. This is also the difference between the highest degree of a secondary and the bound given by the degrees of the primary invariants.Cat::FiniteGroupInvariantRing(R) represents the category of invariant rings of finite groups of matrices over a field R, that is ....
Creating the Category
Cat::FiniteGroupInvariantRing(R)
Parameters:
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R: |
A Cat::Field. |
Categories
Cat::Algebra(R)
Details:
A Cat::FiniteGroupInvariantRing(R) is a.
Entries
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"variables" |
returns the list of . |
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"group" |
returns the group. |
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"dimen" |
returns a . |
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"Poly" |
returns the . |
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"isModular" |
this entry states the characteristic |
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"isCohenMacaulay" |
this entry states whether the invariant ring is Cohen-Macaulay, that is, whether the invariant ring is free module over the primary invariants.It can be UNKNOWN. |
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"primaryInvariantsDegrees" |
returns the permutation in cycle representation (list of its cycles). |
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"secondaryInvariantsSeries" |
..... |
primaryInvariantsSeries – generating series of the primary invariants
Cat::FiniteGroupInvariantRing::primaryInvariantsSeries(variable z)
Returns the generating series of the primary invariants, defined by f(z):=sum(z^d, d in degrees of primary invariants)
primaryInvariantsRingSeries – generating series of the ring generated by the primary invariants
Cat::FiniteGroupInvariantRing::primaryInvariantsRingSeries(variable z)
Returns the generating series of the primary invariants, defined by ...
HilbertSeries_Molien – ...
Cat::FiniteGroupInvariantRing::HilbertSeries_Molien(variable z)
Computes the Hilbert series of the invariant ring under the action of the group, using Molien formula.
Reference: Sturmfels Älgorithms in Invariant Theory", p 29 / p 72.
HilbertSeries_FromSecondary – ...
Cat::FiniteGroupInvariantRing::HilbertSeries_FromSecondary(variable z)
Returns the ...
HilbertSeries – ...
Cat::FiniteGroupInvariantRing::HilbertSeries(variable z)
Returns the ...
mu – lowest degree of an invariant
Cat::FiniteGroupInvariantRing::mu()
Returns the lowest degree of an invariant relative to the linear caracter . This is also the difference between the highest degree of a secondary and the bound given by the degrees of the primary invariants.
Reference: Stanley "Invariants of Finite Groups and Their Applications to Combinatorics" (1979), p. 485.