CC BY
30
MSG 20C30, 20C32, 20C15
Indecomposable characters of the group of rational rearrangements of the segment 1
© E. E. Goryachko
S.-Petersburg State University, S.-Petersburg, Russia
Let R be the group of rational rearrangements of the half-interval [0,1). We present a countable family of characters of the group R. These characters are indecomposable
Keywords: symmetric groups, measure spaces, automorphisms, inductive limits, characters
§ 1. Introduction
Consider the half-interval [0,1) (below it is called “the segment”) equipped with the Lebesgue measure and all automorphisms of this measure space such that the action of each of them is a “rearrangement” of a finite number of half-intervals with rational endpoints that form a partition of [0,1),
The formal definition of these automorphisms is as follows.
Definition 11 A bijeetion g: [0,1) ^ [0,1) is called a rational rearrangement of the segment if there exist a number n E N a permutation u E Sn, and a partition |[xo,xi),..., [xn-1,xn)} of [0,1) into n half-intervals with rational endpoints such, that g(x) = x — xi + xu(i) for every i E {0,... ,n — 1} = n and x E [x^xi+1) (here and below the group Sn is defined as the group of all bijections of the set n).
It is easy to show that the set of all rational rearrangements is a dense subgroup in the automorphism group of the segment as a measure space; denote this subgroup by R, This group can also be described algebraically in terms of the symmetric groups and the operations on them defined below.
Definition 12 The direct product u n v E S1m of permutations u E Si and v E Sm (where l,m E N is the permutation defined for any z E lm by the formula
u n v(z) = mu(z div m) + v(z mod m).
Definition 13 The periodic embedding of the group Si to the group Sim is the map that sends any permutation u E Si to the permutation u n idm E Stm.
1Partially supported by the grant NSh-2460.2008.1 of the President of Russian Federation for support of leading scientific schools and by the grant RFBR 08-01-00379-a
1646
Sn n E N
N
bv the divisibility relation,
R
Sn n E N
Sn R u E Sn
goes to the rearrangement g E R that moves the half-intervals [k/n, (k + 1)/n), k = 0,1,..., n — 1, according to the aetion of u; hence
. . u( I xn I) + Ixn) r
g(x)= (L J) { } , x E [0,1). (1)
n
Any partition of [0,1) into half-intervals with rational endpoints can be refined to
R
obtained as above. Finally, it is easy to check that the constructed embeddings are
Sn n E N
the proof, □
R
and, in particular, its indecomposable characters. This group is an inductive limit of symmetric groups, so we can apply to it the general methods of the representation theory of inductive limits of finite groups, which are stated in the papers [6], [7]; these papers are fundamental for us. The analogous approach is possible for other examples of groups of the same type, such as the infinite symmetric group S^ (see [4], [5], [7]) and various groups of matrices over countable fields of finite characteristic
R
first started in the paper [1], This paper also contains detailed proofs of certain statements used in the present renewed and abridged account.
The author is grateful to Professor Anatoly Vershik for setting the problem and useful discussions.
§ 2. Preliminaries on characters and the K0-functor
In this section, we formulate some general definitions and facts related to the representation theory of groups that are inductive limits of finite groups; then we
R
Let a group G be an inductive limit of finite groups Gi; i E I, where I is an at most countable directed set. Under this general assumption, we can define the following main objects of our study,
GG
G
G
1647
Definition 15 The K0-/uncior K0(G) of the group G is the ordered Abelian group with order identity formed by all equivalence classes of finitely generated projective
G
K0 K0
G
K0(G)
of Abelian groups between K0(G) and C, is a trace on the K0-funetor of the group G if it is nonnegative and equal to 1 at the order identity of K0(G) (more precisely, it is a finite trace, but we do not consider other traces),
G
on K0(G) are Choquet simpliees in the topological vector space of complex-valued central functions on G and complex-valued homomorphisms on K0(G), respectively. The Choquet boundaries of these simpliees consist of their indecomposable points, i, e,, of indecomposable characters and indecomposable traces (recall that a point of a Choquet simplex is called indecomposable if it cannot be decomposed into a nontrivial convex combination of other points of the simplex). Denote by exX(G) the
G
of the set X(G) of all characters of G),
Further, as shown in [1], there exists an isomorphism of topological vector spaces between the mentioned spaces that also establishes a homeomorphism between the sets of indecomposable characters of G and indecomposable traces on K0(G), so the problems of describing these sets are equivalent,
G
given in terms of the elementary representation theory. Namely, the indecomposable characters of a finite group are exactly all normalized (divided by the dimension) traces of its irreducible representations.
The general idea of the representation theory of inductive limits of finite groups is to approximate different objects related to a limit group by similar objects related to its finite subgroups,
K0(G)
category of ordered Abelian groups with order identities) of the groups K0(Gi), iEI
operation of induction of representations.
For G = R, this fact implies that K0(R) is isomorphic to the quotient of the group ®neN K0(Sn) by its subgroup generated by the elements
n — ind|^m(n) foralll,m E N and n E K0(Si) (2)
(here we assume that Si is periodically embedded into Sim), The set exX (R), which is
exX(Sn) n E
N
characters, which will be stated below.
1648
§ 3. The structure of a Riesz ring on K0(R) and a series of indecomposable characters of the group R
In this section, we define the structure of a Riesz ring on K0(R), Using it, we formulate a criterion for indeeomposability of traces on K0(R) and describe a series
R
To define a structure of a Riesz ring on K0(R), we need to introduce on it a nonnegative multiplication such that the identity element with respect to this multiplication is the order identity of K0(R); for this we will use the following operation on representations of symmetric groups.
Definition 17 The exterior product n M p E K0(Sim) of representations n E K0(Si) and p E K0(Sm), which corresponds to the operation of the direct product (1) of permutations, is the representation indS^^ (n ® p) (here we assume that the group Si x Sm is embedded into the group Sim using the operation n).
It is easy to see that the extension by linearity of the operation M to the Abelian n N K0 (S n)
unital ring. Further, as we already know, the group K0(R) is isomorphic to the quotient of the group ®neN K0(Sn) by its subgroup generated by the elements (2), From [1] it follows that this subgroup is an ideal of the above-mentioned ring. Hence the operation M determines an associative and commutative multiplication on the quotient of the ring ®neN K0(Sn) by this ideal, i. e,, on K0(R), This multiplication defines the required structure of a Riesz ring on K0(R), Further, there is a general criterion for indeeomposability of traces on a Riesz ring (see [7, Sec, 3,1]); below we formulate it in the case of K0 (R),
Theorem 2 Let t be a trace on K0(R). Then t is indecomposable if and only if t is a complex-valued character of the ring K0(R), i. e., a homomorphism of unital rings between K0(R) and C.
Using the above-mentioned homeomorphism between the sets of indecomposable characters of R and indecomposable traces on K0(R), one can reformulate this
R
the following bijeetion. Let x be a character of R; define a function x on the set UneNSn as follows. For u E Sn, set X(u) = x(g) where g E R acts on the segment by formula (1), Obviously, for all l,m E N and u E Si we havh Xls is a character of Si; and X(u) = x(u n idm), Conversely, a function on the set |JneN Sn that has these
R
Corollary 1 Let x be a charader of R. Then x is indecomposable if and only if X is a 'multiplicative function with respect to the operation n, i. eX(u n v) = X(u) X(v) for all l, m E N u E Si; v E Sm.
1649
R
indecomposability.
Definition 18 The natural character xnat of the group R is defined for any rational rearrangement g E R by the formula xnat(g) = M{x E [0,1) | g(x) = x}),
xnat R
all k E N U {0,
Proof. Clearly, xnat = 1 is the trivial character of R, and xi^t = limk^^ xnat = ^id is the character of its regular representation. Now, for any k, n E N, let us compute the value at u E Sn of the function xnat corresponding to xnat (here and below Fix(u)
u
From the obtained formula we see that the restriction of the function xLt to the group Sn is the normalized character of the fcth tensor power of the natural representation of the group Sn, hence it is a character of this group, and, therefore, the function xLt is a character of the group R,
It can easily be proved that |Fix(u n v)| = |Fix(u)| |Fix(v)| for all 1,m G N,
u G S^^d v G Sm, thus the function xLt, is multiplicative, and, therefore, the character xLt is indecomposable. □
R
obtain some general properties of indecomposable characters of this group, see [1]. Unfortunately, it seems too difficult to obtain their complete description in that way, so that below we will consider quite another approach.
§ 4. The ergodic method for computing indecomposable characters
In this section, we describe the ergodic method for computing indecomposable characters of an inductive limit of finite groups in the case where this limit is the R
Let us introduce the main notion of the ergodic method (also applied to the R
1650
Definition 19 A net (Xn)neN G üneN exX(Sn) of indecomposable characters of the symmetric groups is called weakly convergent with respect to the periodic embeddings if there exists a complex-valued function ^ on the set |JneN Sn such that for all l G N and u G Si the net (xim (u n idm)) N tends to ^(u); the function ^ is called the limit of (xn)neN- Here N is ordered by the divisibility relation.
Clearly, if ^ is the limit of the net (xn)neN then ^(u) = ^(u n idm) for all
l,m G N and u G Sl; so that the function ^ gives rise to a well-defined function on the group R, Further, we exactly formulate how the set exX (R) can be described in terms of the finite sets exX (Sn), n G N.
R
convergent net of indecomposable characters of the symmetric groups. Then this
Rx
R
of indecomposable characters of the symmetric groups that tends to the function x
x
Proof. This statement is a special case of a general theorem on approximation of indecomposable characters. This theorem is proved for analogous cases in [5, § 4]
□
At the present time, our main problem is to prove, using the ergodic method,
R
Xnat; below we show that this conjecture is a corollary of the following estimation,
nGN
group Sn is parameterized by the set Yn of Young diagrams with n cells. If À G Yn, then denote by xA the normalized character of the irreducible representation of Sn corresponding to À. Denote by ri(À) and ci(À) the number of cells in the first row À
Conjecture. For all l G N and u G Sl there exists a constant C depending only on u
for all m E N and A E with r^A) ^ Oi(A),
Corollary 2 A net (xAn )n€N E nneN exX(Sn) of indecomposable characters of the symmetric groups is weakly convergent if and only if the net
m
C
(3)
(n - max{ri (An) , Ci (An) })neN
has a limit k E N U {0, to}; in the case where the limit of the net (xAn)neN exists, it determines the character xnat on the group R.
1651
Proof. Using the operation of transposition of diagrams, it is easy to see that for all l E N and u E Aj the conjectured estimation (3) implies the similar estimation (3) where r1(A) has to be replaced by max{r1(A), c1(A)} for all m E N and A E Y1m, Obviously, from this we see that the existence of limm xXlm (u n idm) is equivalent to the existence of limTO(/m — max(r1(A1m), c1(A1m)^; finally, if u E Si \ A1; then replace it by u n id2 E A21, □
References
1, E. E. Goryachko. The K0-functor and characters of the group of rational rearrangements of the segment, J. Math, Sci, (N, Y,), 2009, vol. 158, No, 6, 838844.
2, K. P. Kokhas. The classification of complex factor-representations of the threedimensional Heisenberg group over a countable group field of finite characteristic, J. Math. Sci. (N. Y.), 2004, vol. 121, No. 3, 2371-2379.
3, H.-L. Skudlarek. Die unzerlegbaren Charaktere einiger diskreter Gruppen, Math. Ann., 1976, vol. 223, 213-231.
4, A. M. Vershik, S. V. Kerov, Characters and factor-representations of the infinite symmetric group, Soviet Math, Dokl,, 1981, vol. 23, No, 2, 389-392,
5, A, M, Vershik, S, V, Kerov, Asymptotic theory of characters of the symmetric group, Funct, Anal, Appl,, 1981, vol. 15, No, 4, 246-255,
6, A, M, Vershik, S, V, Kerov, Locally semisimple algebras. Combinatorial theory
K0
7, A, M, Vershik, S, V, Kerov, The Grothendieck group of the infinite symmetric
K0
AF-algebras), in: Representation of Lie Groups and Related Topics (Adv. Stud, Contemp, Math., 7). Gordon and Breach, New York, 1990, 39-117.
1652
