How to write down the action of trivial representation and sign representation of $U_q(gl_n)$?
The algebra $U_q(gl_n)$ is generated by $E_i, F_i, K_i$, $i=1,2,\ldots, n-1$.
Let $\mathbb{C} \cong \mathbb{C}v$ be the trivial representation of $U_q(gl_n)$. Then $x.v=v$ for all $x \in U_q(gl_n)$.
What is the formula for the $U_q(gl_n)$-action in the case when $V$ is the sign representation?
Thank you very much.