Let us consider, for example, the standard irreducible $\mathfrak{sl}_3$-module
$\Gamma_{1,1}$ with the highest weight $(1,1),$ $\dim \Gamma_{1,1}=8.$
The set of all weight of $\dim \Gamma_{1,1}$ is $(0,0),(1,1),(2,-1),(1,-2),(-2,1),(-1,2),(-1,-1).$
Question. What is the Gelfand-Tsetlin basis for $\Gamma_{1,1}$?
As I understood from literature (Zhelobenko ) there is a combinatorial structure $\Lambda$ depended of $(1,1)$ such that a basis of $\dim \Gamma_{1,1}$ can be labeled via the $\Lambda$ but I cant do it. Anybody can help?
Best Answer
It's more common to talk about the GT basis of $\mathfrak{gl}_n$-modules. In the case of the adjoint representation, the first row of the GT scheme is $(1,0,\ldots,0,-1)$ and each subsequent row satisfies the interlacing condition, which implies that the left (resp, right) edge consists of a string of $1$s (resp $-1$s), followed by a string of $0$s (possibly empty), except that the bottom entry can be $1$ (resp $-1$) if all entries on the left (resp right) are $1$s (resp $-1$), and all other entries are zeros. It follows that the whole scheme can be reconstructed from the positions of the lowest $1$ along the left edge of the triangle and the lowest $-1$ along the right edge. They may occupy any of the $n$ possible rows/positions each, except that both cannot occur in the $n$th row, corresponding to the impossibility of the bottom entry being simultaneously $1$ and $-1.$
In your special case $n=3$ the diagrams will look like this:
$$ \begin{array}{rrrrr} 1 & & 0 & & -1\\ & 1 & & 0 & \\ & & 1 & & \end{array} \quad $$
(this is the highest weight; there are 7 more).