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Linear representation theory of dihedral group:D8

110 bytes added, 23:08, 29 June 2011
Summary information
| trivial || 1 || -- || any || remains the same || whole group || [[trivial group]] || 1 || 1 || --
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| sign representation with kernel cyclic of order four || 1 || -- || any || remains the same || [[cyclic maximal subgroup of dihedral group:D8]] : <math>\langle a \rangle</math> || [[cyclic group:Z2]] || 1 || 1 || There are no ''bad'' characteristics, but it is noteworthy that in characteristic two, this representation is the same as the trivial representation.
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| sign representation with kernel a Klein four-subgroup || 2 || -- || any || remains the same || [[Klein four-subgroups of dihedral group:D8]] : <math>\langle a^2, x \rangle</math> or <math>\langle a^2, ax \rangle</math> || [[cyclic group:Z2]] || 1 || 1 || There are no ''bad'' characteristics, but it is noteworthy that in characteristic two, this representation is the same as the trivial representation.
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| two-dimensional irreducible || 1 || 2 || any || remains the same || trivial subgroup, i.e., it is a [[faithful linear representation]] || [[dihedral group:D8]] || 2 || 1 || The exact form of the new representation depends on the choice of matrices before we go mod 2, but the kernel becomes one of the [[Klein four-subgroups of dihedral group:D8]], and we thus get a representation of [[cyclic group:Z2]] in characteristic two that sends the non-identity element to <math>\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}</math>. This has an invariant one-dimensional subspace and is not irreducible.
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