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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)=?

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5  
I find your notation hard to follow. However, the Weyl Dimension Formula en.wikipedia.org/wiki/… should answer your questions. –  David Speyer Sep 30 '10 at 1:35
    
@David: It looks like the vector gives the coefficients of fundamental weights, probably the inputs from above are for the software package LiE (so uses their indexing). –  Steven Sam Sep 30 '10 at 4:36
    
@Steven: Yes, you're right. My notation follows the LiE program. –  Osiris Sep 30 '10 at 5:43
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1 Answer

up vote 5 down vote accepted

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} ) $

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