\\ Left Cell Data for A4 (CC 2012, B. Binegar) \\ \\ This file contains a table of data for the 26 left cells of the Weyl \\ group W of A4. Each line below corresponds to a particular left cell \\ of W. The datums held in the (colon-delimited) columns are as follows: \\ Column Contents \\ 1 KLatlas index (order of discovery by KLatlas, counting from 0) \\ 2 number of Weyl group elements in the left cell \\ 3 Weyl group elements contained in the left cell (as enumerated by KLatlas, counting from 0) \\ 4 KLatlas indices of the left cells immediately below the left cell in the W-graph of W \\ 5 representation of W carried by left cell (in terms of partitions for the classical Weyl groups and \\ or R. Carter's notation for the irreducible representations of the exceptional Weyl groups) \\ 6 special representation of W attached to the cell \\ 7 special nilpotent orbit attached to the left cell (in terms of partitions for the nilpotent orbits of \\ classical groups or Bala-Carter notation for the nilpotent orbits of the exceptional groups \\ 8 KLatlas block index of the unique Duflo involution contained in the left cell \\ 9 KLatlas cell index of the unique Duflo involution contained in the left cell \\ 10 reduced word expression for the unique Duflo involution contained in the left cell \\ 11 A-value of the unique Duflo involution contained in the left cell \\ 12 tau-infinity invariant of primitive ideal attached to the left cell \\ 0 : 26 : 0 : [0, 0] : phi[5] : phi[5] : [5] : 0 : 0 : e : 0 : [{{}}, {}, {}] 1 : 26 : 1 : [1, 0] : phi[4,1] : phi[4,1] : [4, 1] : 1 : 0 : 4 : 1 : [{{4}}, {{3}}, {{2}, {4}}] 2 : 26 : 2 : [2, 0] : phi[4,1] : phi[4,1] : [4, 1] : 2 : 0 : 3 : 1 : [{{3}}, {{2}, {4}}, {{1}, {3}}] 3 : 26 : 3 : [3, 0] : phi[4,1] : phi[4,1] : [4, 1] : 3 : 0 : 2 : 1 : [{{2}}, {{1}, {3}}, {{2}, {4}}] 4 : 26 : 4 : [4, 0] : phi[4,1] : phi[4,1] : [4, 1] : 4 : 0 : 1 : 1 : [{{1}}, {{2}}, {{1}, {3}}] 5 : 26 : 8 : [5, 0] : phi[3,2] : phi[3,2] : [3, 2] : 8 : 0 : 24 : 2 : [{{2, 4}}, {{2}, {3}, {1, 4}}, {{1, 3}, {2, 4}}] 6 : 26 : 11 : [6, 0] : phi[3,2] : phi[3,2] : [3, 2] : 11 : 0 : 14 : 2 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}] 7 : 26 : 12 : [7, 0] : phi[3,2] : phi[3,2] : [3, 2] : 12 : 0 : 13 : 2 : [{{1, 3}}, {{2}, {3}, {1, 4}}, {{1, 3}, {2, 4}}] 8 : 26 : 31 : [8, 1] : phi[3,2] : phi[3,2] : [3, 2] : 31 : 1 : 3243 : 2 : [{{3}}, {{2, 4}}, {{2}, {3}, {1, 4}}] 9 : 26 : 15 : [9, 0] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 15 : 0 : 343 : 3 : [{{3, 4}}, {{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}] 10 : 26 : 39 : [10, 1] : phi[3,2] : phi[3,2] : [3, 2] : 39 : 1 : 2132 : 2 : [{{2}}, {{1, 3}}, {{2}, {3}, {1, 4}}] 11 : 26 : 20 : [11, 0] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 20 : 0 : 232 : 3 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 12 : 26 : 28 : [12, 0] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 28 : 0 : 121 : 3 : [{{1, 2}}, {{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}] 13 : 26 : 53 : [13, 2] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 53 : 2 : 23432 : 3 : [{{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}] 14 : 26 : 67 : [14, 3] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 67 : 3 : 12321 : 3 : [{{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}] 15 : 26 : 99 : [15, 5] : phi[3,1,1] : phi[3,1,1] : [3, 1, 1] : 99 : 5 : 1234321 : 3 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 16 : 26 : 41 : [16, 0] : phi[2,2,1] : phi[2,2,1] : [2, 2, 1] : 41 : 0 : 1343 : 4 : [{{1, 3, 4}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 17 : 26 : 47 : [17, 0] : phi[2,2,1] : phi[2,2,1] : [2, 2, 1] : 47 : 0 : 1214 : 4 : [{{1, 2, 4}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 18 : 26 : 77 : [18, 1] : phi[2,2,1] : phi[2,2,1] : [2, 2, 1] : 77 : 1 : 213432 : 4 : [{{2, 4}}, {{2, 3}, {1, 3, 4}}, {{1, 3}, {2, 4}}] 19 : 26 : 82 : [19, 1] : phi[2,2,1] : phi[2,2,1] : [2, 2, 1] : 82 : 1 : 132143 : 4 : [{{1, 3}}, {{2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}] 20 : 26 : 108 : [20, 4] : phi[2,2,1] : phi[2,2,1] : [2, 2, 1] : 108 : 4 : 21321432 : 4 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 21 : 26 : 74 : [21, 0] : phi[2,1,1,1] : phi[2,1,1,1] : [2, 1, 1, 1] : 74 : 0 : 232432 : 6 : [{{2, 3, 4}}, {{1, 3, 4}}, {{1, 2, 4}, {2, 3, 4}}] 22 : 26 : 90 : [22, 0] : phi[2,1,1,1] : phi[2,1,1,1] : [2, 1, 1, 1] : 90 : 0 : 121321 : 6 : [{{1, 2, 3}}, {{1, 2, 4}}, {{1, 3, 4}, {1, 2, 3}}] 23 : 26 : 110 : [23, 1] : phi[2,1,1,1] : phi[2,1,1,1] : [2, 1, 1, 1] : 110 : 1 : 12324321 : 6 : [{{1, 3, 4}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}] 24 : 26 : 112 : [24, 2] : phi[2,1,1,1] : phi[2,1,1,1] : [2, 1, 1, 1] : 112 : 2 : 12134321 : 6 : [{{1, 2, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}] 25 : 26 : 119 : [25, 0] : phi[1,1,1,1,1] : phi[1,1,1,1,1] : [1, 1, 1, 1, 1] : 119 : 0 : 1213214321 : 10 : [{{1, 2, 3, 4}}, {}, {}]