Representations Associated to the Group Matrix
For a finite group G = {g_0 = 1, g_1,. . ., g_{n-1}} , we can associate independent variables x_0, x_1, . . ., x_{n-1} where x_i = x_{g_i}. There is a natural action of Aut(G) on C[x_0, . . . ,x_{n-})]. Let C_1, . . . , C_r be the conjugacy classes of G. If C = {g_{i_1}, g_{i_2}, . . . , g_{i_u }} i...
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| Format: | Others |
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BYU ScholarsArchive
2014
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| Online Access: | https://scholarsarchive.byu.edu/etd/3902 https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=4901&context=etd |
| Summary: | For a finite group G = {g_0 = 1, g_1,. . ., g_{n-1}} , we can associate independent variables x_0, x_1, . . ., x_{n-1} where x_i = x_{g_i}. There is a natural action of Aut(G) on C[x_0, . . . ,x_{n-})]. Let C_1, . . . , C_r be the conjugacy classes of G. If C = {g_{i_1}, g_{i_2}, . . . , g_{i_u }} is a conjugacy class, then let x(C) = x_{i_1} + x_{i_2} + . . . + x_{i_u}. Let ρG be the representation of Aut(G) on C[x_0, . . . , x_(n-1)]/〈x(C_1), . . . , x(C_r) 〉 and let Χ_G be the character afforded by ρ_G. If G is a dihedral group of the form D_2p, D_4p or D_{2p^2}, with p an odd prime, I show how Χ_G splits into irreducible constituents. I also show how the module C[x_0, . . . ,x_{n-1}]/ decomposes into irreducible submodules. This problem is motivated by results of Humphries [2] relating to random walks on groups and the group determinant. |
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