# Частично центральные состояния на бесконечной симметрической группе

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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 & gt- 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-
© 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
& lt-^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^& gt-:
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& gt- 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* (& amp-&-,) 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 & amp-n. Let (а, в) be Thoma parameters, denote by пав the GNS-representation of & amp-n,^ corresponding to the finite character хав- F°r g є Єп and h є & amp-n,™ pu t Tл, а в (gh) = ^(g) ® Пав (h) • Let both sgqugїісєз, а and в be finite and
3 & lt-^k + 3 вk = 1, kk
then we have the following two properties:
(i) the representation ПЛав of induced by the representationав of the subgroup & amp-n ¦ & amp-n ^ is a ІІте-factor representation of
(ii) the faithful semi-finite trace т on factor ПЛав (& amp-<-х)"- defines, by the formula т (ПЛав (A)) = тЛав (A), where A є C* (& amp-^) semifinite trace тЛав on C* (& amp-x).
The converse of this statement is true. Namely, for any semifinite trace x on C* (& amp-^) 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 (c)TO x (c)TO
The theory of finite characters on (c)O is a special case of a general theory of admissible representations of the group (c)O x (c)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 (kgk2) = 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 ((c)O x (c)O, diag (c)O) and the set of finite characters on (c)o-
G, I, Olshanski [3] initiated the study of a more general class of representations for the group (c)O x (c)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 (c)nTC is dense in H then n is called tame [1],
n=l
Definition 7 A unitary representation n of (c)O x (c)O is called admissible if its restriction to diag (c)O is tame.
Obviously, a spherical representation of a pair ((c)O x (c)O, diag (c)O) is admissible,
(c)O x (c)O
(c)O
gave a construction of examples admissible representations and a conjectural full classification. A, Yu, Okounkov proved Olshanski’s conjecture. We notice that a
(c)O x e
or e x (c)O gives new examples of factor representations for (c)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 (c)O. We say that n is associated with an admissible irreducible representation n (2) of (c)O x (c)O if n arises as the restriction of n (2) to the subgroup (c)O x e of the group (c)O x (c)O, Denote by FA the class of all such representations n of (c)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& quot- = (S'-)'-.
Definition 8 Unitary representations n1 and n2 of a group G are called quasiequivalent, if there exists isomorphism a: n1 (G)& quot- ^ n2 (G)& quot- 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& quot- be a state from n (G)*. Denote u (g) = u& quot-(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 ne a factor-representation of (c)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 ((c)nO)-central state
p G n ((c)o)*-
Remark 1. If cd (n) = 0 then ^ ^ ^^^^^^ratation of the type IIi,
Proposition 11 If factor representations n1 and n2 of (c)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 (c)O x (c)O, and let n1 and n2 be its restrictions to (c)O x e and e x (c)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 (c)O with n = cd (n) & lt- ro, and let u G n ((c)O)'-l be an Ad n ((c)nO)-central -state. For m G n ((c)O)'-'- put
Let E denote the support of u and Eg = Ad n (g) (E). Then
i) for each pair (g, h) G (c)O x (c)O, we have Eg = Eh or Eg^Eh-
ii) {g G (c)o: Eg = E} = {g G (c)o: g (i) ^ i for all i ^ n} = (c)n ¦ (c)no/
Theorem 14 The representation nXa^ of (c)O, defined in the same manner as in Proposition 4, is a factor representation.
§ 4. Quasiequivalence in the case aj + Y1
fa & lt-1
Here we have the next surprising result.
Theorem 15 If (aj + faj) & lt- 1, then for any partition X the representation
na/3 is quasiequivalent to Thoma’s representation nap. In particular, the w*-algebra nXap ((c)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 nap and n^Ys are not quasiequivalent.
Corollary 17 The central depth of the factor representation na/3 is equal to |X|.
iii) the algebra En ((c)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* ((c)ro)
We suppose that J2ai + fai = 1 and |X| ^ 1. Then, bv Theorems 14 and 16,
i i
w*-algebra nXap ((c)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 ((c)O)'-'- are isomorphic to F, we can assume nXali ((c)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* ((c)O) such that
0 & lt- Tr (nxa/3(a)) & lt- 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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