\\ Left Cell Data for B4 (CC 2012, B. Binegar) \\ \\ This file contains a table of data for the 50 left cells of the Weyl \\ group W of B4. 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 : 50 : 0 : [0, 0] : phi[[4],[]] : phi[[4],[]] : [9] : 0 : 0 : e : 0 : [{{}}, {}, {}, {}, {}] 1 : 50 : 1 : [1, 0] : phi[[3, 1],[]]+phi[[3],[1]] : phi[[3],[1]] : [7, 1, 1] : 1 : 0 : 1 : 1 : [{{1}}, {{2}}, {{1}, {3}}, {{2}, {4}}, {{1}, {3}}] 2 : 50 : 2 : [2, 0] : phi[[3, 1],[]]+phi[[3],[1]] : phi[[3],[1]] : [7, 1, 1] : 2 : 0 : 2 : 1 : [{{2}}, {{1}, {3}}, {{2}, {4}}, {{1}, {3}}, {{2}, {4}}] 3 : 50 : 3 : [3, 0] : phi[[3, 1],[]]+phi[[3],[1]] : phi[[3],[1]] : [7, 1, 1] : 3 : 0 : 3 : 1 : [{{3}}, {{2}, {4}}, {{1}, {3}}, {{2}, {4}}, {{1}, {3}}] 4 : 50 : 4 : [4, 0] : phi[[3],[1]]+phi[[],[4]] : phi[[3],[1]] : [7, 1, 1] : 4 : 0 : 4 : 1 : [{{4}}, {{3}}, {{2}, {4}}, {{1}, {3}}, {{2}, {4}}] 5 : 50 : 8 : [5, 0] : phi[[2, 2],[]]+phi[[2],[2]] : phi[[2],[2]] : [5, 3, 1] : 8 : 0 : 31 : 2 : [{{1, 3}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}] 6 : 50 : 11 : [6, 0] : phi[[2],[2]]+phi[[1],[3]] : phi[[2],[2]] : [5, 3, 1] : 11 : 0 : 41 : 2 : [{{1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}] 7 : 50 : 12 : [7, 0] : phi[[2],[2]]+phi[[1],[3]] : phi[[2],[2]] : [5, 3, 1] : 12 : 0 : 42 : 2 : [{{2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}] 8 : 50 : 32 : [8, 1] : phi[[2, 2],[]]+phi[[2],[2]] : phi[[2],[2]] : [5, 3, 1] : 32 : 1 : 2312 : 2 : [{{2}}, {{1, 3}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}] 9 : 50 : 15 : [9, 0] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 15 : 0 : 212 : 3 : [{{1, 2}}, {{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}] 10 : 50 : 41 : [10, 1] : phi[[2],[2]]+phi[[1],[3]] : phi[[2],[2]] : [5, 3, 1] : 41 : 1 : 3423 : 2 : [{{3}}, {{4}, {2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}, {{2}, {3}, {1, 4}}] 11 : 50 : 20 : [11, 0] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 20 : 0 : 323 : 3 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}] 12 : 50 : 61 : [12, 2] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 61 : 2 : 32123 : 3 : [{{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}] 13 : 50 : 121 : [13, 4] : phi[[2],[2]]+phi[[1],[3]] : phi[[2],[2]] : [5, 3, 1] : 121 : 4 : 434234 : 2 : [{{4}}, {{3}}, {{4}, {2, 4}}, {{2}, {3}, {1, 4}}, {{4}, {1, 3}, {2, 4}}] 14 : 50 : 80 : [14, 2] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 80 : 2 : 43234 : 3 : [{{2, 4}}, {{2}, {3}, {1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}] 15 : 50 : 155 : [15, 5] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 155 : 5 : 4321234 : 3 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}] 16 : 50 : 299 : [16, 7] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 299 : 7 : 23432123432 : 3 : [{{2}}, {{1, 3}}, {{2}, {3}, {1, 4}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}] 17 : 50 : 45 : [17, 0] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 45 : 0 : 4212 : 4 : [{{1, 2, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}] 18 : 50 : 147 : [18, 4] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 147 : 4 : 3432343 : 3 : [{{3}}, {{2, 4}}, {{2}, {3}, {1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}] 19 : 50 : 53 : [19, 0] : phi[[2],[1, 1]]+phi[[],[3, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 53 : 0 : 4343 : 4 : [{{3, 4}}, {{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 20 : 50 : 231 : [20, 6] : phi[[2, 1],[1]] : phi[[2, 1],[1]] : [5, 2, 2] : 231 : 6 : 343212343 : 3 : [{{1, 3}}, {{2}, {3}, {1, 4}}, {{1, 3}, {2, 4}}, {{2}, {3}, {1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}] 21 : 50 : 99 : [21, 1] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 99 : 1 : 342123 : 4 : [{{1, 3}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}] 22 : 50 : 192 : [22, 3] : phi[[2],[1, 1]]+phi[[],[3, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 192 : 3 : 41234321 : 4 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 23 : 50 : 205 : [23, 3] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 205 : 3 : 43421234 : 4 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}] 24 : 50 : 338 : [24, 5] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 338 : 5 : 343234312343 : 4 : [{{3}}, {{2, 4}}, {{3}, {1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}] 25 : 50 : 84 : [25, 0] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 84 : 0 : 43431 : 5 : [{{1, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}] 26 : 50 : 112 : [26, 1] : phi[[2],[1, 1]]+phi[[],[3, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 112 : 1 : 423432 : 4 : [{{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}] 27 : 50 : 178 : [27, 2] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 178 : 2 : 32343123 : 4 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}] 28 : 50 : 330 : [28, 8] : phi[[2, 1, 1],[]]+phi[[2],[1, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 330 : 8 : 234321234321 : 4 : [{{1, 2}}, {{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 29 : 50 : 271 : [29, 7] : phi[[2, 1, 1],[]]+phi[[2],[1, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 271 : 7 : 3431234321 : 4 : [{{1, 3}}, {{1, 4}, {1, 2}, {2, 3}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}] 30 : 50 : 303 : [30, 5] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 303 : 5 : 34231234321 : 5 : [{{1, 3}}, {{1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}] 31 : 50 : 94 : [31, 0] : phi[[1, 1, 1],[1]]+phi[[1, 1],[1, 1]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 94 : 0 : 323123 : 6 : [{{1, 2, 3}}, {{2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}] 32 : 50 : 191 : [32, 5] : phi[[2, 1, 1],[]]+phi[[2],[1, 1]] : phi[[2],[1, 1]] : [5, 1, 1, 1, 1] : 191 : 5 : 34323432 : 4 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {1, 2}, {2, 3}, {3, 4}}] 33 : 50 : 284 : [33, 4] : phi[[1, 1],[2]] : phi[[1, 1],[2]] : [3, 3, 3] : 284 : 4 : 4323431234 : 4 : [{{2, 4}}, {{3}, {1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{3}, {1, 4}, {2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}}] 34 : 50 : 152 : [34, 1] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 152 : 1 : 4234312 : 5 : [{{2, 4}}, {{2, 3}, {1, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}] 35 : 50 : 236 : [35, 3] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 236 : 3 : 421234321 : 5 : [{{1, 2, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}] 36 : 50 : 363 : [36, 7] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 363 : 7 : 4342341234321 : 5 : [{{1, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}] 37 : 50 : 200 : [37, 1] : phi[[1, 1, 1],[1]]+phi[[1, 1],[1, 1]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 200 : 1 : 43231234 : 6 : [{{1, 2, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 38 : 50 : 228 : [38, 2] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 228 : 2 : 342343123 : 5 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}] 39 : 50 : 273 : [39, 2] : phi[[1, 1, 1],[1]]+phi[[1, 1],[1, 1]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 273 : 2 : 3432312343 : 6 : [{{1, 3}}, {{2, 3}, {1, 2, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}] 40 : 50 : 322 : [40, 5] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 322 : 5 : 43423431234 : 5 : [{{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}}] 41 : 50 : 332 : [41, 4] : phi[[1, 1, 1],[1]]+phi[[1, 1],[1, 1]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 332 : 4 : 323432123432 : 6 : [{{2, 3}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 42 : 50 : 368 : [42, 7] : phi[[1],[2, 1]] : phi[[1],[2, 1]] : [3, 3, 1, 1, 1] : 368 : 7 : 4343234312343 : 5 : [{{3, 4}}, {{2, 4}}, {{1, 4}, {2, 3}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}, {3, 4}, {1, 2, 4}, {1, 3, 4}}] 43 : 50 : 351 : [43, 6] : phi[[1, 1],[1, 1]]+phi[[],[2, 2]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 351 : 6 : 434323412343 : 6 : [{{1, 3, 4}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}] 44 : 50 : 375 : [44, 7] : phi[[1, 1],[1, 1]]+phi[[],[2, 2]] : phi[[1, 1],[1, 1]] : [3, 2, 2, 1, 1] : 375 : 7 : 43423432123432 : 6 : [{{2, 4}}, {{2, 3}, {1, 3, 4}}, {{1, 3}, {2, 4}}, {{2, 3}, {1, 2, 4}, {1, 3, 4}}, {{1, 3}, {2, 4}, {1, 2, 3}}] 45 : 50 : 258 : [45, 0] : phi[[1],[1, 1, 1]]+phi[[],[2, 1, 1]] : phi[[1],[1, 1, 1]] : [3, 1, 1, 1, 1, 1, 1] : 258 : 0 : 434323432 : 9 : [{{2, 3, 4}}, {{1, 3, 4}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}] 46 : 50 : 324 : [46, 1] : phi[[1],[1, 1, 1]]+phi[[],[2, 1, 1]] : phi[[1],[1, 1, 1]] : [3, 1, 1, 1, 1, 1, 1] : 324 : 1 : 43431234321 : 9 : [{{1, 3, 4}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}] 47 : 50 : 360 : [47, 2] : phi[[1],[1, 1, 1]]+phi[[],[2, 1, 1]] : phi[[1],[1, 1, 1]] : [3, 1, 1, 1, 1, 1, 1] : 360 : 2 : 4234321234321 : 9 : [{{1, 2, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}] 48 : 50 : 354 : [48, 1] : phi[[1, 1, 1, 1],[]]+phi[[1],[1, 1, 1]] : phi[[1],[1, 1, 1]] : [3, 1, 1, 1, 1, 1, 1] : 354 : 1 : 3234321234321 : 9 : [{{1, 2, 3}}, {{1, 2, 4}}, {{1, 3, 4}, {1, 2, 3}}, {{1, 2, 4}, {2, 3, 4}}, {{1, 3, 4}, {1, 2, 3}}] 49 : 50 : 383 : [49, 0] : phi[[],[1, 1, 1, 1]] : phi[[],[1, 1, 1, 1]] : [1, 1, 1, 1, 1, 1, 1, 1, 1] : 383 : 0 : 4343234321234321 : 16 : [{{1, 2, 3, 4}}, {}, {}, {}, {}]