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.
i1 : R = ZZ; |
i2 : n = 7; |
i3 : M = matrix{{3, 3, 6, 7},{2, 3, 5, 7}}; 2 4 o3 : Matrix ZZ <--- ZZ |
i4 : print LRrule(n,M); PHCv2.4.77 released 2020-06-12 writing data to file /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/2PHCinput running phc -e, writing output to /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/3PHCoutput opening output file /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/3PHCoutput [ 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].
If the right hand side of the equation returned by LRrule consists of one bracket of consecutive natural numbers starting at 1 (as above), then there are finitely many solutions. Otherwise, the problem may be underdetermined, consider the example:
i5 : LRrule(7, matrix{{2,3,6,7},{1,3,5,7},{1,2,5,7}}) PHCv2.4.77 released 2020-06-12 writing data to file /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/5PHCinput running phc -e, writing output to /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/6PHCoutput opening output file /var/folders/11/t6mmkf11455d4nt0fn639c6h0000gn/T/M2-95707-0/6PHCoutput o5 = [ 3 6 7 ]^2*[ 3 5 7 ]*[ 2 5 7 ] = +2[1 2 4]+4[1 2 4]+2[1 2 4] |
Littlewood-Richardson homotopies work only for Schubert problems, when the sum of the codimensions of the Schubert varieties equals the dimension of the Grassmannian.
The object LRrule is a method function.