# 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 map between two free modules
• Outputs:
• F, , 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.

## See also

• schurModule -- creates Schur module from a partition and free module

## Ways to use schur :

• "schur(List,Matrix)"

## For the programmer

The object schur is .