schubertPolynomial -- computes the Schubert polynomial of a permutation

Usage:

schubertPolynomial w
Inputs:
- w, a list,
Optional inputs:
- Algorithm => a string, default value "DividedDifference", algorithm "Transition" also available

Description

Given a permutation in 1-line notation, finds its (single) Schubert polynomial. Two algorithms are impliemented: DividedDifference (which is the default) and Transition (which makes use of the transition equations for Schubert polynomials).

i1 : w = {2,1,5,4,3}

o1 = {2, 1, 5, 4, 3}

o1 : List

i2 : schubertPolynomial w

      3      2 2    3       2          2      2 2        2    3      2      
o2 = x x  + x x  + x x  + 2x x x  + x x x  + x x  + x x x  + x x  + x x x  +
      1 2    1 2    1 3     1 2 3    1 2 3    1 3    1 2 3    1 4    1 2 4  
     ------------------------------------------------------------------------
        2      2                     2
     x x x  + x x x  + x x x x  + x x x
      1 2 4    1 3 4    1 2 3 4    1 3 4

o2 : QQ[x ..x ]
         1   5

i3 : schubertPolynomial (w,Algorithm=>"Transition")

      3      2 2    3       2          2      2 2        2    3      2      
o3 = x x  + x x  + x x  + 2x x x  + x x x  + x x  + x x x  + x x  + x x x  +
      1 2    1 2    1 3     1 2 3    1 2 3    1 3    1 2 3    1 4    1 2 4  
     ------------------------------------------------------------------------
        2      2                     2
     x x x  + x x x  + x x x x  + x x x
      1 2 4    1 3 4    1 2 3 4    1 3 4

o3 : QQ[x ..x ]
         1   5

Ways to use schubertPolynomial:

schubertPolynomial(List)

For the programmer

The object schubertPolynomial is a method function with options.

The source of this document is in MatrixSchubert/permutationMethodsDOC.m2:479:0.