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HighestWeights :: Example 6

Example 6 -- The coordinate ring of the spinor variety

Consider the complex Lie group SO(10) of type D5. We denote V(ω) the highest weight representation of SO(10) with highest weight ω. We denote by ω1,...,ω5 the fundamental weights in the root system of type D5.

We will obtain the minimal free resolution of the coordinate ring of the spinor variety of type D5 (see Rincon - Isotropical linear spaces and valuated delta-matroids, Sec. 2, for a concise introduction to spinor varieties). The affine cone over the spinor variety of type D5 lives in the representation V(ω5), the 5th fundamental representation of SO(10), considered as an affine space. Polynomial functions on this affine space are given by the symmetric algebra over the dual representation, i.e., V(ω4).

Following the description in Fulton, Harris - Representation Theory, Ch. 20.1, we can construct V(ω4) as 0 E ⊕∧2 E ⊕∧4 E, where E is a 5 dimensional complex vector space. Let {e0,...,e4} be a basis of E. Then a basis of V(ω4) is given by the exterior products eJ = ej1 ∧... ∧ej2r, for all subsets J={j1,...,j2r} of even cardinality of {0,..., 4}. Denote by xJ the variable corresponding to eJ in R.

The spinor variety of type D5 is cut out by quadratic equations which represent all possible relations among the sub Pfaffians of a 5×5 generic skew symmetric matrix. A general description can be found for example in Manivel - On Spinor Varieties and Their Secants.

i1 : R=QQ[x_{}, x_{0,1}, x_{0,2}, x_{1,2}, x_{0,3}, x_{1,3}, x_{2,3}, x_{0,4}, x_{1,4}, x_{2,4}, x_{3,4}, x_{0,1,2,3}, x_{0,1,2,4}, x_{0,1,3,4}, x_{0,2,3,4}, x_{1,2,3,4}]

o1 = R

o1 : PolynomialRing
i2 : I=ideal(x_{}*x_{0,1,2,3}-x_{0,1}*x_{2,3}+x_{0,2}*x_{1,3}-x_{0,3}*x_{1,2},
         x_{}*x_{0,1,2,4}-x_{0,1}*x_{2,4}+x_{0,2}*x_{1,4}-x_{0,4}*x_{1,2},
         x_{}*x_{0,1,3,4}-x_{0,1}*x_{3,4}+x_{0,3}*x_{1,4}-x_{0,4}*x_{1,3},
         x_{}*x_{0,2,3,4}-x_{0,2}*x_{3,4}+x_{0,3}*x_{2,4}-x_{0,4}*x_{2,3},
         x_{}*x_{1,2,3,4}-x_{1,2}*x_{3,4}+x_{1,3}*x_{2,4}-x_{1,4}*x_{2,3},
         x_{0,1}*x_{0,2,3,4}-x_{0,2}*x_{0,1,3,4}+x_{0,3}*x_{0,1,2,4}-x_{0,4}*x_{0,1,2,3},
         -x_{0,1}*x_{1,2,3,4}+x_{1,2}*x_{0,1,3,4}-x_{1,3}*x_{0,1,2,4}+x_{1,4}*x_{0,1,2,3},
         x_{0,2}*x_{1,2,3,4}-x_{1,2}*x_{0,2,3,4}+x_{2,3}*x_{0,1,2,4}-x_{2,4}*x_{0,1,2,3},
         -x_{0,3}*x_{1,2,3,4}+x_{1,3}*x_{0,2,3,4}-x_{2,3}*x_{0,1,3,4}+x_{3,4}*x_{0,1,2,3},
         x_{0,4}*x_{1,2,3,4}-x_{1,4}*x_{0,2,3,4}+x_{2,4}*x_{0,1,3,4}-x_{3,4}*x_{0,1,2,4});

o2 : Ideal of R
i3 : RI=res I; betti RI

            0  1  2  3  4 5
o4 = total: 1 10 16 16 10 1
         0: 1  .  .  .  . .
         1: . 10 16  .  . .
         2: .  .  . 16 10 .
         3: .  .  .  .  . 1

o4 : BettiTally

The root system of type D5 is contained in 5. It is easy to express the weight of each variable of the ring R with respect to the coordinate basis of 5. The weight of xJ is a vector (a1,...,a5)∈ℝ5, with ak = 1/2 if k∈J and ak = -1/2 otherwise.

i5 : ind = apply(gens R,g->(baseName g)#1)

o5 = {{}, {0, 1}, {0, 2}, {1, 2}, {0, 3}, {1, 3}, {2, 3}, {0, 4}, {1, 4}, {2,
     ------------------------------------------------------------------------
     4}, {3, 4}, {0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1,
     ------------------------------------------------------------------------
     2, 3, 4}}

o5 : List
i6 : makeWeight = J -> apply(5,i->if member(i,J) then 1/2 else -1/2)

o6 = makeWeight

o6 : FunctionClosure
i7 : W'=apply(ind,makeWeight)

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

o7 : List

Now we convert these weights into the basis of fundamental weights. To achieve this we make each previous weight into a column vector and join all column vectors into a matrix. Then we multiply on the left by the matrix M expressing the change of basis from the coordinate basis of 5 to the base of simple roots of D5 (as described in Humphreys - Introduction to Lie Algebras and Representation Theory, Ch. 12.1). Finally we multiply the resulting matrix on the left by N, the transpose of the Cartan matrix of D5, which expresses the change of basis from the simple roots to the fundamental weights of D5. The columns of the matrix thus obtained are the desired weights, so they can be attached to the ring R.

i8 : M=inverse promote(matrix{{1,0,0,0,0},{-1,1,0,0,0},{0,-1,1,0,0},{0,0,-1,1,1},{0,0,0,-1,1}},QQ)

o8 = | 1   0   0   0   0    |
     | 1   1   0   0   0    |
     | 1   1   1   0   0    |
     | 1/2 1/2 1/2 1/2 -1/2 |
     | 1/2 1/2 1/2 1/2 1/2  |

              5        5
o8 : Matrix QQ  <--- QQ
i9 : D=dynkinType{{"D",5}}

o9 = DynkinType{{D, 5}}

o9 : DynkinType
i10 : N=transpose promote(cartanMatrix(rootSystem(D)),QQ)

o10 = | 2  -1 0  0  0  |
      | -1 2  -1 0  0  |
      | 0  -1 2  -1 -1 |
      | 0  0  -1 2  0  |
      | 0  0  -1 0  2  |

               5        5
o10 : Matrix QQ  <--- QQ
i11 : W=entries transpose lift(N*M*(transpose matrix W'),ZZ)

o11 = {{0, 0, 0, 0, -1}, {0, 1, 0, 0, -1}, {1, -1, 1, 0, -1}, {-1, 0, 1, 0,
      -----------------------------------------------------------------------
      -1}, {1, 0, -1, 1, 0}, {-1, 1, -1, 1, 0}, {0, -1, 0, 1, 0}, {1, 0, 0,
      -----------------------------------------------------------------------
      -1, 0}, {-1, 1, 0, -1, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 0, 1}, {0, 0,
      -----------------------------------------------------------------------
      0, 1, 0}, {0, 0, 1, -1, 0}, {0, 1, -1, 0, 1}, {1, -1, 0, 0, 1}, {-1, 0,
      -----------------------------------------------------------------------
      0, 0, 1}}

o11 : List
i12 : setWeights(R,D,W)

o12 = Tally{{0, 0, 0, 1, 0} => 1}

o12 : Tally

At this stage, we can issue the command to decompose the resolution.

i13 : highestWeightsDecomposition(RI)

o13 = HashTable{0 => HashTable{{0} => Tally{{0, 0, 0, 0, 0} => 1}}}
                1 => HashTable{{2} => Tally{{1, 0, 0, 0, 0} => 1}}
                2 => HashTable{{3} => Tally{{0, 0, 0, 0, 1} => 1}}
                3 => HashTable{{5} => Tally{{0, 0, 0, 1, 0} => 1}}
                4 => HashTable{{6} => Tally{{1, 0, 0, 0, 0} => 1}}
                5 => HashTable{{8} => Tally{{0, 0, 0, 0, 0} => 1}}

o13 : HashTable

We deduce that the resolution has the following structure

R ←V(ω1) ⊗R(-2) ←V(ω5) ⊗R(-3) ←V(ω4) ⊗R(-5) ←V(ω1) ⊗R(-6) ←R(-8) ←0

Let us also decompose some graded components of the quotient R/I.

i14 : highestWeightsDecomposition(R/I,0,4)

o14 = HashTable{0 => Tally{{0, 0, 0, 0, 0} => 1}}
                1 => Tally{{0, 0, 0, 1, 0} => 1}
                2 => Tally{{0, 0, 0, 2, 0} => 1}
                3 => Tally{{0, 0, 0, 3, 0} => 1}
                4 => Tally{{0, 0, 0, 4, 0} => 1}

o14 : HashTable

We deduce that, for d∈{0,...,4}, (R/I)d = V(dω4).