What does the notation for a group ${A_n}^*$ mean? I was on a website that catalouges lattices. I was looking through the alternating group lattices and the dihedral lattices and there were two kinds for each. For example, there was $A_2$ and ${A_2}^*$ as well as $D_3$ and ${D_3}^*$.
I can't seem to find any info on what these other groups are. What does that $^*$ indicate?

'*' indicates dual or weighted lattices. See this pdf. But, to fulfill this answer, I may assure you that $A_n\;^*$ means the lattice of vectors having integral inner
products with all vectors in $A_n:A_n\;^*=\{x\in R^n |(x,r)\in Z \;for\;all\;r\in A_n\}$. And, that,$A_n\;^*\;^*=A_n$.
