Transformation as a doublet under SO

[LCSphysicist]

Homework Statement

Let 5 objects, S11, S22, S12, S13, S23, furnish the representation 5 of SO(3). Suppose we restrict ourselves to a subgroup SO(2), a rotation about the z axis. How can i show that (S12, S11 − S22) transform like a doublet? And what exactly means transform like a doublet?

Relevant Equations

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$$S'^{12} = R^{1k}R^{2l}S^{kl},

S'^{12} = R^{11}R^{21}S^{11} + R^{11}R^{22}S^{12} + R^{12}R^{21}S^{21} + R^{12}R^{22}S^{22},

S'^{12} = R^{11}R^{21}(S^{11}-S^{22}) + (R^{11}R^{22} + R^{12}R^{21})S^{12}$$

Is this enough to say that (S12, S11 − S22) transform like a doublet? To be pretty honest, i am a little confused about what exactly a doublet means in this tensor context. I used the Symmetry of S and Rij = -Rji for j different from i

 

[Adrmay]

to get the above result.A:Yes, this is enough to say that $(S^{12}, S^{11}-S^{22})$ transforms like a doublet. A doublet is a pair of two quantities which transform in exactly the same way under a certain symmetry transformation. In your case, you can see that both $S^{12}$ and $S^{11}-S^{22}$ transform in the same way since they have the same coefficients in front of them when written in terms of the $R_{ij}$'s.