flagMatroid -- construct a flag matroid

Synopsis

Usage:

FM = flagMatroid(ML)

FM = flagMatroid(A,L)
Inputs:
- ML, a list, of Matroids on a common ground set
- A, a matrix,
- L, a list, of integers between 1 and the number of rows of A (inclusive)
Outputs:
- FM, a flag matroid,

Description

Given a list $ML$ of matroids on a common ground set, this method stores the data as a FlagMatroid.

i1 : ML = {uniformMatroid(2,6),matroid completeGraph 4}

o1 = {a "matroid" of rank 2 on 6 elements, a "matroid" of rank 3 on 6
     ------------------------------------------------------------------------
     elements}

o1 : List

i2 : FM = flagMatroid(ML)

o2 = a "flag matroid" with rank sequence {2, 3} on 6 elements 

o2 : FlagMatroid

i3 : isWellDefined FM

o3 = true

For $A$ an $r\times n$ matrix over a field and $L = \{r_1, \ldots, r_k}$ a list of integers, let $M_i$ be the Matroid defined by the columns of the matrix obtained by the first $r_i$ rows of $A$. These matroids form a flag matroid $\mathbf M = \{M_1, \ldots, M_k\}$. This method creates this FlagMatroid.

i4 : A = random(QQ^2,QQ^4)

o4 = | 9/2 9/4 1   3/2 |
     | 1/2 1/2 3/4 3/4 |

              2       4
o4 : Matrix QQ  <-- QQ

i5 : FM = flagMatroid(A,{1,2})

o5 = a "flag matroid" with rank sequence {1, 2} on 4 elements 

o5 : FlagMatroid

Caveat

When a list of matroids is given as input, this method does not check if the flag matroid is well-defined.

Ways to use flagMatroid :

flagMatroid(List)
flagMatroid(Matrix,List)

For the programmer