Question #23880

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. How about RG and the real representations of G?

Expert's answer

Tensoring Q*G **∼* Q *× *Q(*ω*) *×*M_{3}(Q(*√−*7))*,*up to R, we get R*G **∼* R *× *C *×*M_{3}(C)*,*which means that each of the irreducible Q-representations above remainsirreducible over R (and these give all irreducible R-representations). We see

that the 6-dimensional irreducible

rational (resp. real) representationof*G *tensors up into the direct sum of the two 3-dimensional complexrepresentations of *G*. (Of course, this is also easy to see by checkingdirectly that *χ**K*= *χ*4 + *χ*5.)

that the 6-dimensional irreducible

rational (resp. real) representationof

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