Ice Models and Classical Groups Yulia Alexandr 1 Patricia Commins 2 Alexandra Embry 3 Sylvia Frank 4 Yutong Li 5 Alexander Vetter 6 1 Wesleyan University 2 Carleton College 5 Indiana University 4 Amherst College 5 Haverford College 6 Villanova University July 30th, 2018 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 1 / 48
GL(n) Case 1 SO(2n+1) Case 2 Sundaram Tableaux Koike-Terada Tableau Proctor Tableaux Alexandr et al. Ice Models and Classical Groups July 30th, 2018 2 / 48
GL(n) Case: Tableaux Let λ = ( λ 1 , λ 2 , · · · , λ n ) x x · · · x x λ 1 Our alphabet is (1 , 2 , · · · , n ) x x · · · x To form a Young tableau, fill the λ 2 tableaux with elements of the . . . . . . alphabet such that: Weakly increasing along rows 1 x x λ n Strictly increasing along 2 columns Alexandr et al. Ice Models and Classical Groups July 30th, 2018 3 / 48
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 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 4 / 48
Gelfand-Tsetlin Patterns GL ( n ) ↓ GL ( n − 1) Gelfand-Tsetlin pattern rules: Rows weakly decreasing 1 Interleaving 2 1 1 1 3 3 5 3 2 3 3 2 2 2 3 3 3 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 5 / 48
Gelfand-Tsetlin Patterns Gelfand-Tsetlin pattern rules: Rows weakly decreasing 1 Interleaving 2 5 3 2 1 1 1 3 3 3 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 6 / 48
Gelfand-Tsetlin Patterns Gelfand-Tsetlin pattern rules: Rows weakly decreasing 1 Interleaving 2 1 1 1 5 3 2 3 3 3 2 2 2 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 7 / 48
Gelfand-Tsetlin Patterns Gelfand-Tsetlin pattern rules: Rows weakly decreasing 1 Interleaving 2 1 1 1 3 3 5 3 2 3 3 2 2 2 3 3 3 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 8 / 48
Strict Gelfand-Tsetlin Patterns ρ = ( n − 1 , ..., 0) + ρ = (2 , 1 , 0) − → 5 3 2 7 4 2 + ρ = (1 , 0) − → 3 3 4 3 + ρ = (0) − → 3 3 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 9 / 48
Tokuyama’s Formula (1 + t ) S ( T ) t L ( T ) z wt ( T ) = � � ( z i + tz j ) s λ ( z 1 , ... z n ) T ∈ SGT ( λ + ρ ) i < j Alexandr et al. Ice Models and Classical Groups July 30th, 2018 10 / 48
GL ( n ) Shifted Tableaux 7 4 2 1 1 1 2 3 3 3 4 3 2 2 2 3 3 3 3 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 11 / 48
GL ( n ) Ice Models: Boundary Conditions 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 12 / 48
GL ( n ) Ice Models: Gelfand-Tsetlin Pattern 1 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 13 / 48
GL ( n ) Ice Models: Gelfand-Tsetlin Pattern 2 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 14 / 48
GL ( n ) Ice Models: Gelfand-Tsetlin Pattern 3 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 15 / 48
GL ( n ) Ice Models: Gelfand-Tsetlin Pattern 4 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 16 / 48
GL ( n ) Ice Models: Gelfand-Tsetlin Pattern 5 7 6 5 4 3 2 1 0 3 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 17 / 48
Boltzmann Weights j j j j j j 1 zi ( ti + 1) 1 zi ti zi Figure: Boltzmann weights for Gamma Ice Alexandr et al. Ice Models and Classical Groups July 30th, 2018 18 / 48
Introduction Group tableaux Gelfand- shifted Tsetlin-type tableaux patterns strict GT-type ice models patterns Alexandr et al. Ice Models and Classical Groups July 30th, 2018 19 / 48
Branching Rule for Sundaram Tableaux � s sp s so λ = ( µ ) µ ⊆ λ Sp (2 n ) ↓ Sp (2 n − 2) ⊗ U (1) Alexandr et al. Ice Models and Classical Groups July 30th, 2018 20 / 48
Sundaram Tableaux Partition: λ = ( λ 1 ≥ λ 2 ≥ . . . λ n ≥ 1) Alphabet: { 1 < ¯ 1 < · · · < n < ¯ n < 0 } Rows are weakly increasing. 1 Columns are strictly increasing, but 0s do not violate this condition. 2 No row contains multiple 0s. 3 In row i, all entries are greater than or equal to i. 4 1 0 1 2 0 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 21 / 48
Sundaram Gelfand-Tsetlin-type patterns Gelfand-Tsetlin-type pattern rules: Rows weakly decreasing 1 Interleaving 2 Difference between top rows ≤ 1 3 Even rows cannot end in 0 4 3 2 0 1 0 1 3 1 2 1 2 0 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 22 / 48
Sundaram Strict Gelfand-Tsetlin-Type Patterns 3 2 0 5 3 0 3 1 add ρ − → 5 2 2 1 4 2 2 3 1 2 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 23 / 48
Sundaram Shifted Tableaux 5 3 0 ¯ ¯ 1 1 2 2 0 4 3 3 2 ¯ 2 2 2 2 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 24 / 48
Sundaram Ice Models: Boundary Conditions 3 2 1 0 ¯ 2 2 ¯ 1 1 Figure: Alexandr et al. Ice Models and Classical Groups July 30th, 2018 25 / 48
Sundaram Ice Models: Modeling GT-Type Pattern 3 2 1 0 3 2 0 ¯ 2 3 1 2 1 2 2 1 ¯ 1 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 26 / 48
Sundaram Ice Models: Full Model 3 2 1 0 ¯ 2 3 2 0 3 1 2 2 1 2 ¯ 1 1 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 27 / 48
Sundaram Boltzmann Weights j j j j j j ∆ Ice Even rows 1 tzi 1 zi zi ( t + 1) 1 j j j j j j Γ Ice Odd rows z − 1 z − 1 z − 1 1 t ( t + 1) 1 i i i Figure: Boltzmann weights for ∆ and Γ Sundaram Ice z − 1 t i Figure: Boltzmann Weights for Sundaram Bends Alexandr et al. Ice Models and Classical Groups July 30th, 2018 28 / 48
Sundaram Botlzmann Weights Alternate Bend Weights for B Deformation: � � � � (1 + tz i ) (1 + tz i ) z − 1 z − n + i − 1 z − n + i − 1 · t · i i i (1 + tz 2 (1 + tz 2 i ) i ) Alexandr et al. Ice Models and Classical Groups July 30th, 2018 29 / 48
Branching Rule for Koike-Terada Tableaux SO (2 n + 1) ↓ SO (2 n − 1) ⊗ GL (1) Alexandr et al. Ice Models and Classical Groups July 30th, 2018 30 / 48
Koike-Terada Tableaux Partition: λ = ( λ n ≥ λ n − 1 ≥ · · · ≥ λ 1 ≥ 0) Alphabet: { 1 < ¯ 1 < 1 < 2 < ¯ 2 < 2 · · · n < ¯ n < n } . Let T i , j be the entry of the tableau in the i -th row and the j -th column. Then: Rows are weakly increasing 1 Columns are strictly increasing 2 k can only appear in T k , 1 3 T i , j ≥ i 4 1 2 2 1 2 2 2 2 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 31 / 48
Koike-Terada Gelfand-Tsetlin-type pattern The pattern has 3 n rows. Label these rows 1 , ¯ 1 , 1 , · · · , n , ¯ n , n , starting from 1 the bottom of the pattern. Rows i , ¯ i , and i must have i entries. a i , j − 1 ≥ a i , j ≥ a i , j +1 ≥ 0 2 a i − 1 , j ≥ a i , j ≥ a i − 1 , j +1 3 Row i must end in a 1 or a 0 (for i ∈ { 1 , · · · , n } ) 4 Each entry in row i (for i ∈ { 1 , · · · , n } ) must be left-leaning. 5 5 3 2 ¯ 4 2 2 2 1 2 2 1 ¯ 1 1 0 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 32 / 48
Koike-Terada Shifted Tableaux Rows are weakly increasing. 1 Columns are weakly increasing. 2 Diagonals are strictly increasing. 3 The first entry in row k is k , k or k . 4 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 33 / 48
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-vertex models, the vertices being SW, NW, and NE. Alexandr et al. Ice Models and Classical Groups July 30th, 2018 34 / 48
Koike-Terada Ice: Boundary Conditions 2 1 2 ¯ 2 The following depicts the boundary 2 conditions for an ice model with top row λ = (2 , 1). 1 ¯ 1 1 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 35 / 48
Koike-Terada Ice: Full Model 2 1 A 2 1 D 2 1 2 0 2 2 C 1 O Alexandr et al. Ice Models and Classical Groups July 30th, 2018 36 / 48
Theorem (1) The following are equivalent: Koike-Terada Gelfand-Tsetlin-type pattern rules 4 and 5 are satisfied. 1 Each ice row labeled k ∈ { 1 , · · · , n } has no NS, SE, or EW configurations, 2 and tie boundary conditions are satsified. Alexandr et al. Ice Models and Classical Groups July 30th, 2018 37 / 48
Branching Rule for Proctor Tableaux SO (2 n + 1) ↓ SO (2 n − 1) ⊗ SO (2) Alexandr et al. Ice Models and Classical Groups July 30th, 2018 38 / 48
Proctor Tableaux Rows are weakly increasing 1 Columns are strictly increasing 2 Follows the 2c orthogonal condition 3 Follows the 2 m protection condition 4 Alexandr et al. Ice Models and Classical Groups July 30th, 2018 39 / 48
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