Hi,  

The complex simple algebraic group $Sp\_{m,\mathbb{C}}$ of $2m$-dimensional space $V$ has, for $m≥2$, an irreducible representation of dimension $m(2m−1)−1$ in a subspace of codimension $1$ of the space $\Lambda^2V$. Is it the irreducible representation of smallest dimension after $V$ itself?


Thank you.



Non-trivial representation of second-smallest dimension
The irreducible complex representations of the simply connected simple group $G=Sp\_{r,{\mathbb C}}$ of type $C\_r$,
for $r>1$, of dimension $n<{\rm dim}\ G$
are listed in the [paper of Andreev, Vinberg, and Elashvili](http://link.springer.com/content/pdf/10.1007%2FBF01076005),
Table 1 (see also the [Russian version](http://www.mathnet.ru/links/419af2a1d9d33839a49ff8898866b056/faa2839.pdf)). 
They are the fundamental irreducible representations $R(\pi\_1)$ of dimension $2r$, $R(\pi\_2)$ of dimension $2r^2-r-1$,
and, for $r=3$, $R(\pi\_3)$ of dimension 14. 
For all $r\ge 2$, $r\neq 3$, we have ${\rm dim}\ R(\pi\_1)=2r<2r^2-r-1={\rm dim}\ R(\pi\_2)$, 
hence $R(\pi\_2)$ is the nontrivial irreducible representation of second smallest dimension.
For $r=3$, as Jim Humphreys noted, the dimensions are $6,14,14$, so ${\rm dim}\ R(\pi\_2)={\rm dim}\ R(\pi\_3)>{\rm dim}\ R(\pi\_1)$, and 
$R(\pi\_2)$ is *a* nontrivial irreducible representation of second smallest dimension.


