundaram Ice Models: Full Model
2
1
1
2
0
3 2 1
3 2 03 1
2 12
1
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Sundaram Boltzmann Weights
j j j j j j
∆ IceEven rows
1 tzi 1 zi zi (t + 1) 1
j j j j j j
Γ IceOdd rows
1 z−1i
t z−1i
z−1i
(t + 1) 1
Figure: Boltzmann weights for ∆ and Γ Sundaram Ice
z−1i
t
Figure: Boltzmann Weights for Sundaram Bends
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Sundaram Botlzmann Weights
Alternate Bend Weights for B Deformation:
z−1i ·
(z−n+i−1i
(1 + tzi )
(1 + tz2i )
)t ·
(z−n+i−1i
(1 + tzi )
(1 + tz2i )
)
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Branching Rule for Koike-Terada Tableaux
SO(2n + 1) ↓ SO(2n − 1)⊗ GL(1)
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Koike-Terada Tableaux
Partition: λ = (λn ≥ λn−1 ≥ · · · ≥ λ1 ≥ 0)Alphabet: {1 < 1 < 1 < 2 < 2 < 2 · · · n < n < n}.Let Ti,j be the entry of the tableau in the i-th row and the j-th column. Then:
1 Rows are weakly increasing
2 Columns are strictly increasing
3 k can only appear in Tk,1
4 Ti,j ≥ i
1 1 2 2 2
2 2 2
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Koike-Terada Gelfand-Tsetlin-type pattern
1 The pattern has 3n rows. Label these rows 1, 1, 1, · · · , n, n, n, starting fromthe bottom of the pattern. Rows i , i , and i must have i entries.
2 ai,j−1 ≥ ai,j ≥ ai,j+1 ≥ 0
3 ai−1,j ≥ ai,j ≥ ai−1,j+1
4 Row i must end in a 1 or a 0 (for i ∈ {1, · · · , n})5 Each entry in row i (for i ∈ {1, · · · , n}) must be left-leaning.
5 3 2
4 2 22 1 2
2 1
1 10 1
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Koike-Terada Shifted Tableaux
1 Rows are weakly increasing.
2 Columns are weakly increasing.
3 Diagonals are strictly increasing.
4 The first entry in row k is k , k or k .
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Koike-Terada Ice: Bends and Ties
To connect rows k and k for each k ∈ {1, · · · , n}:
A B C
For rows k, where k k ∈ {1, · · · , n}, there are 3 possible ”ties”:
U D O
Note: Along with ties U, D and O, rows k k ∈ {1, · · · , n} are three-vertexmodels, the vertices being SW, NW, and NE.
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Koike-Terada Ice: Boundary Conditions
The following depicts the boundaryconditions for an ice model with toprow λ = (2, 1).
1
1
1
2
2
2
2 1
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Koike-Terada Ice: Full Model
A
D
C
O
2 1
2 12 1
2 02
21
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Theorem (1)
The following are equivalent:
1 Koike-Terada Gelfand-Tsetlin-type pattern rules 4 and 5 are satisfied.
2 Each ice row labeled k ∈ {1, · · · , n} has no NS, SE, or EW configurations,and tie boundary conditions are satsified.
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Branching Rule for Proctor Tableaux
SO(2n + 1) ↓ SO(2n − 1)⊗ SO(2)
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Proctor Tableaux
1 Rows are weakly increasing
2 Columns are strictly increasing
3 Follows the 2c orthogonal condition
4 Follows the 2m protection condition
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Proctor Tableaux
2c Orthogonal Condition: Less than or equal to 2c entries that are less than orequal to 2c in the first two columns.
1 1 3 5
x 3 4
5
1 1 3 5
3 3 4
5
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Proctor Tableaux
2c Orthogonal Condition: Less than or equal to 2c entries that are less than orequal to 2c in the first two columns.
1 1 3 5
x 3 4
5
1 1 3 5
3 3 4
5
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Proctor Tableaux
2m Protection Condition: For a 2m-1 entry in the first column, specified 2mentries must be ”protected” by 2m-1 entries.
1 1 x 5
3 x 4
5
1 1 3 5
3 3 4
5
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Proctor Tableaux
2m Protection Condition: For a 2m-1 entry in the first column, specified 2mentries must be ”protected” by 2m-1 entries.
1 1 x 5
3 x 4
5
1 1 3 5
3 3 4
5
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Proctor Gelfand-Tsetlin-type Patterns
n=4 shape
x x x x (9)x x x x (8)
x x x x (7)x x x x (6)
x x x x (5)x x x x (n = 4)
x x x (3)x x (2)
x (1)
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Proctor Gelfand-Tsetlin-type Patterns
2c Orthogonal Condition
6 5 3 16 5 3 1 = 2 + 2 + 2 + 1 (8)
6 4 2 14 3 1 0 = 2 + 2 + 1 (6)
4 2 1 02 1 0 0 = 2 + 1 (4)
2 1 01 0 = 1 (2)
1
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Proctor Gelfand-Tsetlin-type Patterns
2m Protection Condition:Add in 0s to make all rows length n
6 5 3 26 5 3 2
6 4 2 14 3 1 0
4 2 1 02 1 0 0
2 1 0 01 0 0 0
1 0 0 00 0 0 0
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Proctor Gelfand-Tsetlin-type Patterns
2m Protection Condition:Identify non-left-leaning 0s in even rows
6 5 3 26 5 3 2
6 4 2 14 3 1 0
4 2 1 02 1 0 0
2 1 0 01 0 0 0
1 0 0 00 0 0 0
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Proctor Gelfand-Tsetlin-type Patterns
2m Protection Condition
Check Example:
For 0, if 2 > 1
Check:
2 ≥ 2
2 > 1
0 ≤ 1
6 5 3 26 5 3 2
6 4 2 14 3 1 0
4 2 1 02 1 0 0
2 1 0 01 0 0 0
1 0 0 00 0 0 0
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Proctor Strict Gelfand-Tsetlin-type Patterns
Change to be Strict and check Orthogonal Condition:
6 5 3 16 5 3 1
6 4 2 14 3 1 0
4 2 1 03 2 1 0 = 2 + 2 + 1 (4)
2 1 01 0
1
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Acknowledgements:
Prof. Ben Brubaker and Katy Weber
University of Minnesota, Twin Cities REU
NSF
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.
Questions?
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