Example 2 -- Symbolic powers of star configurations

In this example, we identify the Betti characters of the third symbolic power of a monomial star configuration. The action of the symmetric group on the resolution of this ideal is described in Example 6.5 of J. Biermann, H. De Alba, F. Galetto, S. Murai, U. Nagel, A. O'Keefe, T. Römer, A. Seceleanu - Betti numbers of symmetric shifted ideals, and belongs to the larger class of symmetric shifted ideals.

We start by constructing the ideal, and compute a minimal free resolution and its Betti table.

i1 : R=QQ[x_1..x_6]

o1 = R

o1 : PolynomialRing

i2 : I=intersect(apply(subsets(gens R,4),x->(ideal x)^3))

                                                                       
o2 = ideal (x x x x x , x x x x x , x x x x x , x x x x x , x x x x x ,
             2 3 4 5 6   1 3 4 5 6   1 2 4 5 6   1 2 3 5 6   1 2 3 4 6 
     ------------------------------------------------------------------------
                    2 2 2     2 2 2     2 2 2   2   2 2   2   2 2   2   2 2 
     x x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      1 2 3 4 5   3 4 5 6   2 4 5 6   1 4 5 6   3 4 5 6   2 4 5 6   1 4 5 6 
     ------------------------------------------------------------------------
        2 2 2     2 2 2   2   2 2   2   2 2     2 2 2   2   2 2   2 2   2 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      2 3 5 6   1 3 5 6   2 3 5 6   1 3 5 6   1 2 5 6   1 2 5 6   3 4 5 6 
     ------------------------------------------------------------------------
      2 2   2   2 2   2   2 2   2   2 2   2   2 2   2     2 2 2     2 2 2 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      2 4 5 6   1 4 5 6   2 3 5 6   1 3 5 6   1 2 5 6   2 3 4 6   1 3 4 6 
     ------------------------------------------------------------------------
      2   2 2   2   2 2     2 2 2   2   2 2   2 2   2   2 2   2   2 2   2 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      2 3 4 6   1 3 4 6   1 2 4 6   1 2 4 6   2 3 4 6   1 3 4 6   1 2 4 6 
     ------------------------------------------------------------------------
        2 2 2   2   2 2   2 2   2   2 2 2     2 2 2     2 2 2     2 2 2   
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      1 2 3 6   1 2 3 6   1 2 3 6   3 4 5 6   2 4 5 6   1 4 5 6   2 3 5 6 
     ------------------------------------------------------------------------
      2 2 2     2 2 2     2 2 2     2 2 2     2 2 2     2 2 2       2 2 2 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      1 3 5 6   1 2 5 6   2 3 4 6   1 3 4 6   1 2 4 6   1 2 3 6   2 3 4 5 
     ------------------------------------------------------------------------
        2 2 2   2   2 2   2   2 2     2 2 2   2   2 2   2 2   2   2 2   2 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      1 3 4 5   2 3 4 5   1 3 4 5   1 2 4 5   1 2 4 5   2 3 4 5   1 3 4 5 
     ------------------------------------------------------------------------
      2 2   2     2 2 2   2   2 2   2 2   2   2 2 2     2 2 2     2 2 2   
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , x x x x ,
      1 2 4 5   1 2 3 5   1 2 3 5   1 2 3 5   2 3 4 5   1 3 4 5   1 2 4 5 
     ------------------------------------------------------------------------
      2 2 2       2 2 2   2   2 2   2 2   2   2 2 2     3 3 3   3 3 3 
     x x x x , x x x x , x x x x , x x x x , x x x x , x x x , x x x ,
      1 2 3 5   1 2 3 4   1 2 3 4   1 2 3 4   1 2 3 4   4 5 6   3 5 6 
     ------------------------------------------------------------------------
      3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3 
     x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x ,
      2 5 6   1 5 6   3 4 6   2 4 6   1 4 6   2 3 6   1 3 6   1 2 6   3 4 5 
     ------------------------------------------------------------------------
      3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3   3 3 3
     x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x )
      2 4 5   1 4 5   2 3 5   1 3 5   1 2 5   2 3 4   1 3 4   1 2 4   1 2 3

o2 : Ideal of R

i3 : RI=res I

      1      86      230      210      65
o3 = R  <-- R   <-- R    <-- R    <-- R   <-- 0
                                               
     0      1       2        3        4       5

o3 : ChainComplex

i4 : betti RI

            0  1   2   3  4
o4 = total: 1 86 230 210 65
         0: 1  .   .   .  .
         1: .  .   .   .  .
         2: .  .   .   .  .
         3: .  .   .   .  .
         4: .  6   5   .  .
         5: .  .   .   .  .
         6: . 60 165 150 45
         7: .  .   .   .  .
         8: . 20  60  60 20

o4 : BettiTally

Next we set up the group action on the resolution. The group is the symmetric group on 6 elements. Its conjugacy classes are determined by cycle types, which are in bijection with partitions of 6. Once the action is set up, we compute the Betti characters.

i5 : G = for p in partitions(6) list (
         L := gens R;
         g := for u in p list (
             l := take(L,u);
             L = drop(L,u);
             rotate(1,l)
             );
         matrix { flatten g }
         )

o5 = {| x_2 x_3 x_4 x_5 x_6 x_1 |, | x_2 x_3 x_4 x_5 x_1 x_6 |, | x_2 x_3 x_4
     ------------------------------------------------------------------------
     x_1 x_6 x_5 |, | x_2 x_3 x_4 x_1 x_5 x_6 |, | x_2 x_3 x_1 x_5 x_6 x_4 |,
     ------------------------------------------------------------------------
     | x_2 x_3 x_1 x_5 x_4 x_6 |, | x_2 x_3 x_1 x_4 x_5 x_6 |, | x_2 x_1 x_4
     ------------------------------------------------------------------------
     x_3 x_6 x_5 |, | x_2 x_1 x_4 x_3 x_5 x_6 |, | x_2 x_1 x_3 x_4 x_5 x_6 |,
     ------------------------------------------------------------------------
     | x_1 x_2 x_3 x_4 x_5 x_6 |}

o5 : List

i6 : A=action(RI,G)

o6 = ChainComplex with 11 actors

o6 : ActionOnComplex

i7 : elapsedTime c=character A
 -- 0.52044 seconds elapsed

o7 = HashTable{0 => GradedCharacter{{0} => Character{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}      }
               1 => GradedCharacter{{5} => Character{0, 1, 0, 2, 0, 1, 3, 0, 2, 4, 6}  }
                                    {7} => Character{0, 0, 0, 0, 0, 1, 3, 0, 4, 16, 60}
                                    {9} => Character{0, 0, 0, 0, 2, 2, 2, 0, 4, 8, 20}
               2 => GradedCharacter{{6} => Character{-1, 0, -1, 1, -1, 0, 2, -1, 1, 3, 5}  }
                                    {8} => Character{0, 0, -1, -1, 0, 0, 6, -3, 1, 33, 165}
                                    {10} => Character{0, 0, 0, 0, 0, 1, 3, 0, 4, 16, 60}
               3 => GradedCharacter{{9} => Character{0, 0, 0, -2, 0, -1, 3, 0, -6, 20, 150}}
                                    {11} => Character{0, 0, 0, 0, 0, -1, 3, 0, -4, 8, 60}
               4 => GradedCharacter{{10} => Character{0, 0, 1, -1, 0, 0, 0, 3, -3, 3, 45}}
                                    {12} => Character{0, 0, 0, 0, 2, 0, 2, 0, -4, 0, 20}

o7 : HashTable

To make sense of these characters we decompose them against the character table of the symmetric group, which can be computed using the function characterTable provided by the package SpechtModule. First we form a hash table with the irreducible characters. Then we define a function that computes the multiplicities of the irreducible characters in a given character. Finally, we apply this function to the Betti characters computed above.

i8 : needsPackage "SpechtModule"

o8 = SpechtModule

o8 : Package

i9 : irrReps = new HashTable from pack(2,
         mingle {
             partitions 6,
             apply(entries (characterTable 6)#values, r -> mutableMatrix{r})
             }
         )

o9 = HashTable{Partition{1, 1, 1, 1, 1, 1} => | -1 1 1 -1 1 -1 1 -1 1 -1 1 |}
               Partition{2, 1, 1, 1, 1} => | 1 0 -1 -1 -1 0 2 1 1 -3 5 |
               Partition{2, 2, 1, 1} => | 0 -1 1 1 0 0 0 -3 1 -3 9 |
               Partition{2, 2, 2} => | 0 0 -1 1 2 -1 -1 3 1 -1 5 |
               Partition{3, 1, 1, 1} => | -1 0 0 0 1 1 1 2 -2 -2 10 |
               Partition{3, 2, 1} => | 0 1 0 0 -2 0 -2 0 0 0 16 |
               Partition{3, 3} => | 0 0 -1 -1 2 1 -1 -3 1 1 5 |
               Partition{4, 1, 1} => | 1 0 0 0 1 -1 1 -2 -2 2 10 |
               Partition{4, 2} => | 0 -1 1 -1 0 0 0 3 1 3 9 |
               Partition{5, 1} => | -1 0 -1 1 -1 0 2 -1 1 3 5 |
               Partition{6} => | 1 1 1 1 1 1 1 1 1 1 1 |

o9 : HashTable

i10 : multiplicities = c -> select(
          applyValues(irrReps,
              v -> innerProduct(6,v,mutableMatrix{new List from c})),
          v -> v!=0
          )

o10 = multiplicities

o10 : FunctionClosure

i11 : applyValues(c#0, v -> multiplicities v)

o11 = GradedCharacter{{0} => HashTable{Partition{6} => 1}}

o11 : GradedCharacter

i12 : applyValues(c#1, v -> multiplicities v)

o12 = GradedCharacter{{5} => HashTable{Partition{5, 1} => 1}   }
                                       Partition{6} => 1
                      {7} => HashTable{Partition{3, 2, 1} => 1}
                                       Partition{3, 3} => 1
                                       Partition{4, 1, 1} => 1
                                       Partition{4, 2} => 2
                                       Partition{5, 1} => 2
                                       Partition{6} => 1
                      {9} => HashTable{Partition{3, 3} => 1}
                                       Partition{4, 2} => 1
                                       Partition{5, 1} => 1
                                       Partition{6} => 1

o12 : GradedCharacter

i13 : applyValues(c#2, v -> multiplicities v)

o13 = GradedCharacter{{6} => HashTable{Partition{5, 1} => 1}      }
                      {8} => HashTable{Partition{3, 1, 1, 1} => 1}
                                       Partition{3, 2, 1} => 3
                                       Partition{3, 3} => 2
                                       Partition{4, 1, 1} => 4
                                       Partition{4, 2} => 4
                                       Partition{5, 1} => 4
                                       Partition{6} => 1
                      {10} => HashTable{Partition{3, 2, 1} => 1}
                                        Partition{3, 3} => 1
                                        Partition{4, 1, 1} => 1
                                        Partition{4, 2} => 2
                                        Partition{5, 1} => 2
                                        Partition{6} => 1

o13 : GradedCharacter

i14 : applyValues(c#3, v -> multiplicities v)

o14 = GradedCharacter{{9} => HashTable{Partition{3, 1, 1, 1} => 2} }
                                       Partition{3, 2, 1} => 3
                                       Partition{3, 3} => 1
                                       Partition{4, 1, 1} => 4
                                       Partition{4, 2} => 3
                                       Partition{5, 1} => 2
                      {11} => HashTable{Partition{3, 1, 1, 1} => 1}
                                        Partition{3, 2, 1} => 1
                                        Partition{4, 1, 1} => 2
                                        Partition{4, 2} => 1
                                        Partition{5, 1} => 1

o14 : GradedCharacter

i15 : applyValues(c#4, v -> multiplicities v)

o15 = GradedCharacter{{10} => HashTable{Partition{3, 1, 1, 1} => 1}}
                                        Partition{3, 2, 1} => 1
                                        Partition{4, 1, 1} => 1
                                        Partition{4, 2} => 1
                      {12} => HashTable{Partition{3, 1, 1, 1} => 1}
                                        Partition{4, 1, 1} => 1

o15 : GradedCharacter

The description provided in J. Biermann, H. De Alba, F. Galetto, S. Murai, U. Nagel, A. O'Keefe, T. Römer, A. Seceleanu - Betti numbers of symmetric shifted ideals uses representations induced from products of smaller symmetric groups. In order, to compare with the results obtained here one may use the Littlewood-Richardson rule to decompose induced representations into a direct sum of irreducibles.