How faithfully the symmetric group acts on a tensor power

The symmetric group S_d acts on (Rⁿ)^{⊗d} by permuting tensor factors, and that action extends to an algebra map R[S_d] → End((Rⁿ)^{⊗d}). Over a nontrivial commutative ring, this file measures how much of R[S_d] survives: the algebra map is injective exactly when d ≤ n.

Above the threshold the tensor power is too small to separate permutations linearly. For n < d every monomial basis vector of (Rⁿ)^{⊗d} repeats one of its factors, so the full antisymmetrizer ∑_σ sgn(σ) σ acts as 0 over every commutative ring, and is nonzero when the ring is nontrivial. This is why Schur--Weyl duality is stated for the image of R[S_d] in End((Rⁿ)^{⊗d}) rather than for R[S_d] itself.

Over a nontrivial commutative semiring, the group-level question has a different, coarser answer, recorded here as well: the permutation representation itself is faithful as soon as Rⁿ has two independent coordinate directions with which to tell tensor slots apart, that is, exactly when 2 ≤ n or d ≤ 1.

Both answers are read off the monomial basis tensorPowerBasis of (Rⁿ)^{⊗d}, indexed by the functions Fin d → Fin n, on which a permutation acts by precomposition with its inverse.

Main results

• TauCeti.permTensorAction_tensorPowerBasis: a permutation acts on the monomial basis by precomposition with its inverse.
• TauCeti.permTensorActionAlgHom_injective_iff: the algebra map R[S_d] → End((Rⁿ)^{⊗d}) is injective if and only if d ≤ n.
• TauCeti.permTensorAction_injective_iff: the permutation representation itself is faithful if and only if 2 ≤ n or d ≤ 1.

References

• Schur--Weyl roadmap, Layer 8, “The double centralizer (image-level)”: the faithfulness refinement recording that the symmetric-group image is faithful once the dimension is at least the tensor degree, and that the group algebra maps on as a proper quotient otherwise.

theorem TauCeti.permTensorActionAlgHom_injective_of_le {R : Type u} {n d : ℕ} [CommSemiring R] (h : d ≤ n) : Function.Injective ⇑(permTensorActionAlgHom R n d)

Faithfulness of the symmetric-group image. If the tensor degree does not exceed the dimension, the algebra map R[S_d] → End((Rⁿ)^{⊗d}) is injective.

theorem TauCeti.permTensorActionAlgHom_sum_sign_eq_zero {R : Type u} {n d : ℕ} [CommRing R] (h : n < d) : (permTensorActionAlgHom R n d) (∑ σ : Equiv.Perm (Fin d), MonoidAlgebra.single σ ↑↑(Equiv.Perm.sign σ)) = 0

The antisymmetrizer dies below the threshold. If the dimension is smaller than the tensor degree, then every monomial basis vector of (Rⁿ)^{⊗d} repeats a factor, and the full antisymmetrizer ∑_σ sgn(σ) σ annihilates the whole tensor power.

theorem TauCeti.permTensorActionAlgHom_not_injective_of_lt {R : Type u} {n d : ℕ} [CommRing R] [Nontrivial R] (h : n < d) : ¬Function.Injective ⇑(permTensorActionAlgHom R n d)

Non-faithfulness above the threshold. If the dimension is smaller than the tensor degree, the algebra map R[S_d] → End((Rⁿ)^{⊗d}) has a nonzero kernel, so the symmetric-group image is a proper quotient of the group algebra.

theorem TauCeti.permTensorActionAlgHom_injective_iff {R : Type u} {n d : ℕ} [CommRing R] [Nontrivial R] : Function.Injective ⇑(permTensorActionAlgHom R n d) ↔ d ≤ n

The faithfulness threshold. The algebra map R[S_d] → End((Rⁿ)^{⊗d}) is injective exactly when the tensor degree does not exceed the dimension.

theorem TauCeti.permTensorAction_eq_one_of_le_one {R : Type u} {n d : ℕ} [CommSemiring R] (h : n ≤ 1) (σ : Equiv.Perm (Fin d)) : (permTensorAction R n d) σ = 1

With at most one coordinate direction the tensor slots cannot be told apart, and the permutation action on (Rⁿ)^{⊗d} is trivial.

theorem TauCeti.permTensorAction_injective_of_two_le {R : Type u} {n d : ℕ} [CommSemiring R] [Nontrivial R] (h : 2 ≤ n) : Function.Injective ⇑(permTensorAction R n d)

With at least two coordinate directions the permutation action on (Rⁿ)^{⊗d} is faithful: the monomial basis vector marking a single tensor slot already detects the permutation.

theorem TauCeti.permTensorAction_injective_iff {R : Type u} {n d : ℕ} [CommSemiring R] [Nontrivial R] : Function.Injective ⇑(permTensorAction R n d) ↔ 2 ≤ n ∨ d ≤ 1

When the permutation representation itself is faithful. Unlike the group-algebra image, the representation of S_d on (Rⁿ)^{⊗d} is faithful for every tensor degree once 2 ≤ n; it fails only when the dimension collapses the action while there is more than one slot to permute.