Let \begin{equation} X= \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}, \qquad Y= \begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}, \qquad H= \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{bmatrix}\tag{1} \end{equation} If $V_m$ is the $(m+1)$-dimensional complex representation of $\text{sl}(2,\mathbb{C})$, then we know that there exists a basis $u_m,u_{m-2},...,u_{-m}$ such that $u_k$ is the eigenvector of $H$ with eigenvalue $k$ and $Y u_k = u_{k-2}$.
In physics, on the other hand, we write $S_{\pm} =X,Y$ and $S_z = H/2$ and use the basis $|s,m_s \rangle$ such that $$\begin{align} S_z |s,m_s \rangle &= m_s |s,m_s \rangle \\ S_{\pm} |s,m_s \rangle &= \sqrt{s(s+1)-m_s (m_s \pm 1)} |s,m_s \pm 1\rangle \end{align}\tag{2}$$ and that $|s,m_s \rangle$ are orthogonal.
Is there a deeper reason why we put the coefficients in front of $|s,m_s \rangle$, when $S_{\pm}$ acts on it and why $|s,m_s \rangle$ are orthogonal?