BettiCharacters 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.

First, we construct the ideal and compute its minimal free resolution and 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. Representatives for the conjugacy classes of the symmetric group acting on a polynomial ring by permuting the variables can be obtained via symmetricGroupActors. After setting up the action, we compute the Betti characters.

i5 : S6 = symmetricGroupActors R

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,S6)

o6 = ChainComplex with 11 actors

o6 : ActionOnComplex

i7 : elapsedTime c=character A
 -- 1.09388 seconds elapsed

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

o7 : Character

Next, we decompose the characters against the character table of the symmetric group, which can be computed using the function symmetricGroupTable. The irreducible characters are indexed by the partitions of 6, which are written using a compact notation (the exponents indicate how many times a part is repeated).

i8 : T = symmetricGroupTable R

o8 = Character table over R
      
              |  120  144  90  90  40  120  40  15  45  15   1
     ---------+-----------------------------------------------
         (6)  |    1    1   1   1   1    1   1   1   1   1   1
       (5,1)  |   -1    0  -1   1  -1    0   2  -1   1   3   5
       (4,2)  |    0   -1   1  -1   0    0   0   3   1   3   9
          2   |                                             
      (4,1 )  |    1    0   0   0   1   -1   1  -2  -2   2  10
          2   |                                             
        (3 )  |    0    0  -1  -1   2    1  -1  -3   1   1   5
     (3,2,1)  |    0    1   0   0  -2    0  -2   0   0   0  16
          3   |                                             
      (3,1 )  |   -1    0   0   0   1    1   1   2  -2  -2  10
          3   |                                             
        (2 )  |    0    0  -1   1   2   -1  -1   3   1  -1   5
       2  2   |                                             
     (2 ,1 )  |    0   -1   1   1   0    0   0  -3   1  -3   9
          4   |                                             
      (2,1 )  |    1    0  -1  -1  -1    0   2   1   1  -3   5
          6   |                                             
        (1 )  |   -1    1   1  -1   1   -1   1  -1   1  -1   1

o8 : CharacterTable

i9 : decomposeCharacter(c,T)

o9 = Decomposition table
      
                |                         2     2                3
                |  (6)  (5,1)  (4,2)  (4,1 )  (3 )  (3,2,1)  (3,1 )
     -----------+--------------------------------------------------
      (0, {0})  |    1      0      0       0     0        0       0
      (1, {5})  |    1      1      0       0     0        0       0
      (1, {7})  |    1      2      2       1     1        1       0
      (1, {9})  |    1      1      1       0     1        0       0
      (2, {6})  |    0      1      0       0     0        0       0
      (2, {8})  |    1      4      4       4     2        3       1
     (2, {10})  |    1      2      2       1     1        1       0
      (3, {9})  |    0      2      3       4     1        3       2
     (3, {11})  |    0      1      1       2     0        1       1
     (4, {10})  |    0      0      1       1     0        1       1
     (4, {12})  |    0      0      0       1     0        0       1

o9 : CharacterDecomposition

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. To compare that description with the results obtained here, one may use the Littlewood-Richardson rule to decompose induced representations into a direct sum of irreducibles.