Let E=C3 with coordinate basis {e1,e2,e3}, F=C4 with coordinate basis {f1,...,f4} and H=C2 with coordinate basis {h1,h2}. Denote R the symmetric algebra Sym((E⊕F)⊗H); R is a polynomial ring with variables xi,j = ei⊗hj and yi,j = fi⊗hj. We take the variables xi,j to have degree (1,0) and the variables yi,j to have degree (0,1). Let G be a 2×7 generic matrix of variables in R and I the ideal generated by the 2×2 minors of G. The minimal free resoluiton of I is an example of an Eagon-Northcott complex (see Eisenbud - Commutative Algebra, Appendix A2.6).
i1 : R=QQ[x_(1,1)..x_(3,2),y_(1,1)..y_(4,2),Degrees=>{6:{1,0},8:{0,1}}] o1 = R o1 : PolynomialRing |
i2 : G=genericMatrix(R,2,3)|genericMatrix(R,y_(1,1),2,4) o2 = | x_(1,1) x_(2,1) x_(3,1) y_(1,1) y_(2,1) y_(3,1) y_(4,1) | | x_(1,2) x_(2,2) x_(3,2) y_(1,2) y_(2,2) y_(3,2) y_(4,2) | 2 7 o2 : Matrix R <--- R |
i3 : I=minors(2,G); o3 : Ideal of R |
i4 : EN=res I; betti EN 0 1 2 3 4 5 6 o5 = total: 1 21 70 105 84 35 6 0: 1 . . . . . . 1: . 21 70 105 84 35 6 o5 : BettiTally |
Notice how the first three columns of G involve only the xi,j variables while the other columns involve only the yi,j variables. In fact, G is the matrix of the map
φ: E⊗R(-1,0) ⊕F⊗R(0,-1) →H* ⊗R, ei ⊗1 ↦∑k=12 hk* ⊗xi,k, fj ⊗1 ↦∑k=12 hk* ⊗yj,k
with respect to the bases {e1⊗1,...,e3⊗1,f1⊗1,...,f4⊗1} of the domain and {h1*⊗1,h2*⊗1} of the codomain.The ring R carries a degree compatible action of SL3 (C) ×SL4 (C) ×SL2 (C). Notice how the SL3 (C) factor acts non trivially on E, i.e., on the variables xi,j, and trivially on F, i.e., on the variables yi,j. The opposite holds for the SL4 (C) factor. The map φ is G-equivariant and so the ideal generated by the 2×2 of φ inherits the G-action on R.
The weight of xi,j = ei⊗hj is obtained by concatenating the weight of ei with the trivial weight {0,0,0} for the action of SL4 (C) and then with the weight of hj. Similarly the weight of yi,j = fi⊗hj is obtained by concatenating the trivial weight {0,0} for the action of SL3 (C) with the weight of fi and then with the weight of hj. As in Example 2, we automatize the process as illustrated below and then we attach the list of weights to the ring R.
i6 : e={{1,0},{-1,1},{0,-1}} o6 = {{1, 0}, {-1, 1}, {0, -1}} o6 : List |
i7 : f={{1,0,0},{-1,1,0},{0,-1,1},{0,0,-1}} o7 = {{1, 0, 0}, {-1, 1, 0}, {0, -1, 1}, {0, 0, -1}} o7 : List |
i8 : h={{1},{-1}} o8 = {{1}, {-1}} o8 : List |
i9 : W=(flatten table(e,h,(u,v)->u|{0,0,0}|v))|(flatten table(f,h,(u,v)->{0,0}|u|v)) o9 = {{1, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, -1}, {-1, 1, 0, 0, 0, 1}, {-1, 1, ------------------------------------------------------------------------ 0, 0, 0, -1}, {0, -1, 0, 0, 0, 1}, {0, -1, 0, 0, 0, -1}, {0, 0, 1, 0, 0, ------------------------------------------------------------------------ 1}, {0, 0, 1, 0, 0, -1}, {0, 0, -1, 1, 0, 1}, {0, 0, -1, 1, 0, -1}, {0, ------------------------------------------------------------------------ 0, 0, -1, 1, 1}, {0, 0, 0, -1, 1, -1}, {0, 0, 0, 0, -1, 1}, {0, 0, 0, 0, ------------------------------------------------------------------------ -1, -1}} o9 : List |
i10 : D=dynkinType{{"A",2},{"A",3},{"A",1}}; setWeights(R,D,W) o11 = Tally{{0, 0, 1, 0, 0, 1} => 1} {1, 0, 0, 0, 0, 1} => 1 o11 : Tally |
Now we decompose the resolution of I.
i12 : highestWeightsDecomposition(EN) o12 = HashTable{0 => HashTable{{0, 0} => Tally{{0, 0, 0, 0, 0, 0} => 1}}} 1 => HashTable{{0, 2} => Tally{{0, 0, 0, 1, 0, 0} => 1}} {1, 1} => Tally{{1, 0, 1, 0, 0, 0} => 1} {2, 0} => Tally{{0, 1, 0, 0, 0, 0} => 1} 2 => HashTable{{0, 3} => Tally{{0, 0, 0, 0, 1, 1} => 1}} {1, 2} => Tally{{1, 0, 0, 1, 0, 1} => 1} {2, 1} => Tally{{0, 1, 1, 0, 0, 1} => 1} {3, 0} => Tally{{0, 0, 0, 0, 0, 1} => 1} 3 => HashTable{{0, 4} => Tally{{0, 0, 0, 0, 0, 2} => 1}} {1, 3} => Tally{{1, 0, 0, 0, 1, 2} => 1} {2, 2} => Tally{{0, 1, 0, 1, 0, 2} => 1} {3, 1} => Tally{{0, 0, 1, 0, 0, 2} => 1} 4 => HashTable{{1, 4} => Tally{{1, 0, 0, 0, 0, 3} => 1}} {2, 3} => Tally{{0, 1, 0, 0, 1, 3} => 1} {3, 2} => Tally{{0, 0, 0, 1, 0, 3} => 1} 5 => HashTable{{2, 4} => Tally{{0, 1, 0, 0, 0, 4} => 1}} {3, 3} => Tally{{0, 0, 0, 0, 1, 4} => 1} 6 => HashTable{{3, 4} => Tally{{0, 0, 0, 0, 0, 5} => 1}} o12 : HashTable |
We show what part of the resolution looks like in terms of Schur functors:
R ←(∧2 F ⊗R(0,-2)) ⊕(E ⊗F ⊗R(-1,-1)) ⊕(∧2 E ⊗R(-2,0)) ←(∧3 F ⊗H ⊗R(0,-3)) ⊕(E ⊗∧2 F ⊗H ⊗R(-1,-2)) ⊕(∧2 E ⊗F ⊗H ⊗R(-2,-1)) ⊕(H ⊗R(-3,0)) ←...
Finally we decompose some graded components in the quotient ring R/I. Unlike with ℤ-gradings, when working in the multigraded setting it is not possible to decompose a range of degrees but only one multidegree at a time.
i13 : Q=R/I o13 = Q o13 : QuotientRing |
i14 : highestWeightsDecomposition(Q,{2,0}) o14 = Tally{{2, 0, 0, 0, 0, 2} => 1} o14 : Tally |
i15 : highestWeightsDecomposition(Q,{1,1}) o15 = Tally{{1, 0, 1, 0, 0, 2} => 1} o15 : Tally |
i16 : highestWeightsDecomposition(Q,{0,2}) o16 = Tally{{0, 0, 2, 0, 0, 2} => 1} o16 : Tally |
We have Q(2,0) = S2 E ⊗S2 H, Q(1,1) = E ⊗F ⊗S2 H and Q(0,2) = S2 F ⊗S2 H.