Let $I$ be an ideal of $\mathbb{C}[GL_n]$ generated by a set of elements in $\mathbb{C}[GL_n]$. Are there some convenient method or software which can check that whether $I$ is a coideal or not? Thank you very much.
For example, let I be the ideal of $\mathbb{C}[GL_3]$ generated by the following elements (the indices start from $0$):
\begin{align}
& c_{1,2} {c_{0,0}}^2 - 2 c_{1,1} c_{0,0} c_{0,1} - c_{1,0} c_{0,2} c_{0,0} + 2 c_{1,0} {c_{0,1}}^2, \\
& c_{0,0} c_{0,1} c_{1,2} - 2 c_{0,0} c_{0,2} c_{1,1} + c_{0,1} c_{1,0} c_{0,2}, \\
& 2 c_{2,1} {c_{0,0}}^2 - c_{1,1} c_{0,0} c_{1,0} - 2 c_{0,1} c_{2,0} c_{0,0} + c_{0,1} {c_{1,0}}^2, \\
& c_{2,2} {c_{0,0}}^2 - c_{0,0} {c_{1,1}}^2 - c_{0,2} c_{2,0} c_{0,0} + c_{0,1} c_{1,0} c_{1,1}, \\
& 2 c_{0,0} c_{0,1} c_{2,2} - 2 c_{0,0} c_{0,2} c_{2,1} - c_{0,0} c_{1,1} c_{1,2} + c_{0,1} c_{1,0} c_{1,2}, \\
& 2 c_{1,2} {c_{0,1}}^2 - 2 c_{1,1} c_{0,1} c_{0,2} + c_{1,0} {c_{0,2}}^2 - c_{0,0} c_{1,2} c_{0,2}, \\
& - 4 c_{2,0} {c_{0,1}}^2 + 4 c_{0,0} c_{2,1} c_{0,1} + c_{0,2} {c_{1,0}}^2 - c_{0,0} c_{1,2} c_{1,0}, \\
& 2 c_{0,0} c_{0,1} c_{2,2} - c_{0,0} c_{1,1} c_{1,2} - 2 c_{0,1} c_{0,2} c_{2,0} + c_{1,0} c_{0,2} c_{1,1}, \\
& 4 c_{2,2} {c_{0,1}}^2 - 4 c_{0,2} c_{2,1} c_{0,1} - c_{0,0} {c_{1,2}}^2 + c_{1,0} c_{0,2} c_{1,2}, \\
& c_{0,0} c_{1,0} c_{2,1} - 2 c_{0,0} c_{1,1} c_{2,0} + c_{0,1} c_{1,0} c_{2,0}, \\
& c_{0,0} c_{1,0} c_{2,2} - 2 c_{0,0} c_{1,1} c_{2,1} - c_{0,0} c_{2,0} c_{1,2} + 2 c_{0,1} c_{1,0} c_{2,1}, \\
& c_{0,1} c_{1,0} c_{2,2} - c_{0,0} c_{1,2} c_{2,1}, \\
& c_{0,0} c_{1,0} c_{2,2} - 2 c_{0,0} c_{1,1} c_{2,1} + 2 c_{0,1} c_{1,1} c_{2,0} - c_{1,0} c_{0,2} c_{2,0}, \\
& c_{1,0} c_{0,2} c_{2,1} - c_{0,1} c_{2,0} c_{1,2}, \\
& 2 c_{0,1} c_{1,1} c_{2,2} - c_{0,0} c_{1,2} c_{2,2} - 2 c_{0,1} c_{1,2} c_{2,1} + c_{1,0} c_{0,2} c_{2,2}, \\
& c_{2,1} {c_{1,0}}^2 - c_{1,1} c_{1,0} c_{2,0} + 2 c_{0,1} {c_{2,0}}^2 - 2 c_{0,0} c_{2,1} c_{2,0}, \\
& c_{2,2} {c_{1,0}}^2 - c_{2,0} c_{1,2} c_{1,0} - 4 c_{0,0} {c_{2,1}}^2 + 4 c_{0,1} c_{2,0} c_{2,1}, \\
& 2 c_{0,1} c_{2,0} c_{2,2} - 2 c_{0,0} c_{2,1} c_{2,2} + c_{1,0} c_{1,1} c_{2,2} - c_{1,0} c_{1,2} c_{2,1}, \\
& 2 c_{0,0} c_{0,2} c_{2,1} - c_{0,1} c_{1,0} c_{1,2} - 2 c_{0,1} c_{0,2} c_{2,0} + c_{1,0} c_{0,2} c_{1,1}, \\
& - c_{2,0} {c_{0,2}}^2 + c_{0,2} {c_{1,1}}^2 + c_{0,0} c_{2,2} c_{0,2} - c_{0,1} c_{1,2} c_{1,1}, \\
& - 2 c_{2,1} {c_{0,2}}^2 + c_{1,1} c_{0,2} c_{1,2} + 2 c_{0,1} c_{2,2} c_{0,2} - c_{0,1} {c_{1,2}}^2, \\
& c_{0,0} c_{2,0} c_{1,2} - 2 c_{0,1} c_{1,0} c_{2,1} + 2 c_{0,1} c_{1,1} c_{2,0} - c_{1,0} c_{0,2} c_{2,0}, \\
& c_{0,0} c_{1,2} c_{2,2} - 2 c_{0,1} c_{1,1} c_{2,2} + 2 c_{0,2} c_{1,1} c_{2,1} - c_{0,2} c_{2,0} c_{1,2}, \\
& 2 c_{0,2} c_{1,1} c_{2,2} - c_{0,1} c_{1,2} c_{2,2} - c_{0,2} c_{1,2} c_{2,1}, \\
& - {c_{1,1}}^2 c_{2,0} + c_{1,0} c_{2,1} c_{1,1} + c_{0,2} {c_{2,0}}^2 - c_{0,0} c_{2,2} c_{2,0}, \\
& c_{1,0} c_{1,1} c_{2,2} - 2 c_{0,0} c_{2,1} c_{2,2} + 2 c_{0,2} c_{2,0} c_{2,1} - c_{1,1} c_{2,0} c_{1,2}, \\
& {c_{1,1}}^2 c_{2,2} - c_{1,2} c_{2,1} c_{1,1} - c_{0,0} {c_{2,2}}^2 + c_{0,2} c_{2,0} c_{2,2}, \\
& 2 c_{0,1} c_{1,2} c_{2,1} - c_{1,0} c_{0,2} c_{2,2} - 2 c_{0,2} c_{1,1} c_{2,1} + c_{0,2} c_{2,0} c_{1,2}, \\
& c_{1,0} c_{1,2} c_{2,1} - 2 c_{0,1} c_{2,0} c_{2,2} + 2 c_{0,2} c_{2,0} c_{2,1} - c_{1,1} c_{2,0} c_{1,2}, \\
& - c_{2,0} {c_{1,2}}^2 + c_{1,0} c_{2,2} c_{1,2} + 4 c_{0,2} {c_{2,1}}^2 - 4 c_{0,1} c_{2,2} c_{2,1}, \\
& - c_{2,1} {c_{1,2}}^2 + c_{1,1} c_{1,2} c_{2,2} - 2 c_{0,1} {c_{2,2}}^2 + 2 c_{0,2} c_{2,1} c_{2,2}, \\
& - c_{1,2} {c_{2,0}}^2 + 2 c_{1,1} c_{2,0} c_{2,1} + c_{1,0} c_{2,2} c_{2,0} - 2 c_{1,0} {c_{2,1}}^2, \\
& 2 c_{1,1} c_{2,0} c_{2,2} - c_{1,0} c_{2,1} c_{2,2} - c_{2,0} c_{1,2} c_{2,1}, \\
& - 2 c_{1,2} {c_{2,1}}^2 + 2 c_{1,1} c_{2,1} c_{2,2} - c_{1,0} {c_{2,2}}^2 + c_{2,0} c_{1,2} c_{2,2}.
\end{align}