## The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebrasPart IA conjecture on Weyl's dimension polynomial

Last update: 2 April 2014

## § 3. Evidence in low ranks

Given below are the explicit expressions for the polynomial $D\left(\lambda \right)=D\left({x}_{1},{x}_{2},\dots ,{x}_{l}\right)$ in $\lambda ={x}_{1}{\lambda }_{1}+{x}_{2}{\lambda }_{2}+\dots +{x}_{l}{\lambda }_{l}\in E$ 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 $x={x}_{1},$ $y={y}_{2},$ $z={x}_{3}\text{.}$ The integer coefficient in front of $D\left(x,\dots \right)$ comes from the normalization $D\left(1,\dots ,1\right)=1\text{.}$ The Coxeter diagrams of $\stackrel{\sim }{W}\prime$ 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 ${D}_{\sigma }$ for $\sigma ={R}_{{i}_{1}}{R}_{{i}_{2}}\dots {R}_{{i}_{k}}\in W$ can be read off explicitly from the column headed by the juxtaposition ${i}_{1}{i}_{2}\dots {i}_{k}$ according to the following rule: In the first column ${D}_{x\dots x}$ (resp. ${D}_{y\dots y}\text{),}$ with the subscripts appearing $i$ times, stands for ${\left(\frac{\partial }{\partial x}\right)}^{i}D$ (resp. ${\left(\frac{\partial }{\partial y}\right)}^{i}D\text{)}$ 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 $W\text{-harmonic}$ polynomials. (The mixed partial derivative ${D}_{x\dots xy\dots y}$ can be computed in terms of this from the differential equation $∂2D∂x∂y= ∂2D∂x2+ 1c∂2D∂y2$ satisfied by the polynomial $D$ in these cases, where $c$ equals 1, 2 or 3 according as $\Delta$ is of type ${A}_{2},$ ${B}_{2}$ or ${G}_{2}$ (the number of lines in the Dynkin diagram); the invariant differential (Casimir) operator ${\left({\partial }_{x}\right)}^{2}+{c}^{-1}{\left({\partial }_{y}\right)}^{2}-{\partial }_{x}{\partial }_{y}$ is precisely the one corresponding to the second-degree Killing invariant.) Now the column ${i}_{1}{i}_{2}\dots {i}_{k}$ (which stands for $\sigma ={R}_{{i}_{1}}{R}_{{i}_{2}}\dots {R}_{{i}_{k}}\text{)}$ simply gives the expansion of the inhomogeneous polynomial ${D}_{\sigma }$ in terms of the basis of $W\text{-harmonics}$ 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 ${D}_{\sigma }$ is simply the inner product of the column under $\sigma$ 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 ${D}_{\sigma }\text{'s}$ in these tables; and the values of ${b}_{\sigma }\text{'s}$ 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 ${G}_{2}$ was obtained from a computer print-out of the ${D}_{\sigma }\text{'s.}$ $TABLE FOR TYPE A2 ⏞Ω0R1R2R1 ⏞Ω0 σ= 12112 1221id. D-1-1-1111 Dx11-1-1 Dy1-11-1 12Dxx=12y-3111 12Dyy=12x-3111 11 bσ=111111 ⏟ ⏟$ $TABLE FOR TYPE B2 ⏞Ω0·R2 ⏞Ω0·R1R2 ⏞Ω0·R2R1R2 ⏞Ω0 σ= 12122 112 21212 121id. D1-1-111-1-11 Dx-1-21-211 Dy-11-11-11 12Dxx31-1-1 12Dyy21-1-1 16Dxxx=16y-1341411 16Dyyy=16x-1121212 11 bσ=11112222 ⏟ ⏟ ⏟ ⏟$ $TABLE FOR TYPE G2 σ= 1212122 12212121221212 121121212112121id. D1-11-11-1-11-11-11 Dx -1 -3 -3 -3 -3 1 2 2 1 1 Dy -1 1 2 2 1 1 -1 -1 -1 -1 12Dxx 3 -3 -3 3 3 1 -1 -1 16Dyy 5 3 -3 -3 -1 -1 1 1 16Dyyy=16x(x+3y)(2x+3y) -3 1 -1 -1 1 1 16Dxxx=16y(x+y)(x+2y) -10 1 -1 -1 1 1 124Dxxxx=124y(2x+3y) 39 9 9 -9 -9 1 -1 -1 1216Dyyyy=124x(x+2y) 55 9 -9 -9 1 1 -1 -1 11080Dyyyyy=160x -149 9 18 18 9 9 1 1 1 1 1120Dxxxxx=160y -211 27 27 27 27 1 2 2 1 1 11 bσ= 1 1 1 3 2 4 1 1 3 9 4 2$

At the top of the tables for types ${A}_{2}$ and ${B}_{2},$ are indicated the right-cosets of ${\Omega }_{0}$ in $W,$ on each of which the weighting function is constant. In the adjoining Tessellation Diagrams (Figs 1—3) are also shown the numbers ${b}_{\sigma }$ for $\sigma \in W$ in bold numerals inside the corresponding ${w}_{\sigma }\text{-simplices}$ (which fill up ${E}^{♯}\text{),}$ as also are shown the points ${\lambda }^{\left(\sigma \right)}$ for $\lambda \in \text{interior}\left({E}^{*}\right)$ and $\sigma \in W\text{.}$ The points ${\lambda }^{\left(\sigma \right)}={\lambda }^{{w}_{\sigma }}$ ${\alpha }_{1}=2{\lambda }_{1}-{\lambda }_{2} {\alpha }_{2}=-{\lambda }_{1}+2{\lambda }_{2} {\lambda }_{1}=\frac{1}{3}\left(2{\alpha }_{1}+{\alpha }_{2}\right) {\lambda }_{2}=\frac{1}{3}\left({\alpha }_{1}+2{\alpha }_{2}\right) -{\alpha }_{0}={\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}^{\vee }={\alpha }_{1}^{\vee }+{\alpha }_{2}^{\vee } {n}_{1}=1={n}_{2} 0 {\alpha }_{1} {\alpha }_{2} {\lambda }_{1} {\lambda }_{2} \delta {\lambda }^{\left(1\right)} {\lambda }^{\left(2\right)} {\lambda }^{\left(12\right)} {\lambda }^{\left(21\right)} {\lambda }^{\left(121\right)} = {\lambda }^{0} 1 1 Fig. 1. Tessellation diagram for type A2$ ${\alpha }_{1}=2{\lambda }_{1}-{\lambda }_{2} {\alpha }_{2}=-{\lambda }_{1}+2{\lambda }_{2} {\lambda }_{1}={\alpha }_{1}+\frac{1}{2}{\alpha }_{2} {\lambda }_{2}={\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}={\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}^{\vee }={\alpha }_{1}^{\vee }+{\alpha }_{2}^{\vee } {n}_{1}=1, {n}_{2}=2 0 {\alpha }_{1} {\alpha }_{2} {\lambda }_{1} \frac{{\lambda }_{2}}{2} {\lambda }_{2} \delta {\lambda }^{\left(1\right)} {\lambda }^{\left(2\right)} {\lambda }^{\left(12\right)} {\lambda }^{\left(21\right)} {\lambda }^{\left(121\right)} {\lambda }^{\left(212\right)} {\lambda }^{\left(1212\right)} = = = {\lambda }^{0} {\lambda }^{20} {\lambda }^{120} 1 1 2 2 Fig. 2. Tessellation diagram for type B2$ for $\sigma \in W,$ comprising the intersection of the $\stackrel{\sim }{W}\prime \text{-orbit}$ of $\lambda$ with ${E}^{♯},$ are shown linked through dotted lines joining pairs of elements that are adjacent with respect to the partial-order $\leqq \text{.}$ As in the tables we have followed again the juxtaposition convention in these Diagrams, so that ${\lambda }^{\left(\sigma \right)}$ (resp. ${\lambda }^{w}\text{)}$ for $\sigma \in W$ (resp. $w\in {\stackrel{\sim }{W}}^{♯}\text{)}$ given by a reduced expression ${R}_{{i}_{1}}{R}_{{i}_{2}}\dots {R}_{{i}_{k}}$ with the indices ${i}_{1},{i}_{2},\dots ,{i}_{k}$ ranging in $\left\{1,2\right\}$ (resp. $\left\{0,1,2\right\}\text{),}$ has been abbreviated to $λ(i1i2⋯ik) (resp. λi1i2⋯ik).$

Lastly, we come to the case of type ${A}_{3}\text{.}$ Here we omit the explicit values of the 24 polynomials ${D}_{\sigma }$ 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 ${b}_{\sigma }$ for $\sigma \in {W}^{♯},$ the table on p. 676, showing also (schematically) the "linkage diagram" for the partial-order on $\stackrel{\sim }{W}$ (analogous to the linkage formed by $\left\{{\lambda }^{w} | w\in {W}^{♯}\right\}$ in the rank-two diagrams above, shown by the dotted lines there). $0 1 1 1 1 1 2 2 3 3 4 4 9 {\alpha }_{1} {\alpha }_{2} \frac{{\lambda }_{1}}{2} {\lambda }_{1} \frac{{\lambda }_{2}}{3} {\lambda }_{2} \lambda {\lambda }^{0} {\lambda }^{10} {\lambda }^{210} {\lambda }^{1210} = {\lambda }^{w} {\lambda }^{2w} {\lambda }^{0w} {\lambda }^{02w} {\lambda }^{102w} {\lambda }^{2102w} {\lambda }^{12102w} {\lambda }^{012102w} \delta {\alpha }_{1}=2{\lambda }_{1}-{\lambda }_{2} {\alpha }_{2}=-3{\lambda }_{1}+2{\lambda }_{2} {\lambda }_{1}=2{\alpha }_{1}+{\alpha }_{2} {\lambda }_{2}=3{\alpha }_{1}+2{\alpha }_{2} -{\alpha }_{0}=2{\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}^{\vee }=2{\alpha }_{1}^{\vee }+3{\alpha }_{2}^{\vee } {n}_{1}=2, {n}_{2}=3 w\stackrel{\text{def}}{=}1210 {\alpha }_{1}=2{\lambda }_{1}-{\lambda }_{2} {\alpha }_{2}=-{\lambda }_{1}+2{\lambda }_{2} {\lambda }_{1}={\alpha }_{1}+\frac{1}{2}{\alpha }_{2} {\lambda }_{2}={\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}={\alpha }_{1}+{\alpha }_{2} -{\alpha }_{0}^{\vee }={\alpha }_{1}^{\vee }+{\alpha }_{2}^{\vee } {n}_{1}=1, {n}_{2}=2 0 {\alpha }_{1} {\alpha }_{2} {\lambda }_{1} \frac{{\lambda }_{2}}{2} {\lambda }_{2} \delta {\lambda }^{\left(1\right)} {\lambda }^{\left(2\right)} {\lambda }^{\left(12\right)} {\lambda }^{\left(21\right)} {\lambda }^{\left(121\right)} {\lambda }^{\left(212\right)} {\lambda }^{\left(1212\right)} = = = {\lambda }^{0} {\lambda }^{20} {\lambda }^{120} 1 1 2 2 σ 121212 2 12 212 1212 21212 1 21 121 2121 12121 id wσ 012102w 12102w 2102w 102w 02w 0w 2w 1210=w 210 10 0 id Fig. 2. Tessellation diagram for type G2$ In the Tessellation Diagram for type ${G}_{2}$ we also exhibit, incidentally, all the $w\prime \text{-simplices}$ for $w\prime \in {\stackrel{\sim }{W}}^{+}$ satisfying $w\prime \leqq {w}_{0}\text{.}$ The four triangles shown by broken lines correspond to the four such values of $w\prime$ not lying in ${\stackrel{\sim }{W}}^{♯},$ viz. $R1R2w, R0R1R2w, R1R0R1R2w, and R2R1R0R1R2w,$ where $w$ stands for ${R}_{1}{R}_{2}{R}_{1}{R}_{0}$ given by the juxtaposition $w=1210$ which has been used throughout this diagram for the sake of clarity in printing. $TABLE FOR TYPE A3 σ∈W♯ bσ λ(σ) wσ linkage R1R3 1 (1-x,x+y+z,1-z) w0=R2R1R3R0 R2R1R3 1 (1-x-y,x+y+z,1-y-z) R1R3R0 R1R2R1R3 1 (1-y,y+z,1-x-y-z) R3R0 R3R2R1R3 1 (1-x-y-z,x+y,1-y) R1R0 R1R3R2R1R3 2 (1-y-z,y,1-x-y) R0 id. 0 (x,y,z) id.$ In this case there are two elements of ${\stackrel{\sim }{W}}^{+}$ that lie below ${w}_{0}$ with respect to $\leqq$ but do not belong to ${\stackrel{\sim }{W}}^{♯},$ viz. ${R}_{2}{R}_{1}{R}_{0}$ and ${R}_{2}{R}_{3}{R}_{0}\text{.}$

## 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).