On subfactors arising from asymptotic representations of symmetric groups

We consider the infinite symmetric group and its infinite index subgroup given as the stabilizer subgroup of one element under the natural action on a countable set. This inclusion of discrete groups induces a hyperfinite subfactor for each finite factorial representation of the larger group. We compute subfactor invariants of this construction in terms of the Thoma parameter.

The representations of S ∞ has also been an important example in the theory of operator algebras, as evidenced by the construction of a hyperfinite II 1 factor using its regular representation by Murray-von Neumann, which was among the pioneering works of the subject. When one considers the structure of the representations of S ∞ through operator algebras, the first important phenomenon is the uniqueness of the hyperfinite (AFD) II 1 factor. It implies that all the II 1 factorial representations of the locally finite group S ∞ generate isomorphic von Neumann algebras.
Regarding the structure of the hyperfinite II 1 factor, the theory of subfactors initiated by Jones [5] was very fruitful and revealed many surprising combinatorial aspects of the structure of inclusion between such factors. One should note that the infinite symmetric group also has a similar structure of inclusions, as it contains a subgroup isomorphic to itself, given as the stabilizer of a single point under the natural representation on a countable set. This inclusion of discrete groups induces an inclusion of II 1 factors for each II 1 factorial representation of S ∞ .
In this paper we investigate the hyperfinite subfactors obtained this way and investigate how subfactor invariants reflect the Thoma parameter. First we obtain the characterization of irreducibility in terms of the Thoma parameter (Theorem 1), next the characterization of being an infinite index inclusion as the faithfulness of the corresponding state on the group algebra (Theorem 3), and then an estimate of the subfactor entropy (Theorem 4).
Finally, we note that Gohm-Köstler [3] recently obtained the same characterization of irreducibility as well as the new proof of Thoma's classification result.

Preliminaries
For each positive integer n, we identify the n-th symmetric group S n with the group of the bijections on the set {0, 1, . . . , n − 1}. These groups form an increasing sequence by letting the elements of S n act on {n, . . . , m − 1} trivially when n < m. Given a sequence m 1 , . . . , m n of distinct numbers, we write (m 1 , m 2 , . . . , m n ) for the cyclic permutation s defined by s(m k ) = m k+1 for k = 1, . . . , n−1 and s(m n ) = m 1 .
The infinite symmetric group S ∞ is defined to be the union of the groups S n for n ∈ N >0 . It is identified with the set of the permutations of N = {0, 1, . . .} which move only a finite number of elements. It is a countable infinite conjugacy class group, meaning that the conjugacy classes are infinite sets except for the trivial class of the neutral element.
For each n ∈ N, let S n≤ denote the subgroup of S ∞ consisting of the elements which fix the numbers in {0, . . . , n − 1}. Thus we have a decreasing sequence of infinite groups S ∞ ⊃ S 1≤ ⊃ S 2≤ · · · , all isomorphic to S ∞ and having infinite index at each inclusion.
The unitary representations of S ∞ whose image generate (hyperfinite) II 1 factors are classified by Thoma (in the following we follow the treatment of Wassermann [17,Chapter III]). We briefly recall their classification. They are parametrized by the Thoma parameter consisting of two nonincreasing sequences of nonnegative real numbers α 1 ≥ α 2 ≥ · · · , β 1 ≥ β 2 ≥ · · · , and another nonnegative number γ satisfying Let κ be a Thoma parameter as above. Then we construct a positive definite function τ κ on S ∞ by putting when s is a cyclic permutation of n numbers, and setting τ κ (s 0 s 1 · · · s m ) = τ κ (s k ) when s 0 , s 1 , . . . , s m are cyclic permutations with nonintersecting supports. Thus, under the above notation we have for n distinct integers m 1 , . . . , m n . The function τ κ defines an extremal tracial state on the group C * -algebra C * S ∞ of S ∞ , hence the associated Gelfand-Naimark-Segal representation of C * S ∞ generates a hyperfinite II 1 factor unless α 1 = 1 or β 1 = 1. Moreover, these representations of S ∞ for different values of κ are never mutually quasi-equivalent.
Let M κ denote the II 1 factor generated by S ∞ under the GNS representation with respect to τ κ and let N κ denote the subfactor of M κ generated by the image of S 1≤ . In the rest of the paper we analyze the inclusion of von Neumann algebras N κ ⊂ M κ .

Characterization of Irreducibility
Let T n be the permutation group of the set {1, . . . , n − 1}. The group S 1≤ is the union of the increasing sequence of the groups T n . The conditional expectation onto the relative commutant of T n inside M κ can be computed using the following lemma. Proof. For an arbitrary element x of M κ , one has Under the conjugation action by T n , the stabilizer subgroup of (0, 1) is the subgroup of T n consisting of the permutations of the set {2, . . . , n − 1}, and the conjugates of (0, 1) are (0, j) for j = 1, . . . , n − 1. Hence we obtain E πκ(Tn) ′ ∩Mκ (π κ ((0, 1))) = 1 n − 1 n−1 j=1 π κ ((0, j)), which proves the assertion.
Recall that a subfactor N ⊂ M is said to be irreducible when the relative commutant N ′ ∩ M equals C. We have the following characterization of irreducibility for the inclusion N κ ⊂ M κ . Theorem 1. The following conditions are equivalent: (c) One has α i = β i = 0 for any i (hence γ = 1).
Proof. (i) ⇒ (ii): Suppose that the inclusion N κ ⊂ M κ is irreducible. As in Lemma 1, let s be the image of the transposition (0, 1) in M κ and E N ′ κ ∩Mκ be the conditional expectation of M κ onto N ′ κ ∩ M κ . By the irreducibility assumption and the general principle must be equal to the scalar τ κ (s). By Lemma 1, we have Hence we obtain On the other hand, we have E N ′ κ ∩Mκ (s) = lim n→∞ E πκ(Tn) ′ ∩Mκ (x) in the 2-norm topology. Since E πκ(Tn) ′ ∩Mκ (s) 2 2 converges to τ κ ((0, 1, 2)) as n → 0 by (1), one has to have τ κ (s) 2 = τ κ ((0, 1, 2)) that is equivalent to Let (γ i ) i∈Z\{0} be the sequence defined by γ i = α i for i > 0 and by γ i = β −i for i < 0. Then we have Combining this with the inequality it follows that (2) holds if and only if the the above inequalities are actually equalities.
On one hand, in order to have the equality in (3), one has to have and there has to be at most two values in including the mandatory 0. On the other hand, in order to have the equality in (4), We first consider the case (iia). In this case one can realize M κ as the fixed point algebra of the adjoint action of a compact group on the infinite tensor product of matrix algebras (an ITPFI action, [5, Section 5.3; 17, Section III.4]) as follows.
Let V be a Hilbert space of dimension n. The permutation of tensors There is an action Ad of the projective unitary group PU(V ) of V on B(V ) by the conjugation. It induces the tensor product action Ad ⊗m of PU(V ) on B(V ) ⊗m , and the image of C * S m is the fixed point algebra of this action. Hence we have a compatible system Moreover, M admits a subfactor N generated by the union of the subalgebras 1 ⊗ B(V ⊗m−1 ) of B(V ⊗m ) for m ∈ N >0 , which is stable under the action of PU(V ). The fixed point subalgebra N PU(V ) of N , which agrees with the intersection of M κ and N , is equal to N κ . For such a construction one has the irreducibility The case (iib) also follows from the above argument. Indeed, the mapping θ : σ → (−1) |σ| σ on C[S ∞ ] defines an * -algebra automorphism of C * S ∞ . If the parameter κ is of the form (iib), let κ ′ denote the parameter of the form (iia) obtained by exchanging α i and β i in κ for each i ∈ N. Then one has τ κ ′ = τ κ • θ, and θ extends to an isomorphism σ : Hence it reduces to the case (iia), and we also have N ′ κ ∩ M κ = C in this case. It remains to consider the case (iic). In this case the trace τ κ is the standard trace τ ( s∈S∞ c s λ s ) = c e of the discrete group S ∞ . Hence the GNS representation obtained from τ κ can be identified with the left regular representation of S ∞ . The conjugation by the elements of S 1≤ on S ∞ defines an equivalence relation whose equivalence classes are infinite sets except for the trivial class {e} consisting of the neutral element. The operators in the relative commutant of S 1≤ inside the left regular von Neumann algebra LS ∞ should be represented by the functions that are constant on each equivalence class of this relation.
Remark 1. Gohm-Köstler [3] also obtained the content of the above theorem with a different method.
For any pair (i, i ′ ) of elements in X α,β , let e i,i ′ denote the partial isometry δ j → δ i,j δ i ′ in B(ℓ 2 X α,β ). We also write e i to denote the projection e i,i .
With the above constructions, one has the following refinement of Theorem 1 in this situation.
Lemma 2. In the above notation, we have Proof. We prove this by estimating the 2-norm of with respect to φ ⊗∞ α,β as k → ∞. By Lemma 1, the square of the 2-norm of (5) is equal to ⊗∞ in which the leading sign of the left hand side is determined according to that of i. Hence (6) can be computed as This converges to 0 as k → ∞. We have thus obtained which proves the assertion.
On the other hand, we have the double coset decomposition of S ∞ with respect to S 1≤ . Hence the operator E P Q (π κ (s)) ∈ Q actually belongs to N κ for any s ∈ S ∞ . For each 1 ≤ k ≤ p let f k denote the projection i∈I k e i ∈ B(ℓ 2 X α,β ). Analogously we define f l = j∈J l e j for each −q ≤ l ≤ −1.
When one applies E B(ℓ 2 X α,β ) to (8), the only surviving terms are the ones corresponding to x ∈ X n α,β which are constant on the orbits of s. Let ω 0 denote the orbit of 0 ∈ {0, . . . , n − 1} under s and let ω 1 , . . . , ω r be the other nontrivial orbits.
Suppose that x is constant on each orbit ω q , and let x q be the value of x on ω q . Then one has τ (s, x) = q=0,...,r xq<0 Thus the image of π κ (s) under E B(ℓ 2 X α,β ) is equal to  Hence M κ contains e n 11 x n e n 22 = e n 12 e n+1 21 , etc., which allows us to realize the representative α 2 e n 11 e n+1 22 + α 1 e n 22 e n+1 11 + √ α 1 α 2 (e n 12 e n+1 21 + e n 21 e n+1 12 ) inside M κ of the Jones projections corresponding to the index (α 1 α 2 ) −1 .
Remark 2. The group algebra C * S ∞ is an AF-algebra, being the inductive limit of the increasing sequence of the algebras C * S n . The subalgebra C * S 1≤ is also the inductive limit of the subalgebras C * T n of C * S n . Hence we obtain squares of finite dimensional algebras The cases where these squares become commuting (i.e. [E C * Tn+1 ] = 0) with respect to the trace τ κ happens to agree with the ones of Theorem 1. Indeed, if the above square is commuting for n = 2, the image of the transposition If that is the case, we will have τ κ ((0 1)(1 2)) = τ κ (0 1)τ κ (1 2), which is equivalent to the formula (2).
Proof. The natural logarithm of the above expression is equal to for any positive integer a. Hence (10) has the estimate from above by This can be reorganized as the sum of and k(k + 1) 2 log(k + 2) − log √ k(k + 1) 2 .
All of the above go to −∞ as k → ∞, which shows that (10) converges to −∞.
Lemma 6. Let τ be a faithful tracial state on C * S ∞ . For any positive real number ǫ, there exists a positive integer n and a projection e ∈ C * S n such that 0 < τ (e) < ǫ n .
Proof. We consider the Young diagrams with the isosceles right triangle shape, i.e. the ones having rows of length k, k − 1, . . . , 1 exactly once for each for k ∈ N, with the total number of boxes equal to k(k + 1)/2. By the hook-length formula, the irreducible representation of S k(k+1)/2 corresponding to this diagram has dimension .
Hence a minimal projection e k belonging to the factor of this representation satisfies Applying Lemma 5 to R = 2ǫ −1 , we obtain for large enough k. For such k, taking n = k(k + 1)/2 and the corresponding projection e = e k in C * S n as above, we have τ (e) < ǫ n . Remark 3. The trace τ κ is faithful in the following cases: 1) α i > 0 for any i, 2) β i > 0 for any i, and 3) γ > 0. Otherwise it is not, as the corresponding infinite Young tableaux can be taken to have a bounded number of rows and columns [17,Theorem III.5].
Proof of the theorem. Suppose that the trace τ κ is faithful. For each i ∈ N, let L τκ S i≤ be the subalgebra of M κ generated by S i≤ . Since there is an isomorphism σ n : S ∞ → S n≤ satisfying σ n (S 1≤ ) = S n+1≤ and τ κ (s) = τ κ (σ n (s)), the inclusions L τκ S i+1≤ ⊂ L τκ S i≤ are all isomorphic to N κ ⊂ M κ . In particular, we have [M κ : We also note that the image of C * S n in M κ is contained in the relative commutant of L τκ S n≤ for any n ∈ N >0 . Let ǫ be an arbitrary positive real number. By Lemma 6, there exists an integer n and a projection e ∈ C * S n satisfying τ κ (e) < ǫ n . This means that the image E Lτ κ S n≤ (e) of e under the conditional expectation onto L τκ S n≤ is a positive scalar smaller than ǫ n . By the Pimsner-Popa inequality, we have [M κ : L τκ S n≤ ] > ǫ −n . Thus [M κ : N κ ] > ǫ −1 for any ǫ, which shows [M κ : N κ ] = ∞ when τ κ is faithful.
Next, suppose that τ κ is not faithful. By Remark 3, this implies that γ = 0 and only finitely many terms in the sequences (α i ) i∈N>0 and that (β j ) j∈N>0 are nonzero. Hence we can apply the construction of Section 3.1 to this situation, and there are algebras Q ⊂ P and a commuting square as in (7) by Lemma 3. Again by the Pimsner-Popa inequality, in order to conclude [M κ : N κ ] < ∞ it is enough to show that there is a constant C > 0 satisfying E P Q (y) ≥ Cy for any y ∈ P + . Put δ = min(α mα , β m β ) so that we have φ α,β − δ tr ≥ 0 on B(ℓ 2 X α,β ). Then we have the estimate E P Q (y) ≥ δτ ⊗ Id(y) ≥ δ(m α + m β ) −2 y for y ∈ P + . This shows that [M κ : N κ ] < ∞ when τ κ is not faithful.

Relative Entropy.
In the rest of the paper we consider the relative entropy of the inclusion N κ ⊂ M κ . Recall that it is defined by where η(t) = −t log(t). We follow the conventions of Neshveyev-Størmer [8, Chapter 10] in the following.
Theorem 4. We have the estimate of the relative entropy The equality holds when γ = 0 and there is no duplicate in each of the sequences (α i ) i∈N and (β j ) j∈N other than 0.
Proof. We are going to use the AF-structure of the inclusion σ(M κ ) = N κ ⊂ M κ given by the squares (9). Although they are not commuting in general, we still have H τκ (M κ | N κ ) ≤ lim H τκ (C * S n | C * T n ) by [11,Proposition 3.4].
Let G be the group of finitely supported permutations of the set Z. Its group algebra admits 'the shift automorphism' σ characterized by σ : (n, m) → (n+1, m+ 1) on the transpositions. For any Thoma parameter κ, the trace τ κ on C * S ∞ admits a unique extension as a trace (which we still denote by τ κ ) to C * G.