The formulas hold modulo typos :-)
It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in the next paper I opened :-) that the highest weight dimension formula for $G_2(i,j)$ is ([x] denotes the quantum integer x)
$[3+3*i]*[1+j]*[4+3*i+j]*[5+3*i+2*j]*[6+3*i+3*j]*[9+6*i+3*j])
/([3]*[4]*[5]*[6]*[9])$
The obvious question is whether analogous formulas for the other Lie algebras hold, for example $A_2(i,j)=[i+1]*[j+1]*[i+j+2]/[2]$. (Would be nice to have e.g. $A_k$ and quantum binomials!) And no, to my best understanding Landsberg/Manivel give general formulas for the powers of special irreps but not as a function of highest weight.
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asked Jun 1, 2015 at 14:42
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$\begingroup$ It seems you want to transcribe the standard Weyl dimension formula into quantum format? This seems straightforward. For instance, the $A_2$ Weyl dimension is $(i+1)(j+1)(i+j+2)/2$ if the highest weight has coordinates $(i,j)$. This is the prototype for what you write here in terms of quantum integers. But I have trouble deciphering your $G_2$ formula. $\endgroup$– Jim HumphreysCommented Jun 1, 2015 at 15:28
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$\begingroup$ G2: Some integers cancel out and I possibly use q^3 where I should use q or so :-) (Cf. "GLOBAL DIMENSIONS FOR LIE GROUPS AT LEVEL k AND THEIR CONFORMALLY EXCEPTIONAL QUANTUM SUBGROUPS" by R. Coquereaux, p.23., which was the "next" paper mentioned above. More precise there.) It is not 100% trivial, though, since e.g. i+1 in the A2 formula could translate to [2*i+2]/[2] or [3*i+3]/[3] in quantum integers. Do you have a handy reference to look up the Weyl dimensions of all the simple Lie algebras? $\endgroup$– Hauke ReddmannCommented Jun 2, 2015 at 11:19
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$\begingroup$ Concerning references, most books on Lie groups and Lie algebras give the Weyl dimension formula in its explicit general format. But details about each Lie type require input on the coefficients of all positive roots. Examples are often given, as in 7.6 of the old but concise notes by J. Tits (Springer Lecture Notes in Math. 40, 1967) and Lecture 24 in the Springer GTM volume (1991) by Fulton & Harris, etc. But what you are doing obviously gets more complicated. Also, sources vary a lot in notation, terminology, etc. $\endgroup$– Jim HumphreysCommented Jun 2, 2015 at 18:10
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