Canonical Names for Irreps of the Exceptional Weyl Groups This table provides a translation between the numbering of the irreps of the exceptional Weyl groups used by the coxeter package and the coordinate-free, canonical names described in the paper "Explicit matrices for irreducible representations of Weyl groups." Each representation is labeled [N, t, e] or [N, t1, t2, e], where N = the dimension, t = the trace of a reflection (or if there are two classes of reflections, the traces are t1 and t2, and t1 is the trace of s[1]), and e = the sign of the trace of the longest element -- either +, 0, or -. This distinguishes all irreps of the exceptional groups, aside from two representations of W(F4) that both have the label [6,0,0,+]. The last column in the table reports the dimension of the 1-eigenspace of s[1], expressed as a fraction of the dimension. This is a useful parameter related to positivity issues in the spherical unitary dual. G2 1 [ 1, 1, 1, +] 1.00 2 [ 1, -1, -1, +] 0.00 3 [ 1, -1, 1, -] 0.00 4 [ 1, 1, -1, -] 1.00 5 [ 2, 0, 0, -] 0.50 6 [ 2, 0, 0, +] 0.50 F4 1 [ 1, 1, 1, +] 1.00 2 [ 1, -1, -1, +] 0.00 3 [ 1, 1, -1, +] 1.00 4 [ 1, -1, 1, +] 0.00 5 [ 2, -2, 0, +] 0.00 6 [ 2, 2, 0, +] 1.00 7 [ 2, 0, -2, +] 0.50 8 [ 2, 0, 2, +] 0.50 9 [ 4, -2, -2, -] 0.25 10 [ 4, -2, 2, -] 0.25 11 [ 4, 2, -2, -] 0.75 12 [ 4, 2, 2, -] 0.75 13 [ 4, 0, 0, +] 0.50 14 [ 6, 0, 0, +] 0.50 15 [ 6, 0, 0, +] 0.50 16 [ 8, 4, 0, -] 0.75 17 [ 8, -4, 0, -] 0.25 18 [ 8, 0, -4, -] 0.50 19 [ 8, 0, 4, -] 0.50 20 [ 9, 3, -3, +] 0.67 21 [ 9, 3, 3, +] 0.67 22 [ 9, -3, -3, +] 0.33 23 [ 9, -3, 3, +] 0.33 24 [ 12, 0, 0, +] 0.50 25 [ 16, 0, 0, -] 0.50 E5 1 [ 1, 1, +] 1.00 2 [ 4, 2, +] 0.75 3 [ 5, 1, +] 0.60 4 [ 6, 0, +] 0.50 5 [ 5, -1, +] 0.40 6 [ 4, -2, +] 0.25 7 [ 1, -1, +] 0.00 8 [ 5, 3, -] 0.80 9 [ 15, 3, -] 0.60 10 [ 10, 0, -] 0.50 11 [ 15, -3, -] 0.40 12 [ 5, -3, -] 0.20 13 [ 10, 4, +] 0.70 14 [ 20, 2, +] 0.55 15 [ 10, -2, +] 0.40 16 [ 10, 2, +] 0.60 17 [ 20, -2, +] 0.45 18 [ 10, -4, +] 0.30 E6 1 [ 1, 1, +] 1.00 2 [ 1, -1, +] 0.00 3 [ 6, -4, -] 0.17 4 [ 6, 4, -] 0.83 5 [ 10, 0, -] 0.50 6 [ 15, 5, +] 0.67 7 [ 15, -5, +] 0.33 8 [ 15, 5, -] 0.67 9 [ 15, -5, -] 0.33 10 [ 20, 10, +] 0.75 11 [ 20, -10, +] 0.25 12 [ 20, 0, +] 0.50 13 [ 24, 4, +] 0.58 14 [ 24, -4, +] 0.42 15 [ 30, 10, -] 0.67 16 [ 30, -10, -] 0.33 17 [ 60, 10, -] 0.58 18 [ 60, -10, -] 0.42 19 [ 60, 0, +] 0.50 20 [ 64, -16, O] 0.38 21 [ 64, 16, O] 0.63 22 [ 80, 0, -] 0.50 23 [ 81, -9, +] 0.44 24 [ 81, 9, +] 0.56 25 [ 90, 0, -] 0.50 E7 1 [ 1, 1, +] 1.00 2 [ 1, -1, -] 0.00 3 [ 7, -5, +] 0.14 4 [ 7, 5, -] 0.86 5 [ 15, -5, +] 0.33 6 [ 15, 5, -] 0.67 7 [ 21, -11, +] 0.24 8 [ 21, 11, -] 0.76 9 [ 21, 9, +] 0.71 10 [ 21, -9, -] 0.29 11 [ 27, 15, +] 0.78 12 [ 27, -15, -] 0.22 13 [ 35, 15, +] 0.71 14 [ 35, -15, -] 0.29 15 [ 35, -5, +] 0.43 16 [ 35, 5, -] 0.57 17 [ 56, -24, +] 0.29 18 [ 56, 24, -] 0.71 19 [ 70, -10, +] 0.43 20 [ 70, 10, -] 0.57 21 [ 84, 4, +] 0.52 22 [ 84, -4, -] 0.48 23 [ 105, -35, +] 0.33 24 [ 105, 35, -] 0.67 25 [ 105, 5, +] 0.52 26 [ 105, -5, -] 0.48 27 [ 105, 25, +] 0.62 28 [ 105, -25, -] 0.38 29 [ 120, 40, +] 0.67 30 [ 120, -40, -] 0.33 31 [ 168, 40, +] 0.62 32 [ 168, -40, -] 0.38 33 [ 189, -39, +] 0.40 34 [ 189, 39, -] 0.60 35 [ 189, -51, +] 0.37 36 [ 189, 51, -] 0.63 37 [ 189, 21, +] 0.56 38 [ 189, -21, -] 0.44 39 [ 210, 10, +] 0.52 40 [ 210, -10, -] 0.48 41 [ 210, 50, +] 0.62 42 [ 210, -50, -] 0.38 43 [ 216, -24, +] 0.44 44 [ 216, 24, -] 0.56 45 [ 280, 40, +] 0.57 46 [ 280, -40, -] 0.43 47 [ 280, -40, +] 0.43 48 [ 280, 40, -] 0.57 49 [ 315, -45, +] 0.43 50 [ 315, 45, -] 0.57 51 [ 336, -16, +] 0.48 52 [ 336, 16, -] 0.52 53 [ 378, -30, +] 0.46 54 [ 378, 30, -] 0.54 55 [ 405, 45, +] 0.56 56 [ 405, -45, -] 0.44 57 [ 420, 20, +] 0.52 58 [ 420, -20, -] 0.48 59 [ 512, 0, +] 0.50 60 [ 512, 0, -] 0.50 E8 1 [ 1, 1, +] 1.00 2 [ 1, -1, +] 0.00 3 [ 8, -6, -] 0.13 4 [ 8, 6, -] 0.88 5 [ 28, 14, +] 0.75 6 [ 28, -14, +] 0.25 7 [ 35, -21, +] 0.20 8 [ 35, 21, +] 0.80 9 [ 50, -20, +] 0.30 10 [ 50, 20, +] 0.70 11 [ 56, -14, -] 0.38 12 [ 56, 14, -] 0.63 13 [ 70, 0, +] 0.50 14 [ 84, 42, +] 0.75 15 [ 84, -42, +] 0.25 16 [ 112, -56, -] 0.25 17 [ 112, 56, -] 0.75 18 [ 160, 64, -] 0.70 19 [ 160, -64, -] 0.30 20 [ 168, 0, +] 0.50 21 [ 175, -35, +] 0.40 22 [ 175, 35, +] 0.60 23 [ 210, -84, +] 0.30 24 [ 210, 84, +] 0.70 25 [ 300, -90, +] 0.35 26 [ 300, 90, +] 0.65 27 [ 350, -70, +] 0.40 28 [ 350, 70, +] 0.60 29 [ 400, 120, -] 0.65 30 [ 400,-120, -] 0.35 31 [ 420, 0, +] 0.50 32 [ 448, 0, -] 0.50 33 [ 448,-112, -] 0.38 34 [ 448, 112, -] 0.63 35 [ 525,-105, +] 0.40 36 [ 525, 105, +] 0.60 37 [ 560, 196, -] 0.68 38 [ 560,-196, -] 0.33 39 [ 567, 189, +] 0.67 40 [ 567,-189, +] 0.33 41 [ 700,-210, +] 0.35 42 [ 700, 210, +] 0.65 43 [ 700, -70, +] 0.45 44 [ 700, 70, +] 0.55 45 [ 840, 126, -] 0.58 46 [ 840,-126, -] 0.43 47 [ 840, -84, +] 0.45 48 [ 840, 84, +] 0.55 49 [ 972, 162, +] 0.58 50 [ 972,-162, +] 0.42 51 [1008,-252, -] 0.38 52 [1008, 252, -] 0.63 53 [1050,-210, +] 0.40 54 [1050, 210, +] 0.60 55 [1134, 0, +] 0.50 56 [1296,-216, -] 0.42 57 [1296, 216, -] 0.58 58 [1344, 336, +] 0.63 59 [1344,-336, +] 0.38 60 [1344, 0, -] 0.50 61 [1400, 210, -] 0.58 62 [1400,-210, -] 0.43 63 [1400,-280, +] 0.40 64 [1400, 280, +] 0.60 65 [1400,-350, -] 0.38 66 [1400, 350, -] 0.63 67 [1400, 0, +] 0.50 68 [1575,-315, +] 0.40 69 [1575, 315, +] 0.60 70 [1680, 0, +] 0.50 71 [2016, 0, -] 0.50 72 [2100,-210, +] 0.45 73 [2100, 210, +] 0.55 74 [2100, 0, +] 0.50 75 [2240,-336, +] 0.43 76 [2240, 336, +] 0.58 77 [2268, 378, +] 0.58 78 [2268,-378, +] 0.42 79 [2400, 120, -] 0.53 80 [2400,-120, -] 0.48 81 [2688, 0, +] 0.50 82 [2800, 280, -] 0.55 83 [2800,-280, -] 0.45 84 [2835,-189, +] 0.47 85 [2835, 189, +] 0.53 86 [3150, 0, +] 0.50 87 [3200,-160, +] 0.48 88 [3200, 160, +] 0.53 89 [3240, 594, -] 0.59 90 [3240,-594, -] 0.41 91 [3360, 336, -] 0.55 92 [3360,-336, -] 0.45 93 [4096, 512, +] 0.56 94 [4096,-512, +] 0.44 95 [4096,-512, -] 0.44 96 [4096, 512, -] 0.56 97 [4200,-420, +] 0.45 98 [4200, 420, +] 0.55 99 [4200, 210, -] 0.53 100 [4200,-210, -] 0.48 101 [4200, 0, +] 0.50 102 [4480, 0, +] 0.50 103 [4536,-378, -] 0.46 104 [4536, 378, -] 0.54 105 [4536, 0, +] 0.50 106 [5600, 0, -] 0.50 107 [5600, 280, -] 0.53 108 [5600,-280, -] 0.48 109 [5670, 0, +] 0.50 110 [6075,-405, +] 0.47 111 [6075, 405, +] 0.53 112 [7168, 0, -] 0.50