# Table of products for Lie algebra inner product of roots and weights

For a simple Lie algebra $$\frak{g}$$, it is usual to scale the inner product so that the shortest simple root has length $$2$$. With this conventions, where can I find a table (online) of the following values for a general simple $$\frak{g}$$:

i) $$(\alpha_i,\alpha_j)$$, for all $$i,j=1,\dots, r:=\mathrm{rank}(\frak{g})$$

ii) $$(\alpha_i,\pi_j)$$, for $$\pi_j$$ a fundamental weight

iii) $$(\pi_i,\pi_j)$$?

• For individual simple types, look at the planches at the end of Bourbaki, Chap; 4-6 (or English translation published by Springer). Google Scholar permits an online search. Jan 25 '20 at 19:57

This information is available using the Atlas of Lie Groups and Representations software http://www.liegroups.org/software. Here is an example, comments are in braces.

atlas> set G=Sp(4)
Variable G: RootDatum
atlas> set f=G.invariant_form {W-invariant bilinear form}
Added definition [3] of f: (ratvec,ratvec->rat)
atlas> set alpha=highest_short_root(G)
Variable alpha: vec
atlas> f(alpha,alpha)
Value: 6/1  {need to normalize it differently}
atlas> set g(ratvec v,ratvec w)=f(v,w)/3
Added definition [3] of g: (ratvec,ratvec->rat)
atlas> g(alpha,alpha)  {normalized invariant form}
Value: 2/1
atlas> void:for alpha in G.posroots do
for beta in G.posroots do
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[  1, -1 ] [  1, -1 ] 2/1
[  1, -1 ] [ 0, 2 ] -2/1
[  1, -1 ] [ 1, 1 ] 0/1
[  1, -1 ] [ 2, 0 ] 2/1
[ 0, 2 ] [  1, -1 ] -2/1
[ 0, 2 ] [ 0, 2 ] 4/1
[ 0, 2 ] [ 1, 1 ] 2/1
[ 0, 2 ] [ 2, 0 ] 0/1
[ 1, 1 ] [  1, -1 ] 0/1
[ 1, 1 ] [ 0, 2 ] 2/1
[ 1, 1 ] [ 1, 1 ] 2/1
[ 1, 1 ] [ 2, 0 ] 2/1
[ 2, 0 ] [  1, -1 ] 2/1
[ 2, 0 ] [ 0, 2 ] 0/1
[ 2, 0 ] [ 1, 1 ] 2/1
[ 2, 0 ] [ 2, 0 ] 4/1
atlas> void:for alpha in G.posroots do for beta in
G.fundamental_weights do
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[  1, -1 ] [ 1, 0 ]/1 1/1
[  1, -1 ] [ 1, 1 ]/1 0/1
[ 0, 2 ] [ 1, 0 ]/1 0/1
[ 0, 2 ] [ 1, 1 ]/1 2/1
[ 1, 1 ] [ 1, 0 ]/1 1/1
[ 1, 1 ] [ 1, 1 ]/1 2/1
[ 2, 0 ] [ 1, 0 ]/1 2/1
[ 2, 0 ] [ 1, 1 ]/1 2/1
atlas> void:for alpha in G.fundamental_weights do
for beta in G.fundamental_weights do
prints(alpha, " ", beta, " ", g(alpha,beta)) od od
[ 1, 0 ]/1 [ 1, 0 ]/1 1/1
[ 1, 0 ]/1 [ 1, 1 ]/1 1/1
[ 1, 1 ]/1 [ 1, 0 ]/1 1/1
[ 1, 1 ]/1 [ 1, 1 ]/1 2/1