invariants(DiagonalAction) -- computes the generating invariants of a group action

Description

This function is provided by the package InvariantRing. It implements an algorithm to compute a minimal set of generating monomial invariants for a diagonal action of an abelian group $(k^*)^r \times \mathbb{Z}/d_1 \times \cdots \times \mathbb{Z}/d_g$ on a polynomial ring $R = k[x_1, \dots, x_n]$. Saying the action is diagonal means that $(t_1,\ldots,t_r) \in (k^*)^r$ acts by $$(t_1,\ldots,t_r) \cdot x_j = t_1^{w_{1,j}}\cdots t_r^{w_{r,j}} x_j$$ for some integers $w_{i,j}$ and the generators $u_1, \dots, u_g$ of the cyclic abelian factors act by $$u_i \cdot x_j = \zeta_i^{w_{r+i,j}} x_j$$ for $\zeta_i$ a primitive $d_i$-th root of unity. The integers $w_{i,j}$ comprise the weight matrix W. In other words, the $j$ -th column of W is the weight vector of $x_j$.

The algorithm combines a modified version of an algorithm for tori due to Derksen and Kemper which can be found in:

• Derksen, H. & Kemper, G. (2015).Computational Invariant Theory. Heidelberg: Springer. pp 159-164

together with an algorithm for finite abelian groups due to Gandini which can be found in:

• Gandini, F. Ideals of Subspace Arrangements. Thesis (Ph.D.)-University of Michigan. 2019. ISBN: 978-1392-76291-2. pp 29-34.

Version 2.5 includes a faster algorithm to compute invariants of elementary abelian $p$-groups which is used by default when possible, i.e., when there is no torus action, all cyclic factors have the same prime order, and the weight matrix has maximal rank. For more information, see:

• Arasha S., Cassell M., Dolorfino M., Gandini F., Novak G., Qin D., and Strom S. (2026). An algorithm for invariants of elementary abelian groups. arXiv:2605.19158

Here is an example of a one-dimensional torus acting on a two-dimensional vector space:

i1 : R = QQ[x_1,x_2]

o1 = R

o1 : PolynomialRing

i2 : W = matrix{{1,-1}}

o2 = | 1 -1 |

              1       2
o2 : Matrix ZZ  <-- ZZ

i3 : T = diagonalAction(W, R)

            *
o3 = R <- QQ  via 

     | 1 -1 |

o3 : DiagonalAction

i4 : invariants T

o4 = {x x }
       1 2

o4 : List

Here is an example of a product of two cyclic groups of order 3 acting on a three-dimensional vector space:

i5 : R = QQ[x_1..x_3]

o5 = R

o5 : PolynomialRing

i6 : d = {3,3}

o6 = {3, 3}

o6 : List

i7 : W = matrix{{1,0,1},{0,1,1}}

o7 = | 1 0 1 |
     | 0 1 1 |

              2       3
o7 : Matrix ZZ  <-- ZZ

i8 : A = diagonalAction(W, d, R)

o8 = R <- ZZ/3 x ZZ/3 via 

     | 1 0 1 |
     | 0 1 1 |

o8 : DiagonalAction

i9 : invariants A

       3   3   3       2   2 2
o9 = {x , x , x , x x x , x x x }
       3   2   1   1 2 3   1 2 3

o9 : List

The example above uses the new algorithm for elementary abelian $p$-groups introduced in version 2.4. To call the older general-purpose algorithm, use the option Strategy=>"DerksenGandini".

i10 : invariants(A,Strategy=>"DerksenGandini")

        3   3   3       2   2 2
o10 = {x , x , x , x x x , x x x }
        3   2   1   1 2 3   1 2 3

o10 : List

Here is an example of a diagonal action by the product of a two-dimensional torus with a cyclic group of order 3 acting on a two-dimensional vector space:

i11 : R = QQ[x_1, x_2]

o11 = R

o11 : PolynomialRing

i12 : d = {3}

o12 = {3}

o12 : List

i13 : W1 = matrix{{1,-1}, {-1,1}}

o13 = | 1  -1 |
      | -1 1  |

               2       2
o13 : Matrix ZZ  <-- ZZ

i14 : W2 = matrix {{1,0}}

o14 = | 1 0 |

               1       2
o14 : Matrix ZZ  <-- ZZ

i15 : D = diagonalAction(W1, W2, d, R)

              * 2
o15 = R <- (QQ )  x ZZ/3 via 

      (| 1  -1 |, | 1 0 |)
       | -1 1  |

o15 : DiagonalAction

i16 : invariants D

        3 3
o16 = {x x }
        1 2

o16 : List

Caveat

See also

• diagonalAction -- diagonal group action via weights
• invariantRing -- the ring of invariants of a group action
• isInvariant -- check whether a polynomial is invariant under a group action