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
aram@unimelb.edu.au

Last update: 2 April 2014

§ 3. Evidence in low ranks

Given below are the explicit expressions for the polynomial $D\left(\lambda \right)=D(x_1,x_2,\dots ,x_l)$ 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: $$\begin{array}{c} α1 α2 α1 α2 α1 α2 α1 α2 α3 \end{array}$$ $$\begin{array}{ccc}{A}_2\text{:}& 2·D(x,y)=xy(x+y),& \left|W\right|=6\text{;}\\ B_2\text{:}& 6·D(x,y)=xy(x+y)(x+2y),& \left|W\right|=8\text{;}\\ G_2\text{:}& 120·D(x,y)=xy(x+y)(x+2y)(x+3y)(2x+3y),& \left|W\right|=12\text{;}\\ A_3\text{:}& 12·D(x,y,z)=xyz(x+y)(y+z)(x+y+z),& \left|W\right|=24\text{;}\end{array}$$ where $x=x_1,$ $y=y_2,$ $z=x_3\text{.}$ The integer coefficient in front of $D(x,\dots )$ comes from the normalization $D(1,\dots ,1)=1\text{.}$ The Coxeter diagrams of $\stackrel{\sim }{W}\prime$ are then as follows: $$\begin{array}{c} A2: R1 R2 R0 B2: R1 R2 R0 4 4 G2: R0 R1 R2 6 A3: R1 R2 R0 R3 \end{array}$$

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 $$\frac{{\partial }^{2}D}{\partial x\partial y}=\frac{{\partial }^{2}D}{\partial x^2}+\frac{1}{c}\frac{{\partial }^{2}D}{\partial y^2}$$ 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 $\left|W\right|\times \left|W\right|$ 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 $12\times 12$ matrix in the case of type $G_2$ was obtained from a computer print-out of the $D_{\sigma }\text{'s.}$ $$\begin{array}{c}\text{TABLE FOR TYPE}\hspace{0.17em}{A}_2\\ \begin{array}{ccccccc}& \multicolumn{3}{c}{\underset{\stackrel{{\Omega }_0{R}_1{R}_2{R}_1}{⏞}}{}}& \multicolumn{3}{c}{\underset{\stackrel{{\Omega }_0}{⏞}}{}}\\ \sigma =& 121& 1& 2& 12& 21& \text{id.}\\ D& -1& -1& -1& 1& 1& 1\\ D_x& 1& 1& -1& -1\\ D_y& 1& -1& 1& & -1\\ \frac{1}{2}{D}_{xx}=\frac{1}{2}y& -3& 1& 1& & 1\\ \frac{1}{2}{D}_{yy}=\frac{1}{2}x& -3& 1& 1& 1\\ 1& 1\\ b_{\sigma }=& 1& 1& 1& 1& 1& 1\\ & \multicolumn{3}{c}{\stackrel{⏟}{}}& \multicolumn{3}{c}{\stackrel{⏟}{}}\end{array}\end{array}$$ $$\begin{array}{c}\text{TABLE FOR TYPE}\hspace{0.17em}{B}_2\\ \begin{array}{ccccccccc}& \multicolumn{2}{c}{\underset{\stackrel{{\Omega }_0·R_2}{⏞}}{}}& \multicolumn{2}{c}{\underset{\stackrel{{\Omega }_0·R_1{R}_2}{⏞}}{}}& \multicolumn{2}{c}{\underset{\stackrel{{\Omega }_0·R_2{R}_1{R}_2}{⏞}}{}}& \multicolumn{2}{c}{\underset{\stackrel{{\Omega }_0}{⏞}}{}}\\ \sigma =& 1212& 2& 1& 12& 21& 212& 121& \text{id.}\\ D& 1& -1& -1& 1& 1& -1& -1& 1\\ D_x& -1& -2& 1& -2& 1& & 1\\ D_y& -1& 1& -1& 1& -1& 1\\ \frac{1}{2}{D}_{xx}& 3& & 1& & -1& & -1\\ \frac{1}{2}{D}_{yy}& 2& 1& & -1& & -1\\ \frac{1}{6}{D}_{xxx}=\frac{1}{6}y& -13& 4& 1& 4& 1& & 1\\ \frac{1}{6}{D}_{yyy}=\frac{1}{6}x& -11& 2& 1& 2& 1& 2\\ 1& 1\\ b_{\sigma }=& 1& 1& 1& 1& 2& 2& 2& 2& & \\ & \multicolumn{2}{c}{\stackrel{⏟}{}}& \multicolumn{2}{c}{\stackrel{⏟}{}}& \multicolumn{2}{c}{\stackrel{⏟}{}}& \multicolumn{2}{c}{\stackrel{⏟}{}}\end{array}\end{array}$$ $$\begin{array}{c}\text{TABLE FOR TYPE}\hspace{0.17em}{G}_2\\ \begin{array}{ccccccccccccc}\sigma =& 121212& 2& 12& 212& 1212& 21212& 1& 21& 121& 2121& 12121& \text{id.}\\ D& 1& -1& 1& -1& 1& -1& -1& 1& -1& 1& -1& 1\\ D_x& -1& -3& -3& -3& -3& & 1& 2& 2& 1& 1\\ D_y& -1& 1& 2& 2& 1& 1& -1& -1& -1& -1\\ \frac{1}{2}{D}_{xx}& 3& -3& -3& 3& 3& & 1& & & -1& -1\\ \frac{1}{6}{D}_{yy}& 5& 3& & & -3& -3& -1& -1& 1& 1\\ \frac{1}{6}{D}_{yyy}=\frac{1}{6}x(x+3y)(2x+3y)& -3& 1& -1& -1& 1& 1\\ \frac{1}{6}{D}_{xxx}=\frac{1}{6}y(x+y)(x+2y)& -10& & & & & & 1& -1& -1& 1& 1\\ \frac{1}{24}{D}_{xxxx}=\frac{1}{24}y(2x+3y)& 39& 9& 9& -9& -9& & 1& & & -1& -1\\ \frac{1}{216}{D}_{yyyy}=\frac{1}{24}x(x+2y)& 55& 9& & & -9& -9& 1& 1& -1& -1\\ \frac{1}{1080}{D}_{yyyyy}=\frac{1}{60}x& -149& 9& 18& 18& 9& 9& 1& 1& 1& 1\\ \frac{1}{120}{D}_{xxxxx}=\frac{1}{60}y& -211& 27& 27& 27& 27& & 1& 2& 2& 1& 1\\ 1& 1\\ b_{\sigma }=& 1& 1& 1& 3& 2& 4& 1& 1& 3& 9& 4& 2\end{array}\end{array}$$

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 ${\lambda }^{\left(\sigma \right)}={\lambda }^{w_{\sigma }}$ $$\begin{array}{c}\begin{array}{c} α1=2λ1-λ2 α2=-λ1+2λ2 λ1=13(2α1+α2) λ2=13(α1+2α2) -α0=α1+α2 -α0∨=α1∨+α2∨ n1=1=n2 0 α1 α2 λ1 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) = λ0 1 1 \end{array}\\ \text{Fig. 1. Tessellation diagram for type}\hspace{0.17em}{A}_2\end{array}$$ $$\begin{array}{c}\begin{array}{c} α1=2λ1-λ2 α2=-λ1+2λ2 λ1=α1+12α2 λ2=α1+α2 -α0=α1+α2 -α0∨=α1∨+α2∨ n1=1, n2=2 0 α1 α2 λ1 λ22 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) λ(212) λ(1212) = = = λ0 λ20 λ120 1 1 2 2 \end{array}\\ \text{Fig. 2. Tessellation diagram for type}\hspace{0.17em}{B}_2\end{array}$$ 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 $\{1,2\}$ (resp. $\{0,1,2\}\text{),}$ has been abbreviated to $${\lambda }^{\left(i_1{i}_2\cdots i_k\right)}\text{(resp.}\hspace{0.17em}{\lambda }^{i_1{i}_2\cdots i_k}\text{).}$$

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 $\left\{{\lambda }^w\hspace{0.17em}\right|\hspace{0.17em}w\in W^{♯}\}$ in the rank-two diagrams above, shown by the dotted lines there). $$\begin{array}{c}\begin{array}{c} 0 1 1 1 1 1 2 2 3 3 4 4 9 α1 α2 λ12 λ1 λ23 λ2 λ λ0 λ10 λ210 λ1210 = λw λ2w λ0w λ02w λ102w λ2102w λ12102w λ012102w δ α1=2λ1-λ2 α2=-3λ1+2λ2 λ1=2α1+α2 λ2=3α1+2α2 -α0=2α1+α2 -α0∨=2α1∨+3α2∨ n1=2, n2=3 w=def1210 α1=2λ1-λ2 α2=-λ1+2λ2 λ1=α1+12α2 λ2=α1+α2 -α0=α1+α2 -α0∨=α1∨+α2∨ n1=1, n2=2 0 α1 α2 λ1 λ22 λ2 δ λ(1) λ(2) λ(12) λ(21) λ(121) λ(212) λ(1212) = = = λ0 λ20 λ120 1 1 2 2 \end{array}\\ \begin{array}{ccccccccccccc}\sigma & 121212& 2& 12& 212& 1212& 21212& 1& 21& 121& 2121& 12121& \text{id}\\ w_{\sigma }& 012102w& 12102w& 2102w& 102w& 02w& 0w& 2w& 1210=w& 210& 10& 0& \text{id}\end{array}\\ \text{Fig. 2. Tessellation diagram for type}\hspace{0.17em}{G}_2\end{array}$$ 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. $$R_1{R}_{2}w,R_0{R}_1{R}_{2}w,R_1{R}_0{R}_1{R}_{2}w,\text{and}{R}_2{R}_1{R}_0{R}_1{R}_{2}w,$$ 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. $$\begin{array}{c}\text{TABLE FOR TYPE}\hspace{0.17em}{A}_3\\ \begin{array}{ccccc}\sigma \in W^{♯}& b_{\sigma }& {\lambda }^{\left(\sigma \right)}& w_{\sigma }& \text{linkage}\\ R_1{R}_3& 1& (1-x,x+y+z,1-z)& w_0=R_2{R}_1{R}_3{R}_0& \begin{array}{c} \end{array}\\ R_2{R}_1{R}_3& 1& (1-x-y,x+y+z,1-y-z)& R_1{R}_3{R}_0\\ R_1{R}_2{R}_1{R}_3& 1& (1-y,y+z,1-x-y-z)& R_3{R}_0\\ R_3{R}_2{R}_1{R}_3& 1& (1-x-y-z,x+y,1-y)& R_1{R}_0\\ R_1{R}_3{R}_2{R}_1{R}_3& 2& (1-y-z,y,1-x-y)& R_0\\ \text{id.}& 0& (x,y,z)& \text{id.}\end{array}\end{array}$$ 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).

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