0
$\begingroup$

Intro (tldr-ish):
In the meantime, in the literature I dug up the formulae not only for the dimension D of a $G_2$ module, but also its quadratic Casimir C2 (eigenvalue). After some playing, I obtained a remarkable symmetry:
$D(G_2(i,j))=([3+3*i]*[1+j]*[4+3*i+j]*[5+3*i+2*j]*[6+3*i+3*j]*[9+6*i+3*j]) /([3]*[4]*[5]*[6]*[9])$
$W(G_2(i,j))=(]3+3*i[*]1+j[*]4+3*i+j[*]5+3*i+2*j[*]6+3*i+3*j[*]9+6*i+3*j[) /(]3[*]4[*]5[*]6[*]9[)$
where [x] denotes quantum integer $x_q$ and ]x[ $r^{x^2}$. W is the writhe factor, which is $q^{C2}$ (My definition - C2 is obviously defined only up to a constant factor, virtually no two people agree on the factor, and you have to retrofit $r=q^l$ so that the formula comes out right). Example:
$D(G_2(1,0))=1+q^{10}+q^8+q^2+q^{-2}+q^{-8}+q^{-10}$
$W(G_2(1,0))=r^{144}=q^6$ so $l=24$. (But all Casimirs are OK after that.)

I computed the analogous formulae for $A_n$ and again, they have identical form. Then I went berzerk and computed the Casimir for all simply laced Lie algebras. Behold:
Laced[Z_,L1_,L2_,L3_]:=Sum[(1+i+j+k+Z+Sum[L1[[ii]],{ii,1,i}]+Sum[L2[[jj]],{jj,1,j}]+Sum[L3[[kk]],{kk,1,k}])^2,{i,0,Length[L1]},{j,0,Length[L2]},{k,0,Length[L3]}]-(Length[L2]*Length[L3]-1)*Sum[(1+io-iu+Sum[L1[[i]],{i,iu,io}])^2,{io,1,Length[L1]},{iu,1,io}]-(Length[L1]*Length[L3]-1)*Sum[(1+jo-ju+Sum[L2[[j]],{j,ju,jo}])^2,{jo,1,Length[L2]},{ju,1,jo}]-(Length[L1]*Length[L2]-1)*Sum[(1+ko-ku+Sum[L3[[k]],{k,ku,ko}])^2,{ko,1,Length[L3]},{ku,1,ko}];
How this MATHEMATICA snippet works: Z is the weight at the central 3-junction and L1,L2,L3 the weights of the legs (as lists) going outward. So e.g. Laced[0,{1,0,0},{0,0},{0}]=165 (dim 27664).
Phew :-)
Question 1: Of course I now like to have the dimension formulae for $E_k, k=6,7,8$ (quantum or not), but in a paper of 2011 ("Partial differential equation approach to F4") I found a formula for $F_4(0,0,i,j)$ which somewhat implies that this is so complicated it wasn't done explicitly yet (as otherwise, why didn't he give the whole $F_4(g,h,i,j)$ formula?). Jim Humphreys suggested Tits and Fulton-Harris, but I only found $B_3$ in the former and nothing in the latter.
Question 2: I have no doubt that the snippet also works for the relevant affine Lie algebras, anyone knows a source for their Casimirs? Gould's paper looks very relevant but as usual, only general formulas are given.

$\endgroup$
1
$\begingroup$

Quantum dimension You have asked previously about quantum dimensions in:

General quantum highest-weights dimension formulas

In the response to that question you were directed to the Weyl character formula and the Weyl dimension formula. This gives the quantum dimension of any representation.

You won't find this written explicitly in a textbook as you have done for $G_2$ because the number of terms is the number of positive roots. For $G_2$ this number is six but for $E_6$, $E_7$, $E_8$ it is 36, 63, 120.

Casimir The formula for the quadratic Casimir is $C_2(\lambda)=\langle \lambda+2\rho,\lambda\rangle$ where $\rho$ is half the sum of positive roots (in the basis of fundamental weights all components are 1).

I cannot make sense of your expression for the Casimir and nor could I make sense of your previous question on the Casimir in:

Pictorial explanation of Dynkin index and quadratic Casimir?

These two formulae are in several well-written and accessible text books so your question is not asking about research level mathematics but is asking for tuition. I recommend that you read one of these text books.

$\endgroup$
2
  • $\begingroup$ In fact, this questio $\endgroup$ Aug 10 '15 at 17:22
  • $\begingroup$ In fact, this question can be closed: a) I found a Chinese (:-) site which wonderfully explained how to apply the Weyl formula step by step, and I immediately understood everything. It IS simple but I never would have understood your standard abstract textbook. And b), the formula for the quantum dimension will be automatically correct for the Casimir by the replacements I described, but not in reverse - you can do nice fully general Casimir formulas for laced Dynkin diagrams OR Bn+Cn+F4, but unfortunately, this doesn't hold for the quantum dimensions and my work was a bit pointless. $\endgroup$ Aug 10 '15 at 17:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.