Clones of Irreps of Exceptional Weyl Groups
A nontrivial clone of a representation rho of W is a representation
distinct from rho that restricts to the same representation of the
parabolic subgroup generated by s[1],...,s[n-1].
For the Weyl groups of F4 and E4 through E8, we list each (nontrivial)
clone of each irreducible representation. For example, the 9th irrep
of E5 has two clones, one of which is the sum of the 3rd and 16th irrep,
and the other is the sum of the 2nd, 3rd and 4th irrep.
F4
`5 has clones`, [2, 4]
`6 has clones`, [1, 3]
`13 has clones`, [7, 8]
`16 has clones`, [11, 12]
`17 has clones`, [9, 10]
`24 has clones`, [14, 15]
`25 has clones`, [18, 19], [13, 24], [13, 14, 15], [7, 8, 24], [7, 8, 14, 15]
E4
E5
`8 has clones`, [1, 2]
`9 has clones`, [3, 16], [2, 3, 4]
`10 has clones`, [3, 5]
`11 has clones`, [5, 15], [4, 5, 6]
`12 has clones`, [6, 7]
`13 has clones`, [3, 8], [1, 2, 3]
`14 has clones`, [10, 16], [5, 9], [3, 5, 16], [2, 4, 10], [2, 3, 4, 5]
`15 has clones`, [4, 6]
`16 has clones`, [2, 4]
`17 has clones`, [10, 15], [4, 6, 10], [3, 11], [3, 5, 15], [3, 4, 5, 6]
`18 has clones`, [5, 12], [5, 6, 7]
E6
E7
`59 has clones`, [60]
`60 has clones`, [59]
E8
`109 has clones`, [55, 105]
`112 has clones`, [81, 102]