Are there some good references about representations of classical groups? I am looking for some references similar to the following. What are the fundamental representations of classical groups of type $B, D$? Thank you very much.

For type C classical groups, the fundamental representations are given by the following.

Let $\mathfrak{g}=\mathfrak{s}\mathfrak{p}(2r,\mathbb{C})=\langle \mathfrak{h},e_i,f_i(i\in I)\rangle$ and $\mathbb{B}^{(r)}:= \{v_i,v_{\overline i}|i=1,2,\cdots,r\}$. Define a vector space $V(\omega_1)$ as $V(\omega_1):=\bigoplus_{v\in\mathbb{B}^{(r)}}\mathbb{C} v$. The weights of $v_i$, $v_{\overline{i}}$ $(i=1,\cdots,r)$ are as follows: \begin{equation}\label{wtv} {\rm wt}(v_i)=\omega_i-\omega_{i-1},\quad {\rm wt}(v_{\overline{i}})=\omega_{i-1}-\omega_{i}, \end{equation} where $\omega_0=0$. Define the $\mathfrak{s}\mathfrak{p}(2r,\mathbb{C})$-action on $V(\omega_1)$ as follows: \begin{eqnarray} && h v_j=\langle h,{\rm wt}(v_j)\rangle v_j\ \ (h\in P^*,\ j\in J), \\ &&f_iv_i=v_{i+1},\ f_iv_{\overline{i+1}}=v_{\overline i},\quad e_iv_{i+1}=v_i,\ e_iv_{\overline i}=v_{\overline{i+1}} \quad(1\leq i<r),\label{c-f1}\\ &&f_r v_r=v_{\overline r},\qquad e_r v_{\overline r}=v_r,\label{c-f2} \end{eqnarray} and the other actions are trivial ($0$ action).

The fundamental representation $V(\omega_i)$ $(1\leq i\leq r)$ is embedded in $\wedge^i V(\omega_1)$ with multiplicity free. The explicit form of the highest weight vector $v_{\omega_i}$ of $V(\omega_i)$ is realized in $\wedge^i V(\omega_1)$ as follows: \begin{equation} \begin{array}{ccc}\displaystyle v_{\omega_i}&=&v_1\wedge v_2\wedge\cdots\wedge v_i. \end{array} \label{h-l} \end{equation} Thus, we have $V(\omega_i)\cong \mathfrak{g}v_{\omega_i}=\mathfrak{n}_- v_{\omega_i}$. Here, $\mathfrak{n}_-$ is sub Lie algebra of $\mathfrak{g}$ generated by $f_i$ ($i\in I$).

The action of $f_j$ on $u_1\wedge \cdots \wedge u_i\in \wedge^i V(\omega_1)$ are \begin{align} f_j(u_1\wedge \cdots \wedge u_i)=\sum^{i}_{k=1}u_1\wedge \cdots \wedge u_{k-1}\wedge f_ju_k\wedge u_{k+1}\wedge \cdots \wedge u_i. \end{align}

Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$?

I would like to know explicit formulas of the actions of the Lie algebras ( Lie groups ) on fundamental representations.

Thank you very much.