= Solution
The <C2v molecular point group> has four one-dimensional <irreducible representations>. In the operation order used above, the <character table> rows are $A_1=(1,1,1,1)$, $A_2=(1,1,-1,-1)$, $B_1=(1,-1,1,-1)$ and $B_2=(1,-1,-1,1)$. Their <character inner products> with $\chi_\Gamma=(6,0,2,0)$ give
$$
m_{A_1}=\frac{6+2}{4}=2,\quad m_{A_2}=\frac{6-2}{4}=1,\quad m_{B_1}=2,\quad m_{B_2}=1.
$$
Hence \b[$\Gamma=2A_1\oplus A_2\oplus2B_1\oplus B_2$], whose dimensions sum to six. The labels $B_1,B_2$ refer to the source axes; swapping $x$ and $y$ swaps these names.
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