${R}_{1}(u){R}_{2}(\mathrm{uv}){R}_{1}(v)={R}_{2}(v){R}_{1}(\mathrm{uv}){R}_{2}(u)$ |
${R}_{1}(x){R}_{2}(x+y){R}_{1}(y)={R}_{2}(y){R}_{1}(x+y){R}_{2}(x)$ |
$-2/3$ | $0$ | $1$ | $2$ | $4$ | $6$ | $8$ | |
$\backslash mathbbR$ | $0$ | $0$ | ${A}_{1}$ | ${A}_{2}$ | ${C}_{3}$ | ${C}_{3}.{H}_{14}$ | ${F}_{4}$ |
$\backslash mathbbC$ | $0$ | ${T}_{2}$ | ${A}_{2}$ | $2{A}_{2}$ | ${A}_{5}$ | ${A}_{5}.{H}_{20}$ | ${E}_{6}$ |
$\backslash mathbbH$ | ${A}_{1}$ | $3{A}_{1}$ | ${C}_{3}$ | ${A}_{5}$ | ${D}_{6}$ | ${D}_{6}.{H}_{32}$ | ${E}_{7}$ |
$\backslash mathbbS$ | ${A}_{1}.{H}_{4}$ | $(3{A}_{1}).{H}_{8}$ | ${C}_{3}.{H}_{14}$ | ${A}_{5}.{H}_{20}$ | ${D}_{6}.{H}_{32}$ | ${D}_{6}.{H}_{32}.{H}_{44}$ | ${E}_{7}.{H}_{56}$ |
$\backslash mathbbO$ | ${G}_{2}$ | ${D}_{4}$ | ${F}_{4}$ | ${E}_{6}$ | ${E}_{7}$ | ${E}_{7}.{H}_{56}$ | ${E}_{8}$ |
$m$ | -2 | -2/3 | 0 | 1 | 2 | 4 | 6 | 8 |
$G$ | ${A}_{2}$ | 0 | ${T}_{2}$ | ${A}_{2}$ | ${A}_{2}\oplus {A}_{2}$ | ${A}_{5}$ | ${A}_{5}.{H}_{20}$ | ${E}_{6}$ |
$\Lambda}^{2}(V)={V}_{2}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{S}^{2}(V)={V}^{*}\oplus {V}^{2$ |
${A}_{2}$ | ${A}_{2}$ | $2{A}_{2}$ | ${A}_{5}$ | ${E}_{6}$ | |
$V$ | $-[1,0]$ | $[2,0]$ | $[0,1|1,0]$ | $[0,1,0,0,0]$ | $[1,0,0,0,0,0]$ |
$\backslash mathfrakg$ | $[1,1]$ | $[1,1]$ | $[1,1|0,0]$ | $[1,0,0,0,1]$ | $[0,1,0,0,0,0]$ |
${V}_{2}$ | $-[2,0]$ | $[2,1]$ | $[1,0|2,0]$ | $[1,0,1,0,0]$ | $[0,0,1,0,0,0]$ |
0 | 1 | 2 | |
$\backslash mathfrakg$ | ${\lambda}_{1}+{\lambda}_{5}$ | ${\lambda}_{3}$ | 1 |
$V$ | ${\lambda}_{2}$ | ${\lambda}_{5}$ | |
${V}^{*}$ | ${\lambda}_{4}$ | ${\lambda}_{1}$ | |
${V}^{2}$ | $2{\lambda}_{2}$ | $({\lambda}_{2}+{\lambda}_{5})$ | ${\lambda}_{5}$ |
${V}_{2}$ | $({\lambda}_{1}+{\lambda}_{3})$ | $({\lambda}_{2}+{\lambda}_{5})\oplus {\lambda}_{1}$ | ${\lambda}_{4}$ |
$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{dim}{}_{q}(V)}& =\hfill & \frac{[3m/2][m+1]}{[m/2]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}({V}^{2})}& =\hfill & \frac{[m+2][m+1][3m/2+1][3m/2]}{[2][m/2+1][m/2]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}({V}_{2})}& =\hfill & \frac{[m+1][m-2][3m/2][3m/2+1]}{[2][m/2][m/2-1]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}(\backslash mathfrakg)}& =\hfill & \frac{[m-2][m][m+1][3m/2+1]}{[m/2-1][m/2][m/2+2]}\hfill \end{array}$ |
${V}^{p}$ | $4{p}^{2}/3+2\mathrm{pm}$ |
${V}_{2}{V}^{p}$ | $4{p}^{2}/3+2p(3m+5)/3+4(3m+1)/3$ |
${V}^{*}{V}^{p}$ | $4{p}^{2}/3+2p(3m+2)/3+2(3m+2)/3$ |
$\backslash mathfrakg{V}^{p}$ | $4{p}^{2}/3+2p(m+1)+3m$ |
$V}^{p+1}\stackrel{2p+2}{\to}{V}_{2}{V}^{p-1}\stackrel{2m+2p-2}{\to}{V}^{*}{V}^{p-1$ |
$V}^{*}{V}^{p}\stackrel{m+2p+2}{\to}\backslash mathfrakg{V}^{p-1}\stackrel{3m+2p-2}{\to}{V}^{p-1$ |
$m$ | $-2/3$ | 0 | 1 | 2 | 4 | 6 | 8 |
$G$ | ${A}_{1}$ | $3{A}_{1}$ | ${C}_{3}$ | ${A}_{5}$ | ${D}_{6}$ | ${D}_{6}.{H}_{32}$ | ${E}_{7}$ |
$m$ | -3 | -8/3 | -5/2 | -2 | -4/3 | -1 |
$G$ | ${D}_{5}$ | ${B}_{3}$ | ${G}_{2}$ | $2{A}_{1}$ | 0 | $D(2,1,\alpha )$ |
$\Lambda}^{2}(V)=1\oplus {V}_{2}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{S}^{2}(V)=\backslash mathfrakg\oplus {V}^{2$ |
${A}_{1}$ | $3{A}_{1}$ | ${C}_{3}$ | ${A}_{5}$ | ${D}_{6}$ | ${E}_{7}$ | |
$V$ | $[3]$ | $[1,1,1]$ | $[0,0,1]$ | $[0,0,1,0,0]$ | $[0,0,0,0,0,1]$ | $[0,0,0,0,0,0,1]$ |
${V}_{2}$ | $[4]$ | $[2,2,0]$ | $[0,2,0]$ | $[0,1,0,1,0]$ | $[0,0,0,1,0,0]$ | $[0,0,0,0,0,1,0]$ |
0 | 1 | 2 | |
$\backslash mathfrakg$ | ${\lambda}_{2}$ | ${\lambda}_{5}$ | 1 |
$V$ | ${\lambda}_{6}$ | ${\lambda}_{1}$ | |
${V}^{2}$ | $2{\lambda}_{6}$ | $({\lambda}_{1}+{\lambda}_{6})$ | $2{\lambda}_{1}$ |
${V}_{2}$ | ${\lambda}_{4}$ | $({\lambda}_{1}+{\lambda}_{6})$ | ${\lambda}_{2}$ |
${D}_{5}$ | ${B}_{3}$ | ${G}_{2}$ | $2{A}_{1}$ | |
$V$ | $-[1,0,0,0,0]$ | $-[0,0,1]$ | $-[1,0]$ | $-[1]\otimes [1]$ |
$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{dim}{}_{q}(\backslash mathfrakg)}& =\hfill & \frac{[2m+3][3m/2+2][3m/2]}{[m/2][m/2+2]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}(V)}& =\hfill & \frac{[m+2][3m/2+2][2m+2]}{[m/2+1][m+1]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}({V}^{2})}& =\hfill & \frac{[m+3][3m/2+2][3m/2+3][2m+2][2m+3]}{[2][m/2+1][m/2+2][m+1]}\hfill \\ \multicolumn{1}{c}{\mathrm{dim}{}_{q}({V}_{2})}& =\hfill & \frac{[2m+3][2m+2][3m/2][3m/2+3]}{[2][m/2][m/2+1]}\hfill \end{array}$ |
$V}^{p-1}\stackrel{2m+p+1}{\to}\backslash mathfrakg{V}^{p-1}\stackrel{m+p+1}{\to}{V}_{2}{V}^{p-1}\stackrel{p+1}{\to}{V}^{p+1$ |
${V}^{p}$ | $3{p}^{2}/4+p(3m+3)/2$ |
$\backslash mathfrakg{V}^{p}$ | $3{p}^{2}/4+p(3m+5)/2+2m+2$ |
${V}_{2}{V}^{p}$ | $3{p}^{2}/4+p(3m+7)+3m+4$ |
$\Lambda}^{2}(V)=\backslash mathfrakg\oplus {V}_{2}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{S}^{2}(V)=1\oplus {V}^{2$ |
$m$ | -5/2 | -3/2 | -4/3 | -1 | -2/3 | 0 | 1 | 2 | 4 | 6 | 8 |
$G$ | $1$ | $\mathrm{SOSp}(2,1)$ | ${A}_{1}$ | ${A}_{2}$ | ${G}_{2}$ | ${D}_{4}$ | ${F}_{4}$ | ${E}_{6}$ | ${E}_{7}$ | ${E}_{7}.{H}_{56}$ | ${E}_{8}$ |
${\Lambda}^{2}(\backslash mathfrakg)=\backslash mathfrakg\oplus X\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{S}^{2}(\backslash mathfrakg)=1\oplus Y(\alpha )\oplus Y(\beta )\oplus Y(\gamma )$ |
$G$ | $\backslash mathfrakg$ | $X$ | $Y(\alpha )$ | $Y(\beta )$ | $Y(\gamma )$ |
$\backslash spl(n)$ | ${\lambda}_{1}+{\lambda}_{n}$ | ${\lambda}_{2}+2{\lambda}_{n}$ | $2{\lambda}_{1}+2{\lambda}_{2}$ | ${\lambda}_{1}+{\lambda}_{n}$ | ${\lambda}_{2}+{\lambda}_{n-1}$ |
$\backslash so(n)$ | ${\lambda}_{2}$ | ${\lambda}_{1}+{\lambda}_{3}$ | $2{\lambda}_{2}$ | ${\lambda}_{4}$ | $2{\lambda}_{2}$ |
$\backslash syp(2n)$ | $2{\lambda}_{1}$ | $2{\lambda}_{1}+{\lambda}_{2}$ | $2{\lambda}_{2}$ | $4{\lambda}_{1}$ | ${\lambda}_{2}$ |
$\mathrm{dim}{}_{q}(\backslash mathfrakg)=\frac{[\alpha +2\beta +2\gamma ][2\alpha +\beta +2\gamma ][2\alpha +2\beta +\gamma ]}{[\alpha ][\beta ][\gamma ]}$ |
$S}^{2}(\backslash mathfrakg)=1\oplus Y\oplus \backslash mathfrak{g}^{2$ |
$\alpha $ | $\beta $ | $\gamma $ | ||
$\backslash spl(n)$ | -2 | 2 | $n$ | $\alpha +\beta =0$ |
$\backslash so(n)$ | -2 | 4 | $n-2$ | $2\alpha +\beta =0$ |
$\backslash syp(2n)$ | -2 | 4 | $-2n-2$ | $2\alpha +\beta =0$ |
$\backslash mathbbO$ | -2 | $m+4$ | $2m+4$ | $2\alpha +2\beta -\gamma =0$ |
$\backslash mathbbH$ | -2 | $m$ | $m+4$ | $2\alpha -\beta +\gamma =0$ |
0 | 1 | 2 | 3 | 4 | |
$\backslash mathfrakg$ | $2{\lambda}_{1}$ | ${\lambda}_{1}$ | 1 | ||
$X$ | $(2{\lambda}_{1}+{\lambda}_{2})$ | $({\lambda}_{1}+{\lambda}_{2})\oplus 3{\lambda}_{1}$ | ${\lambda}_{2}\oplus 2{\lambda}_{1}$ | ${\lambda}_{1}$ | |
$Y(\alpha )$ | $2{\lambda}_{2}$ | $({\lambda}_{1}+{\lambda}_{2})$ | $2{\lambda}_{1}$ | $\lambda -1$ | 1 |
$Y(\beta )$ | $4{\lambda}_{1}$ | $3{\lambda}_{1}$ | |||
$Y(\gamma )$ | ${\lambda}_{2}$ | ${\lambda}_{1}$ |
$\begin{array}{ccc}\multicolumn{1}{c}{\mathrm{dim}(\backslash mathfrakg)}& =\hfill & n(n+1)/2\hfill \\ \multicolumn{1}{c}{\mathrm{dim}(X)}& =\hfill & (n-2)n(n+1)(n+3)/8\hfill \\ \multicolumn{1}{c}{\mathrm{dim}(Y(\alpha ))}& =\hfill & (n-2)(n-1)n(n+3)/12\hfill \\ \multicolumn{1}{c}{\mathrm{dim}(Y(\beta ))}& =\hfill & n(n+1)(n+2)(n+3)/24\hfill \\ \multicolumn{1}{c}{\mathrm{dim}(Y(\gamma ))}& =\hfill & (n-2)(n+1)/2\hfill \end{array}$ |
0 | 1 | 2 | 3 | 4 | $\mathrm{dim}$ | |
$\backslash mathfrakg$ | ${\lambda}_{1}$ | ${\lambda}_{7}$ | 1 | 190 | ||
$X$ | ${\lambda}_{3}$ | $({\lambda}_{1}+{\lambda}_{7})\oplus {\lambda}_{2}$ | ${\lambda}_{1}\oplus {\lambda}_{6}$ | ${\lambda}_{7}$ | 17765 | |
$\backslash mathfrak{g}^{2}$ | $2{\lambda}_{1}$ | $({\lambda}_{1}+{\lambda}_{7})$ | ${\lambda}_{1}\oplus 2{\lambda}_{7}$ | ${\lambda}_{7}$ | 1 | 15504 |
$Y$ | ${\lambda}_{6}$ | ${\lambda}_{2}\oplus {\lambda}_{7}$ | ${\lambda}_{1}$ | 2640 |
0 | 1 | 2 | 3 | 4 | $\mathrm{dim}$ | |
$\backslash mathfrakg$ | ${\lambda}_{2}$ | ${\lambda}_{5}$ | 1 | 99 | ||
$X$ | $({\lambda}_{1}+{\lambda}_{3})$ | $({\lambda}_{2}+{\lambda}_{5})\oplus ({\lambda}_{1}+{\lambda}_{6})$ | ${\lambda}_{2}\oplus {\lambda}_{4}$ | ${\lambda}_{5}$ | 4752 | |
$Y(\alpha )$ | $2{\lambda}_{2}$ | $({\lambda}_{1}+{\lambda}_{5})$ | ${\lambda}_{2}\oplus 2{\lambda}_{5}$ | ${\lambda}_{5}$ | 1 | 3927 |
$Y(\beta )$ | ${\lambda}_{4}$ | ${\lambda}_{5}\oplus ({\lambda}_{1}+{\lambda}_{6})$ | ${\lambda}_{2}$ | 945 | ||
$Y(\gamma )$ | $2{\lambda}_{1}$ | 77 |
0 | 1 | 2 |
78 | 64+13 | 1 |
$R(x)={P}_{X}\oplus \left(\frac{\alpha +x}{\alpha -x}\right){P}_{Y(\alpha )}\oplus \left(\frac{\beta +x}{\beta -x}\right){P}_{Y(\beta )}\oplus \left(\frac{\gamma +x}{\gamma -x}\right){P}_{Y(\gamma )}\oplus {P}_{\backslash mathfrakg}\oplus {P}_{1}$ |
$\begin{array}{ccc}\multicolumn{1}{c}{a}& =\hfill & \alpha +\beta +\gamma \hfill \\ \multicolumn{1}{c}{b}& =\hfill & \alpha \beta \gamma \hfill \\ \multicolumn{1}{c}{h(x)}& =\hfill & (x-\alpha )(x-\beta )(x-\gamma )\hfill \end{array}$ |
${P}_{\backslash mathfrakg}=\frac{1}{h(x)}\left(\begin{array}{ccc}\hfill -h(x)+2b\hfill & \hfill -2x\hfill & \hfill 0\hfill \\ \hfill 2\mathrm{abx}\hfill & \hfill -h(-x)+2b\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill h(x)\hfill \end{array}\right)$ |
${P}_{1}=\frac{1}{h(x)}\left(\begin{array}{cc}\hfill -h(x)+2b\hfill & \hfill \mathrm{bx}/(x-a)\hfill \\ \hfill 4\mathrm{ax}(x+a)\hfill & \hfill (h(-x)+2b)(x+a)/(x-a)\hfill \end{array}\right)$ |
$\begin{array}{ccc}\multicolumn{1}{c}{{V}_{a}(Y)}& =\hfill & Y\oplus \backslash mathfrakg\oplus 1\hfill \\ \multicolumn{1}{c}{{V}_{a}(X)}& =\hfill & X\oplus {Y}^{*}\oplus 2\backslash mathfrakg\oplus 1\hfill \end{array}$ |
$\begin{array}{ccc}\multicolumn{1}{c}{{V}_{a}(Y(\alpha ))}& =\hfill & Y(\alpha )\oplus \backslash mathfrakg\oplus 1\hfill \\ \multicolumn{1}{c}{{V}_{a}(X)}& =\hfill & X\oplus Y(\alpha )\oplus 2\backslash mathfrakg\oplus 1\hfill \end{array}$ |
$m$ | -2/3 | 0 | 1 | 2 | 4 | 6 | 8 |
$G$ | 0 | 0 | ${A}_{1}$ | ${A}_{2}$ | ${C}_{3}$ | ${C}_{3}.{H}_{14}$ | ${F}_{4}$ |
$\Lambda}^{2}(V)=\backslash mathfrakg\oplus {V}_{2}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{S}^{2}(V)=\backslash mathbbC\oplus V\oplus {V}^{2$ |
${A}_{1}$ | ${A}_{2}$ | ${C}_{3}$ | ${F}_{4}$ | |
$V$ | $[4]$ | $[1,1]$ | $[0,1,0]$ | $[0,0,0,1]$ |
$\backslash mathfrakg$ | $[2]$ | $[1,1]$ | $[2,0,0]$ | $[1,0,0,0]$ |
${V}_{2}$ | $[6]$ | $[3,0]$ | $[1,0,1]$ | $[0,0,1,0]$ |
0 | 1 | 2 | |
$\backslash mathfrakg$ | $2{\lambda}_{1}$ | ${\lambda}_{3}$ | 1 |
$V$ | ${\lambda}_{2}$ | ${\lambda}_{1}$ | |
${V}^{2}$ | $2{\lambda}_{2}$ | $({\lambda}_{1}+{\lambda}_{2})$ | ${\lambda}_{3}$ |
${V}_{2}$ | ${\lambda}_{1}+{\lambda}_{3}$ | ${\lambda}_{1}\oplus ({\lambda}_{1}+{\lambda}_{2})$ | ${\lambda}_{2}$ |
$\mathrm{dim}{}_{q}(V)=\frac{[m/4+1]}{[m/2+2]}\frac{[5m/2-2]}{[5m/4-1]}\frac{[m]}{[m/2]}[3m/2+1]$ |
$\begin{array}{ccccc}\hfill 1\hfill & \hfill \backslash mathfrakg\hfill & \hfill {V}^{2}\hfill & \hfill {V}_{2}\hfill & \hfill V\hfill \\ \hfill 0\hfill & \hfill 5m-4\hfill & \hfill 6m+4\hfill & \hfill 6m\hfill & \hfill 3m\hfill \end{array}$ |
${F}_{4}:1\stackrel{36}{\to}\backslash mathfrakg\stackrel{-12}{\to}{V}^{2}\stackrel{24}{\to}{V}_{2}\stackrel{4}{\to}V$ |
$C}_{3}:\begin{array}{ccccccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill V\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \stackrel{12}{\nearrow}\hfill & \hfill \hfill & \hfill \stackrel{-4}{\searrow}\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \stackrel{16}{\to}\hfill & \hfill \backslash mathfrakg\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {V}_{2}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \stackrel{-4}{\searrow}\hfill & \hfill \hfill & \hfill \stackrel{12}{\nearrow}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {V}^{2}\hfill & \hfill \hfill & \hfill \hfill \end{array$ |
$A}_{1}:1\stackrel{1}{\to}\backslash mathfrakg\stackrel{9}{\to}V\stackrel{-4}{\to}{V}_{2}\stackrel{-3}{\to}{V}^{2$ |
$\begin{array}{ccccccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill V\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \stackrel{-2m+4}{\nearrow}\hfill & \hfill \hfill & \hfill \stackrel{3m}{\searrow}\hfill & \hfill \hfill \\ \hfill 1\hfill & \hfill \stackrel{5m-4}{\to}\hfill & \hfill \backslash mathfrakg\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {V}_{2}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \stackrel{m+8}{\searrow}\hfill & \hfill \hfill & \hfill \stackrel{-4}{\nearrow}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {V}^{2}\hfill & \hfill \hfill & \hfill \hfill \end{array}$ |