Minors with prescribed rows or columns

[mahrud]

Is there an easy way to compute the ideal of minors that include a specific set of rows or columns? I tried to think of a way to do this incrementally; e.g. exclude those rows and columns, find the minors, then add in one row or column and update the minors. Perhaps there's a smart way to do this, but I couldn't think of it, so I wrote this piece of code that I think should become a part of the minors command if there isn't a better way to do it:

minors' = method(Options => options minors ++ {Prefix => null})
minors'(ZZ, Matrix) := opts -> (n, m) -> (
    if opts.Prefix === null then return minors(n, m, opts);
    -- rows and columns to force in the minors
    -- TODO: implement general type checking, e.g. instance({2:{ZZ}}, opts.Prefix)
    (rows, cols) := (set opts.Prefix#0, set opts.Prefix#1);
    rowsets := apply(subsets(set(0..numrows m - 1) - rows, n - #rows), r -> toList(r + rows));
    colsets := apply(subsets(set(0..numcols m - 1) - cols, n - #cols), c -> toList(c + cols));
    ideal apply(rowsets ** colsets, coords -> det submatrix(m, coords#0, coords#1)))

Example:

i2 : R = QQ[a..z];

i3 : m = genericMatrix(R, 2, 4)

o3 = | a c e g |
     | b d f h |

             2       4
o3 : Matrix R  <--- R

i4 : minors'(2, m, Prefix => {{}, {0}})

o4 = ideal (- b*c + a*d, - b*e + a*f, - b*g + a*h)

It doesn't take advantage of the engine's rawMinors routine, but still might be useful. It is likely also implemented by others, which is why I'm asking here to see if there is a better implementation.