# 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:
• , 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