LRrule -- computes the product of Schubert classes using geometric Littlewood-Richardson rule

Synopsis

Usage:

s = LRrule(n,M)
Inputs:
- n, an integer, the dimension of the ambient space of the Grassmannian $Gr(k,n)$.
- M, a matrix, whose rows encode Schubert conditions. Each row is a $k+1$-tuple, {m,b}, where $m$ is a nonegative integer and $b$ a bracket (see bracket2partition for details). The bracket $b$ represents a Schubert condition and $m$ is its multiplicity.
Optional inputs:
- Verbose => ..., default value false, request verbose feedback
Outputs:
- s, a string, contains an equation, with at the left the product of Schubert conditions and at the right the result as a formal sum of brackets.

Description

LRrule uses the geometric Littlewood-Richardson rule to compute a product in the Chow ring of the Grassmannian. This writes a product of brackets as a formal sum of brackets, which represents an intersection of Schubert varieties as a formal sum of Schubert varieties. When the input matrix M is a Schubert problem, this gives the number of solutions to that Schubert problem. The command LRnumber calls LRrule and extracts the number of solutions.

i1 : R = ZZ;

i2 : n = 7;

i3 : M = matrix{{3, 3, 6, 7},{2, 3, 5, 7}};

              2        4
o3 : Matrix ZZ  <--- ZZ

i4 : LRrule(n,M)

o4 = [ 3 6 7 ]^3*[ 3 5 7 ]^2 = +10[1 2 3]

The output: [ 3 6 7 ]^3*[ 3 5 7 ]^2 = +10[1 2 3] means that the Schubert problem [ 3 6 7 ]^3*[ 3 5 7 ]^2 in multiplicative form has 10 solution 3-planes. That is, there are 10 3-planes that satisfy three Schubert conditions given by the bracket [3, 6, 7] and two conditions given by the bracket [3, 5, 7].

More generally, this computes a product in the Chow ring:

i5 : LRrule(7, matrix{{2,3,6,7},{1,3,5,7},{1,2,5,7}})

o5 = [ 3 6 7 ]^2*[ 3 5 7 ]*[ 2 5 7 ] = +2[1 2 4]+4[1 2 4]+2[1 2 4]

Caveat

The Littlewood-Richardson homotopy algorithm requires a Schubert problems (sum of codimensions equals the dimension of the Grassmannian).

Ways to use LRrule :

LRrule(ZZ,Matrix)

For the programmer

The object LRrule is a method function with options.