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galois groups are not only symmetry groups of field extensions , they are also symmetry groups in enumerative geometry . this second point was made by jordan in 1870  @xcite who studied some classical problems in enumerative geometry , showing that several had galois groups which were not the full symmetric group , reflecting the intrinsic structure of these problems . earlier , hermite gave a different connection to geometry , showing that the algebraic galois group coincided with a geometric monodromy group  @xcite . harris used this to study the galois group of several problems in enumerative geometry  @xcite . for each he showed that their monodromy groups were the full symmetric groups on their sets of solutions and therefore the problem had no intrinsic structure . the schubert calculus is a well - understood family of problems in enumerative geometry that involve linear subspaces having prescribed positions with respect to other , fixed linear spaces . it provides a laboratory for studying galois groups in enumerative geometry . for example , leykin and sottile  @xcite directly computed monodromy for many schubert problems on small grassmannians involving simple ( codimension one ) schubert conditions and found that each problem had monodromy the full symmetric group . in  @xcite , vakil gave a general combinatorial method based on the principle of specialization and group theory for showing that a problem in enumerative geometry has at least alternating galois group in that its galois / monodromy group contains the alternating group . with this method and his geometric littlewood - richardson rule  @xcite , vakil showed that many schubert problems on small grassmannians had at least alternating galois groups . he also found schubert problems whose galois groups are not the full symmetric group . brooks , et al . @xcite used vakil s combinatorial criterion and some delicate estimates of integrals to show that all schubert problems on grassmannians of 2-planes have at least alternating galois groups . vakil gave another , stronger , criterion for showing that a galois group was at least alternating which requires knowing that it is a doubly transitive permutation group . we study double transitivity of galois groups of schubert problems on grassmannians . vakil s stronger criterion is not our only motivation . there appears to be a siginificant gap in transitivity : every galois group that we have studied is either at least alternating and therefore highly transitive , or else it fails to be doubly transitive and is imprimitive in that it preserves a partition of the solutions . we conjecture that a galois group of a schubert problem is either the full symmetric group on its solutions , or it is imprimitive . one purpose of this paper is to give theoretical and computational evidence supporting this conjecture . a schubert condition on @xmath1-planes in @xmath2-space is if the condition is that the @xmath1-plane meets an @xmath3-plane nontrivially with @xmath4 . a schubert problem is if it only has special conditions . we state our main results . * theorem . *    _ we have the following : _    1 . every special schubert problem has a doubly transitive galois group . every schubert problem in @xmath5 has at least alternating galois group . every schubert problem in @xmath6 has a doubly transitive galois group . there are exactly fourteen schubert problems in @xmath0 whose galois groups are not at least alternating . for each , the galois group is imprimitive . part ( 2 ) was proven in  @xcite . using part ( 1 ) and vakil s stronger criterion , we give a significantly simpler proof of that result . this approach does not generalize to show that all schubert problems involving 3-planes have at least alternating monodromy , for that requires a significantly more delicate analysis of geometric problems involving vakil s checkerboard varieties  @xcite .    by part ( 4 ) , our results on double transitivity can not be extended to grassmannians of 4-planes . one of the fourteen schubert problems with imprimitive galois group was first described in  3.13 of  @xcite , and is due to derksen . we determine exactly the schubert problems on this grassmannian whose galois group is not at least alternating . in each case , we compute the galois group , and demonstrate that the resulting groups are imprimitive . in addition to derksen s example , there are essentially two others which generalize to give two families of schubert problems with imprimitive monodromy . one family contains a problem in which all but two of its schubert conditions are special and the remaining two conditions are dual special schubert conditions . hence our results on double transitivity can not be further extended to simple general statements . this paper is organized as follows . in section  [ s : mg ] , we provide some background on galois / monodromy groups , and explain vakil s work , including his criteria . in section  [ s : sc ] we give basic definitions in the schubert calculus on grassmannians . in section  [ s : dt ] we discuss how to show double transitivity of galois groups using geometry and prove some geometric lemmas . in section  [ s : mr ] we establish parts ( 1 ) and ( 3 ) of our main theorem about special schubert problems and schubert problems involving 3-planes . in section  [ s : g2n ] , we use the double transitivity of special schubert problems to prove part ( 2 ) . we close with section  [ s : im ] , in which we prove part ( 4 ) and study the galois group of every schubert problem on @xmath0 . we provide some background on galois / monodromy groups in enumerative geometry . more information can be found in  @xcite . we work over an algebraically closed field @xmath7 of arbitrary characteristic . suppose that @xmath8 is a proper , generically finite and separable ( i.e.generically tale ) morphism of @xmath7-schemes of degree @xmath9 , and @xmath10 is irreducible . let @xmath11 be the zariski closure in the @xmath9-fold fiber product of the scheme @xmath12 where @xmath13 is the big diagonal . let @xmath14 be a closed point in the open set over which @xmath15 is finite and separable such that the fiber @xmath16 consists of @xmath9 reduced points . ( such a point @xmath17 is a of @xmath15 . ) choose an ordering @xmath18 of the points of @xmath16 . then the of @xmath8 is the subgroup of the symmetric group of permutations of @xmath19 $ ] consisting of those permutations @xmath20 such that the points @xmath21 and @xmath22 in the fiber above @xmath17 are in the same irreducible component of @xmath23 . write , where @xmath24 , for the component of @xmath23 containing the point @xmath25 . the group @xmath26 is well - defined as a subgroup of @xmath27 , up to conjugacy . let @xmath28 be an integer . a permutation group @xmath29 is if any two ordered sets of @xmath30 distinct numbers are mapped onto each other by an element of @xmath31 . we interpret this geometrically for galois / monodromy groups . define @xmath32 to be the zariski closure of the scheme @xmath33 where @xmath13 is the big diagonal as before . let be the union of the irreducible components of @xmath34 that map dominantly onto @xmath10 . each component of @xmath35 has the same dimension as @xmath10 and its projection to @xmath10 has finite fibers over the dense subset of regular values of @xmath15 . the fiber of @xmath35 over the point @xmath17 consists of all @xmath30-tuples @xmath36 where @xmath37 are distinct elements of @xmath18 , taken in every possible order . [ l : t - transitive ] the galois / monodromy group @xmath26 of @xmath8 is @xmath30-transitive if and only if @xmath35 is irreducible . in particular , @xmath26 is transitive if and only if @xmath38 is irreducible . let @xmath14 be a regular value of @xmath15 , order the points of @xmath16 , and set @xmath24 . let @xmath39 be the component of @xmath23 containing @xmath25 and consider its image in @xmath34 under the projection map @xmath40 given by forgetting the last @xmath41 factors in the fiber product . as @xmath17 is a regular value , we have @xmath42 , and so @xmath43 is an irreducible component of @xmath35 . the fiber of @xmath43 over @xmath17 consists of all @xmath30-tuples @xmath44 for @xmath45 . if @xmath26 is @xmath30-transitive , then @xmath46 , which implies that @xmath35 is irreducible . conversely , if @xmath35 is irreducible , then @xmath46 , which implies that @xmath26 is @xmath30-transitive . suppose that we have @xmath8 as above with @xmath38 irreducible and @xmath10 smooth . then the galois / monodromy group @xmath26 is a transitive subgroup of the symmetric group @xmath27 . a subgroup @xmath31 of @xmath27 is if it is either the alternating subgroup of @xmath27 or the full symmetric group @xmath27 . vakil gave several criteria which may be used to show that @xmath26 is at least alternating . we follow the discussion of  @xcite . suppose that we have a fiber diagram @xmath47 where @xmath48 is the closed embedding of a cartier divisor , @xmath10 is smooth in codimension one along @xmath49 , and @xmath50 is a generically finite and separable morphism of degree @xmath9 . when @xmath51 has at most two components , we have the following .    1 . if @xmath51 is irreducible , then there is an inclusion @xmath52 into @xmath26 . if @xmath51 has two components , @xmath53 and @xmath54 , each of which maps dominantly to @xmath49 of respective degrees @xmath55 and @xmath56 , then the monodromy group of @xmath57 is a subgroup of @xmath58 that maps surjectively onto each factor @xmath59 and which includes into @xmath26 ( via @xmath60 ) .    in  @xcite , vakil gave criteria for deducing that @xmath26 is at least alternating . this follows by purely group - theoretic arguments including goursat s lemma . suppose that we have a fiber diagram  @xmath61 . then @xmath26 is at least alternating if one of the following holds .    1 .   if we are in case @xmath62 and @xmath52 is at least alternating . if we are in case @xmath63 , @xmath64 and @xmath65 are at least alternating , and either @xmath66 or @xmath67 . if we are in case @xmath63 , @xmath64 and @xmath65 are at least alternating , one of @xmath55 or @xmath56 is not @xmath68 , and @xmath26 is @xmath69-transitive .    in the introduction , we referred to criterion ( b ) as vakil s combinatorial criterion , and ( c ) as his stronger criterion . [ r : vakil ] these criteria apply to more general inclusions @xmath48 of an irreducible variety into @xmath10 . all that is needed is that @xmath10 is generically smooth along @xmath49 , for then we may replace @xmath10 by an affine open set meeting @xmath49 and there are subvarieties @xmath70 with each inclusion @xmath71 that of a cartier divisor where @xmath72 is smooth in codimension one along @xmath73 , and then apply induction . vakil observed that as the alternating group is least @xmath74-transitive , a consequence of criterion ( c ) is that to show that a galois / monodromy group is @xmath74-transitive , it often suffices to show that it is merely doubly transitive . we develop the schubert calculus for the grassmannian in a form that we will use . more material , including proofs and references , may be found in  @xcite . write for the linear span of a subset @xmath75 of a vector space . for a positive integer @xmath2 , write for the set @xmath76 . for a finite - dimensional @xmath7-vector space @xmath77 , let be the projective space of 1-dimensional linear subspaces of @xmath77 . for @xmath78 , the of @xmath1-dimensional linear subspaces of @xmath77 is a smooth irreducible algebraic variety of dimension @xmath79 . then @xmath80 . we will also write @xmath81 for @xmath82 . the schubert calculus concerns @xmath1-planes in @xmath77 having prescribed positions with respect to other linear subspaces . the prescribed positions are encoded by , which are weakly decreasing sequences of nonnegative integers , @xmath83 for example @xmath84 is a partition for @xmath85 and @xmath86 is a partition for @xmath87 . we typically omit trailing 0s and represent a partition by a left - justified array of boxes with @xmath88 boxes in row @xmath89 . thus these two partitions are @xmath90    a partition prescribes the position of a @xmath1-plane with respect to a ( complete ) of subspaces , which is a sequence @xmath91 of linear subspaces with @xmath92 . given a partition @xmath93 and a flag @xmath94 , the schubert condition of type @xmath93 imposed by the flag @xmath94 on @xmath1-planes @xmath95 is @xmath96 for the partition @xmath97 this is @xmath98 and the other conditions given by the trailing 0s are that @xmath99 for @xmath100 , which always hold . the set of all @xmath95 which satisfy the schubert condition @xmath93 on the flag @xmath94 is a @xmath101 this is an irreducible subvariety of @xmath82 of codimension @xmath102 and thus of dimension @xmath103 . the schubert variety has a distinguished open subset @xmath104 , which is the locus where the inequalities   are all equalities . we will also need schubert varieties in manifolds of partial flags . a partial flag @xmath94 is an increasing sequence of subspaces @xmath105 where @xmath106 . this sequence of dimensions @xmath107 is the of the flag @xmath94 . the set of all flags of type @xmath108 forms a smooth variety of dimension @xmath109 where @xmath110 is taken to be @xmath111 . this flag variety has distinguished schubert varieties , which we describe as follows . flags @xmath94 of type @xmath108 and @xmath112 of type @xmath113 have a , which is encoded by the . this is the @xmath114 array where @xmath115 the set of flags @xmath116 whose position with respect to @xmath112 is at least that of @xmath94 , @xmath117 is a . this irreducible subvariety of @xmath118 has a distinguished dense open subset @xmath119 consisting of @xmath120 for which @xmath121 for all @xmath122 . only subspaces @xmath123 with @xmath124 in a flag @xmath94 are used to define the schubert variety @xmath125  . thus we will often replace complete flags with partial flags of the type occurring in  , that is @xmath126 and @xmath127 ( @xmath128 is the last nonzero part of @xmath93 . ) write for this space of partial flags . it has dimension @xmath129 for example , when @xmath130 and @xmath131 , @xmath132 . when @xmath133 has only one nonzero part , so that it consists of a single row , we will call it a , and simply write it as @xmath134 . dually , when @xmath93 consists of a single column , so that @xmath135 with @xmath136 1s , then it is a , and we write it as @xmath137 . we may omit the subscripts giving the dimensions of subspaces in a flag in @xmath138 . thus for @xmath130 and @xmath131 , @xmath139 means that the partition is @xmath140 and @xmath141 has dimension @xmath142 , and @xmath143 means that the flag is @xmath144 . associating a @xmath1-plane @xmath145 to its annihilator @xmath146 in the dual space @xmath147 of @xmath77 gives an isomorphism between @xmath82 and @xmath148 , where @xmath149 . under this isomorphism , @xmath125 is sent to @xmath150 , where @xmath151 is the transpose of @xmath93 , the partition obtained by interchanging rows and columns , and @xmath152 is the flag whose @xmath89-plane is the annihilator of @xmath153 . in particular , special schubert varieties are sent to dual special schubert varieties . we close this subsection with a dimension calculation . let @xmath154 and @xmath155 be types of @xmath156-step flags . ( note that we allow @xmath157 , so that a flag of type @xmath158 may have repeated subspaces . ) for any @xmath159 , consider the schubert subvariety of @xmath118 defined by @xmath160    [ lem : xeldot ] the schubert variety @xmath161 is nonempty if and only if @xmath162 where @xmath163 . when  @xmath164 is satisfied , @xmath161 has dimension @xmath165    summing   for @xmath166 gives the obvious necessary condition for nonemptiness . we prove this by induction on the length @xmath156 of the flags . suppose first that @xmath167 . then condition   is @xmath168 , and @xmath169 , which is simply @xmath170 and has dimension @xmath171 . this is nonempty if and only if @xmath168 . now suppose that the statement of the lemma holds for flags of length @xmath172 . given a flag @xmath112 of length @xmath156 , let @xmath173 be its truncation to length @xmath172 and @xmath174 , its last subspace . set @xmath175 to be those @xmath176 in @xmath177 where @xmath178 , which is an open subset of @xmath177 . this set is nonempty ( and therefore dense ) only if there exists a flag @xmath94 in @xmath177 with @xmath179 , which is equivalent to the inequality @xmath180 . but this holds by  , as @xmath181 .    for @xmath182 we consider the fiber @xmath183 . this is isomorphic to @xmath184 , which is @xmath185 where @xmath186 , which is @xmath187 , as @xmath182 . we must have that @xmath188 for this to be nonempty , and when this is satisfied , we have @xmath189\,,\ ] ] the dimension of this grassmannian . this completes the proof . a on @xmath81 is a list @xmath190 of partitions with @xmath191 . given a schubert problem @xmath192 and a list of general flags @xmath193 with @xmath194 , the intersection @xmath195 is transverse  @xcite and consists of finitely many points . this number of points is independent of the choice of general flags and the algebraically closed field @xmath7 . the schubert problem is if @xmath196 . by transversality , when the flags are general , all points of the intersection lie in the intersection of the open schubert varieties @xmath197 where @xmath198 is the subset of @xmath125 where the inequalities in   are replaced by equalities . for a schubert problem @xmath199 , consider the family of all intersections  , @xmath200 set @xmath201 , the space of @xmath202-tuples of partial flags for schubert varieties indexed by the partitions of @xmath192 . this family @xmath203 is equipped with two projections @xmath204 and @xmath205 . fibers of the projection to @xmath206 are intersections  , which , when transverse and nonempty , are zero - dimensional . thus we expect that @xmath207 . in fact , for a schubert problem @xmath192 we will always have this equality of dimension . to see this , define the subvariety @xmath208 for @xmath209 to be @xmath210 this is a schubert subvariety of @xmath211 of codimension @xmath212 , and it is irreducible . now consider fibers of the projection @xmath213 . for @xmath95 , we have @xmath214 ( here , as elsewhere , we identify a fiber @xmath215 with its image under the second projection @xmath15 to @xmath206 . ) since any two fibers are isomorphic , @xmath204 realizes @xmath203 as a fiber bundle . the dimension of a fiber @xmath215 is the sum of dimensions of the schubert varieties in the product  , which is @xmath216 from which it follows that @xmath207 . since the base and fibers of the projection @xmath204 are irreducible , we deduce that @xmath203 is irreducible . let @xmath192 be a schubert problem . by the transversality of a general intersection  , the map @xmath217 a proper , generically finite and separable ( i.e. generically tale ) morphism of @xmath7-schemes of degree @xmath218 . furthermore @xmath206 is irreducible and smooth as it is a product of flag manifolds . the of the schubert problem @xmath192 is the galois group of this family of all instances of the schubert problem @xmath192 , that is , @xmath219 which acts transitively , as @xmath203 is irreducible .      a schubert problem is reduced , if , roughly , it is not equivalent to a schubert problem on a smaller grassmannian . every schubert problem @xmath192 is equivalent to a reduced schubert problem @xmath220 , and we have @xmath221 . thus , when proving that a galois group has a given property , it suffices to assume that the schubert problem is reduced . we give a definition of reduced schubert problems and sketch a proof of these facts . [ d : reduced ] a schubert problem @xmath192 on @xmath81 is if for every pair of partitions @xmath222 from @xmath192 , none of the following hold . 1 .   @xmath223 . 2 .   @xmath224 . 3 .   there is some @xmath225 with @xmath226 . there is some @xmath227 with @xmath228 . if the schubert problem is nontrivial , then ( c ) can hold at most with equality , for if @xmath229 and the flags @xmath230 are in general position , then @xmath231 . [ prop : reduced ] any nontrivial schubert problem @xmath192 is equivalent to a reduced schubert problem @xmath220 having an isomorphic galois group , @xmath232 . we indicate how a nonreduced schubert problem @xmath192 is equivalent to a schubert problem on a smaller grassmannian . this process may be iterated to obtain an equivalent reduced schubert problem . the arguments given in remark 4 of  @xcite generalize to show that galois groups are preserved by this reduction process . suppose that @xmath233 in @xmath234 with @xmath235 . then @xmath236 thus if condition ( a ) of definition  [ d : reduced ] holds , then @xmath237 and if condition ( b ) holds , then @xmath238 . if ( a ) holds , then @xmath239 , which is a schubert variety in @xmath240 , where @xmath241 is the partition @xmath242 and @xmath243 is the image of the flag @xmath112 in @xmath244 , specifically @xmath245 . if @xmath246 where @xmath94 is a flag in general position with respect to @xmath112 , then @xmath247 , where @xmath248 is the image of the flag @xmath94 in @xmath244 , specifically , @xmath249 . if ( b ) holds , then @xmath250 , which is a schubert variety in @xmath251 , where @xmath252 is the partition @xmath253 and @xmath254 is the flag in @xmath255 induced by @xmath112 , specifically , @xmath256 . if @xmath246 where @xmath94 is a flag in general position with respect to @xmath112 , then @xmath257 , where @xmath258 . the point of these reductions is that the original schubert problem @xmath192 is equivalent to a schubert problem @xmath220 in a smaller grassmannian . all partitions of @xmath220 are from @xmath192 , except that the partition @xmath259 has been changed in that either its first row of length @xmath260 has been removed or all of its columns of height @xmath1 have been removed .    for condition ( c ) in definition  [ d : reduced ] , note that if @xmath261 , then for every @xmath225 , @xmath262 since @xmath145 has dimension @xmath1 , these conditions imply that @xmath263 if @xmath264 are in general position and @xmath265 , then   is impossible as @xmath266 . thus for the schubert problem @xmath192 to have solutions , we must have @xmath267 . suppose that @xmath268 . then @xmath269 is a 1-dimensional linear space contained in @xmath145 that we will call and @xmath270 . furthermore , if @xmath271 and @xmath272 are obtained from @xmath259 and @xmath273 by omitting the parts @xmath274 and @xmath275 and @xmath276 , @xmath277 are the images of @xmath112 and @xmath94 in @xmath278 , then @xmath279 . as before , if @xmath280 with @xmath281 in general position with respect to @xmath112 and @xmath94 , then @xmath282 .    finally , note that if @xmath261 , we have   as well as @xmath283 . if @xmath112 and @xmath94 are in general position , this implies that @xmath284 if we have @xmath228 , then this sum @xmath51 of subspaces of the flags has codimension @xmath285 in @xmath77 . then @xmath286 , where @xmath271 is obtained from @xmath259 by subtracting @xmath285 from each of the first @xmath89 parts of @xmath259 and @xmath272 is obtained from @xmath273 by subtracting @xmath285 from each of the first @xmath287 parts of @xmath273 . we give an example of this , where @xmath288 and @xmath289 with @xmath290 and @xmath291 . then if @xmath292 , @xmath293 and @xmath294 . drawing @xmath259 together with @xmath273 ( rotated by @xmath295 ) in the @xmath296 box shows that there are two ( @xmath297 ) columns of height three covered . removing those columns from @xmath259 and @xmath273 gives @xmath298 and @xmath299 . @xmath300 as before , if @xmath280 in @xmath82 with @xmath281 in general position , then @xmath301 in @xmath302 . we also note that any schubert problem @xmath192 on @xmath81 is equivalent to a dual schubert problem on the dual grassmannian @xmath303 which has an isomorphic galois group . we develop tools for showing that the galois / monodromy group @xmath304 of a schubert problem @xmath192 on the grassmannian @xmath81 is doubly transitive . we will assume that @xmath305 , for otherwise double transitivity is an empty condition . by proposition  [ prop : reduced ] , we may assume that @xmath192 is reduced . by lemma  [ l : t - transitive ] , it suffices to show that @xmath306 is irreducible . for this , we investigate the projection @xmath307 . here , @xmath308 is a subset of the fiber product @xmath309 , but @xmath310 is simply the ordinary product of grassmannians . unlike the projection @xmath311 , this is not a fiber bundle as @xmath312 depends upon the relative position of @xmath313 and @xmath314 . however , over the locus where @xmath315 it is a fiber bundle . since @xmath306 is a union of components of @xmath316 that project dominantly to @xmath206 , we will study @xmath317 and determine the components @xmath318 with @xmath319 . from our description of such components , we prove lemma  [ lem : mainlemma ] , which gives a numerical condition for @xmath318 that is equivalent to @xmath320 . in section  [ s : mr ] we use lemma  [ lem : mainlemma ] to show that @xmath306 is irreducible when @xmath192 is a special schubert problem , or a schubert problem in @xmath6 . we first determine the set - theoretic fibers @xmath312 of @xmath307 . [ l:2-fiber ] let @xmath199 be a schubert problem on @xmath81 and @xmath321 . if @xmath322 is the projection , then @xmath323    as before , we are identifying @xmath312 with its projection to @xmath206 in  .    by the definition of fiber product , @xmath324 since @xmath246 if and only if @xmath325 , we see that @xmath326 which implies  . [ r : dominant_components ] a component @xmath318 of @xmath317 that projects dominantly to @xmath206 must contain points @xmath327 where the flags @xmath328 are in general position so that @xmath329 . this implies that @xmath330 . consequently , in our arguments we may replace @xmath331 by any subset containing @xmath332 .    by lemma  [ l:2-fiber ] , we must first understand the set - theoretic intersection @xmath333 when @xmath334 . we begin with the case when the partition is a special schubert condition , @xmath335 , abbreviated . flags of type @xmath134 are elements @xmath141 of @xmath336 . for @xmath145 of dimension @xmath1 and @xmath141 of dimension @xmath337 write @xmath338 and @xmath339 for the corresponding special schubert varieties , @xmath340 and note that @xmath341 . set @xmath342 , so that @xmath343 . we expect that @xmath344 . [ lem : speccomponents ] let @xmath334 with @xmath345 . if @xmath346 or @xmath347 then the intersection @xmath348 has dimension @xmath349 and it is irreducible if @xmath350 or @xmath347 . when @xmath351 and @xmath352 , it consists of two components , one of dimension @xmath349 , and the other of dimension @xmath353 . this second , possibly excess , component consists of those @xmath141 with @xmath354 . set @xmath355 , which is @xmath356-dimensional . since @xmath357 , we have that @xmath358 . the intersection is a disjoint union @xmath359 where consists of those @xmath360 with @xmath361 and consists of those @xmath141 with @xmath362 . we compute the dimensions of @xmath363 and @xmath364 by studying incidence varieties which map birationally to each . set @xmath365 . this projects birationally to @xmath363 , and its image in the last two factors @xmath366 is the open subset @xmath367 with fiber over @xmath368 those @xmath141 which contain the 2-dimensional linear span @xmath369 but do not meet @xmath370 . this is a zariski open subset of the grassmannian @xmath371 thus the dimension of @xmath372 ( and thus of @xmath363 ) is @xmath373 for @xmath364 , define @xmath374 . this projects birationally to @xmath364 . the second projection @xmath375 is surjective with fiber over @xmath376 the set of those @xmath141 which contain @xmath377 , which is @xmath378 . thus the dimension of @xmath379 ( and thus of @xmath364 ) is @xmath380 note that @xmath381 , the expected dimension of @xmath382 . when @xmath347 , @xmath383 , and this intersection is irreducible . if @xmath384 , then @xmath385 and so @xmath364 lies in the closure of @xmath363 . when @xmath386 , @xmath387 , and so its closure is a component of the intersection , which has the expected dimension @xmath388 only when @xmath389 . we introduce notation for the sets @xmath363 and @xmath364 in the decomposition  . write for the set @xmath363 and for the set @xmath364 . we now consider intersections @xmath390 , where @xmath93 is any schubert condition . let @xmath391 with @xmath345 and set @xmath392 , so that @xmath393 . since we assume that our schubert problem is reduced , we will have @xmath394 and @xmath395 for some @xmath396 . we decompose @xmath397 into @xmath398 disjoint subsets and then compute the dimension of those that are nonempty . [ def : u_s ] for @xmath399 $ ] , define @xmath400 to be @xmath401 \cap s|\,,\ j=1,\dotsc , m \}\,.\ ] ] recall that @xmath402:=\{1,\dotsc , j\}$ ] . observe that we must have @xmath403 , and that @xmath404 indeed , for each @xmath405 and @xmath406 , we have @xmath407 , so that @xmath408 . furthermore , if @xmath409 , then @xmath410 . all possibilities for the dimensions of @xmath411 are given by @xmath412\cap s|$ ] for subsets @xmath413 of @xmath414 $ ] of cardinality at most @xmath356 . [ lem : components ] when @xmath415 , the set @xmath416 is irreducible of dimension @xmath417 \smallsetminus s|\right)\,.\ ] ]    the intersections of a flag @xmath418 with @xmath313 , @xmath314 , and @xmath370 are flags @xmath419 in these spaces with prescribed incidences . specifically , for each @xmath406 and @xmath405 , we have @xmath420 \cap s|}\ : = \     f_{n - k+j-\lambda_j } \cap l\,.\ ] ] then we have @xmath421 the flags @xmath422 instantiate the conditions for @xmath94 to belong to @xmath423 and the flag @xmath424 measures how the flags @xmath425 , @xmath426 , and @xmath94 meet @xmath427 . let consist of triples @xmath428 of flags , where @xmath429 \cap s|}\,.\ ] ] mapping @xmath418 to its intersections @xmath430 with @xmath313 , @xmath314 , and @xmath370 gives a projection @xmath431 . we compute the dimension of the fiber over @xmath432 . set @xmath433 , which has dimension @xmath434\cap s|$ ] . then the fiber @xmath435 is identified with its projection to @xmath138 which is an open subset of the schubert variety @xmath436 of lemma  [ lem : xeldot ] . in the notation of that lemma , @xmath437 and @xmath438\cap s|$ ] . then the fiber @xmath439 has dimension @xmath440 as @xmath441 if @xmath442 and @xmath443 if @xmath444 . we compute the dimension of @xmath445 . let @xmath446 be the projection forgetting @xmath425 and @xmath426 . for @xmath447 , the fiber @xmath448 is an open dense subset of @xmath449 where @xmath161 is the schubert variety of lemma  [ lem : xeldot ] . since @xmath450 has dimension @xmath451 and in the notation of lemma  [ lem : xeldot ] , @xmath452 and @xmath453\cap s|$ ] , this fiber @xmath448 has dimension @xmath454 finally , as @xmath455 has dimension @xmath456 \cap s|)\ = \ \sum_{j\in s } ( d - j+|[j]\smallsetminus s|)\,,\ ] ] we have @xmath457 \smallsetminus s|)\,.\ ] ] since all sets and fibers that we considered are irreducible , @xmath416 is irreducible . the decomposition of @xmath458 in definition  [ def : u_s ] as a disjoint union of sets @xmath459 gives a decomposition of the product of the intersections @xmath460 . for each @xmath461 , let @xmath462 be the index of the last nonzero part of @xmath463 . for each @xmath464 and each sequence @xmath465 where @xmath466 is a subset of @xmath467 $ ] of cardinality at most @xmath356 , define @xmath468 we write the product of the intersections @xmath460 as a disjoint union @xmath469 where @xmath470 ranges over all such sequences of subsets .    for each choice of @xmath470 , the sets @xmath471 for @xmath334 form a fiber bundle over @xmath472 , which we write as . the following proposition is clear . [ p : xtwocomponents ] @xmath306 is a subset of the closure of @xmath473 .    only sets @xmath474 of dimension equal to that of @xmath206 could map dominantly to @xmath206 with finite fibers over an open subset , and so it is a necessary condition that @xmath475 for the closure of @xmath474 to be a component of @xmath306 . we extract a numerical condition which is equivalent to this dimension condition . [ lem : mainlemma ] let @xmath192 be a schubert problem on @xmath81 with at least two solutions , @xmath356 an integer between @xmath111 and @xmath476 , and @xmath465 with @xmath477 $ ] and @xmath478 . then @xmath479 if and only if @xmath480 \smallsetminus s^i| )      \   = \   d(n-2k+d ) \,.\ ] ]    by lemma  [ lem : components ] , @xmath481 has dimension @xmath482 \smallsetminus s^i|)\bigr)\\     \ = \ n({{\boldsymbol{\lambda } } } ) - 2k(n - k)\ + \      \sum_{i=1}^s\sum_{j \in s^i } ( \lambda^i_j - k + d + |[j ] \smallsetminus s^i|)\ , .    \qquad \end{gathered}\ ] ] since @xmath483 , and @xmath484 , we have @xmath485 \smallsetminus s^i|)\,.\ ] ] since @xmath486 , the result follows . we use lemma  [ lem : mainlemma ] to show that if a schubert problem @xmath192 on @xmath81 is either special or if @xmath487 , then its galois group @xmath304 is doubly transitive . a is a schubert problem all of whose conditions are special , that is , a list @xmath488 in which each partition consists of a single row . writing @xmath489 for @xmath461 , the non - negative integers @xmath490 satisfy the numerical condition @xmath491 we identify @xmath192 with the sequence @xmath492 . thus , @xmath490 is always understood to be @xmath493 . [ th : gspec_double ] every special schubert problem has doubly transitive galois group .    let @xmath134 be a special schubert condition on @xmath81 and @xmath494 with @xmath495 . by lemma  [ lem : speccomponents ] , the intersection @xmath382 has possibly two components @xmath496 and @xmath497 , which are the closures of the sets @xmath498 and @xmath499 of lemma  [ lem : components ] . we adapt the notation of proposition  [ p : xtwocomponents ] to that of lemma  [ lem : speccomponents ] . the fiber @xmath312 has a decomposition into sets of the form @xmath500 where @xmath501 and @xmath502 $ ] . this is a mild change in notation from lemma  [ lem : mainlemma ] , where @xmath503 was indexed by a sequence @xmath504 , with @xmath505 when @xmath506 , and @xmath507 otherwise . let @xmath508 be the subset of @xmath317 whose fiber over @xmath509 is @xmath510 . then the condition   that @xmath511 becomes @xmath512    let @xmath513 so that @xmath514 is zariski dense in @xmath310 . we show that the only set @xmath508 which maps dominantly onto @xmath206 is the unique component of @xmath515 . we do this by first showing that @xmath515 has dimension @xmath516 and then that no component of @xmath317 for @xmath517 which has dimension @xmath516 maps dominantly to @xmath206 . this implies that @xmath306 is irreducible and thus @xmath304 is doubly transitive . first suppose that @xmath518 . by lemma  [ lem : speccomponents ] , the @xmath89th factor @xmath519 in   has a unique component which has dimension @xmath520 . indeed , if @xmath521 so that @xmath522 , then by lemma  [ lem : speccomponents ] , @xmath519 is irreducible of dimension @xmath520 . if @xmath523 , then @xmath524 . since @xmath192 is a reduced special schubert problem , we have @xmath525 and so @xmath526 and again lemma  [ lem : speccomponents ] implies that @xmath519 is irreducible of dimension @xmath520 . hence @xmath312 is irreducible of dimension @xmath527 since @xmath528 , we have that @xmath529 . we now show that if @xmath517 , then no set of the form @xmath508 of dimension @xmath516 maps dominantly to @xmath206 . suppose that @xmath517 and @xmath502 $ ] satisfies   so that @xmath530 . we claim that the image @xmath531 of @xmath508 in @xmath206 is not dense . suppose not . then @xmath532 meets every open subset of @xmath206 . for @xmath334 , if @xmath533 , then the definition of @xmath510 ( via @xmath534 for @xmath535 ) , implies that if @xmath535 , then @xmath536 meets the @xmath356-plane @xmath392 nontrivially . since @xmath537 , we see that @xmath370 lies in the intersection of schubert varieties in @xmath538 , @xmath539 and so this intersection is nonempty . since @xmath532 meets every open subset of @xmath206 , it meets the subset consisting of @xmath540 which are in sufficiently general position in that the intersection   has the expected codimension @xmath541 . suppose that @xmath509 is a point of @xmath472 with @xmath533 , which is in this general position . since   is nonempty , its codimension is at most the dimension of @xmath538 . that is , @xmath542 rewrite the sum as @xmath543 and then use   to obtain @xmath544 thus @xmath545 . if we set @xmath546 , then we have @xmath547 whence @xmath548 . thus there is some @xmath535 for which @xmath549 , contradicting our assumption that every @xmath550 and @xmath192 is reduced . [ th : gr3n_double ] every schubert problem in @xmath6 has a doubly transitive galois group . let @xmath192 be a schubert problem in @xmath6 . we may assume that @xmath192 is reduced . if not , then by proposition  [ prop : reduced ] it is equivalent to a reduced schubert problem on either a @xmath551 , a @xmath552 , or a @xmath553 with @xmath554 . all reduced schubert problems on @xmath555 have a single solution . theorem  [ th : gr3n_double ] covers the first case of @xmath551 and theorem  [ th : gspec_double ] covers the second case of @xmath552 , for every reduced schubert problem on @xmath552 is special  @xcite .    since @xmath556 is reduced , for every two partitions @xmath557 , we have @xmath558 in particular , these imply that at most one partition @xmath259 in @xmath192 has @xmath559 . indices @xmath470 of sets @xmath474 in the decomposition of @xmath317 are lists @xmath465 of subsets @xmath560 where @xmath561 if @xmath562 and @xmath478 . for @xmath563 define @xmath564\mid s^i = . then we can rewrite   as @xmath565    we prove theorem  [ th : gr3n_double ] by showing that there is at most one set @xmath474 with @xmath470 and @xmath356 satisfying  , ( and thus having dimension @xmath516 ) which maps dominantly to @xmath206 . since @xmath566 , this set maps dominantly to @xmath567 , and so it is dense in @xmath568 . this implies that @xmath569 is irreducible , and thus @xmath304 is doubly transitive . we consider each case @xmath570 separately in the following three lemmas . the set @xmath206 consists of @xmath202-tuples of partial flags @xmath571 where , if the @xmath89th partition is special , @xmath562 and so @xmath572 , then @xmath573 is omitted as it is not needed to define the schubert variety indexed by @xmath463 . however , to simplify the arguments that follow , we will write these flags as if they are all two - step flags . the resulting ambiguity may be remedied by setting @xmath574 when @xmath562 . [ lem : gr3n_part1 ] if there is a partition @xmath259 in @xmath192 with @xmath559 , then @xmath575 . in all other cases , @xmath576 is irreducible and has dimension @xmath516 . first suppose that there is a partition @xmath259 in @xmath192 with @xmath559 . if @xmath577 , then @xmath578 and @xmath579 . in particular this means that if @xmath580 , then @xmath581 and so @xmath582 . thus @xmath575 . suppose now that every partition @xmath259 in @xmath192 has @xmath583 . the only index @xmath470 for @xmath347 is when @xmath584 . then @xmath585 satisfies   and so @xmath586 has dimension @xmath516 . as the index @xmath470 is unique , @xmath576 is irreducible and has dimension @xmath516 . [ lem : gr3n_part2 ] there is a set @xmath587 of dimension @xmath516 if and only if there is a partition @xmath259 in @xmath192 with @xmath559 , and in that case the index @xmath470 is unique .    first note that if there is a partition @xmath259 in @xmath192 with @xmath559 , then it is unique , say @xmath588 . consider the index @xmath465 , where @xmath589 , and @xmath590 for all @xmath591 . then @xmath470 satisfies   with @xmath592 , and so @xmath587 has dimension @xmath516 . conversely , suppose that the index @xmath470 satisfies   for @xmath592 . one of the sets @xmath593 , @xmath594 , or @xmath595 must be nonempty . since @xmath592 , @xmath596 , because @xmath597 . let @xmath598 and set @xmath599 , which has dimension 1 . suppose that @xmath600 and that the flags are in general position . then by the definition of @xmath601 and @xmath602 , we have @xmath603 thus @xmath604 and so @xmath605 subtracting   from this and dividing by 2 gives @xmath606 . if @xmath607 then @xmath608 and   implies that @xmath609 , and thus @xmath192 is not a reduced schubert problem , which is a contradiction . thus @xmath610 and we have @xmath611 . if @xmath612 with @xmath613 , then   implies that @xmath614 , which implies that @xmath192 is not reduced , a contradiction . we are left with the case of @xmath615 . then   implies that @xmath616 , which completes the proof . [ lem : gr3n_part3 ] no set @xmath617 of dimension @xmath516 maps dominantly to @xmath206 . we first state an auxiliary lemma which will be used in the proof of lemma  [ lem : gr3n_part3 ] . [ lem : codim ] let @xmath93 be a schubert condition for @xmath6 , @xmath618 , and @xmath619 . setting @xmath620 and @xmath621 , then @xmath370 and @xmath622 satisfy schubert conditions @xmath259 and @xmath273 , respectively , with respect to the flag @xmath623 depending upon which subset @xmath459 they belong to according to the following table . @xmath624 here , @xmath625 . suppose that the index @xmath470 satisfies   for @xmath626 . we will show that the projection of @xmath617 to @xmath627 is not dense . let @xmath618 and suppose that @xmath628 . setting @xmath629 and @xmath630 then these lie in intersections of schubert varieties given by flags in @xmath631 as detailed in lemma  [ lem : codim ] . if we assume that the flags in @xmath632 are in general position so that those schubert varieties intersect in the expected dimensions , then we show that the nonemptiness of those intersections gives an inconsistent system of linear equations and inequalities , which proves the lemma . the condition that the intersection involving @xmath370 is nonempty is @xmath633 subtracting   from this and dividing by 2 gives @xmath634 the condition that the intersection involving @xmath622 is nonempty is @xmath635 adding   to this and dividing by two gives @xmath636 since @xmath192 is a schubert problem on @xmath6 , @xmath637 and so we obtain @xmath638 , which contradicts  . thus the projection of the set @xmath617 to @xmath206 can not meet the subset consisting of @xmath202-tuples of flags that are in general position for these two intersections of schubert varieties . we determine the dimensions of @xmath639 , @xmath640 , @xmath641 , and @xmath642 and then apply the definition   to obtain the partitions encoding those conditions . recall that @xmath643 and @xmath644 , where @xmath645 . we use the notation of the proof of lemma  [ lem : components ] , for @xmath406 , setting @xmath646 and @xmath647 . then the conditions on @xmath648 are that @xmath649 and @xmath650 , for each @xmath406 . if we set @xmath433 for @xmath651 , then @xmath652\cap s|$ ] and similarly @xmath653\cap s|$ ] . these are lower bounds on the dimension of @xmath641 and @xmath642 . suppose first that @xmath654 so that @xmath655 , and so @xmath656 and @xmath657 . then the flags @xmath623 impose no conditions on @xmath370 so that @xmath658 . if @xmath659 , so that @xmath660 , then @xmath661 and @xmath273 is @xmath662 . however , if @xmath663 so that @xmath664 then @xmath665 , and we have @xmath666 . suppose that @xmath667 so that @xmath668 and this has dimension 1 , so that @xmath669 . then @xmath670 and @xmath671 . as in the previous case , the dimension of @xmath641 depends upon whether or not @xmath663 , and so we get @xmath672 . suppose that @xmath673 so that @xmath674 , but @xmath675 . then @xmath676 . we have that @xmath677 and @xmath678 , so that @xmath679 . finally , suppose that @xmath680 so that @xmath681 and @xmath682 . then @xmath683 and @xmath684 . the main result of  @xcite is that every schubert problem on @xmath5 has at least alternating galois group . the proof relied on vakil s criterion ( b ) and used elementary , but very involved , estimates of trigonometric integrals to establish an inequality between two kostka numbers . we use the stronger and simpler vakil s criterion ( c ) ( which applies , by theorem  [ th : gspec_double ] ) to give a much simpler and purely combinatorial proof of that result . [ thm : g2nalternating ] every schubert problem on @xmath5 has at least alternating galois group . it suffices to prove this for reduced schubert problems , by proposition  [ prop : reduced ] . by definition [ d : reduced ] , a reduced schubert problem on @xmath5 has the form @xmath685 with @xmath686 and @xmath687 for each @xmath122 , in particular , a reduced schubert problem is special . let @xmath688 be the number of solutions to a special schubert problem @xmath685 on @xmath5 . this is a kostka number , which counts the number of young tableaux of shape @xmath689 and content @xmath685 . these are fillings of the young diagram of shape @xmath689 with @xmath690 ones , @xmath691 twos , through @xmath692 s s , such that the filling is weakly increasing along the rows and strictly increasing down the columns . section 2.1 of  @xcite shows that there is a fiber diagram  , such that @xmath693 and @xmath694 are isomorphic to the galois groups of the schubert problems @xmath695 and @xmath696 , respectively . this gives , @xmath697    the following proposition is proven in  @xcite using a combinatorial injection . [ prop : injection ] let @xmath698 be a reduced schubert problem on @xmath5 such that @xmath699 with @xmath700 . then @xmath701    we compute some kostka numbers . [ lem : multparts ] let @xmath702 be positive integers with @xmath703 . then    1 . @xmath704 , 2 . @xmath705 , 3 . @xmath706 and @xmath707 , where @xmath708 is even , 4 . @xmath709 , 5 . @xmath710 when @xmath711 . statements ( 1)(3 ) are from lemmas 8 and 9 of  @xcite . statement ( 4 ) is simple , as @xmath712 is the number of triples @xmath713 of nonnegative integers whose sum is @xmath134 . a young tableau of shape @xmath689 and content @xmath714 is determined by its second row which consists of @xmath715 fives and @xmath134 remaining numbers which are some twos , threes , and fours .    for ( 5 ) , we construct tableaux of shape @xmath689 and content @xmath716 from tableaux of shape @xmath717 and content @xmath718 and of shape @xmath719 and content @xmath720 . let @xmath711 and consider a tableau @xmath721 of shape @xmath717 and content @xmath722 . for each @xmath723 , construct a tableau @xmath724 of @xmath689 and content @xmath716 as follows . the second row of @xmath721 ends in @xmath715 entries of @xmath725 . remove these to get a tableau @xmath726 of shape @xmath727 and content @xmath720 , which we extend to get @xmath724 as follows . place @xmath89 entries of @xmath725 and @xmath728 entries of @xmath729 at the end of the first row of @xmath726 and then @xmath728 entries of @xmath725 and @xmath89 entries of @xmath729 in the second row , followed by @xmath715 entries of @xmath730 to obtain @xmath724 . @xmath731 this gives @xmath732 tableaux of shape @xmath689 and content @xmath716 . given a tableau @xmath721 of shape @xmath719 and content @xmath720 , add @xmath134 entries of @xmath725 and @xmath134 of @xmath729 to the first row and @xmath715 entries of @xmath730 to get a tableau of shape @xmath689 and content @xmath716 . @xmath733 this gives @xmath734 tableaux of shape @xmath689 and content @xmath716 , all of which are different from the others we constructed , which proves statement  ( 5 ) . we prove the result by induction on @xmath202 and @xmath2 . let @xmath735 be a reduced schubert problem . we may assume that @xmath736 , for otherwise @xmath737 and there is nothing to prove . since @xmath304 is doubly transitive by theorem  [ th : gspec_double ] , vakil s criterion ( c ) and schubert s recursion   imply that @xmath304 is at least alternating if one of @xmath738 or @xmath739 is not six , for some reordering of the list @xmath740 . suppose first that not all the @xmath490 are equal and that they are ordered so that @xmath699 with @xmath700 . by  , @xmath688 is equal to either the sum , @xmath741 or the sum @xmath742 by proposition  [ prop : injection ] , @xmath743 , so they can not both be six , which implies @xmath304 is at least alternating . assume now @xmath744 for some positive integer @xmath134 . there are only two values of @xmath745 for which @xmath746 , and for both of these @xmath747 , which implies that @xmath748 is at least alternating .    indeed , by lemma  [ lem : multparts ] , if @xmath749 , then @xmath750 . if @xmath751 , then @xmath752 only for @xmath753 , and then @xmath754 . if @xmath755 , then @xmath756 only for @xmath757 , and then @xmath758 . we show that if @xmath759 , then @xmath760 . by lemma  [ lem : multparts ] ( 5 ) , @xmath761 so it suffices to show @xmath760 for @xmath762 . when @xmath763 , this becomes @xmath764 and when @xmath765 , @xmath766 is even , and we have @xmath767 this completes the proof . using vakil s criteria and the frobenius algorithm of  @xcite ( a symbolic method to prove that a galois group is full symmetric by computing cycle types of elements ) , we study the galois group of every reduced schubert problem on @xmath0 . a nonreduced problem in @xmath0 is equivalent to a problem in some @xmath81 with @xmath768 and @xmath769 , and vakil earlier showed that these problems have at least alternating galois groups . there are 2987 reduced schubert problems on @xmath0 having two or more solutions . all except fourteen have at least alternating galois group with many known to be the full symmetric group . each of these fourteen have imprimitive galois group , which we determine , and they fall into three families according to their geometry . much of this computation , except for the determination of the galois groups of the last thirteen schubert problems , was done by vakil in  @xcite . vakil wrote a maple script based on his geometric littlewood - richardson rule and his criterion ( b ) to test which schubert problems had galois groups that are at least alternating  @xcite . we altered it to only test reduced schubert problems on @xmath0 and found 28 problems for which it was inconclusive . we list them , using a compact product notation in which the schubert problem @xmath770 with 32 solutions is written @xmath771 .                              in subsection  [ s : largesp ] we show that the two schubert problems in the first row have at least alternating galois groups . we used the frobenius algorithm  @xcite to show that the next twelve have full symmetric galois group . the remaining fourteen on the last four lines have imprimitive galois groups . they are grouped by similar geometry , which we indicate by strings of equalities . we describe one problem from each family in the remaining three subsections . the schubert problem @xmath789 is special and therefore has doubly transitive galois group by theorem  [ th : gspec_double ] . it asks for the 4-planes @xmath145 in @xmath790 that meet 16 general 4-planes @xmath791 nontrivially . if @xmath792 so that @xmath793 is a 3-plane and @xmath794 is a 5-plane , then lemma  [ lem : speccomponents ] shows that @xmath795 thus if @xmath796 and @xmath797 is the restriction of @xmath203 to @xmath798 ( as in  ) , then @xmath799 where @xmath800 and @xmath801 are ( essentially ) total spaces of the schubert problems @xmath802 these are equivalent dual problems and each was found to have at least alternating galois group by vakil s maple script . as the original problem @xmath772 is doubly transitive , vakil s criterion ( c ) and remark  [ r : vakil ] implies that its galois group is at least alternating . the schubert problem @xmath803 also has doubly transitive galois group . to see this note that by lemma  [ lem : components ] the subset @xmath804 of @xmath805 has dimension at most @xmath806 unless @xmath807 and @xmath808 , and that @xmath602 has dimension 9 . since @xmath809 has dimension @xmath810 for @xmath495 , we see that the only set @xmath474 having dimension @xmath516 is when @xmath347 and each component of @xmath470 is @xmath811 . the special position   gives a subset @xmath812 with the restriction of @xmath203 to @xmath798 having two components , each essentially the total space of one of the schubert problems @xmath813 these equivalent dual schubert problems were found to have at least alternating galois group by vakil s maple script . as the original problem @xmath773 is doubly transitive , vakil s criterion ( c ) implies that its galois group is at least alternating . derksen discovered that this schubert problem has galois group @xmath814 and it was described by vakil  @xcite . an instance is given by four @xmath815-planes @xmath816 in general position in @xmath790 . its solutions are those @xmath817 for which @xmath818 for @xmath819 . consider the _ auxiliary _ problem @xmath820 in @xmath821 given by @xmath816 . this asks for those @xmath822 with @xmath823 for @xmath819 . there are four solutions @xmath824 to this problem , and its galois group is the full symmetric group @xmath814 . each of the 4-planes @xmath825 will meet each @xmath536 in a 2-plane , and so they are solutions to the original problem . in fact , they are the only solutions . it follows that the galois group of @xmath785 is @xmath814 acting on the pairs @xmath826 . this is an imprimitive permutation group as it preserves the partition @xmath827 of the solutions . we also see that @xmath569 has two components . exactly two sets @xmath828 and @xmath829 have dimension @xmath516 . the structure of this problem shows that if the @xmath536 are real , then either two or all six of the solutions will be real . indeed , if all four solutions @xmath830 to the auxiliary problem are real , than all six solutions @xmath831 will also be real . if however , two or four of the @xmath830 occur in complex conjugate pairs , then exactly two of the @xmath831 will be real . let @xmath833 , @xmath834 , and @xmath835 be general . these give an instance of this schubert problem , @xmath836 any solution @xmath145 meets each of the 6-planes @xmath837 in a 3-plane and therefore their four - dimensional intersection @xmath838 in a 2-plane , @xmath839 . then @xmath839 must meet the 2-planes @xmath377 , @xmath840 , @xmath841 , and @xmath842 , so it is a solution to the problem @xmath843 in @xmath844 . as the subspaces are in general position , this has two solutions @xmath845 and @xmath846 . any solution @xmath145 to our original problem also meets each of the 2-planes @xmath847 and @xmath848 in a 1-plane , and therefore meets their span , @xmath849 , in a 2-plane , @xmath156 . as @xmath850 , they are in direct sum and @xmath145 is the span of @xmath156 and one of the @xmath830 . fix @xmath851 and suppose that @xmath852 . then @xmath145 meets each of @xmath853 for @xmath651 in a 3-plane and therefore @xmath145 meets each of the 2-planes @xmath854 and @xmath855 . thus @xmath856 is a solution to the problem @xmath857 in @xmath858 . this has two solutions , @xmath859 and @xmath860 . the four 4-planes @xmath861 for @xmath862 are the solutions to our schubert problem . since each element of the galois group either fixes @xmath845 and @xmath846 or it interchanges them , it preserves the partition @xmath863 and so it is imprimitive . in fact , it is a subgroup of the dihedral group @xmath864 of symmetries of the square whose diagonals are the partition . we verified that it was the dihedral group @xmath864 by reducing modulo primes @xmath865 and computing the cycles type of the resulting frobenius elements ( this method is described in  @xcite ) and found elements of the galois group with cycle types @xmath866 as the only subgroup of @xmath867 having elements of these cycle types is @xmath864 itself , we conclude that the original schubert problem had galois group @xmath864 . it is an exercise to verify that the three problems @xmath868 , @xmath869 , and @xmath870 , have nearly the same geometry behind their solutions and also have galois group @xmath864 . this last problem was mentioned by billey and vakil  @xcite , who asked if its galois group was indeed @xmath864 . an instance of this problem is given by the choice of two 6-planes @xmath872 , two 2-planes @xmath873 and four 4-planes @xmath816 , all in general position . solutions will be those @xmath817 such that @xmath874 for @xmath406 and @xmath875 . consider the first four conditions in  . let @xmath876 , the linear span of @xmath877 and @xmath878 , which is isomorphic to @xmath879 . then @xmath880 is two - dimensional . if we set @xmath881 and @xmath882 , then @xmath883 , and so @xmath884 meets each of the four two - planes @xmath885 , so @xmath839 is a solution to the problem @xmath886 in @xmath887 given by @xmath885 , and therefore there are two solutions , @xmath845 and @xmath846 . now set @xmath888 , which is four - dimensional , and fix one of the solutions to the problem of the previous paragraph . for each @xmath875 , set @xmath889 , which is two - dimensional . these four two - planes are in general position and therefore give an instance of @xmath886 in @xmath890 . let and be the two solutions to this problem , so that @xmath891 for each @xmath892 . then the four subspaces @xmath893 are solutions to the original schubert problem . indeed , since @xmath894 for @xmath875 and @xmath895 , we have @xmath896 , and so @xmath897 satisfies the first four conditions of  . since @xmath898 , @xmath899 does not meet @xmath900 for @xmath875 , so @xmath901 , which implies that @xmath902 , and shows that @xmath897 is a solution . the galois group preserves the partition @xmath903 and so it is imprimitive . the same arguments as before show that @xmath864 is its galois group . it is also an exercise to show that each of the following eight schubert problems has a solution whose description is nearly identical to @xmath871 , and therefore has galois group @xmath864 .
we investigate double transitivity of galois groups in the classical schubert calculus on grassmannians . 
 we show that all schubert problems on grassmannians of 2- and 3-planes have doubly transitive galois groups , as do all schubert problems involving only special schubert conditions . 
 we use these results to give a new proof that schubert problems on grassmannians of 2-planes have galois groups that contain the alternating group . 
 we also investigate the galois group of every schubert problem on @xmath0 , finding that each galois group either contains the alternating group or is an imprimitive permutation group and therefore fails to be doubly transitive . 
 these imprimitive examples show that our results are the best possible general results on double transitivity of schubert problems .