-- This file conains the column "M" of Table 2 in our article:

-- "FORMATS OF 6×6 SKEW MATRICES OF LINEAR FORMS 
--  WITH VANISHING PFAFFIAN"

-- the matrices S_a..S_f are included for testing purposes only
-- and are explained in our article:

-- "MODULI SPACES OF 6×6 SKEW MATRICES OF LINEAR FORMS ON P4 
--  WITH A VIEW TOWARDS INTERMEDIATE JACOBIANS OF CUBIC THREEFOLDS"

restart
K = QQ
R = K[l_0..l_4]

-------------
-- Table 2 --
-------------

--------------
-- case (a) --
--------------

M_a = matrix {
    {   0, l_3,   0,   0, l_0, l_1}, 
    {-l_3,   0,   0,-l_0,   0, l_2}, 
    {   0,   0,   0,-l_1,-l_2,   0}, 
    {   0, l_0, l_1,   0, l_4,   0}, 
    {-l_0,   0, l_2,-l_4,   0,   0},
    {-l_1,-l_2,   0,   0,   0,   0}
      }
  
S_a = matrix {
    { l_2,   0}, 
    {-l_1,   0}, 
    { l_0, l_4}, 
    {   0, l_2}, 
    {   0,-l_1}, 
    { l_3, l_0}
    }

assert (0==M_a*S_a)
 
--------------
-- case (b) --
--------------

M_b = matrix {
    {   0, l_0, l_1,   0,   0,   0}, 
    {-l_0,   0, l_2,   0,   0,   0}, 
    {-l_1,-l_2,   0,   0,   0,   0}, 
    {   0,   0,   0,   0, l_2, l_3}, 
    {   0,   0,   0,-l_2,   0, l_4}, 
    {   0,   0,   0,-l_3,-l_4,   0}
    }

S_b = matrix {
    { l_2,   0}, 
    {-l_1,   0}, 
    { l_0,   0}, 
    {   0, l_4},
    {   0,-l_3}, 
    {   0, l_2}
    }

assert (0==M_b*S_b)


--------------
-- case (c) --
--------------

M_c = matrix {
    {   0, l_0, l_1,  0,   0,   0}, 
    {-l_0,   0, l_2,  0,   0,   0}, 
    {-l_1,-l_2,   0,  0,   0,   0}, 
    {   0,   0,   0,  0, l_1, l_2}, 
    {   0,   0,   0,-l_1,  0, l_3}, 
    {   0,   0,   0,-l_2,-l_3,  0}}

S_c = matrix {
    { l_2,   0}, 
    {-l_1,   0}, 
    { l_0,   0}, 
    {   0, l_3},
    {   0,-l_2}, 
    {   0, l_1}}


assert (0==M_c*S_c)

--------------
-- case (d) --
--------------

M_d = matrix{
    {   0,   0,   0,   0, l_0, l_1},
    {   0,   0,   0,-l_0,   0, l_2},
    {   0,   0,   0,-l_1,-l_2,   0},
    {   0, l_0, l_1,   0, l_3, l_4},
    {-l_0,   0, l_2,-l_3,   0,   0},
    {-l_1,-l_2,   0,-l_4,   0,   0}
    }

S_d = matrix {
    { l_2,   0}, 
    {-l_1,-l_4}, 
    { l_0, l_3}, 
    {   0, l_2},
    {   0,-l_1}, 
    {   0, l_0}}

assert (0==M_d*S_d)

--------------
-- case (e) --
--------------

M_e = matrix{
    {   0,   0,   0,   0, l_0, l_1},
    {   0,   0,   0,-l_0,   0, l_2},
    {   0,   0,   0,-l_1,-l_2,   0},
    {   0, l_0, l_1,   0, l_3,   0},
    {-l_0,   0, l_2,-l_3,   0,   0},
    {-l_1,-l_2,   0,   0,   0,   0}
    }

S_e = matrix {
    { l_2,   0}, 
    {-l_1,   0}, 
    { l_0, l_3}, 
    {   0, l_2},
    {   0,-l_1}, 
    {   0, l_0}}

assert (0==M_e*S_e)


--------------
-- case (f) --
--------------

M_f = matrix{
    {   0,   0,   0,   0, l_0, l_1},
    {   0,   0,   0,-l_0,   0, l_2},
    {   0,   0,   0,-l_1,-l_2,   0},
    {   0, l_0, l_1,   0,   0,   0},
    {-l_0,   0, l_2,   0,   0,   0},
    {-l_1,-l_2,   0,   0,   0,   0}
    }

S_f = matrix {
    { l_2,   0}, 
    {-l_1,   0}, 
    { l_0,   0}, 
    {   0, l_2},
    {   0,-l_1}, 
    {   0, l_0}}

assert (0==M_f*S_f)

-- the cases in the table
cases = {a,b,c,d,e,f}

-- TEST: is M_i skew?
apply(cases,i->assert (0==M_i+transpose M_i));

-- TEST: is M_iS_i = 0 iff i==j?
apply(cases,i->apply(cases,j->(
    if i==j then assert (0 == M_i*S_j);
    if i!=j then assert (0 != M_i*S_j);
    )));