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}};
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i3 : D=dynkinType{{"A",2}};
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i4 : setWeights(R,D,L)
o4 = Tally{{1, 0} => 1}
o4 : Tally
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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
| 0 12x2 0 0 6xy 0 0 6xz 0 0 2y2 0 0 2yz 0
| 0 0 12x2 0 0 6xy 0 0 6xz 0 0 2y2 0 0 2yz
| 0 0 0 0 6x2 0 -12x2 0 0 0 8xy 0 -8xy 4xz 0
| 0 0 0 0 0 6x2 0 -3x2 0 0 0 8xy 0 -2xy 4xz
| 0 0 0 0 0 0 0 0 0 0 0 2x2 0 -x2 0
------------------------------------------------------------------------
2z2 0 0 0 0 0 0 0 0 0 0 0 |
0 2z2 0 0 0 0 0 0 0 0 0 0 |
0 0 2z2 0 0 0 0 0 0 0 0 0 |
-16xz 0 0 6y2 0 4yz 0 2z2 0 0 0 0 |
0 -4xz 0 0 6y2 -y2 4yz -2yz 2z2 0 0 0 |
2x2 0 0 0 6xy -2xy 2xz -2xz 0 12y2 6yz 2z2 |
6 27
o6 : Matrix R <--- R
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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 $m_0*L_0+m_1*L_1+m_2*L_2$, where $m_i$ 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
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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
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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
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We conclude that with the action of $SL(E)$ the complex has the following structure: $$S_{2,2} E \otimes R \leftarrow S_{4,2} E \otimes R(-2) \leftarrow S_{4,3} E \otimes R(-3) \leftarrow E \otimes R(-6) \leftarrow 0$$ where $S_\lambda$ denotes the Schur functor associated with the partition $\lambda$.