schur -- creates a map between Schur modules

Synopsis

Usage:

schur(lambda,f)
Inputs:
- lambda, a list, a list of numbers representing a partition
- f, a module, a map between two free modules
Outputs:
- F, a matrix, the result of applying the Schur functor associated to lambda to f

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.

Ways to use `schur` :

"schur(List,Matrix)"

For the programmer

The object schur is a method function.