sl3 interlacing character formula (${\chi }_{a,b}={\sum }_{b\le p\le a+b}{\sum }_{0\le q\le b}{\sum }_{q\le r\le p}{x}_1^r{x}_2^{p+q-r}{x}_3^{a+2b-p-q}$)

For the special linear Lie algebra ${\mathfrak{sl}}_3$, the irreducible highest-weight representation with Dynkin labels $(a,b)$ has the displayed formal character on diagonal matrices with $x_1{x}_2{x}_3=1$. The interlacing rows $(a+b,b,0)$, $(p,q)$ and $(r)$ enumerate a Gelfand–Tsetlin basis; each triple contributes the exponent differences $(r-(p+q-r),(p+q-r)-(a+2b-p-q))$ as its weight. This gives exact integer weight multiplicities, including repeated weights, without floating-point character evaluation.