BettiCharacters Example 5 -- semidirect product of torus and symmetric group

We present the example in the introduction of S. Murai, C. Raicu - An equivariant Hochster’s formula for $\mathfrak{S}_n$-invariant monomial ideals.

Consider the ideal $I$ in three variables generated by monomials whose exponent vectors are permutations of $(4,1,1)$ or $(5,2,0)$. This ideal is clearly stable under the permutation action of $\mathfrak{S}_3$. Moreover, $I$ is compatible with the fine grading on $R = \Bbbk [x_1,x_2,x_3]$ given by $\deg (x_i) = e_i \in \mathbb{Z}^3$. We compute a minimal free resolution of $R/I$ and show its Betti diagram.

i1 : R = QQ[x_1..x_3,Degrees=>{{1,0,0},{0,1,0},{0,0,1}}]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(x_1^4*x_2*x_3,x_1*x_2^4*x_3,x_1*x_2*x_3^4,
         x_1^5*x_2^2,x_1^5*x_3^2,x_1^2*x_2^5,x_1^2*x_3^5,x_2^5*x_3^2,x_2^2*x_3^5)

             4         4         4   5 2   5 2   2 5   2 5   5 2   2 5
o2 = ideal (x x x , x x x , x x x , x x , x x , x x , x x , x x , x x )
             1 2 3   1 2 3   1 2 3   1 2   1 3   1 2   1 3   2 3   2 3

o2 : Ideal of R

i3 : RI = res I

      1      9      12      4
o3 = R  <-- R  <-- R   <-- R
                            
     0      1      2       3

o3 : Complex

i4 : betti RI

            0 1  2 3
o4 = total: 1 9 12 4
         0: 1 .  . .
         1: . .  . .
         2: . .  . .
         3: . .  . .
         4: . .  . .
         5: . 3  . .
         6: . 6  6 .
         7: . .  3 .
         8: . .  3 3
         9: . .  . 1

o4 : BettiTally

Next, we set up the action of the semidirect product $(\Bbbk^\times)^3 \rtimes \mathfrak{S}_3$ where $\mathfrak{S}_3$ acts on $(\Bbbk^\times)^3$ by permuting entries. This results in $\mathfrak{S}_3$ acting on the grading group $\mathbb{Z}^3$ (the character group of $(\Bbbk^\times)^3$) by permuting the entries of the degree vectors. Thus, the orbit of a degree $d\in \mathbb{Z}^3$ consists of all permutations of $d$; we fix the nonincreasing permutation of $d$ as the distinguished representative of this orbit. See Semidirect for details.

i5 : S3 = symmetricGroupActors(R)

o5 = {| x_2 x_3 x_1 |, | x_2 x_1 x_3 |, | x_1 x_2 x_3 |}

o5 : List

i6 : A = action(RI,S3,Semidirect=>{uniquePermutations,rsort})

o6 = Complex with 3 actors

o6 : ActionOnComplex

i7 : c = character A

o7 = Character over QQ
      
     (0, {0, 0, 0})  |  1   1  1
     (1, {4, 1, 1})  |  0   1  3
     (1, {5, 2, 0})  |  0   0  6
     (2, {4, 4, 1})  |  0  -1  3
     (2, {5, 2, 1})  |  0   0  6
     (2, {5, 5, 0})  |  0  -1  3
     (3, {4, 4, 4})  |  1  -1  1
     (3, {5, 5, 1})  |  0  -1  3

o7 : Character

To match the description of the paper, which resolves the ideal $I$ instead of the quotient $R/I$, we remove the component in homological degree 0, then shift the complex to the left. Finally, the resulting character is decomposed against the character table of $\mathfrak{S}_3$.

i8 : c = (c - c_0)[1]

o8 = Character over QQ
      
     (0, {4, 1, 1})  |  0   1  3
     (0, {5, 2, 0})  |  0   0  6
     (1, {4, 4, 1})  |  0  -1  3
     (1, {5, 2, 1})  |  0   0  6
     (1, {5, 5, 0})  |  0  -1  3
     (2, {4, 4, 4})  |  1  -1  1
     (2, {5, 5, 1})  |  0  -1  3

o8 : Character

i9 : T = symmetricGroupTable(3,QQ)

o9 = Character table over QQ
      
            |   2   3  1
     -------+-----------
       (3)  |   1   1  1
     (2,1)  |  -1   0  2
        3   |          
      (1 )  |   1  -1  1

o9 : CharacterTable

i10 : decomposeCharacter(c,T)

o10 = Decomposition table
       
                      |                3
                      |  (3)  (2,1)  (1 )
      ----------------+------------------
      (0, {4, 1, 1})  |    1      1     0
      (0, {5, 2, 0})  |    1      2     1
      (1, {4, 4, 1})  |    0      1     1
      (1, {5, 2, 1})  |    1      2     1
      (1, {5, 5, 0})  |    0      1     1
      (2, {4, 4, 4})  |    0      0     1
      (2, {5, 5, 1})  |    0      1     1

o10 : CharacterDecomposition

The irreducible representations found above match our expectations as can be verified by applying Pieri's rule to the description in Example 1.4 of Murai and Raicu's paper.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/BettiCharacters.m2:3859:0.