# -------------------------------------------------------------------------------------------------- # Monk code: version of April 26, 2003. # Author: Alexander Yong # # Written to accompany: # "On combinatorics of cohomology classes related to Schubert calculus" # by Alexander Yong, a proposed PhD dissertation at the University of Michigan, Ann Arbor # # This version requires Maple V,VI,VII and J.R. Stembridge's Coxeter package # available at his website www.math.lsa.umich.edu/~jrs # # This is ** not ** a Maple worksheet # # To run this, load Coxeter and this file in a Maple session # Theorem 5 for the Exceptional Lie types is verified by running "verify_thm()" # which also prints out the resulting classes in each type # Note that the labelings of the simple roots in Coxeter and in the thesis differ, # the code below is written to # # Acknowledgements: I thank John Stembridge for explaining how to use Coxeter to obtain a reduced # word corresponding to a given positive root (which we use in "Monk") # -------------------------------------------------------------------------------------------------- Monk:=proc(root_sys, b_root_position, weyl_elt) local the_b_roots, the_pos_roots, transposition, v0, ans, ii,jj,temp_root,weyl_elt_length,prod_temp,n_alpha_beta,beta; the_b_roots:=base(root_sys): beta:=op(b_root_position,the_b_roots): the_pos_roots:=pos_roots(root_sys): ans:=0: weyl_elt_length:=nops(weyl_elt): for ii from 1 to nops(the_pos_roots) do v0:=interior_pt(root_sys): temp_root:=op(ii,the_pos_roots): vec2fc(reflect(temp_root,v0),root_sys,'transposition'): prod_temp:=reduce([op(weyl_elt),op(transposition)],root_sys): if((nops(prod_temp)-weyl_elt_length)=1) then n_alpha_beta:=op(b_root_position,root_coords(temp_root,root_sys)): ans:=ans+n_alpha_beta*(iprod(beta,beta)/iprod(temp_root,temp_root))*X[prod_temp]: fi: od: RETURN(ans): end: # This applies Monk to a linear combo of Schubert classes. Monk_poly:=proc(root_system,i,p) # For each element of the polynomial'' p we do the Monk calculation, and # collect terms. local num_terms,counter1, p_temp,p_temp1,num_pos_roots,beta_coords, alpha_coords,n_alpha_beta,u,s,us,outcoef; # bust out if p=0 if(p=0) then RETURN(0): fi: p_temp:=0; # num terms is a little tricky: b/c of the way nops works if(type(p,+)=true) then num_terms:=nops(p): else num_terms:=1: fi: for counter1 from 1 to num_terms by 1 do #we have to keep track of the coefficient that appears in the polynomial if(num_terms=1) then if(nops(p)=1) then outcoef:=1: else outcoef:=op(1,p): fi: else if(nops(op(counter1,p))=1) then outcoef:=1: else outcoef:=op(1,op(counter1,p)): fi: fi: # slightly tricky extracting the right thing here, b/c of the way op works # we need to determine if it is a sum of monomials, or just a single monomial if(num_terms>1) then u:=op(counter1,p): if(nops(u)=1) then u:=op(u): else u:=op(op(2,u)): fi: else if(nops(p)=1) then u:=op(p): else u:=op(op(2,p)): fi: fi: p_temp1:=Monk(root_system,i,u); p_temp:=p_temp+outcoef*p_temp1; od: RETURN(p_temp); end: # checking type G2 # we only have to compute \tau_{i,d} for d=1,2,3,4,5 iterate_MonkG2_1:=proc() local iii,SC,ASD; iii:=2: SC[1]:=X[[1]]: print(X[[1]]); ASD:=X[[1]]: while(iii<6) do ASD:=Monk_poly(G2,1,ASD): SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: iii:=iii+1: od: RETURN(SC); end: # checking type F4 # Coxeter uses the labelling 1--2=<=3--4, we use 1--4=>=3--2 in the thesis, translate accordingly # By looking at the max(i) values, we only have to check for \tau_{2,d} and \tau_{3,d} iterate_MonkF4_1:=proc() # i.e, computing \tau_{2,d} local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[1]]: print(X[[1]]); ASD:=X[[1]]: while(iii<10) do ASD:=Monk_poly(F4,1,ASD): SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkF4_2:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkF4_1 dumped out to F4_1 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[2]]: print(X[[2]]); ASD:=X[[2]]: while(iii<8) do ASD:=Monk_poly(F4,2,ASD)-Monk_poly(F4,1,ASD)+F4_1[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: # checking type E6 # Coxeter uses: We use in the thesis # 2 5 # | | # 1--3--4--5--6 6--4--3--2--1 # so we translate accordingly # Note, we only have to check for \tau_{1,d},\tau_{2,d},\tau_{3,d},\tau_{4,d} # also, we don't have max(i) values to work with, so we compute until we get 0 iterate_MonkE6_6:=proc() # i.e, computing \widehat{\tau_{1,d}} local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[6]]: print(X[[6]]); ASD:=X[[6]]: while(ASD<>0) do ASD:=Monk_poly(E6,6,ASD): SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE6_5:=proc() # i.e., compute \tau_{2,d} # assumes iterate_MonkE6_6 dumped out to E6_6 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[5]]: print(X[[5]]); ASD:=X[[5]]: while(ASD<>0) do ASD:=Monk_poly(E6,5,ASD)-Monk_poly(E6,6,ASD)+E6_6[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE6_4:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkE6_5 dumped out to E6_5 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[4]]: print(X[[4]]); ASD:=X[[4]]: while(ASD<>0) do ASD:=Monk_poly(E6,4,ASD)-Monk_poly(E6,5,ASD)+E6_5[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE6_3:=proc() # i.e., compute \tau_{4,d} # assumes iterate_MonkE6_4 dumped out to E6_4 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[3]]+X[[2]]: print(X[[3]]+X[[2]]); ASD:=X[[3]]+X[[2]]: while(ASD<>0) do ASD:=Monk_poly(E6,3,ASD)+Monk_poly(E6,2,ASD)-Monk_poly(E6,4,ASD)+E6_4[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: # checking type E7 # Coxeter uses: We use in the thesis # 2 6 # | | # 1--3--4--5--6--7 7--5--4--3--2--1 # so we translate accordingly # Note, we only have to check for \tau_{1,d},\tau_{2,d},\tau_{3,d},\tau_{4,d},\tau_{5,d} # also, we don't have max(i) values to work with, so we compute until we get 0 iterate_MonkE7_7:=proc() # i.e, computing \widehat{\tau_{1,d}} local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[7]]: print(X[[7]]); ASD:=X[[7]]: while(ASD<>0) do ASD:=Monk_poly(E7,7,ASD): SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE7_6:=proc() # i.e., compute \tau_{2,d} # assumes iterate_MonkE7_7 dumped out to E7_7 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[6]]: print(X[[6]]); ASD:=X[[6]]: while(ASD<>0) do ASD:=Monk_poly(E7,6,ASD)-Monk_poly(E7,7,ASD)+E7_7[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE7_5:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkE7_6 dumped out to E7_6 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[5]]: print(X[[5]]); ASD:=X[[5]]: while(ASD<>0) do ASD:=Monk_poly(E7,5,ASD)-Monk_poly(E7,6,ASD)+E7_6[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE7_4:=proc() # i.e., compute \tau_{4,d} # assumes iterate_MonkE7_5 dumped out to E7_5 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[4]]: print(X[[4]]); ASD:=X[[4]]: while(ASD<>0) do ASD:=Monk_poly(E7,4,ASD)-Monk_poly(E7,5,ASD)+E7_5[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE7_3:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkE7_4 dumped out to E7_4 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[3]]+X[[2]]: print(X[[3]]+X[[2]]); ASD:=X[[3]]+X[[2]]: while(ASD<>0) do ASD:=Monk_poly(E7,3,ASD)+Monk_poly(E7,2,ASD)-Monk_poly(E7,4,ASD)+E7_4[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: # checking type E8 # Coxeter uses: We use in the thesis # 2 7 # | | # 1--3--4--5--6--7--8 8--6--5--4--3--2--1 # so we translate accordingly # Note, we only have to check for \tau_{1,d},\tau_{2,d},\tau_{3,d},\tau_{4,d},\tau_{5,d},\tau_{6,d} # also, we don't have max(i) values to work with, so we compute until we get 0 iterate_MonkE8_8:=proc() # i.e, computing \widehat{\tau_{1,d}} local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[8]]: print(X[[8]]); ASD:=X[[8]]: while(ASD<>0) do ASD:=Monk_poly(E8,8,ASD): SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE8_7:=proc() # i.e., compute \tau_{2,d} # assumes iterate_MonkE8_8 dumped out to E8_8 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[7]]: print(X[[7]]); ASD:=X[[7]]: while(ASD<>0) do ASD:=Monk_poly(E8,7,ASD)-Monk_poly(E8,8,ASD)+E8_8[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE8_6:=proc() # i.e., compute \tau_{2,d} # assumes iterate_MonkE8_7 dumped out to E8_7 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[6]]: print(X[[6]]); ASD:=X[[6]]: while(ASD<>0) do ASD:=Monk_poly(E8,6,ASD)-Monk_poly(E8,7,ASD)+E8_7[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE8_5:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkE8_6 dumped out to E8_6 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[5]]: print(X[[5]]); ASD:=X[[5]]: while(ASD<>0) do ASD:=Monk_poly(E8,5,ASD)-Monk_poly(E8,6,ASD)+E8_6[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE8_4:=proc() # i.e., compute \tau_{4,d} # assumes iterate_MonkE8_5 dumped out to E8_5 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[4]]: print(X[[4]]); ASD:=X[[4]]: while(ASD<>0) do ASD:=Monk_poly(E8,4,ASD)-Monk_poly(E8,5,ASD)+E8_5[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: iterate_MonkE8_3:=proc() # i.e., compute \tau_{3,d} # assumes iterate_MonkE8_4 dumped out to E8_4 local iii,SC,ASD,max_nops; max_nops:=1: iii:=2: SC[1]:=X[[3]]+X[[2]]: print(X[[3]]+X[[2]]); ASD:=X[[3]]+X[[2]]: while(ASD<>0) do ASD:=Monk_poly(E8,3,ASD)+Monk_poly(E8,2,ASD)-Monk_poly(E8,4,ASD)+E8_4[iii]: SC[iii]:=ASD; print(SC[iii]); if(searchtext("-",convert(SC[iii],string))>0) then print(COUNTEREXAMPLE); fi: if(nops(SC[iii])>max_nops) then max_nops:=nops(SC[iii]): fi: iii:=iii+1: od: print("Max nops was:"); print(max_nops); RETURN(SC); end: # Verify theorem procedure # If positivity fails, "COUNTEREXAMPLE" should get written out verify_thm:=proc() global F4_1,E6_6,E6_5,E6_4,E6_3,E7_7,E7_6,E7_5,E7_4,E7_3,E8_8,E8_7,E8_6,E8_5,E8_4,E8_3; print("Remember that Coxeter2_2v and our thesis conventions on root labellings differ."); print("We have internally accounted for this, see the source code"); print("Type G2 check"); print("For i=1"); iterate_MonkG2_1(); print("Type F4 check"); print("For i=2"); F4_1:=iterate_MonkF4_1(); print("For i=3"); iterate_MonkF4_2(); print("Type E6 check"); print("For i=1"); E6_6:=iterate_MonkE6_6(); print("For i=2"); E6_5:=iterate_MonkE6_5(); print("For i=3"); E6_4:=iterate_MonkE6_4(); print("For i=4"); E6_3:=iterate_MonkE6_3(); print("Type E7 check"); print("For i=1"); E7_7:=iterate_MonkE7_7(); print("For i=2"); E7_6:=iterate_MonkE7_6(); print("For i=3"); E7_5:=iterate_MonkE7_5(); print("For i=4"); E7_4:=iterate_MonkE7_4(); print("For i=5"); E7_3:=iterate_MonkE7_3(); print("Type E8 check"); print("For i=1"); E8_8:=iterate_MonkE8_8(); print("For i=2"); E8_7:=iterate_MonkE8_7(); print("For i=3"); E8_6:=iterate_MonkE8_6(); print("For i=4"); E8_5:=iterate_MonkE8_5(); print("For i=5"); E8_4:=iterate_MonkE8_4(); print("For i=6"); E8_3:=iterate_MonkE8_3(); RETURN(0); end: