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For a partition $\lambda$ let $x_{\lambda}$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $p_{\lambda}=\sum\limits_{\pi \in S_n}^{}{x_\lambda( \pi) x_{1 \pi(1)} ... x_{n \pi(n)}}$ be the immanent corresponding to $\lambda$. (For the sign representation we will just get the determinant for example). This is a polynomial in the $n^2$ variables $x_{i,j}$ over $\mathbb{Z}$.

My question is how can I obtain the immanent given a parition $\lambda$ using Sage?

My first problem is already that we need the polynomial ring in the $n^2$ variables $x_{i,j}$ and I am not sure how to define this in Sage depending on $n$.

For a partition $\lambda$ let $x_{\lambda}$ $y_{\lambda}$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $p_{\lambda}=\sum\limits_{\pi \in S_n}^{}{x_\lambda( S_n}^{}{y_\lambda( \pi) x_{1 \pi(1)} ... x_{n \pi(n)}}$ be the immanent corresponding to $\lambda$. (For the sign representation we will just get the determinant for example). This is a polynomial in the $n^2$ variables $x_{i,j}$ over $\mathbb{Z}$.

My question is how can I obtain the immanent given a parition $\lambda$ using Sage?

My first problem is already that we need the polynomial ring in the $n^2$ variables $x_{i,j}$ and I am not sure how to define this in Sage depending on $n$.