The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part I
A conjecture on Weyl's dimension polynomial

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 2 April 2014

§ 3. Evidence in low ranks

Given below are the explicit expressions for the polynomial D(λ)=D(x1,x2,,xl) in λ=x1λ1+x2λ2++xlλlE 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: α1 α2 α1 α2 α1 α2 α1 α2 α3 A2: 2·D(x,y)= xy(x+y), |W|=6; B2: 6·D(x,y)= xy(x+y) (x+2y), |W|=8; G2: 120·D(x,y)= xy(x+y) (x+2y) (x+3y) (2x+3y), |W|=12; A3: 12·D(x,y,z)= xyz(x+y) (y+z) (x+y+z), |W|=24; where x=x1, y=y2, z=x3. The integer coefficient in front of D(x,) comes from the normalization D(1,,1)=1. The Coxeter diagrams of W are then as follows: A2: R1 R2 R0 B2: R1 R2 R0 4 4 G2: R0 R1 R2 6 A3: R1 R2 R0 R3

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 Dσ for σ=Ri1Ri2RikW can be read off explicitly from the column headed by the juxtaposition i1i2ik according to the following rule: In the first column Dxx (resp. Dyy), with the subscripts appearing i times, stands for (x)iD (resp. (y)iD) 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 W-harmonic polynomials. (The mixed partial derivative Dxxyy can be computed in terms of this from the differential equation 2Dxy= 2Dx2+ 1c2Dy2 satisfied by the polynomial D in these cases, where c equals 1, 2 or 3 according as Δ is of type A2, B2 or G2 (the number of lines in the Dynkin diagram); the invariant differential (Casimir) operator (x)2+c-1(y)2-xy is precisely the one corresponding to the second-degree Killing invariant.) Now the column i1i2ik (which stands for σ=Ri1Ri2Rik) simply gives the expansion of the inhomogeneous polynomial Dσ in terms of the basis of W-harmonics 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 Dσ is simply the inner product of the column under σ 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 |W|×|W| matrices showing the expansion of the Dσ's in these tables; and the values of bσ's given in the bottom rows verify the second part. Our verifications were done manually, except that the 12×12 matrix in the case of type G2 was obtained from a computer print-out of the Dσ's. TABLE FOR TYPEA2 Ω0R1R2R1 Ω0 σ= 12112 1221id. D-1-1-1111 Dx11-1-1 Dy1-11-1 12Dxx=12y-3111 12Dyy=12x-3111 11 bσ=111111 TABLE FOR TYPEB2 Ω0·R2 Ω0·R1R2 Ω0·R2R1R2 Ω0 σ= 12122 112 21212 121id. D1-1-111-1-11 Dx-1-21-211 Dy-11-11-11 12Dxx31-1-1 12Dyy21-1-1 16Dxxx=16y-1341411 16Dyyy=16x-1121212 11 bσ=11112222 TABLE FOR TYPEG2 σ= 1212122 12212121221212 121121212112121id. D1-11-11-1-11-11-11 Dx -1 -3 -3 -3 -3 1 2 2 1 1 Dy -1 1 2 2 1 1 -1 -1 -1 -1 12Dxx 3 -3 -3 3 3 1 -1 -1 16Dyy 5 3 -3 -3 -1 -1 1 1 16Dyyy=16x(x+3y)(2x+3y) -3 1 -1 -1 1 1 16Dxxx=16y(x+y)(x+2y) -10 1 -1 -1 1 1 124Dxxxx=124y(2x+3y) 39 9 9 -9 -9 1 -1 -1 1216Dyyyy=124x(x+2y) 55 9 -9 -9 1 1 -1 -1 11080Dyyyyy=160x -149 9 18 18 9 9 1 1 1 1 1120Dxxxxx=160y -211 27 27 27 27 1 2 2 1 1 11 bσ= 1 1 1 3 2 4 1 1 3 9 4 2

At the top of the tables for types A2 and B2, are indicated the right-cosets of Ω0 in W, on each of which the weighting function is constant. In the adjoining Tessellation Diagrams (Figs 1—3) are also shown the numbers bσ for σW in bold numerals inside the corresponding wσ-simplices (which fill up E), as also are shown the points λ(σ) for λinterior(E*) and σW. The points λ(σ)=λwσ α1=2λ1-λ2 α2=-λ1+2λ2 λ1=13(2α1+α2) λ2=13(α1+2α2) -α0=α1+α2 -α0=α1+α2 n1=1=n2 0 α1 α2 λ1 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) = λ0 1 1 Fig. 1. Tessellation diagram for typeA2 α1=2λ1-λ2 α2=-λ1+2λ2 λ1=α1+12α2 λ2=α1+α2 -α0=α1+α2 -α0=α1+α2 n1=1,n2=2 0 α1 α2 λ1 λ22 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) λ(212) λ(1212) = = = λ0 λ20 λ120 1 1 2 2 Fig. 2. Tessellation diagram for typeB2 for σW, comprising the intersection of the W-orbit of λ with E, 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 λ(σ) (resp. λw) for σW (resp. wW) given by a reduced expression Ri1Ri2Rik with the indices i1,i2,,ik ranging in {1,2} (resp. {0,1,2}), has been abbreviated to λ(i1i2ik) (resp.λi1i2ik).

Lastly, we come to the case of type A3. Here we omit the explicit values of the 24 polynomials Dσ 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 E (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 bσ for σW, the table on p. 676, showing also (schematically) the "linkage diagram" for the partial-order on W (analogous to the linkage formed by {λw|wW} in the rank-two diagrams above, shown by the dotted lines there). 0 1 1 1 1 1 2 2 3 3 4 4 9 α1 α2 λ12 λ1 λ23 λ2 λ λ0 λ10 λ210 λ1210 = λw λ2w λ0w λ02w λ102w λ2102w λ12102w λ012102w δ α1=2λ1-λ2 α2=-3λ1+2λ2 λ1=2α1+α2 λ2=3α1+2α2 -α0=2α1+α2 -α0=2α1+3α2 n1=2,n2=3 w=def1210 α1=2λ1-λ2 α2=-λ1+2λ2 λ1=α1+12α2 λ2=α1+α2 -α0=α1+α2 -α0=α1+α2 n1=1,n2=2 0 α1 α2 λ1 λ22 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) λ(212) λ(1212) = = = λ0 λ20 λ120 1 1 2 2 σ 121212 2 12 212 1212 21212 1 21 121 2121 12121 id wσ 012102w 12102w 2102w 102w 02w 0w 2w 1210=w 210 10 0 id Fig. 2. Tessellation diagram for typeG2 In the Tessellation Diagram for type G2 we also exhibit, incidentally, all the w-simplices for wW+ satisfying ww0. The four triangles shown by broken lines correspond to the four such values of w not lying in W, viz. R1R2w, R0R1R2w, R1R0R1R2w, and R2R1R0R1R2w, where w stands for R1R2R1R0 given by the juxtaposition w=1210 which has been used throughout this diagram for the sake of clarity in printing. TABLE FOR TYPEA3 σW bσ λ(σ) wσ linkage R1R3 1 (1-x,x+y+z,1-z) w0=R2R1R3R0 R2R1R3 1 (1-x-y,x+y+z,1-y-z) R1R3R0 R1R2R1R3 1 (1-y,y+z,1-x-y-z) R3R0 R3R2R1R3 1 (1-x-y-z,x+y,1-y) R1R0 R1R3R2R1R3 2 (1-y-z,y,1-x-y) R0 id. 0 (x,y,z) id. In this case there are two elements of W+ that lie below w0 with respect to but do not belong to W, viz. R2R1R0 and R2R3R0.

Notes and References

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).

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