Частично центральные состояния на бесконечной симметрической группе Текст научной статьи по специальности «Математика»

MSC 20C32, 20C30, 22D25

Partly central states on the infinite symmetric

group

© A. V. Dudko, N. I. Nessonov, A. M. Vershik

Phis.-Techn. Inst. Low Temp., Kharkiv, Ukraina;

S.-Petersburg State Univ., S.-Petersburg, Russia

Let 6 ^ be the group of all finite bijections N ^ N Denote bv 6^ the set of all unitary irreducible admissible representations of 6^ = 6^ x 6^. We study the factor representations of 6^ that are the restrictions of the representations from 63^ to 6^ x e, where e is the unit element of 6^. It turn out that these representations are of type I , IIi or II^. The full description for the classes of the quasiequivalent representations is given.

Keywords: infinite symmetric group, factor representations, quasiequivalentness, unitary irreducible admissible representations

1. Characters and traces

Let N be the set of the natural numbers. By definition, a bijeetion s : N ^ N is called finite if the set of i E N such that s(i) = i is finite. Define a group 6^ as the group of all finite bijections N ^ N. For n E N U {0} we have two subgroups: 6n consisting of s such that s(i) = i for all i > n and 6n,^ consisting of s such that s(k) = k for all k ^ n. In particular, 60 is the identity subgroup and 60,^ coincides with 6^,

Definition 2 A function 0 on the group G is called a finite character, if it has the following properties:

(a) 0 is central, that is, 0 (g1g2) = 0 (g2g1), g1,g2 E G;

(b) 0 is positive definite, that is, for all g1,g2,... ,gn the matrix (0 (g^g-1)) is nonnegative;

(c) 0 is normalized, that is, 0 (e) = 1, where e is the unit element of G.

0

of G corresponding to 0 (according to the GNS (Gelfand-Naimark-Segal) construction) is a factor representation.

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A fundamental result of representation theory for Єis a complete description of finite indecomposable characters. To state it we need some notation,

(а, в)

positive numbers а = {а1 ^ а2 ^ ...} and в = {в1 ^ в2 ^ ...} such that

<^k + вk ^ 1.

kk

Denote by А the set of all such pairs (а, в)■

Let us write a permutation s є as a product of disjoint cycles:

S = C1 c2 . . . ct

with lengths l1,l2 ..., lt respectively greater than 1, To any (а, в) є А we assign a function хав on Є^>:

хав (s) = Ц I 3 (аАk - (-вk)lk)

m=1 \ k

In 1964 E, Thoma proved the next important statement

Theorem 3 The functions хав> where (а, в) ranges over А, are exactly the finite indecomposable characters of the group Єте.

The full description of the properly semifinite (non finite) traces on C* (&&,) was obtained by A, M, Vershik and S, V, Kerov [2]. The next proposition contains the corresponding result.

Proposition 4 Let \ be a partition of n є N and 7Л the corresponding irreducible representation of &n. Let (а, в) be Thoma parameters, denote by пав the GNS-representation of &n,^ corresponding to the finite character хав- F°r g є Єп and h є &n,™ pu t Tл а в (gh) = ^(g) ® Пав (h) • Let both sgqugїісєз а and в be finite and

3 <^k + 3 вk = 1, kk

then we have the following two properties:

(i) the representation ПЛав of induced by the representation ^ав of the subgroup &n ■ &n ^ is a ІІте-factor representation of

(ii) the faithful semi-finite trace т on factor ПЛав (&<х)" defines, by the formula т (ПЛав (A)) = тЛав(A), where A є C* (&^)\ semifinite trace тЛав on C* (&x).

The converse of this statement is true. Namely, for any semifinite trace x on C* (&^) there exist n є N a partition X of n and Thoma parameters (а, в) with

x = тЛ а в

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§ 2. Admissible representations of the group ©TO x ©TO

The theory of finite characters on ©O is a special case of a general theory of admissible representations of the group ©O x ©O developed bv G, I, Olshanski and A, Yu, Okounkov,

Let us consider a countable group G and its subgroup K. Denote by SP the set of positive definite functions p on G with p (e) = 1 that are K-biinvariant, This means that p (k\gk2) = p (g) for all g G G and k\,k2 G K, If is a GNS-representation corresponding to p G S^^d is a unit cyclic vector such that p(g) = (nv(g)£v, £p), then nv(k)£v = for all k G K. The set SP is convex.

The following properties are equivalent

p SP

(b) the representation is irreducible.

Let n be a unitary representation of G acting on a Hilbert space H Denote by HK

the subspace of vectors fixed for K, If dim HK = 1, the irreducible representation n

pG

as p(g) = (n(g)£,£), where n is a spherical representation, £ G HK. Hence spherical

SP

Proposition 5 There exists a natural one-to-one correspondence between the set of spherical functions of the pair (©O x ©O, diag ©O) and the set of finite characters on ©o-

G, I, Olshanski [3] initiated the study of a more general class of representations for the group ©O x ©O, To state it we need some notation.

Definition 6 Let n be a unitary representation of G acting on a Hilbert space H

CO

If U H©nTC is dense in H then n is called tame [1],

n=l

Definition 7 A unitary representation n of ©O x ©O is called admissible if its restriction to diag ©O is tame.

Obviously, a spherical representation of a pair (©O x ©O, diag ©O) is admissible,

©O x ©O

©O

gave a construction of examples admissible representations and a conjectural full classification. A, Yu, Okounkov proved Olshanski’s conjecture. We notice that a

©O x e

or e x ©O gives new examples of factor representations for ©O which are dilferent from discussed in Theorem 3 and Proposition 4,

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3. The results

Let n be a factor representation of ©O. We say that n is associated with an admissible irreducible representation n(2) of ©O x ©O if n arises as the restriction of n(2) to the subgroup ©O x e of the group ©O x ©O, Denote by FA the class of all such representations n of ©O,

Let B (H) be the algebra of all bounded operators in a Hilbert space H For a subset S of B (H), its commutant S1 consists of operators T G B (H) such that AT = TA for all A G S, Denote S" = (S')'.

Definition 8 Unitary representations n1 and n2 of a group G are called quasiequivalent, if there exists isomorphism a : n1 (G)" ^ n2 (G)" such that a (n1 (g)) = n2 (g) gGG

P(G) G

p(e) = 1 p G P(G)

GNS-representation is a factor representation. Let PF(G) be the set of all

indecomposable functions from P(G). Let M* stand for the space of all a-weakly continuous functionals on a W*-algebra M. The next important statement is well known.

Proposition 9 Let n be a factor representation of a group G and let u" be a state from n(G)*. Denote u(g) = u"(n(g)). Then u G PF(G) and n is quasiequivalent to the GNS-representation of G.

For a C^^^^bra M denote by Aut M the group of its automorphisms.

Definition 10 Let H be a subgroup of Aut M, A state p on M is called H-central if p(h(m)) = p(m) for all h G H and m G M.

For a unitary u G M define Adu G Aut M by (Adu)(m) = um«*, m G M.

Let n ^e a factor-representation of ©O Define the central depth cd(n) of n as the minimal number n G N U {0} for that there exists an Ad n (©nO)-central state

p G n (©o)*-

Remark 1. If cd(n) = 0 then ^ ^ ^^^^^^ratation of the type IIi,

Proposition 11 If factor representations n1 and n2 of ©O are quasiequivalent, then cd (n1) = cd (n2).

The next statement follows immediately from Definition 7,

Proposition 12 Let n(2) be an admissible irreducible representation of ©O x ©O, and let n1 and n2 be its restrictions to ©O x e and e x ©O respectively. Then n1 and n2 are factor representations with a finite central depth.

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We state here the next important result.

Theorem 13 Let n be a factor-representation of ©O with n = cd(n) < ro, and let u G n (©O)'l be an Ad n (©nO)-central -state. For m G n (©O)'' put

Let E denote the support of u and Eg = Ad n (g) (E). Then

i) for each pair (g, h) G ©O x ©O, we have Eg = Eh or Eg^Eh;

ii) {g G ©o : Eg = E} = {g G ©o : g(i) ^ i for all i ^ n} = ©n ■ ©no/

Theorem 14 The representation nXa^ of ©O, defined in the same manner as in Proposition 4, is a factor representation.

§ 4. Quasiequivalence in the case aj + Y1

fa <1

Here we have the next surprising result.

Theorem 15 If (aj + faj) < 1, then for any partition X the representation

n\a/3 is quasiequivalent to Thoma’s representation nap. In particular, the w*-algebra nXap (©o)” is a IIi-factor.

Let Y be the set of all Young diagrams. Denote bv A1 the subset of Thoma’s parameters (a, fa) such that J2aj + faj = 1-

Theorem 16 If (X, (a, fa)) and (^, (j, 5)) belonging to Y x A1 do not coincide, then the representations n\ap and n^Ys are not quasiequivalent.

Corollary 17 The central depth of the factor representation n\a/3 is equal to |X|.

iii) the algebra En (©O)'' E is a finite factor.

5. The case J^aj + faj = 1

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§ 6. Restriction of the normal semifinite traces to C* (©ro)

We suppose that J2ai + fai = 1 and |X| ^ 1. Then, bv Theorems 14 and 16,

i i

w*-algebra nXap (©O)'' is a semifinite factor of type IO or IIO, We notice that nXap has the type IO if and only if the total amount of numbers in the collection a U fa is equal to one. In this case the representation nXap is tame [1], Further we assume that nXap has tvpe IIO,

F

factor of type IIO. Let Tr be a normal semifinite trace on F,

Since for any {X a fa} the factors nXap (©O)'' are isomorphic to F, we can assume nXali (©o)'' = F.

Theorem 18 Suppose that '^2ai + fai = 1 |X| ^ 1 and the set a U fa contains

ii

more than one elements. Then the following conditions are equivalent: i) there exists a self-adjoint projection p G C* (©O) such that

0 < Tr (nxa/3(a)) < ro;

aUfa

References

1, A, Lieberman, The structure of certain unitary representations of infinite symmetric group, Trans, Amer, Math, Soe,, 1972, vol. 164, 189-198,

2, A, M, Vershik and S, V, Kerov, Asymptotic theory of characters of the symmetric groups, Funct, Anal, and its Appl,, 1981, vol. 15, No. 4, 15-27.

(G, K)

infinite symmetric group S(ro), Algebra i Analiz, 1989, vol. 1, No. 4, 178-209. Engl, transl.: Leningrad Math. J,, 1990, vol. 1, No. 4, 983-1014.

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