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HighestWeights :: Example 4

Example 4 -- The Eisenbud-Fløystad-Weyman complex

We construct and decompose the Eisenbud-Fløystad-Weyman complex of type (0,2,3,6) over a polynomial ring in 3 variables. The ring can be identified with Sym(E), where E is a complex vector space of dimension 3. The ring and the complex carry an action of SL(E).

The complex is constructed using the package PieriMaps. For more information on these complexes, we invite the reader to consult the documentation of that package and the accompanying article.

i1 : R=QQ[x,y,z];
i2 : L={{1,0},{-1,1},{0,-1}};
i3 : D=dynkinType{{"A",2}};
i4 : setWeights(R,D,L)

o4 = Tally{{1, 0} => 1}

o4 : Tally
i5 : loadPackage "PieriMaps";
i6 : f=pureFree({0,2,3,6},R)

o6 = | 12x2 0    0    6xy 0   0   6xz   0    0   2y2 0   0   2yz  0    0   2z2   0    0   0   0   0    0   0    0   0    0   0  
     | 0    12x2 0    0   6xy 0   0     6xz  0   0   2y2 0   0    2yz  0   0     2z2  0   0   0   0    0   0    0   0    0   0  
     | 0    0    12x2 0   0   6xy 0     0    6xz 0   0   2y2 0    0    2yz 0     0    2z2 0   0   0    0   0    0   0    0   0  
     | 0    0    0    0   6x2 0   -12x2 0    0   0   8xy 0   -8xy 4xz  0   -16xz 0    0   6y2 0   4yz  0   2z2  0   0    0   0  
     | 0    0    0    0   0   6x2 0     -3x2 0   0   0   8xy 0    -2xy 4xz 0     -4xz 0   0   6y2 -y2  4yz -2yz 2z2 0    0   0  
     | 0    0    0    0   0   0   0     0    0   0   0   2x2 0    -x2  0   2x2   0    0   0   6xy -2xy 2xz -2xz 0   12y2 6yz 2z2
     ----------------------------------------------------------------------------------------------------------------------------
     |
     |
     |
     |
     |
     |

             6       27
o6 : Matrix R  <--- R

The matrix above is a presentation of the module whose resolution is the complex in the title. The rows of the matrix are indexed by standard tableaux of shape (2,2) and entries from {0,1,2}. The weight of one such tableau is m0*L0+m1*L1+m2*L2, where mi is the multiplicity of i in the tableau. The command below generates all the weights.

i7 : W=apply(apply(standardTableaux(3, {2,2}), flatten), i->sum(apply(i,j->L_j)))

o7 = {{0, 2}, {1, 0}, {2, -2}, {-1, 1}, {0, -1}, {-2, 0}}

o7 : List

Next we generate the resolution and obtain its decomposition.

i8 : EFW=res coker f; betti EFW

            0  1  2 3
o9 = total: 6 27 24 3
         0: 6  .  . .
         1: . 27 24 .
         2: .  .  . .
         3: .  .  . 3

o9 : BettiTally
i10 : highestWeightsDecomposition(EFW,0,W)

o10 = HashTable{0 => HashTable{{0} => Tally{{0, 2} => 1}}}
                1 => HashTable{{2} => Tally{{2, 2} => 1}}
                2 => HashTable{{3} => Tally{{1, 3} => 1}}
                3 => HashTable{{6} => Tally{{1, 0} => 1}}

o10 : HashTable

We conclude that with the action of SL(E) the complex has the following structure:

S2,2 E ⊗R ←S4,2 E ⊗R(-2) ←S4,3 E ⊗R(-3) ←E ⊗R(-6) ←0

where Sλ denotes the Schur functor associated with the partition λ.