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schur -- creates a map between Schur modules

Description

Applies the Schur functor associated to lambda to the map f between free modules. The modules source F and target F are Schur modules containing certain data in cache (see schurModule).

i1 : R=QQ[x_1,x_2,x_3]

o1 = R

o1 : PolynomialRing
 
i2 : F=map(R^1,R^3,vars R)

o2 = | x_1 x_2 x_3 |

             1      3
o2 : Matrix R  <-- R
 
i3 : L=schur({2},F) -- 2nd veronese embedding

o3 = | x_1^2 x_1x_2 x_1x_3 x_2^2 x_2x_3 x_3^2 |

             1      6
o3 : Matrix R  <-- R
 
i4 : F=matrix{{1_QQ,2,4},{3,9,27},{4,16,64}}

o4 = | 1 2  4  |
     | 3 9  27 |
     | 4 16 64 |

              3       3
o4 : Matrix QQ  <-- QQ
 
i5 : schur({1,1},F)

o5 = | 3  15 18  |
     | 8  48 64  |
     | 12 84 144 |

              3       3
o5 : Matrix QQ  <-- QQ
 
i6 : minors(2,F)

o6 = ideal (3, 8, 12, 15, 48, 84, 18, 64, 144)

o6 : Ideal of QQ
 
i7 : schur({1,1,1},F) == det F

o7 = true

Caveat

The partition lambda should be a valid nonempty partition.

See also

Ways to use schur:

  • schur(List,Matrix)

For the programmer

The object schur is a method function.

The source of this document is in SchurFunctors.m2:397:0.