Consider the spin states of three electrons \(\uparrow \uparrow \downarrow \downarrow \downarrow \uparrow \downarrow \uparrow \uparrow\). Treating these as basis functions for \(\mathbf{S}_{3}\), develop the \(3 \times 3\) matrix representation of \(\mathbf{P}_{2}\) and show that its character is 1 .