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On the calculation of group characters

Aug, 2000
8 pages
Published in:
  • J.Geom.Phys. 57 (2007) 2533-2538
e-Print:
DOI:
Report number:
  • ITU-HEP-03-2000
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Abstract:
In its most schematic form, the Weyl character formula can be expressed by the ratio PQ{P \over Q} of two multinomials P≡a(1)+a(2)...a(D)P \equiv a(1) + a(2) ... a(D) and Q≡b(1)+b(2)...b(D)Q \equiv b(1) + b(2) ... b(D) where D is the order of Weyl group W(Gr)W(G_r) for a Lie algebra GrG_r of rank r. Each and every one of a(k)a(k)'s and b(k)b(k)'s is obtained by the action of a Weyl reflection, i.e. an element of W(Gr)W(G_r). We, instead, show that there is a way to obtain all these terms without refering to Weyl reflections. For this, the following observation seems to be crucial: It is known that there is a set of r weight vectors λi,(i=1,2,...r)\lambda_i, (i=1,2, ... r) of which their scalar products form the inverse Cartan matrix of GrG_r. These are sometimes called {\bf fundamental dominant weights} of GrG_r. The observation is now that there are some other sets of r weights which satisfy this same condition so their number is just equal to order of Weyl group W(Gr)W(G_r). By specifying these sets of weights completely, we will show that the character of an irreducible representation can be calculated without any reference to Weyl group summations. All these and some useful technical points will be given in detail in the instructive example of G2G_2 Lie algebra.
  • Characters of Lie algebras
  • G2 Lie algebra