{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "read \"public.mpl\":
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Below we compute the unique p
ermuation array for the intersection of three Schubert varieties.  " }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "doo:=permarray([[1, 3, 2, \+
4], [2, 3, 4, 1], [3, 1, 2, 4]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%$dooG7/7&\"\"%\"\"\"F'F(7&F'F'\"\"#F*7&F*F'F'F*7&F'F'F(\"\"$7&F'F-F*F
-7&F-F'F*F-7&F-F-F'F-7&F'F*F-F'7&F-F-F-F'7&F*F'F-F'7&F-F*F'F'7&F*F-F'F
'7&F(F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Here is the 4-dime
nsional permutation array:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "printdotarray(doo, 4,4);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }
{XPPMATH 20 "6&-%'matrixG6#7&7&7\"F(F(F(F'F'7&\"\"%F(F(F(-F$6#7&F'7&F(
F(F(F*F'7&F*F(F(\"\"#-F$6#7&F'F.7&F(F(F*F07&F*F(F0\"\"\"-F$6#7&F.7&F(F
(F*\"\"$7&F(F*F;F07&F*F;F0F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "GenericFlag(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&-%'RTABLEG6
%\")+(\\I\"-%'MATRIXG6#7&7#\"\"\"7#&%#XXG6$F-\"\"#7#&F06$F-\"\"$7#&F06
$F-\"\"%&%'VectorG6#%'columnG-F%6%\")!y\\I\"-F)6#7&7#\"\"!F,7#&F06$F2F
67#&F06$F2F:F;-F%6%\")?)\\I\"-F)6#7&FEFEF,7#&F06$F6F:F;-F%6%\")+*\\I\"
-F)6#7&FEFEFEF,F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "We work wit
h the following three flags in general position as the base flags for \+
the three Schubert varieties that we are intersecting:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "foo:=[op(TransverseFlags(4,2)), Gen
ericFlag(4)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fooG7%7&-%'RTABLEG
6%\")!)*\\I\"-%'MATRIXG6#7&7#\"\"\"7#\"\"!F1F1&%'VectorG6#%'columnG-F(
6%\")g+08-F,6#7&F1F/F1F1F3-F(6%\")S,08-F,6#7&F1F1F/F1F3-F(6%\")?-08-F,
6#7&F1F1F1F/F37&-F(6%\")+.08FFF3-F(6%\")!Q]I\"F@F3-F(6%\")+008F:F3-F(6
%\")!e]I\"F+F37&-F(6%\")+208-F,6#7&F/7#&%#XXG6$F0\"\"#7#&Fin6$F0\"\"$7
#&Fin6$F0\"\"%F3-F(6%\")!y]I\"-F,6#7&F1F/7#&Fin6$F[oF_o7#&Fin6$F[oFcoF
3-F(6%\")g308-F,6#7&F1F1F/7#&Fin6$F_oFcoF3-F(6%\")+408FFF3" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 70 "The reader can play with other flags by u
ncommenting the line below:  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 87 "#foo:=[op(TransverseFlags(4,2)), RandomFlag(4)];  #uncomment for
 another choice of flag" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Now, \+
we do the main computation to see if there is a solution in the inters
ection of these three Schubert varieties.  " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "FlagSolver(doo, foo);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$Q)Testing:6\"7%7&\"\"\"\"\"$\"\"#\"\"%7&F)F(F*F'7&F(F'F
)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q>Solution~Found.~~See~FLAGSOLS
6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%#NVG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 282 "Hoorray, a solution was found!  Note, occasionally \+
a solution is not found when one does exits due to the fact that Maple
 reorders sets every time it computes with them and we have not progra
mmed around this \"feature\".    Next we print a basis for the solutio
n flag(s) found above. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
SolutionFlagBasis(doo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7&-%'RTAB
LEG6%\")!)408-%'MATRIXG6#7&7#\"\"!F-F-7#\"\"\"&%'VectorG6#%'columnG-F&
6%\")?508-F*6#7&7#,$&%#XXG6$\"\"#\"\"$!\"\"7#,&*&F=F0&F>6$F0F@F0FB&F>6
$F0FAF0F-7#,&*&FHF0&F>6$F@\"\"%F0F0*&&F>6$F0FOF0F=F0FBF1-F&6%\")+608-F
*6#7&7#FB7#,$FFFB7#,$FHFB7#,$FQFBF1-F&6%\")!=^I\"-F*6#7&F-F-F/7#&F>6$F
AFOF1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Below we print out the \+
key vectors which correspond to elements in the permutation array doo.
  Note, we print the 1-dim projection first, then the 2-dim projection
 etc.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "printvectors(Pro
jArray(doo,1),FLAGSOLS[1],4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TA
BLEG6#7#/7%\"\"%\"\"\"F)-%'RTABLEG6%\")g708-%'MATRIXG6#7&7#\"\"!F3F37#
F*&%'VectorG6#%'columnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q)vectors:6
\"-%'matrixG6#7&7&7\"F*F*F*F)F)7&-%'RTABLEG6%\")S808-%'MATRIXG6#7&7#\"
\"!F4F47#\"\"\"&%'VectorG6#%'columnGF*F*F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "printvectors(ProjArray(doo,2),FLAGSOLS[1],4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7%/7%\"\"%F)\"\"#-%'RTABLEG
6%\")?908-%'MATRIXG6#7&7#,$&%#XXG6$F*\"\"$!\"\"7#,&*&F5\"\"\"&F66$F=F*
F=F9&F66$F=F8F=7#\"\"!7#,&*&F@F=&F66$F*F)F=F=*&&F66$F=F)F=F5F=F9&%'Vec
torG6#%'columnG/7%F)F=F)-F,6%\")g708-F06#7&FBFBFB7#F=FL/7%F*F)F)-F,6%
\")g908-F06#7&7#F57#,&F<F=F@F9FBFBFL" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$Q)vectors:6\"-%'matrixG6#7&7&7\"F*F*F*7&F*F*F*-%'RTABLEG6%\")+:08-%'
MATRIXG6#7&7#&%#XXG6$\"\"#\"\"$7#,&*&F5\"\"\"&F66$F=F8F=F=&F66$F=F9!\"
\"7#\"\"!FC&%'VectorG6#%'columnGF)7&-F-6%\")S:08-F16#7&FCFCFC7#F=FEF*F
*-F-6%\")!e^I\"-F16#7&7#,$F5FB7#,&F<FBF@F=FC7#,&*&F@F=&F66$F8\"\"%F=F=
*&&F66$F=FjnF=F5F=FBFE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "p
rintvectors(ProjArray(doo,3),FLAGSOLS[1],4);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#-%&TABLEG6#7(/7%\"\"$\"\"%\"\"#-%'RTABLEG6%\")g;08-%'MA
TRIXG6#7&7#,$&%#XXG6$F+F*!\"\"7#,&&F76$\"\"\"F*F>*&F6F>&F76$F>F+F>F97#
,&*&&F76$F>F)F>F6F>F9*&F<F>&F76$F+F)F>F>7#\"\"!&%'VectorG6#%'columnG/7
%F)F)F*-F-6%\")S<08-F16#7&FJ7#F>7#FHFJFL/7%F*F*F>-F-6%\")?=08-F16#7&7#
F97#,$F@F97#,$FEF97#,$F<F9FL/7%F*F)F+-F-6%\")g=08-F16#7&FJFXFY7#F6FL/7
%F*F>F*-F-6%\")g708-F16#7&FJFJFJFXFL/7%F+F*F*-F-6%\")g908-F16#7&FY7#,&
*&FHF>F@F>F>FEF9FJFJFL" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q)vectors:6
\"-%'matrixG6#7&7&7\"F*F*F*7&F*F*F*-%'RTABLEG6%\")S>08-%'MATRIXG6#7&7#
&%#XXG6$\"\"#\"\"$7#,&*&F5\"\"\"&F66$F=F8F=F=&F66$F=F9!\"\"7#\"\"!FC&%
'VectorG6#%'columnG7&F*F*-F-6%\")!)>08-F16#7&FC7#F=F4FCFE-F-6%\")??08-
F16#7&7#,$&F66$F8\"\"%FB7#,&&F66$F=FenF=*&FYF=F>F=FB7#,&*&F@F=FYF=FB*&
FhnF=F5F=F=FCFE7&-F-6%\")+@08-F16#7&FCFCFCFPFEF*-F-6%\")!=_I\"-F16#7&F
CFPF47#FYFE-F-6%\")gA08-F16#7&7#FB7#,$F>FB7#,$F@FB7#,$FhnFBFE" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "printvectors(ProjArray(doo,4
),FLAGSOLS[1],4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7,/7%
\"\"$\"\"#\"\"%-%'RTABLEG6%\")SB08-%'MATRIXG6#7&7#\"\"!F47#\"\"\"F4&%'
VectorG6#%'columnG/7%F)F+F*-F-6%\")g;08-F16#7&7#,$&%#XXG6$F*F+!\"\"7#,
&&FG6$F7F+F7*&FFF7&FG6$F7F*F7FI7#,&*&&FG6$F7F)F7FFF7FI*&FLF7&FG6$F*F)F
7F7F4F8/7%F7F+F+-F-6%\")?C08-F16#7&F6F4F4F4F8/7%F)F)F)-F-6%\")+D08-F16
#7&F47#,$&FG6$F)F+FI7#,&FFF7*&FeoF7FWF7FIF4F8/7%F+F+F7-F-6%\")?=08-F16
#7&7#FI7#,$FOFI7#,$FTFI7#,$FLFIF8/7%F+F*F)-F-6%\")!e_I\"-F16#7&F4F4F67
#FeoF8/7%F+F)F*-F-6%\")g=08-F16#7&F4F67#FW7#FFF8/7%F+F7F+-F-6%\")g708-
F16#7&F4F4F4F6F8/7%F*F+F)-F-6%\")?E08-F16#7&Fgo7#,*FLFIFNF7*(FeoF7FWF7
FOF7FI*&FeoF7FTF7F7F4F4F8/7%F*F)F+-F-6%\")SF08-F16#7&F4F6F4F4F8" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$Q)vectors:6\"-%'matrixG6#7&7&7\"F*F*-%
'RTABLEG6%\")?G08-%'MATRIXG6#7&7#\"\"\"7#\"\"!F5F5&%'VectorG6#%'column
G7&F*F*-F,6%\")+H08-F06#7&F5F3F5F5F7-F,6%\")!)H08-F06#7&7#,&&%#XXG6$\"
\"#\"\"%F4*&&FK6$\"\"$FNF4&FK6$FMFRF4!\"\"7#,*&FK6$F4FNFU*&FJF4&FK6$F4
FMF4F4*(FPF4FSF4FenF4FU*&FPF4&FK6$F4FRF4F4F5F5F77&F*-F,6%\")gI08-F06#7
&F5F5F3F5F7-F,6%\")SJ08-F06#7&F57#,$FPFUFHF5F7-F,6%\")!=`I\"-F06#7&7#,
$FJFU7#,&FXF4FZFU7#,&*&FinF4FJF4FU*&FXF4FSF4F4F5F77&-F,6%\")gK08-F06#7
&F5F5F5F3F7-F,6%\")+L08-F06#7&F5F5F37#FPF7-F,6%\")SL08-F06#7&F5F37#FS7
#FJF7-F,6%\")?M08-F06#7&7#FU7#,$FenFU7#,$FinFU7#,$FXFUF7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "14 0 0" 170 }{VIEWOPTS 
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"$F,F-&F)6$F+F1F-!"",*&F)6$F-F,F4*&F(F-&F)6$F-F+F-F-*(F/F-F2F-F9F-F4*&F/F-&F)6$
F-F1F-F-""!F?6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13052740X*%)anythingG6"6"[gl!#%!!!"%"%""!"""F'F'6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13052820X*%)anythingG6"6"[gl!#%!!!"%"%"""""!F(F(6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13052900X*%)anythingG6"6"[gl!#%!!!"%"%""!"""F'F'6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13052980X*%)anythingG6"6"[gl!#%!!!"%"%,&&%#XXG6$""#""%"""*&&F)6$"
"$F,F-&F)6$F+F1F-!"",*&F)6$F-F,F4*&F(F-&F)6$F-F+F-F-*(F/F-F2F-F9F-F4*&F/F-&F)6$
F-F1F-F-""!F?6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053060X*%)anythingG6"6"[gl!#%!!!"%"%""!F'"""F'6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053140X*%)anythingG6"6"[gl!#%!!!"%"%""!,$&%#XXG6$""$""%!"",&&F*
6$""#F-"""*&F)F3&F*6$F2F,F3F.F'6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053180X*%)anythingG6"6"[gl!#%!!!"%"%,$&%#XXG6$""#""%!"",&&F)6$"
""F,F1*&F(F1&F)6$F1F+F1F-,&*&&F)6$F1""$F1F(F1F-*&F/F1&F)6$F+F9F1F1""!6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053260X*%)anythingG6"6"[gl!#%!!!"%"%""!F'F'"""6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053300X*%)anythingG6"6"[gl!#%!!!"%"%""!F'"""&%#XXG6$""$""%6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053340X*%)anythingG6"6"[gl!#%!!!"%"%""!"""&%#XXG6$""#""$&F*6$F,
""%6"
}
{RTABLE 
M7R0
I5RTABLE_SAVE/13053420X*%)anythingG6"6"[gl!#%!!!"%"%!"",$&%#XXG6$"""""#F',$&F*6
$F,""$F',$&F*6$F,""%F'6"
}