representations listed above. For example, if we wanted to create a representation with one block of dimension 4, one block of dimension 3, two blocks of the other representation of dimension 3, and five blocks of dimension 1, we would get 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , where the first representation of dimension 3 is shown by the black squares and the other representation is shown by the asterisks. We could even have three blocks of dimension 5 and three blocks of dimension 1, which gives us 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . Note that our original matrix would not be conjugate to all of these irreducible representations, but these are examples of what it might be conjugate to. Hence, we see how a representation of a group can be broken into irreducible representations and how the dimensions of those irreducible representations work. The Monster has 194 conjugacy classes, and hence it has 194 irreducible representations. In 1978, Fischer, Livingstone, and Thorne determined the dimensions of these representations. Starting with the trivial representations, the dimensions of the first few irreducible representations are (rn)n=1,··· ,194 = (1, 196 883, 21 296 876, 842 609 326, 18 538 750 076, · · · ). The rest of the sequence can be found in the ATLAS of Finite Groups [3]. We will now temporally leave the Monster, but this sequence of numbers will be important in the discussion that follows. Page 34 Riley • An Overview of Monstrous Moonshine
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