%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Some MetaPost figures by Anthony Phan. % file: vanhaeck.mp (designed for Pol Vanhaecke) % last modification: May 2, 2002. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Licence? Feel-free-to-send-me-a-postcard % % Anthony Phan, % % D\'epartement de Math\'ematiques, % SP2MI, T\'el\'eport 2, % Boulevard Marie et Pierre Curie, % BP 30179, % F-86962 Futuroscope-Chasseneuil cedex. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% input dynkin; verbatimtex \font\tenfrak=eufm10 \font\sevenfrak=eufm7 \font\fivefrak=eufm5 \newfam\frakfam \textfont\frakfam=\tenfrak \scriptfont\frakfam=\sevenfrak \scriptscriptfont\frakfam=\fivefrak \def\frak{\fam\frakfam\tenfrak} \def\mathfrak#1{{\frak#1}} \def\gobble#1{}\gobble\enddef\gobble\enddef\gobble\enddef etex defaultfont:="cmr7"; %unit:=1/3cm; unit:=1cm; e1:=unit*right; e2:=unit*up; % % TABLE 1 % beginfig(1); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=0.5(z3-z2)=z4-z3=e1; z5=0.5[z1,z4]+e2; simple_link(z1,z2); dotted_link(z2,z3); simple_link(z3,z4); simple_link(z4,z5); simple_link(z5,z1); circ_marks(z1,z2,z3,z4,z5); name_point.top(btex $\scriptstyle 0$ etex,z5); name_point.bot(btex $\scriptstyle 1$ etex,z1); name_point.bot(btex $\scriptstyle 2$ etex,z2); name_point.bot(btex $\scriptstyle r-1$ etex,z3); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z4); label(btex $\displaystyle\strut\mathfrak a_r^{(1)}\atop \displaystyle\strut(r>1)$ etex,z1-e1); endfig; % beginfig(2); % z1=origin; z2-z1=e1; % quadruple_link(z1,z2); % circ_marks(z1,z2); % name_point.bot(btex $\scriptstyle 1$ etex,z1); % name_point.bot(btex $\scriptstyle 1$ etex,z2); % label(btex $\mathfrak a_1^{(1)}$ etex,z1-e1); % endfig; beginfig(2); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z1'-z1=z1-z1''=e2; z2-z1=0.5(z3-z2)=z4-z3=e1; simple_link(z1,z1'); simple_link(z1,z1''); simple_link(z1,z2); dotted_link(z2,z3); double_arrow(z3,z4); circ_marks(z1,z1',z1'',z2,z3,z4); name_point.lft(btex $\scriptstyle 1$ etex,z1'); name_point.lft(btex $\scriptstyle 2$ etex,z1); name_point.lft(btex $\scriptstyle 0$ etex,z1''); name_point.bot(btex $\scriptstyle 3$ etex,z2); name_point.bot(btex $\scriptstyle r-1$ etex,z3); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z4); label(btex $\displaystyle\strut\mathfrak b_r^{(1)}\atop \displaystyle\strut(r>2)$ etex,z1-e1); endfig; beginfig(3); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=0.5(z3-z2)=z4-z3=e1; double_arrow(z1,z2); dotted_link(z2,z3); double_arrow(z4,z3); circ_marks(z1,z2,z3,z4); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle r-1$ etex,z3); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z4); label(btex $\displaystyle\strut\mathfrak c_r^{(1)}\atop \displaystyle\strut(r>1)$ etex,z1-e1); endfig; beginfig(4); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=0.5(z3-z2)=z4-z3=e1; z1'-z1=z1-z1''=z4'-z4=z4-z4''=e2; simple_link(z1,z2); dotted_link(z2,z3); simple_link(z3,z4); simple_link(z1,z1'); simple_link(z1,z1''); simple_link(z4,z4'); simple_link(z4,z4''); circ_marks(z1,z1',z1'',z2,z3,z4,z4',z4''); name_point.lft(btex $\scriptstyle 2$ etex,z1); name_point.lft(btex $\scriptstyle 1$ etex,z1'); name_point.lft(btex $\scriptstyle 0$ etex,z1''); name_point.bot(btex $\scriptstyle 3$ etex,z2); name_point.bot(btex $\scriptstyle r-3$ etex,z3); name_point.rt(btex $\scriptstyle r-2$ etex,z4); name_point.rt(btex $\scriptstyle r-1$ etex,z4'); name_point.rt(btex $\scriptstyle r\vphantom1$ etex,z4''); label(btex $\displaystyle\strut\mathfrak d_r^{(1)}\atop \displaystyle\strut(r>3)$ etex,z1-e1); endfig; beginfig(5); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=z4-z3=z5-z4=e1; z3''-z3'=z3'-z3=e2; simple_link(z1,z2); simple_link(z2,z3); simple_link(z3,z4); simple_link(z4,z5); simple_link(z3,z3'); simple_link(z3',z3''); circ_marks(z1,z2,z3,z3',z3'',z4,z5); name_point.bot(btex $\scriptstyle 1$ etex,z1); name_point.bot(btex $\scriptstyle 3$ etex,z2); name_point.bot(btex $\scriptstyle 4$ etex,z3); name_point.rt(btex $\scriptstyle 2$ etex,z3'); name_point.rt(btex $\scriptstyle 0$ etex,z3''); name_point.bot(btex $\scriptstyle 5$ etex,z4); name_point.bot(btex $\scriptstyle 6$ etex,z5); label(btex $\mathfrak e_6^{(1)}$ etex,z1-e1); endfig; beginfig(6); % unit:=1/3cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=z4-z3=z5-z4=z6-z5=z7-z6=e1; z4'-z4=e2; simple_link(z1,z2); simple_link(z2,z3); simple_link(z3,z4); simple_link(z4,z5); simple_link(z5,z6); simple_link(z6,z7); simple_link(z4,z4'); circ_marks(z1,z2,z3,z4,z4',z5,z6,z7); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle 3$ etex,z3); name_point.bot(btex $\scriptstyle 4$ etex,z4); name_point.rt(btex $\scriptstyle 2$ etex,z4'); name_point.bot(btex $\scriptstyle 5$ etex,z5); name_point.bot(btex $\scriptstyle 6$ etex,z6); name_point.bot(btex $\scriptstyle 7$ etex,z7); label(btex $\mathfrak e_7^{(1)}$ etex,z1-e1); endfig; beginfig(7); % unit:=2/7cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=z4-z3=z5-z4=z6-z5=z7-z6=z8-z7=e1; z6'-z6=e2; simple_link(z1,z2); simple_link(z2,z3); simple_link(z3,z4); simple_link(z4,z5); simple_link(z5,z6); simple_link(z6,z7); simple_link(z7,z8); simple_link(z6,z6'); circ_marks(z1,z2,z3,z4,z5,z6,z6',z7,z8); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 8$ etex,z2); name_point.bot(btex $\scriptstyle 7$ etex,z3); name_point.bot(btex $\scriptstyle 6$ etex,z4); name_point.bot(btex $\scriptstyle 5$ etex,z5); name_point.bot(btex $\scriptstyle 4$ etex,z6); name_point.rt(btex $\scriptstyle 2$ etex,z6'); name_point.bot(btex $\scriptstyle 3$ etex,z7); name_point.bot(btex $\scriptstyle 1$ etex,z8); label(btex $\mathfrak e_8^{(1)}$ etex,z1-e1); endfig; beginfig(8); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=z4-z3=z5-z4=z5-z4=e1; simple_link(z1,z2); simple_link(z2,z3); double_arrow(z3,z4); simple_link(z4,z5); circ_marks(z1,z2,z3,z4,z5); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle 2$ etex,z3); name_point.bot(btex $\scriptstyle 3$ etex,z4); name_point.bot(btex $\scriptstyle 4$ etex,z5); label(btex $\mathfrak f_4^{(1)}$ etex,z1-e1); endfig; beginfig(9); % unit:=1cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=e1; simple_link(z1,z2); triple_arrow(z2,z3); circ_marks(z1,z2,z3); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 2$ etex,z2); name_point.bot(btex $\scriptstyle 1$ etex,z3); label(btex $\mathfrak g_1^{(1)}$ etex,z1-e1); endfig; % % TABLE 2 % beginfig(10); % unit:=0.4cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=0.5(z4-z3)=z5-z4=e1; double_arrow(z1,z2); simple_link(z2,z3); dotted_link(z3,z4); double_arrow(z4,z5); circ_marks(z1,z2,z3,z4,z5); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle 2$ etex,z3); name_point.bot(btex $\scriptstyle r-1$ etex,z4); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z5); label(btex $\displaystyle\strut\mathfrak a_{2n}^{(2)}\atop \displaystyle\strut(r>1)$ etex,z1-e1); endfig; beginfig(12); z1=origin; z2-z1=e1; quadruple_arrow(z2,z1); circ_marks(z1,z2); name_point.bot(btex $\scriptstyle 2$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); label(btex $\mathfrak a_2^{(2)}$ etex,z1-e1); endfig; beginfig(11); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=0.5(z3-z2)=z4-z3=e1; double_arrow(z2,z1); dotted_link(z2,z3); double_arrow(z3,z4); circ_marks(z1,z2,z3,z4); name_point.bot(btex $\scriptstyle 0$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle r-1$ etex,z3); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z4); label(btex $\mathfrak d_{n+1}^{(2)}$ etex,z1-e1); endfig; beginfig(12); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z1'-z1=z1-z1''=e2; z2-z1=0.5(z3-z2)=z4-z3=e1; simple_link(z1,z1'); simple_link(z1,z1''); simple_link(z1,z2); dotted_link(z2,z3); double_arrow(z4,z3); circ_marks(z1,z1',z1'',z2,z3,z4); name_point.lft(btex $\scriptstyle 0$ etex,z1'); name_point.lft(btex $\scriptstyle 2$ etex,z1); name_point.lft(btex $\scriptstyle 1$ etex,z1''); name_point.bot(btex $\scriptstyle 3$ etex,z2); name_point.bot(btex $\scriptstyle r-1$ etex,z3); name_point.bot(btex $\scriptstyle r\vphantom1$ etex,z4); label(btex $\displaystyle\strut\mathfrak a_{2n-1}^{(2)}\atop \displaystyle\strut(r>2)$ etex,z1-e1); endfig; beginfig(13); % unit:=0.5cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=z4-z3=z5-z4=z5-z4=e1; simple_link(z1,z2); double_arrow(z2,z3); simple_link(z3,z4); simple_link(z4,z5); circ_marks(z1,z2,z3,z4,z5); name_point.bot(btex $\scriptstyle 1$ etex,z1); name_point.bot(btex $\scriptstyle 2$ etex,z2); name_point.bot(btex $\scriptstyle 3$ etex,z3); name_point.bot(btex $\scriptstyle 4$ etex,z4); name_point.bot(btex $\scriptstyle 0$ etex,z5); label(btex $\strut\mathfrak e_6^{(2)}$ etex,z1-e1); endfig; % % TABLE 3 % beginfig(14); % unit:=1cm; e1:=unit*right; e2:=unit*up; z1=origin; z2-z1=z3-z2=e1; triple_arrow(z1,z2); simple_link(z2,z3); circ_marks(z1,z2,z3); name_point.bot(btex $\scriptstyle 2$ etex,z1); name_point.bot(btex $\scriptstyle 1$ etex,z2); name_point.bot(btex $\scriptstyle 0$ etex,z3); label(btex $\strut\mathfrak d_4^{(3)}$ etex,z1-e1); endfig; end.