Return to Answer

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 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.

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 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.

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 $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},\qquad \lambda:\textrm{partition of dim V},$$ and on p.270 gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.

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 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.

This is going to sound like massive overkill, but it is 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 rational representations as all $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},$$ and on p.270 gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.

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 $$S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z},\qquad \lambda:\textrm{partition of dim V},$$ and on p.270 gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.