# 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, ,
• L, a list, of integers between 1 and the number of rows of A (inclusive)
• Outputs:
• FM, ,

## 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.