The representation of SO(6) is $[i,j,k]$;


The representation of SO(10) is $[i,j,k,m,n]$.


Is there any analytical formula to calculate the dimensions of those representations?


For example,


for SO(6):


dim([1,0,0],D3)=6 


dim([0,0,1],D3)=4


dim([0,1,0],D3)=4


dim([0,1,1],D3)=15


dim([0,0,2],D3)=10


dim([i,j,k],D3)=?


for SO(10):


dim([1,0,0,0,0],D5)=10


dim([0,0,0,0,1],D5)=16


dim([0,0,0,1,0],D5)=16


dim([0,0,1,0,0],D5)=120


dim([0,1,0,0,0],D5)=45


dim([i,j,k,m,n],D5)=?



How to calculate the dimensions of representations of SO(6) and SO(10)?
Let $[x\_1,\ldots,x\_\ell]$ denote the vector corresponding to the highest weight of
$D\_\ell$. Then the dimension of the representation is given by
$\prod\_{1\leq i < j \leq \ell} ( 1+ \frac{x\_i+\cdots +x\_{j-1}}{j-i} )
\times$
$\prod\_{1\leq i \leq \ell-1} ( 1+ \frac{x\_i+\cdots + x\_{\ell-2}+x\_{\ell}}{\ell-i} )$
$\times
\prod\_{1\leq i < j\leq \ell-1} ( 1+ \frac{x\_i+\cdots +x\_{j-1}+2(x\_j+\cdots + x\_{\ell-2})+x\_{\ell-1}+x\_{\ell}}{2\ell-i-j} )
$


