Orbit algebras of permutation groups

DefinitionProduct of the reduced incidence algebra#✓fully proved

Let \(\Omega\) be a set. For functions \(f\) and \(g\) from the finite subsets of \(\Omega\) to \(\mathbb{Q}\), their product \(fg\) is the function on finite subsets given by \((fg)(X) = \sum_{Y \subseteq X} f(Y)\, g(X \setminus Y)\).

If \(f\) vanishes off the \(n\)-element subsets and \(g\) vanishes off the \(m\)-element subsets, this is the product of Kourovka Problem 13.29, and \(fg\) vanishes off the \((n+m)\)-element subsets.

def incMul (f g : Finset Ω → ℚ) : Finset Ω → ℚ :=
  fun X => ∑ Y ∈ X.powerset, f Y * g (X \ Y)

Used by 1

Used byCameron's conjecture on invariant incidence algebras (Kourovka 13.29)

DefinitionElements of the reduced incidence algebra#✓fully proved

Let \(\Omega\) be a set. A function \(f\) from the finite subsets of \(\Omega\) to \(\mathbb{Q}\) lies in the algebra \(A = \bigoplus_n V_n\), where \(V_n\) consists of the functions vanishing off the \(n\)-element subsets, when there is a natural number \(N\) such that \(f(X) = 0\) for every finite subset \(X\) of \(\Omega\) with more than \(N\) elements.

def IsInA (f : Finset Ω → ℚ) : Prop :=
  ∃ N : ℕ, ∀ X : Finset Ω, N < X.card → f X = 0

Used by 1

Used byCameron's conjecture on invariant incidence algebras (Kourovka 13.29)

DefinitionInvariant functions on finite subsets#✓fully proved

Let \(G\) be a group of permutations of a set \(\Omega\). A function \(f\) from the finite subsets of \(\Omega\) to \(\mathbb{Q}\) is \(G\)-invariant when \(f(\sigma(X)) = f(X)\) for every \(\sigma \in G\) and every finite subset \(X\) of \(\Omega\), where \(\sigma(X)\) is the image of \(X\) under \(\sigma\).

def IsInvariant (G : Subgroup (Equiv.Perm Ω)) (f : Finset Ω → ℚ) : Prop :=
  ∀ σ ∈ G, ∀ X : Finset Ω, f (X.map σ.toEmbedding) = f X

Used by 1

Used byCameron's conjecture on invariant incidence algebras (Kourovka 13.29)

DefinitionCameron's conjecture on invariant incidence algebras (Kourovka 13.29)#✓fully proved

The statement that for every infinite set \(\Omega\) and every group \(G\) of permutations of \(\Omega\) all of whose orbits are infinite, the following holds: if \(f\) and \(g\) are nonzero functions from the finite subsets of \(\Omega\) to \(\mathbb{Q}\) that are \(G\)-invariant (\(f(\sigma(X)) = f(X)\) for all \(\sigma \in G\) and all finite \(X\), and likewise for \(g\)) and that each vanish on all subsets of size larger than some bound, then their product \(fg\) in the reduced incidence algebra, given by \((fg)(X) = \sum_{Y \subseteq X} f(Y)\, g(X \setminus Y)\), is not identically zero. In other words, the algebra \(A^G\) of \(G\)-invariant elements of the reduced incidence algebra \(A\) has no zero divisors.

def invariantIncidenceAlgebraIsDomain : Prop :=
  ∀ (Ω : Type) [DecidableEq Ω] [Infinite Ω] (G : Subgroup (Equiv.Perm Ω)),
    (∀ x : Ω, (MulAction.orbit G x).Infinite) →
    ∀ f g : Finset Ω → ℚ, IsInA f → IsInvariant G f → IsInA g → IsInvariant G g →
      f ≠ 0 → g ≠ 0 → incMul f g ≠ 0

Uses 3 · Used by 1

Statement usesElements of the reduced incidence algebra · Invariant functions on finite subsets · Product of the reduced incidence algebra

Used byCameron's conjecture on invariant incidence algebras holds (Pouzet)

TheoremCameron's conjecture on invariant incidence algebras holds (Pouzet)#●ready to prove

Let \(\Omega\) be an infinite set and let \(G\) be a group of permutations of \(\Omega\) all of whose orbits are infinite. Let \(f\) and \(g\) be nonzero functions from the finite subsets of \(\Omega\) to \(\mathbb{Q}\) that are \(G\)-invariant (\(f(\sigma(X)) = f(X)\) for all \(\sigma \in G\) and all finite \(X\), and likewise for \(g\)) and that each vanish on all subsets of size larger than some bound. Then their product in the reduced incidence algebra, \((fg)(X) = \sum_{Y \subseteq X} f(Y)\, g(X \setminus Y)\), is not identically zero. In other words, the algebra \(A^G\) of \(G\)-invariant elements of the reduced incidence algebra \(A\) has no zero divisors.

public theorem_wanted invariantIncidenceAlgebraIsDomain_holds : invariantIncidenceAlgebraIsDomain

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 13.29 (P. J. Cameron); resolution: M. Pouzet, Theor. Inform. Appl. 42 (2008), no. 1, 83-103, Kourovka Archive, Problem 13.29: the conjecture is proved.

Uses 1

Statement usesCameron's conjecture on invariant incidence algebras (Kourovka 13.29)

4 fully proved1 ready to prove

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