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4 typo

This is going to sound like massive overkill, but it is "very well known" that the only 1-dimensional polynomial representations of $GL(V)$ (which is what you're looking at) are the nonegative nonnegative powers of $\mathrm{det}$.

Reference (I assume from the mention of statistics that you are OK working with base field $\mathbf{R}$ or $\mathbf{C}$): e.g. Procesi on p.278 of Lie Groups lists all irreducible rational representations as all $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},$$ where $\lambda$ runs over a certain set of partitions or Young tableaux; and on p.270 he gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.

3 fixed egregious error

This is going to sound like massive overkill, but it is "very well known" that the only 1-dimensional polynomial representations of $GL(V)$ (which is what you're looking at) are the nonegative powers of $\mathrm{det}$.

Reference (I assume from the mention of statistics that you are OK working with base field $\mathbf{R}$ or $\mathbf{C}$): e.g. Procesi's Lie GroupsProcesi on p.278 of Lie Groups lists all irreducible rational representations as all $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},\qquad \lambda:\textrm{partition  S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},$$ where $\lambda$ runs over a certain set of dim V},$$partitions or Young tableaux; and on p.270 he gives a dimension formula for S_\lambda(V) which is >1 unless S_\lambda(V) is trivial. 2 add missing "irreducible" This is going to sound like massive overkill, but it is "very well known" that the only 1-dimensional polynomial representations of GL(V) (which is what you're looking at) are the nonegative powers of \mathrm{det}. Reference (I assume from the mention of statistics that you are OK working with base field \mathbf{R} or \mathbf{C}): e.g. Procesi's Lie Groups on p.278 lists all irreducible rational representations as all$$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},$$k\in\mathbf{Z},\qquad \lambda:\textrm{partition of dim V},$$ and on p.270 gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.

1