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In our work we use two different bases of the vector space \(V_9\), which is the irreducible representation of \(\operatorname{Sp}(6)\) of dimension 14.
The representation \(V_9\) can be constructed as the kernel of the contraction map \( \varphi_3: \Lambda^3 V \rightarrow V\) given by \[ \varphi_3(v_1 \wedge v_2 \wedge v_3) = \sum_{i\lt j} Q(v_i,v_j) (-1)^{i+j-1} v_k, \] where \(V\) is the standard representation and \(Q\) is the symplectic form. See Fulton and Harris, Representation Theory, Lecture 17. (This representation is discussed in Section 17.1, and a general formula for the contraction operator whose kernel is the irreducible representation of \(\mathfrak{sp}(2n)\) with highest weight \(\omega_k\) is given in Section 17.2).
Our first basis for \(V_9\) is given by the following vectors. We write \(e_{i_{1}i_{2}i_{3}}\) for the wedge product \(e_{i_1} \wedge e_{i_2} \wedge e_{i_3}\). \[ \mathscr{B} = \{ e_{123}, e_{234},e_{134}-e_{235},-e_{124}-e_{236}, -e_{135},e_{125}-e_{136}, e_{126}, e_{156}, e_{146}-e_{256}, -e_{145}-e_{356}, -e_{246}, e_{245}-e_{346},e_{345},e_{456}\}. \]
The second basis is the one produced by de Graaf's algorithm where the term order used is the lex term order on the lowering operators in the order they are listed in the Macaulay2 package LieAlgebraRepresentations.
We use the LieAlgebraRepresentations package to describe the de Graaf basis in terms of the wedge products \(e_I\).
Type "help" to see useful commands
i1 : needsPackage("LieAlgebraRepresentations");
i2 : sp6 = simpleLieAlgebra("C",3);
i3 : rhoStd = standardRepresentation(sp6);
i4 : rhoW3Std = exteriorPower(3,rhoStd);
i5 : representationWeights(rhoW3Std)
o5 = {{0, 0, 1}, {-1, 1, 0}, {0, -1, 1}, {-2, 0, 1}, {1, 0, 0}, {2, -2, 1}, {0, -1, 1}, {1, -1, 0}, {-1, 0, 0}, {0, -2, 1}, {0, 2, -1}, {1, 0, 0}, {-1, 1, 0}, {0, 1, -1}, {-2, 2, -1}, {-1, 0, 0}, {2, 0, -1}, {0, 1, -1}, {1, -1, 0}, {0, 0, -1}}
o5 : List
i6 : rhoV001 = deGraafRepresentation({0,0,1},sp6);
Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning: F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
max-lev=14
Finished level 1. {#G,#B}={2, 2}
Finished level 2. {#G,#B}={4, 3}
Finished level 3. {#G,#B}={5, 5}
Finished level 4. {#G,#B}={7, 7}
Finished level 5. {#G,#B}={9, 9}
Finished level 6. {#G,#B}={12, 11}
Finished level 7. {#G,#B}={14, 12}
Finished level 8. {#G,#B}={16, 13}
Finished level 9. {#G,#B}={17, 14}
Finished level 10. {#G,#B}={18, 14}
Finished level 11. {#G,#B}={18, 14}
Finished level 12. {#G,#B}={18, 14}
Finished level 13. {#G,#B}={18, 14}
Finished level 14. {#G,#B}={18, 14}
Compute rho(B_0)
Compute rho(B_1)
Compute rho(B_2)
Compute rho(B_3)
Compute rho(B_4)
Compute rho(B_5)
Compute rho(B_6)
Compute rho(B_7)
Compute rho(B_8)
Compute rho(B_9)
Compute rho(B_10)
Compute rho(B_11)
Compute rho(B_12)
Compute rho(B_13)
Compute rho(B_14)
Compute rho(B_15)
Compute rho(B_16)
Compute rho(B_17)
Compute rho(B_18)
Compute rho(B_19)
Compute rho(B_20)
i7 : hwv = weightMuHighestWeightVectorsInW({0,0,1},rhoW3Std);
20 1
o7 : Matrix QQ <-- QQ
i8 : V001InW3Std=VInWedgekW(rhoV001,3,rhoStd,hwv)
Length 1 complete. 6 new words found
Length 2 complete. 6 new words found
Length 3 complete. 1 new words found
o8 = {p , p , p - p , p + p , -p , - p + p , p , p - p , p , -p , p + p , - p + p , p , p }
{0, 1, 2} {0, 1, 5} {0, 1, 4} {0, 2, 5} {0, 1, 3} {1, 2, 5} {0, 2, 4} {0, 2, 3} {1, 2, 4} {0, 4, 5} {0, 3, 5} {1, 4, 5} {1, 2, 3} {1, 3, 5} {0, 3, 4} {2, 4, 5} {1, 3, 4} {2, 3, 5} {2, 3, 4} {3, 4, 5}
o8 : List
i9 : R = ring(first V001InW3Std);
i10 : S = QQ[apply(subsets({1,2,3,4,5,6},3), i -> e_i)];
i11 : f = map(S,R,gens S);
o11 : RingMap S <-- R
i12 : apply(V001InW3Std, x -> f(x))
o12 = {e , e , e - e , e + e , -e , - e + e , e , e - e , e , -e , e + e , - e + e , e , e }
{1, 2, 3} {1, 2, 6} {1, 2, 5} {1, 3, 6} {1, 2, 4} {2, 3, 6} {1, 3, 5} {1, 3, 4} {2, 3, 5} {1, 5, 6} {1, 4, 6} {2, 5, 6} {2, 3, 4} {2, 4, 6} {1, 4, 5} {3, 5, 6} {2, 4, 5} {3, 4, 6} {3, 4, 5} {4, 5, 6}
o12 : List