Last update: 2 April 2014
Given below are the explicit expressions for the polynomial in in the four cases of rank 2 and 3 in which Conjecture I has been verified, along with the Dynkin diagrams which reveal our convention for the labelling of fundamental roots; the arrows in the diagrams go from a long root to a short one: where The integer coefficient in front of comes from the normalization The Coxeter diagrams of are then as follows:
We shall first give our verification of Conjecture I in the three rank-two cases. This is done in the three tables below, for the respective cases, where the value of for can be read off explicitly from the column headed by the juxtaposition according to the following rule: In the first column (resp. with the subscripts appearing times, stands for (resp. as in elementary calculus, and the 6, 8 or 12 polynomials given in the first column of the respective table form a basis of the space of polynomials. (The mixed partial derivative can be computed in terms of this from the differential equation satisfied by the polynomial in these cases, where equals 1, 2 or 3 according as is of type or (the number of lines in the Dynkin diagram); the invariant differential (Casimir) operator is precisely the one corresponding to the second-degree Killing invariant.) Now the column (which stands for simply gives the expansion of the inhomogeneous polynomial in terms of the basis of given in the first column (which has been so normalized, through suitable denominators, that all the coefficients in these expansions are integers). Thus the value of is simply the inner product of the column under with the left-end column (the part appearing between the two horizontal lines). All blank places stand for the entry "zero".
The first part of Conjecture I asserts merely the non-singularity of the matrices showing the expansion of the in these tables; and the values of given in the bottom rows verify the second part. Our verifications were done manually, except that the matrix in the case of type was obtained from a computer print-out of the
At the top of the tables for types and are indicated the right-cosets of in on each of which the weighting function is constant. In the adjoining Tessellation Diagrams (Figs 1—3) are also shown the numbers for in bold numerals inside the corresponding (which fill up as also are shown the points for and The points for comprising the intersection of the of with are shown linked through dotted lines joining pairs of elements that are adjacent with respect to the partial-order As in the tables we have followed again the juxtaposition convention in these Diagrams, so that (resp. for (resp. given by a reduced expression with the indices ranging in (resp. has been abbreviated to
Lastly, we come to the case of type Here we omit the explicit values of the 24 polynomials because of the enormity of the size of the corresponding table (once again obtained through a computer print-out of these inhomogeneous polynomials), nor do we give a diagrammatic visualization of the tessellation of (it is not very hard to construct a 3-dimensional model of this tessellation, as also a plane projection of this model). We give only the final listing of the weighting function for the table on p. 676, showing also (schematically) the "linkage diagram" for the partial-order on (analogous to the linkage formed by in the rank-two diagrams above, shown by the dotted lines there). In the Tessellation Diagram for type we also exhibit, incidentally, all the for satisfying The four triangles shown by broken lines correspond to the four such values of not lying in viz. where stands for given by the juxtaposition which has been used throughout this diagram for the sake of clarity in printing. In this case there are two elements of that lie below with respect to but do not belong to viz. and
This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).