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add permutation action on FreeAssAlgElems #2792

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Sep 15, 2023
Merged

add permutation action on FreeAssAlgElems #2792

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16 changes: 15 additions & 1 deletion src/Groups/action.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -284,6 +284,7 @@
"""
on_indeterminates(f::GAP.GapObj, p::PermGroupElem)
on_indeterminates(f::MPolyRingElem, p::PermGroupElem)
on_indeterminates(f::FreeAssAlgElem, p::PermGroupElem)
on_indeterminates(f::MPolyIdeal, p::PermGroupElem)
on_indeterminates(f::GAP.GapObj, p::MatrixGroupElem)
on_indeterminates(f::MPolyRingElem{T}, p::MatrixGroupElem{T, S}) where T where S
Expand All @@ -294,7 +295,7 @@
if `p` is a `MatrixGroupElem` then it acts via evaluating `f` at the
vector obtained by multiplying `p` with the (column) vector of indeterminates.

For `MPolyRingElem` and `MPolyIdeal` objects,
For `MPolyRingElem`, `FreeAssAlgElem`, and `MPolyIdeal` objects,
one can also call `^` instead of `on_indeterminates`.

# Examples
Expand Down Expand Up @@ -348,6 +349,17 @@
return finish(g)
end

function on_indeterminates(f::FreeAssAlgElem{T}, s::PermGroupElem) where T
G = parent(s)
S = parent(f)
@assert ngens(S) == degree(G)

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e = exponent_words(f)
c = Vector{T}(collect(coefficients(f)))
es = [on_tuples(x, s) for x in e]
return S(c, es)

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end

function on_indeterminates(f::GapObj, p::MatrixGroupElem)
# We assume that we act on the indeterminates with numbers 1, ..., nrows(p).
# (Note that `f` does not know about a polynomial ring to which it belongs.)
Expand Down Expand Up @@ -379,6 +391,8 @@

^(f::MPolyRingElem, p::PermGroupElem) = on_indeterminates(f, p)

^(f::FreeAssAlgElem, p::PermGroupElem) = on_indeterminates(f, p)

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^(f::MPolyRingElem{T}, p::MatrixGroupElem{T, S}) where T where S = on_indeterminates(f, p)

^(I::MPolyIdeal, p::PermGroupElem) = on_indeterminates(I, p)
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15 changes: 15 additions & 0 deletions test/Groups/action.jl
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Expand Up @@ -74,6 +74,21 @@ end
@test I^gen(g, 1) == on_indeterminates(I, gen(g, 1))
end

@testset "action on elements of free assoc. algebras: permutations" begin
g = symmetric_group(3)
R, vars = free_associative_algebra(QQ, 3);
(x1, x2, x3) = vars
f = x1*x2 + x2*x3

for p in [g(cperm(1:3)), g(cperm(1:2))]
@test f^p == evaluate(f, permuted(vars, p^-1))
end

for x in gens(g), y in gens(g)
@test on_indeterminates(on_indeterminates(f, x), y) == on_indeterminates(f, x*y)
end
end

@testset "action on multivariate polynomials: matrices" begin
g = general_linear_group(3, 5)
R, vars = polynomial_ring(base_ring(g), degree(g))
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