NSC2phc -- dictionary between different notations for Schubert problems.

Synopsis

Usage:

M = NSC2phc(conds,k,n)
Inputs:
- conds, a list, of Schubert conditions, either partitions or brackets, that constitutes a Schubert problem on the Grassmannian $Gr(k,n)$
- k, an integer,
- n, an integer, $k$ and $n$ represent the Grassmannian $Gr(k,n)$
Outputs:
- a matrix, the corresponding Schubert problem in notation for PHCPack implementation of Littlewood-Richardson rule and homotopies. Its rows encode Schubert conditions with multiplicities. 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 in this Schubert intersection problem.

Description

A Schubert problem in the Grassmannian $Gr(k,n)$ is encoded by either a list of partitions or brackets whose codimensions sum to $k(n-k)$. (see bracket2partition for details on brackets and partitions)

The PHCPack implementations of the geometric Littlewood-Richardson rule encode the brackets in a matrix, where each row has the form ${m, b}$ with $m$ the multiplicity of the bracket $b$, which is a strictly increasing sequence of $k$ integers between $1$ and $n$.

i1 : k=4;

i2 : n = 8;

i3 : SchubProbP = {{2,2},{2,2},{2,2},{1},{1},{1},{1}}

o3 = {{2, 2}, {2, 2}, {2, 2}, {1}, {1}, {1}, {1}}

o3 : List

i4 : NSC2phc(SchubProbP,k,n)

o4 = | 3 3 4 7 8 |
     | 4 4 6 7 8 |

              2        5
o4 : Matrix ZZ  <--- ZZ

i5 : k=4;

i6 : n = 8;

i7 : SchubProbB = {{3,4,7,8},{3,4,7,8},{3,4,7,8},{4,6,7,8},{4,6,7,8},{4,6,7,8},{4,6,7,8}}

o7 = {{3, 4, 7, 8}, {3, 4, 7, 8}, {3, 4, 7, 8}, {4, 6, 7, 8}, {4, 6, 7, 8},
     ------------------------------------------------------------------------
     {4, 6, 7, 8}, {4, 6, 7, 8}}

o7 : List

i8 : NSC2phc(SchubProbB,4,8)

o8 = | 3 3 4 7 8 |
     | 4 4 6 7 8 |

              2        5
o8 : Matrix ZZ  <--- ZZ

Ways to use NSC2phc :

NSC2phc(List,ZZ,ZZ)

For the programmer

The object NSC2phc is a method function.

NSC2phc -- dictionary between different notations for Schubert problems.

Synopsis

Description

See also

Ways to use NSC2phc :

For the programmer