-*
Copyright 2020, Luigi Ferraro, Federico Galetto,
Francesca Gandini, Hang Huang, Matthew Mastroeni, Xianglong Ni.
You may redistribute this file under the terms of the GNU General Public
License as published by the Free Software Foundation, either version 2 of
the License, or any later version.
*-
document {
Key => {actionMatrix, (actionMatrix, LinearlyReductiveAction)},
Headline => "matrix of a linearly reductive action",
Usage => "actionMatrix L",
Inputs => {
"L" => LinearlyReductiveAction,
},
Outputs => {
Matrix => {"of a group action on a ring"}
},
"This function is provided by the package ", TO InvariantRing,
".",
PARA {
"Suppose the action ", TT "L", " consists of the linearly
reductive group with coordinate ring ", TT "S/I",
" (where ", TT "S", " is a polynomial ring) acting on
a (quotient of)
a polynomial ring ", TT "R", " via the action matrix ",
TT "M", ". This function returns the action matrix ", TT "M",
".",
},
EXAMPLE {
"S = QQ[z]",
"I = ideal(z^2 - 1)",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}}",
"R = QQ[x,y]",
"L = linearlyReductiveAction(I, M, R)",
"actionMatrix L",
},
}
document {
Key => {groupIdeal, (groupIdeal, LinearlyReductiveAction)},
Headline => "ideal defining a linearly reductive group",
Usage => "groupIdeal L",
Inputs => {
"L" => LinearlyReductiveAction => {"of a group with coordinate ring ", TT "S/I"},
},
Outputs => {
Ideal => {TT "I"}
},
"This function is provided by the package ", TO InvariantRing,
".",
PARA {
"Suppose the action ", TT "L", " consists of the linearly
reductive group with coordinate ring ", TT "S/I",
" (where ", TT "S", " is a polynomial ring) acting on
a (quotient of)
a polynomial ring ", TT "R", " via the action matrix ",
TT "M", ". This function returns the ideal ", TT "I",
".",
},
EXAMPLE {
"S = QQ[z]",
"I = ideal(z^2 - 1)",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}}",
"R = QQ[x,y]",
"L = linearlyReductiveAction(I, M, R)",
"groupIdeal L",
},
}
document {
Key => {hilbertIdeal, (hilbertIdeal, LinearlyReductiveAction)},
Headline => "compute generators for the Hilbert ideal",
Usage => "hilbertIdeal L",
Inputs => {
"L" => LinearlyReductiveAction,
},
Outputs => {
Ideal => {"the ideal generated by all ring elements invariant under the action"}
},
"This function is provided by the package ", TO InvariantRing,
".",
PARA {
"This function computes the Hilbert ideal
for the action of a linearly reductive group
on a (quotient of a) polynomial ring, i.e., the ideal
generated by all ring elements of positive degree
invariant under the action. The algorithm is based on: ",
},
UL {
{"Derksen, H. & Kemper, G. (2015).", EM " Computational Invariant Theory",
". Heidelberg: Springer. pp 159-164"}
},
PARA {
"The next example constructs a cyclic group of order 2
as a set of two affine points. Then it introduces an
action of this group on a polynomial ring in two variables
and computes the Hilbert ideal. The action permutes the
variables of the polynomial ring. Note that the
generators of the Hilbert ideal need not be invariant."
},
EXAMPLE {
"S = QQ[z]",
"I = ideal(z^2 - 1)",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}}",
"sub(M,z=>1),sub(M,z=>-1)",
"R = QQ[x,y]",
"L = linearlyReductiveAction(I, M, R)",
"H = hilbertIdeal L",
"apply(H_*, f -> isInvariant(f,L))"
},
PARA {
"We offer a slight variation on the previous example
to illustrate this method at work on a quotient of a
polynomial ring."
},
EXAMPLE {
"S = QQ[z];",
"I = ideal(z^2 - 1);",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}};",
"Q = QQ[x,y] / ideal(x*y)",
"L = linearlyReductiveAction(I, M, Q)",
"H = hilbertIdeal L"
},
PARA {
"The algorithm performs an elimination using Groebner
bases. The options ", TO DegreeLimit, " and ",
TO SubringLimit, " are standard ", TO gb, " options
that can be used to interrupt the computation
before it is complete, yielding a partial list of
generators for the Hilbert ideal."
},
Caveat => "The generators of the Hilbert ideal computed
by this function need not be invariant."
}
document {
Key => {linearlyReductiveAction,
(linearlyReductiveAction, Ideal, Matrix, PolynomialRing),
(linearlyReductiveAction, Ideal, Matrix, QuotientRing)},
Headline => "Linearly reductive group action",
Usage => "linearlyReductiveAction(I, M, R) \n linearlyReductiveAction(I, M, Q) ",
Inputs => {
"I" => Ideal => {"of a polynomial ring ", TT "S", " defining a group as an affine variety"},
"M" => Matrix => {"whose entries are in ", TT "S", ", that encodes the group action on ", TT "R"},
"R" => PolynomialRing => {"on which the group acts"},
"Q" => QuotientRing => {"on which the group acts"},
},
Outputs => {
LinearlyReductiveAction => {"the linearly reductive action of ", TT "S/I",
" on ", TT "R", " or ", TT "Q", " via the matrix ", TT "M"}
},
"This function is provided by the package ", TO InvariantRing, ".",
PARA {
"In order to encode a linearly reductive group action,
we represent the group as an affine variety.
The polynomial ring ", TT "S", " is the coordinate ring
of the ambient affine space containing the group,
while ", TT "I", " is the ideal of ", TT "S",
" defining the group as a subvariety. In other words,
the elements of the group are the points of the affine
variety with coordinate ring ", TT "S/I", ". ",
"The group acts linearly on the
polynomial ring ", TT "R", " via the matrix ",
TT "M", " with entries in ", TT "S", ".",
},
PARA {
"The next example constructs a cyclic group of order 2
as a set of two affine points. Then it introduces an
action of this group on a polynomial ring in two variables."
},
EXAMPLE {
"S = QQ[z]",
"I = ideal(z^2 - 1)",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}}",
"sub(M,z=>1),sub(M,z=>-1)",
"R = QQ[x,y]",
"L = linearlyReductiveAction(I, M, R)",
},
PARA {
"This function is also used to define linearly reductive
group actions on quotients of polynomial rings.
We illustrate by a slight variation on the previous example."
},
EXAMPLE {
"S = QQ[z];",
"I = ideal(z^2 - 1);",
"M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}};",
"Q = QQ[x,y] / ideal(x*y)",
"L = linearlyReductiveAction(I, M, Q)",
},
}
document {
Key => {LinearlyReductiveAction},
Headline => "the class of all (non finite, non toric) linearly reductive group actions",
"This class is provided by the package ", TO InvariantRing,".",
PARA {
TT "LinearlyReductiveAction", " is the class of all
linearly reductive group actions on (quotients of)
polynomial rings
for the purpose of computing invariants.
It is created using ", TO "linearlyReductiveAction", ".",
" This class should not be used for actions of
tori or finite groups, as its methods for computing
invariants are in general less efficient than
specialized methods for the classes ",
TO "FiniteGroupAction", ", and ", TO "DiagonalAction", "."
},
}