Degenerate affine Hecke algebras and centralizer construction for the symmetric groups

A. I. Molev and G. I. Olshanski

Abstract

In our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We study the ‘stable’ properties of the centralizers of S(n − m) in the group algebra ℂ[S(n)] as n → ∞ with m fixed. We construct a limit centralizer algebra A and describe its algebraic structure. The algebra A turns out to be closely related with the degenerate affine Hecke algebras. We also show that the so-called tame representations of S(∞) yield a class of natural A-modules.

School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexm@maths.usyd.edu.au

Dobrushin Mathematics Laboratory Institute for Problems of Information Transmission Bolshoy Karetny 19, Moscow 101447, GSP-4, Russia olsh@iitp.ru, olsh@glasnet.ru

1 Introduction

The centralizer construction proposed in [29] shows that certain “quantum” algebras can be obtained as projective limits of centralizers in classical enveloping algebras. This approach has been applied to the series of matrix Lie algebras to construct the quantum algebras called the Yangians and twisted Yangians which are originally defined as certain deformations of enveloping algebras; cf. [5]. The type A case is treated in [29], [30] and extended to the B, C, D types in [32] and [18]. A modified version of the A type construction is given in [17].

In the A type case, one considers the centralizer A_{m}(n) of the subalgebra 𝔤𝔩(n−m) in the enveloping algebra U(𝔤𝔩(n)). It turns out that for each pair n > m there is a natural algebra homomorphism A_{m}(n) → A_{m}(n − 1) so that one can define a projective limit algebra A_{m} by using the chain of homomorphisms

⋯ ⟶ A_{m}(n)⟶A_{m}(n − 1) ⟶ ⋯ ⟶ A_{m}(m + 1)⟶A_{m}(m).  (1.1)

The algebra A_{m} has a large center A_{0} which is isomorphic to the algebra of shifted symmetric functions Λ^{∗} (see [24]) and one has an isomorphism

A_{m} ≃ A_{0} ⊗ Y_{m},  (1.2)

where Y_{m} is the Yangian for the Lie algebra 𝔤𝔩(m); [30, Theorem 2.1.15]. In particular, for each n ≥ m there is a natural homomorphism

Y_{m} → A_{m}(n).  (1.3)

This result was used in [2, 22, 17] to study the class of representations of Y_{m} which naturally arises from this construction. A similar result for the B, C, D types [32, 18] was used in [14, 15, 16] to construct weight bases of Gelfand–Tsetlin type for representations of the classical Lie algebras.

Our aim in the present paper is to extend these constructions to the series of the symmetric groups S(n). Denote by B_{m}(n) the centralizer of the subgroup S(n − m) in the group algebra ℂ[S(n)], where S(n − m) consists of the permutations which fix each of the indices 1, 2, …, m. It turns out that no natural analog of the chain (1.1) exists for the algebras B_{m}(n). Indeed, note that B_{n}(n) = B_{n−1}(n) = ℂ[S(n)] and so, by analogy with (1.1) we would have a homomorphism ℂ[S(n)] → ℂ[S(n − 1)] identical on ℂ[S(n − 1)]. But such a homomorphism does not exist for n > 4.

On the other hand, it was observed in [29] that an analog of (1.3) still exists: for any n ≥ m there is a homomorphism H_{m} → B_{m}(n), where H_{m} is the degenerate affine Hecke algebra introduced by Drinfeld [6] and Lusztig [12]. This fact was used by Okounkov and Vershik [25] to develop a new approach to the representation theory of the symmetric groups; see also earlier results by Cherednik [1].

This observation together with the semigroup method [28] allows one to expect that an analog of the centralizer construction for the symmetric group should exist, with the algebras ℂ[S(n)] replaced with the semigroup algebras A(n) = ℂ[Γ(n)], where Γ(n) is the semigroup of partial bijections of the set {1, …, n}. Alternatively, the elements of Γ(n) can be identified with (0, 1)-matrices which have at most one 1 in each row and column. The semigroups Γ(n) are studied in [28] and used to prove Lieberman’s classification theorem [11] for unitary representations of the complete infinite symmetric group.

Taking the centralizer A_{m}(n) of Γ(n − m) in A(n) instead of the algebras B_{m}(n) we do obtain an analog of the chain of homomorphisms (1.1) for the algebras A_{m}(n). The corresponding projective limit algebra A_{m} has a decomposition analogous to (1.2)

A_{m} ≃ A_{0} ⊗ H̃_{m},  (1.4)

where H̃_{m} is a ‘semigroup analog’ of the degenerate affine Hecke algebra H_{m}. The algebra H̃_{m} can be presented by generators and defining relations. Moreover, the algebra H_{m} is a homomorphic image of H̃_{m}. The subalgebra A_{0} ⊆ A_{m} is commutative and isomorphic to the algebra of shifted symmetric functions Λ^{∗}; see [24] for a detailed study of the algebra Λ^{∗}.

The mentioned above homomorphisms H_{m} → B_{m}(n) can be regarded as ‘retractions’ of the homomorphisms H̃_{m} → A_{m}(n) whose existence is provided with the centralizer construction for the symmetric groups.

Finally, the algebra A is defined as the inductive limit of A_{m} as m → ∞. We show that A naturally acts in the so-called tame representations of S(∞) and it can be regarded as the ‘true’ analog of the group algebra ℂ[S(∞)]. Indeed, contrary to the finite case of ℂ[S(n)], the algebra ℂ[S(∞)] has a trivial center while A contains a large center whose elements act by scalar operators in the tame representations of S(∞). Moreover, the central elements separate the irreducible tame representations.

Some other generalizations of the degenerate affine Hecke algebra H_{m} have been constructed by Nazarov [20, 21].

The paper is organized as follows. Section 2 is preliminary. Here we define tame representations of S(∞) and describe the properties of the semigroups Γ(n); most of these results are contained in [28]. In Section 3 we construct the algebras A_{m} as projective limits of the centralizers A_{m}(n). Section 4 describes the algebra A_{0} and establishes its isomorphism with the algebra of shifted symmetric functions Λ^{∗}. The main results are given in Section 5 where we investigate the structure of A_{m} and describe its relationship with the degenerate affine Hecke algebras.

The authors acknowledge the financial support of the Australian Research Council. G.O. was also supported by the Russian Foundation for Basic Research under grant 98-01-00303.

2 Tame representations and semigroups

Here we give constructions of the tame representations of the infinite symmetric group and describe the semigroups of partial bijections. The material of this section is based on [28] and [27, §2]. About applications of the semigroup method to representations of “big” groups, see Olshanski [31], Neretin [23].

2.1 Constructing tame representations

Let ℕ = {1, 2, … } and n ∈ ℕ. We denote by S(n) the group of permutations of the set ℕ_{n} := {1, …, n}. We also regard S(n) as the group of permutations of ℕ fixing n + 1, n + 2, …, and set

S(∞) = ⋃_{n≥1} S(n).

This is the group of all finite permutations of the set ℕ.

For any m ≤ n, we denote by S_{m}(n) the subgroup of permutations in S(n) fixing 1, …, m, and set

S_{m}(∞) = ⋃_{n≥m} S_{m}(n).

Note that the subgroups S(n) and S_{n}(∞) of S(∞) commute. This simple observation will play an important role later.

For a (unitary) representation T, we denote by H(T) its (Hilbert) space.

Let T be a unitary representation of the group S(∞). For n = 0, 1, …, denote by H_{n}(T) the subspace of S_{n}(∞)-invariant vectors in H(T). Since {S_{n}(∞)} is an descending chain of subgroups, {H_{n}(T)} is an ascending chain of subspaces. Let

H_{∞}(T) = ⋃_{n≥0} H_{n}(T).

Since S(n) and S_{n}(∞) commute, the subspace H_{n}(T) is invariant with respect to S(n), and so, H_{∞}(T) is invariant with respect to the whole group S(∞).

Definition 2.1 A unitary representation T of the group S(∞) is said to be tame if H_{∞}(T) is dense in H(T).

Note that for an irreducible representation T, this is equivalent to saying that H_{∞}(T) is nonzero.

Clearly, the trivial representation of S(∞) is tame, and another one–dimensional representation, s ↦ sgn s, is not tame. Less trivial examples follow.

Example 2.2 (i) Let H be the space l_{2} with its canonical basis e_{1}, e_{2}, … and let S(∞) operate in H by permuting the basis vectors. The representation H is tame. It is irreducible and for any n the subspace H_{n} is spanned by e_{1}, …, e_{n}.

(ii) The right (or left) regular representation of the group S(∞) in the Hilbert space l_{2}(S(∞)) is not tame.

For any n = 0, 1, 2, … and any partition λ ⊢ n, we will construct a tame representation T_{λ}. First, if n = 0 then λ = ∅ and T_{∅} is the one-dimensional trivial representation. Let now n ≥ 1 and let π_{λ} denote the irreducible representation of S(n) corresponding to λ. Then T_{λ} is defined as the induced representation

T_{λ} = Ind_{S(n)×S_{n}(∞)}^{S(∞)}(π_{λ} ⊗ 1)  (2.1)

where 1 stands for the trivial representation of S_{n}(∞). The representation T_{λ} can be realized as follows. Let Ω(n) denote the set of injective maps ω : ℕ_{n} → ℕ. Define a right action of S(∞) on Ω(n) by

ω · s = s^{−1} ∘ ω,  s ∈ S(∞),  (2.2)

and a left action of S(n) by

t · ω = ω ∘ t^{−1},  t ∈ S(n).  (2.3)

Note that these two actions commute. Consider the Hilbert space l_{2}(Ω(n), H(π_{λ})) of H(π_{λ})-valued square-integrable functions on Ω(n), and let H(n, λ) be its subspace formed by the functions f(ω) such that

f(t · ω) = π_{λ}(t)f(ω),  t ∈ S(n), ω ∈ Ω(n).  (2.4)

The action of T_{λ} in H(n, λ) is given by

(T_{λ}(s)f)(ω) = f(ω · s).  (2.5)

The space H(T_{λ}) may now be identified with H(n, λ). For any l ≥ n set

Ω(n, l) = {ω ∈ Ω(n) | ω(ℕ_{n}) ⊆ ℕ_{l}}  (2.6)

and note that

Ω(n) = ⋃_{l≥n} Ω(n, l)  (2.7)

Also, set

H′_{l}(T_{λ}) = {f ∈ H(n, λ) | supp f ⊆ Ω(n, l)},  (2.8)

where supp f = {ω ∈ Ω(n) | f(ω) ≠ 0}.

Proposition 2.3 For any n ∈ ℕ we have

H_{l}(T_{λ}) = \begin{cases}\{0\} & if l < n, \\ H′_{l}(T_{λ}) & if l ≥ n. \end{cases}  (2.9)

Proof. Since S_{l}(∞) acts trivially on Ω(n, l), we have H′_{l}(T_{λ}) ⊆ H_{l}(T_{λ}). Conversely, let f ∈ H_{l}(T_{λ}). Then the function ∥f(ω)∥^{2} is constant on any orbit of the subgroup S_{l}(∞) in Ω(n). Since the sum of the ∥f(ω)∥^{2} taken over ω ∈ Ω(n) must be finite, we have ∥f(ω)∥ = 0 unless the orbit containing ω is finite. But if ω ∉ Ω(n, l), then its orbit is clearly infinite, so that f(ω) = 0. This proves the opposite inclusion H_{l}(T_{λ}) ⊆ H′_{l}(T_{λ}).

Proposition 2.4 For any n ∈ ℕ and any λ ⊢ n, the representation T_{λ} of S(∞) is tame and irreducible.

Proof. By Proposition 2.3,

H_{∞}(T_{λ}) = ⋃_{l} H_{l}(T_{λ}) = ⋃_{l} H′_{l}(T_{λ}).  (2.10)

This is the subspace of finitely supported functions in H(T_{λ}) = H(n, λ) which is clearly dense. So, T_{λ} is tame.

The subspace H_{n}(T_{λ}) = H′_{n}(T_{λ}) is both cyclic in H(T_{λ}) and irreducible under the action of the subgroup S(n). This implies that T_{λ} is irreducible.

We shall identify any partition λ with its Young diagram; see e.g. [13]. We write |λ| = n if λ has n boxes. Given two diagrams λ and μ the notation μ ↗ λ will mean that μ can be obtained from λ by removing one box, i.e. μ ⊂ λ and |μ| = |λ| − 1. An infinite tableau is defined as an infinite chain of diagrams

τ = (λ^{(1)} ↗ λ^{(2)} ↗ …),  |λ^{(n)}| = n.  (2.11)

Two infinite tableaux will be called equivalent if the corresponding chains of diagrams differ in a finite number of places only.

Given an infinite tableau τ, we may construct an inductive limit unitary representation Π(τ) of the group S(∞) as follows. By the branching rule for the symmetric groups (see e.g. [8], [25]), for any n the representation π_{λ^{(n)}} occurs in the decomposition of π_{λ^{(n+1)}} ↓ S(n) with multiplicity one. Hence there is an infinite chain of embeddings

π_{λ^{(1)}} ↪ π_{λ^{(2)}} ↪ …  (2.12)

which are defined uniquely up to scalar multiples. So we may set

Π(τ) = lim ind π_{λ^{(n)}},  n → ∞.  (2.13)

One can show that any Π(τ) is irreducible (cf. [26, Theorem 2.1] and [27, §2.7]), and that Π(τ) and Π(τ′) are isomorphic if and only if τ and τ′ are equivalent.

This construction provides a large class of pairwise non-equivalent irreducible representations of the group S(∞). We will be only interested in some special representations of this class. Let λ = (λ_{1}, …, λ_{l}) be an arbitrary diagram. Consider an infinite tableau τ = (λ^{(i)}) such that

λ^{(i)} = (i − |λ|, λ_{1}, …, λ_{l}) for i ≥ |λ| + λ_{1},  (2.14)

and set Π_{λ} = Π(τ). The representation Π_{λ} is well defined since the equivalence class of Π_{λ} does not depend on the choice of λ^{(i)} for small values of i.

Proposition 2.5 The representations T_{λ} and Π_{λ} are isomorphic for any λ.

Proof. For any l ≥ n + λ_{1}, the natural representation of S(l) in the space H_{l}(T_{λ}) is isomorphic to the induced representation

Ind_{S(n)×S(l−n)}^{S(l)}(π_{λ} ⊗ 1)  (2.15)

where S(l − n) is identified with S_{n}(l), and 1 stands for the trivial representation of S(l − n). This follows immediately from (2.9). It is well known [8] that the representation (2.15) is multiplicity free and that its spectrum consists of the representations π_{μ} such that μ ⊢ l and

μ_{i+1} ≤ λ_{i} ≤ μ_{i},  i ≥ 1.  (2.16)

It follows from (2.16) that π_{λ^{(l)}} occurs in the decomposition of (2.15). Let H_{l}^{0}(T_{λ}) denote the corresponding subspace of H_{l}(T_{λ}). It remains to prove that H_{m}^{0}(T_{λ}) is contained in H_{l}^{0}(T_{λ}) for any m < l provided that m is large enough. This follows from the fact that

π_{λ^{(m)}} ⊂ π_{λ^{(l)}}|_{S(m)},  (2.17)

and π_{λ^{(m)}} ⊄ π_{μ}|_{S(m)} if μ satisfies (2.16) and μ ≠ λ^{(l)}. Indeed, property (2.17) follows from the definition of the diagrams λ^{(l)} for large l and the branching rule. Finally, note that there exists i such that λ_{i} > μ_{i+1} (otherwise μ = λ^{(l)}). Applying the branching rule again we complete the proof.

2.2 The semigroup method

Definition 2.6 Let X be a set.

(i) A partial bijection of X is a bijection γ : D → R between two (possibly empty) subsets of X. The subset D ⊆ X is called the domain of γ and denoted by dom γ. The subset R ⊆ X is called the range of γ and denoted by range γ. If x ∈ X belongs to dom γ, then we will say that γ is defined on x. The set of partial bijections of X is denoted by Γ(X).

(ii) Given γ ∈ Γ(X), we define γ^{∗} ∈ Γ(X) as the inverse bijection γ^{−1} : range γ → dom γ, so that dom γ^{∗} = range γ and range γ^{∗} = dom γ. The mapping γ → γ^{∗} is involutive: (γ^{∗})^{∗} = γ.

(iii) Given γ_{1}, γ_{2} ∈ Γ(X) with dom γ_{i} = D_{i} and range γ_{i} = R_{i}, i = 1, 2, their product γ_{1}γ_{2} is a partial bijection on X with D = dom γ_{1}γ_{2} = γ_{2}^{−1}(D_{1} ∩ R_{2}) and R = range γ_{1}γ_{2} = γ_{1}(D_{1} ∩ R_{2}):

γ_{1}γ_{2} = (γ_{1}|_{γ_{2}(D)}) ∘ (γ_{2}|_{D}).  (2.18)

That is, γ_{1}γ_{2} is defined on x ∈ X if and only if γ_{2} is defined on x and γ_{1} is defined on γ_{2}(x); then (γ_{1}γ_{2})(x) = γ_{1}(γ_{2}(x)).

Any γ ∈ Γ(X) may be regarded as a relation on X. The product defined above is the product of relations; see [3]. With this product Γ(X) becomes a semigroup, called the semigroup of partial bijections of X.

The involution γ → γ^{∗} is an anti-homomorphism of Γ(X), so that Γ(X) is an involutive semigroup.

The semigroup Γ(X) possesses the unity 1, which is the identity bijection X → X, and the zero 0, which is the (unique) bijection of the empty subset of X onto itself. Note that

1 γ = γ 1 = γ, 0 γ = γ 0 = 0,  for any γ ∈ Γ(X).  (2.19)

For any subset Y ⊆ X, let 1_{Y} ∈ Γ(X) denote the identity bijection Y → Y. In particular, 1_{X} = 1 and 1_{∅} = 0. Then 1_{Y} is a self-adjoint idempotent, i.e.,

(1_{Y})^{∗} = 1_{Y} , (1_{Y})^{2} = 1_{Y} .  (2.20)

Moreover, all idempotents of this type are pairwise commuting. For any γ ∈ Γ(X) we obviously have

γ^{∗} γ = 1_{dom γ}, γ γ^{∗} = 1_{range γ}.  (2.21)

The subset of those γ ∈ Γ(X) for which dom γ = range γ = X is clearly identified with the group S(X) of all permutations of the set X.

Remark. The semigroup Γ(X) is a model example of an inverse semigroup (each of its elements has an inverse); see [3]. The class of inverse semigroups is very closed to that of groups, and the role of the semigroups of partial bijections Γ(X) is quite similar to that of the symmetric groups S(X). In particular, any inverse semigroup is isomorphic to an involutive subsemigroup of some Γ(X): this is an analog of Cayley theorem; see [3].

There is a convenient realization of partial bijections by (0,1)-matrices, i.e., by the matrices whose entries are 0 or 1. A (0,1)-matrix is said to be monomial if any of its rows or columns contains at most one 1. Given a set X, we will consider (0,1)-matrices whose rows and columns are labeled by the points of X. Then to any γ ∈ Γ(X), we assign a monomial (0,1)-matrix [γ_{xy}] as follows: for x, y ∈ X

γ_{xy} = \begin{cases}1 & if y ∈ dom γ and γ(y) = x, \\ 0 & otherwise. \end{cases}  (2.22)

In particular, the unity 1 ∈ Γ(X) and the zero 0 ∈ Γ(X) are represented by the unit and the zero matrices, respectively. We thus obtain an isomorphism between the semigroup Γ(X) and the semigroup of monomial matrices over X with the usual matrix multiplication. Note that in this matrix realization, the involution γ ↦ γ^{∗} is represented by the matrix transposition.

Definition 2.7 For Y ⊆ X we define three mappings as follows:

(i) The canonical projection θ : Γ(X) → Γ(Y). Let γ ∈ Γ(X). Then θ(γ) is defined at y ∈ Y if γ is defined at y and γ(y) ∈ Y. The image of y with respect to θ(γ) is γ(y). In matrix terms: the matrix of θ(γ) is obtained from that of γ by striking the rows and columns corresponding to points of X \ Y.

(ii) The canonical embedding ϕ : Γ(Y) ↪ Γ(X). Let γ ∈ Γ(Y). Then ϕ(γ) is defined at x ∈ X if and only if x ∈ dom γ ⊆ Y or x ∈ X \ Y. In the former case ϕ(γ) sends x to γ(x), and in the latter case it fixes x. In matrix terms, for x, y ∈ X,

ϕ(γ)_{xy} = \begin{cases}γ_{xy} & if x, y ∈ Y , \\ δ_{xy} & otherwise. \end{cases}  (2.23)

(iii) The 0-embedding ψ : Γ(Y) ↪ Γ(X). Let γ ∈ Γ(Y). Then ψ(γ) is obtained by regarding dom γ and range γ as subsets of X. In matrix terms, for x, y ∈ X,

ψ(γ)_{xy} = \begin{cases}γ_{xy} & if x, y ∈ Y, \\ 0 & otherwise. \end{cases}  (2.24)

For a positive integer n we shall denote by Γ(n) the semigroup Γ(ℕ_{n}). Given a tame representation T of S(∞) we now construct a representation T_{n} of the semigroup Γ(n). Regarding S(∞) as the group of infinite monomial matrices which have only a finite number of 1’s off the diagonal, define the map θ^{(n)} : S(∞) → Γ(n) which assigns to each infinite matrix its upper left n × n submatrix. It is easy to check that the map θ^{(n)} is surjective and it induces a bijection between the set of double cosets S_{n}(∞)\S(∞)/S_{n}(∞) and Γ(n); see [28].

Let T be a tame representation of S(∞). Denote by P_{n} the orthogonal projection P_{n} : H(T) → H_{n}(T). Suppose that n is so large that H_{n}(T) ≠ {0}. For any σ_{1}, σ_{2} ∈ S(∞) we have

θ^{(n)}(σ_{1}) = θ^{(n)}(σ_{2}) ⇒ P_{n}T (σ_{1})P_{n} = P_{n}T (σ_{2})P_{n}.  (2.25)

Therefore there exists a unique map T_{n} : Γ(n) → End H_{n}(T) such that

T_{n}(θ^{(n)}(σ)) = P_{n}T (σ)|_{H_{n}(T)},  σ ∈ S(∞).  (2.26)

Proposition 2.8 The map T_{n} = T_{n}(T) defined by (2.26) is a representation of the semigroup Γ(n) in H_{n}(T).

Proof. We give a sketch of the proof. The details and one more proof can be found in [28]. We show first that the tame representation T of S(∞) can be extended to a representation T of the semigroup Γ(∞) of partial bijections of the set ℕ.

Further, we prove that P_{n} coincides with the operator T (1_{n}) where 1_{n} is the identity bijection of the subset ℕ_{n}.

Finally, consider the 0-embedding ψ : Γ(n) → Γ(∞); see (2.24). We have for any γ ∈ Γ(n)

T_{n}(γ) = P_{n}T (ψ(γ))P_{n} = T (1_{n})T (ψ(γ))T (1_{n}) = T (1_{n}ψ(γ)1_{n}) = T (ψ(γ)). (2.27)

This proves the multiplicativity of T_{n}.

Consider the tame representations T_{λ} of S(∞) constructed in the previous section. Proposition 2.8 yields a representation T_{n}(λ) := T_{n}(T_{λ}) of Γ(n) provided that H_{n}(T_{λ}) ≠ 0, i.e., n ≥ |λ|.

We now outline the proof of the classification theorem for representations of Γ(n); see [28].

Theorem 2.9 The representations T_{n}(λ) where λ is a partition with |λ| ≤ n exhaust all irreducible representations of Γ(n).

Proof. Let T be a representation of Γ(n). Then for any m ≤ n the operator T (1_{m}) is a projection. Denote its image by H_{m}(T ). We let m(T ) denote the minimum value of m such that H_{m}(T ) ≠ {0}.

Further, if T is irreducible then one shows that dim T < ∞. If m = m(T ) then the subspace H_{m}(T ) is invariant under S(m) and irreducible. So, H_{m}(T ) corresponds to a partition λ with |λ| = m. A standard argument proves that T is uniquely determined by λ.

Conversely, given a partition λ with |λ| = m ≤ n one uses the following argument to construct an irreducible representation T of Γ(n) such that m = m(T ) and the representation of S(m) in H_{m}(T ) corresponds to λ.

Denote by Ω(m, n) the set of injective maps ω : ℕ_{m} → ℕ_{n}. We define H(T ) to be the space of functions f : Ω(m, n) → H(π_{λ}) such that

f(t · ω) = π_{λ}(t)f(ω),  t ∈ S(m),  ω ∈ Ω(m, n);  (2.28)

see (2.3). The action of γ ∈ Γ(n) is given by

(T (γ)f)(ω) = \begin{cases}f(γ^{∗}ω) & if ω = (ω_{1}, …, ω_{m}) ⊆ dom γ^{∗}, \\ 0 & otherwise. \end{cases}  (2.29)

One easily checks that T is a representation of Γ(n). Moreover, it is isomorphic to the representation T_{n}(λ).

Note that the representation of Γ(n) corresponding to m = 0 and the empty diagram is the trivial representation sending all elements of Γ(n) to 1.

Theorem 2.9 leads to the following result.

Theorem 2.10 Let λ range over the set of all Young diagrams including the empty diagram. The representations T_{λ} constructed in Section 2.1 exaust, within equivalence, all the irreducible tame representations of the group S(∞). Moreover, any tame representation of S(∞) can be decomposed into a direct sum of irreducible tame representations.

Proof. See Theorem 6.7 and §7.2 in [28].

This is equivalent to Lieberman’s theorem [11] concerning continuous unitary representations of the complete infinite symmetric group (this group consists of all permutations of the set ℕ); see [28, §7].

3 Centralizer construction

We shall denote by θ_{n} the canonical projection Γ(n) → Γ(n − 1); see Definition 2.7. So, if γ ∈ Γ(n) then θ_{n}(γ) is the upper left corner of γ of order n − 1. Here the elements of Γ(n) and Γ(n − 1) are regarded as (0, 1)-matrices of order n and n − 1, respectively.

We set A(n) = ℂ[Γ(n)], the semigroup algebra of Γ(n). The canonical embedding Γ(n−1) ↪ Γ(n) is extended to an embedding A(n−1) ↪ A(n) by linearity. Further, set A(∞) = ℂ[Γ(∞)] = ⋃_{n≥1} A(n), the semigroup algebra of Γ(∞).

For each i = 1, …, n denote by ε_{i} the diagonal n × n-matrix whose ii-th entry is 0 and all other diagonal entries are equal to 1. The corresponding element of Γ(n) is the identity bijection of the set ℕ_{n} \ {i}. The algebra A(n) is obviously generated by S(n) and the elements ε_{i}, i = 1, …, n. We have for any i and any σ ∈ S(n):

σ ε_{i} σ^{−1} = ε_{σ(i)}.  (3.1)

Proposition 3.1 The algebra A(n) is isomorphic to the abstract algebra with generators s_{1}, …, s_{n−1}, ε_{1}, …, ε_{n} and the defining relations

s_{k}^{2} = 1,  s_{k} s_{k+1} s_{k} = s_{k+1} s_{k} s_{k+1},  s_{k} s_{l} = s_{l} s_{k}, |k − l| > 1, (3.2)
ε_{k}^{2} = ε_{k},  ε_{k} ε_{l} = ε_{l} ε_{k},  (3.3)
s_{k} ε_{k} = ε_{k+1} s_{k},  s_{k} ε_{k} ε_{k+1} = ε_{k} ε_{k+1}.  (3.4)

Proof. Denote the abstract algebra by A(n). The assignments s_{k} ↦ (k, k + 1) and ε_{k} ↦ ε_{k} obviously define an algebra epimorphism A(n) → A(n). Note also that (3.2) are defining relations for the symmetric group S(n) and (3.1) holds. To complete the proof we verify that dim A(n) ≤ dim A(n). We have

dim A(n) = | Γ(n)| = ∑_{r=0}^{n} \binom{n}{r}^{2}r! .  (3.5)

On the other hand, we see from the relations (3.3) and (3.4) that

A(n) = ℂ[S(n)] ℂ[ε_{1}, …, ε_{n}].  (3.6)

Here ℂ[ε_{1}, …, ε_{n}] is the subalgebra of A(n) generated by the ε_{k}. It is spanned by the monomials ε_{k_{1}} ⋯ ε_{k_{r}} with k_{1} < ⋯ < k_{r}. Given such a monomial, consider the subspace ℂ[S(n)] ε_{k_{1}} ⋯ ε_{k_{r}} in A(n). Using (3.1) if necessary, we may assume without loss of generality that k_{i} = i for each i. Observe that by (3.4) we have in A(n)

σ S(r) ε_{1} ⋯ ε_{r} = σ ε_{1} ⋯ ε_{r},  σ ∈ S(n).  (3.7)

Hence the dimension of the subspace ℂ[S(n)] ε_{1} ⋯ ε_{r} does not exceed the number of left cosets of S(n) over the subgroup S(r). Therefore, dim A(n) does not exceed

∑_{r=0}^{n} \binom{n}{r}\frac{n!}{r!}  (3.8)

which coincides with (3.5).

Using Proposition 3.1 we shall sometimes identify A(n) with the algebra A(n).

Corollary 3.2 The mapping

(k, k + 1) ↦ (k, k + 1),  ε_{k} ↦ 0  (3.9)

defines an algebra homomorphism A(n) → ℂ[S(n)] which is identical on the subalgebra ℂ[S(n)].

We shall call (3.9) the retraction homomorphism. It can be equivalently defined as follows. For any γ ∈ Γ(n), define its rank, denoted by rank γ, as the number of 1’s in the (0, 1)-matrix representing γ. That is,

rank γ = | dom γ| = | range γ|.  (3.10)

The rank of an element a = ∑ a_{γ}γ ∈ A(n) is defined as the maximum of the ranks rank γ with a_{γ} ≠ 0. Now (3.9) can also be defined by setting for γ ∈ Γ(n)

γ → \begin{cases}γ & if rank γ = n, \\ 0 & if rank γ < n, \end{cases}  (3.11)

and extending this to A(n) by linearity.

For any 0 ≤ m ≤ n denote by Γ_{m}(n) the subsemigroup of Γ(n) which consists of the matrices with first m diagonal entries equal to 1. Set A_{m}(n) = A(n)^{Γ_{m}(n)}, the centralizer of Γ_{m}(n) in the algebra A(n). In particular, A_{0}(n) is the center of A(n).

We extend θ_{n} to a linear map A(n) → A(n − 1).

Proposition 3.3 The restriction of θ_{n} to A_{n−1}(n) ⊆ A(n) defines a unital algebra homomorphism

θ_{n} : A_{n−1}(n) → A(n − 1).  (3.12)

Moreover,

θ_{n}(A_{m}(n)) ⊆ A_{m}(n − 1)  for m = 0, 1, …, n − 1.  (3.13)

Proof. Note that any a ∈ A_{n−1}(n) commutes with ε_{n} because ε_{n} is contained in Γ_{n−1}(n), and A_{n−1}(n) is the centralizer of Γ_{n−1}(n) in A(n). Since ε_{n} is an idempotent we have

θ_{n}(a) = a ε_{n} = ε_{n} a = ε_{n} a ε_{n},  a ∈ A_{n−1}(n).  (3.14)

Now, if a, b ∈ A_{n−1}(n) then

θ_{n}(ab) = ε_{n} ab ε_{n} = ε_{n} a ε_{n} ε_{n} b ε_{n} = θ_{n}(a)θ_{n}(b).  (3.15)

It is clear that (3.12) preserves the unity.

Finally, we need to show that if a ∈ A_{m}(n) and b ∈ Γ_{m}(n − 1) then θ_{n}(a) and b commute. Regard b as an element of Γ(n); then it lies in Γ_{m}(n) and commutes with ε_{n}. This implies that a ε_{n} and b commute, and so do θ_{n}(a) and b.

For γ ∈ Γ(n), set

\begin{aligned}J(γ) & = \{i | 1 ≤ i ≤ n, γ_{ii} = 0\}, \\ deg γ & = | J(γ)|. \end{aligned}  (3.16)

Proposition 3.4 Let γ, γ′ ∈ Γ(n). Then

deg γγ′ ≤ deg γ + deg γ′.  (3.17)

Moreover, the equality in (3.17) implies γγ′ = γ′γ.

Proof. Regard γ and γ′ as partial bijections of the set ℕ_{n} = {1, …, n}. If δ ∈ Γ(n) then ℕ_{n} \ J(δ) is the set of δ-invariant elements in the domain of δ. This implies

(ℕ_{n} \ J(γ)) ∩ (ℕ_{n} \ J(γ′)) ⊆ (ℕ_{n} \ J(γγ′)).  (3.18)

Therefore,

J(γ) ∪ J(γ′) ⊇ J(γγ′),  (3.19)

and (3.17) follows. Finally, the equality in (3.17) implies that J(γ) and J(γ′) are disjoint. But then γ and γ′ must commute.

Using (3.16), we define a filtration of the space A(n):

ℂ = A^{0}(n) ⊆ A^{1}(n) ⊆ … ⊆ A^{n}(n) = A(n)  (3.20)

where A^{M}(n) is spanned by the subset {γ | deg γ ≤ M} ⊆ Γ(n). By Proposition 3.4 this filtration is compatible with the algebra structure of A(n), and the corresponding graded algebra

gr A(n) = ⨁_{M} (A^{M}(n)/A^{M−1}(n))  (3.21)

is commutative. Note that for γ ∈ Γ(n) the degree deg θ_{n}(γ) can be equal either to deg γ or deg γ − 1. Therefore the homomorphisms (3.13) are compatible with the filtration on A(n).

Definition 3.5 For m = 0, 1, 2, … let A_{m} be the projective limit of the infinite sequence

⋯ ⟶ A_{m}(n) ⟶^{θ_{n}} A_{m}(n − 1) ⟶ ⋯ ⟶ A_{m}(m + 1) ⟶^{θ_{m+1}} A_{m}(m)  (3.22)

taken in the category of filtered algebras.

By the definition, an element a ∈ A_{m} is a sequence (a_{n} | n ≥ m) such that

a_{n} ∈ A_{m}(n), θ_{n}(a_{n}) = a_{n−1},  deg a := sup_{n≥m} deg a_{n} < ∞  (3.23)

with the componentwise operations. For n ≥ m we shall denote by θ^{(n)} the projection A_{m} → A_{m}(n) such that

θ^{(n)}(a) = a_{n}.  (3.24)

The M-th term of the filtered algebra A_{m} will be denoted by A_{m}^{M}.

There are natural algebra homomorphisms A_{m} → A_{m+1} defined by

(a_{n} | n ≥ m) ↦ (a_{n} | n ≥ m + 1)  (3.25)

where we use the inclusions A_{m}(n) ⊂ A_{m+1}(n) for n > m. These homomorphisms are injective because a_{m} is uniquely determined by a_{m+1}.

Definition 3.6 The algebra A is defined as the inductive limit (the union) of the algebras A_{m} taken with respect to the embeddings A_{m} ↪ A_{m+1}, m ≥ 0, defined in (3.25).

Since these embeddings preserve the filtration, A is a filtered algebra. We will denote by A^{M} the M-th term of the filtration, so that

A^{M} = ⋃_{m≥0} A_{m}^{M}.  (3.26)

Proposition 3.7 There exists a natural embedding A(∞) ↪ A whose image consists of stable sequences a = (a_{n}) ∈ A.

Proof. Let b ∈ A(∞). There exists m such that b ∈ A(m). Note that b ∈ A_{m}(n) for any n ≥ m since A(m) and Γ_{m}(n) commute. Set

a = (a_{n} | n ≥ m) ∈ A_{m} ⊂ A with a_{n} ≡ b.  (3.27)

The sequence a ∈ A only depends on b and not on the choice of m. The mapping b ↦ a is clearly an algebra embedding.

Corollary 3.8 There is a natural algebra embedding ℂ[S(∞)] ↪ A.

Proposition 3.9 The center of the algebra A coincides with A_{0}.

Proof. Recall that A_{0}(n) is the center of A(n). The subalgebra A_{0} is contained in the center of A since the sequences a = (a_{n}) ∈ A are multiplied componentwise.

Conversely, if a belongs to the center of A then a commutes with the subalgebra A(∞) ⊂ A. This implies that for any n the element a_{n} is contained in A_{0}(n), and so a ∈ A_{0}.

Remark. The same argument shows that the subalgebra A_{m} ⊂ A coincides with the centralizer in A of the subalgebra

⋃_{n≥m} ℂ[Γ_{m}(n)] ⊂ A(∞) ⊂ A.  (3.28)

Note that the centers of both algebras ℂ[S(∞)] and A(∞) are trivial. However, as it will be shown in the next section, the center A_{0} of the algebra A has a rich structure.

Proposition 3.10 For any tame representation T of the group S(∞), the subspace H_{∞}(T) ⊆ H(T) admits a natural structure of an A-module such that for any m the subspace H_{m}(T) is invariant with respect to the subalgebra A_{m} (and hence H_{n}(T) is invariant with respect to A_{m} for n ≥ m).

Proof. Let a ∈ A and h ∈ H_{∞}(T). Choose m such that a ∈ A_{m}. Then we may write a = (a_{n} | n ≥ m). Let us prove that

a_{n}h = a_{n+1}h, n ≥ m.  (3.29)

Consider the family of representations {T_{n}} associated with T, which has been introduced in Section 2.2. Each T_{n} is a representation of the semigroup Γ(n) in the space H_{n}(T) and so it can be extended to a representation of the semigroup algebra A(n) in the same space. Recall that T_{n+1}(ε_{n+1}) projects H_{n+1}(T) onto its subspace H_{n}(T). Since h is already contained in H_{n}(T) (as we assume n ≥ m), we have T_{n+1}(ε_{n+1})h = h, so that

T_{n+1}(1 − ε_{n+1})h = 0.  (3.30)

This implies that h is annihilated by the left ideal I(n + 1) ⊂ A(n + 1). Since a_{n} − a_{n+1} ∈ I(n + 1), this implies (3.29).

Define a mapping

A × H_{∞}(T) → H_{∞}(T),  (a, h) ↦ a_{m}h  (3.31)

where m is so large that a ∈ A_{m} and h ∈ H_{m}(T). Note that under this assumption ah ∈ H_{m}(T).

The mapping (3.31) is clearly bilinear and 1 h = h. The multiplicativity property (ab)h = a(bh) follows from the definition of the multiplication in A_{m}.

Proposition 3.11 If T is an irreducible tame representation of S(∞) then H_{∞}(T) is irreducible as an A-module. In particular, the center A_{0} of A acts by scalar operators.

Proof. The first claim is obvious because H_{∞}(T) is already irreducible as a A(∞)- module (see Proposition 3.7). To prove the second claim consider an element a ∈ A_{0} as an operator in H_{∞}(T). It suffices to show that a has an eigenvalue. The result will then follow by a standard argument using Schur’s lemma.

Assume that a has no eigenvalues. Let us show first that a is algebraically independent over ℂ. Indeed, let P (x) ∈ ℂ[x] be a nonzero polynomial of a minimum degree such that P (a) = 0. Then P (x) = (x−α) Q(x) for a certain α ∈ ℂ and a polynomial Q(x) ∈ ℂ[x]. Since Q(a) is a nonzero operator, there is a vector v ∈ H_{∞}(T) such that w := Q(a)v ≠ 0. Then w is an eigenvector for a with the eigenvalue α. Contradiction.

We note now that the space H_{∞}(T) has countable dimension and then use a version of Dixmier’s argument [4] as follows.

Since a is algebraically independent over ℂ, the field ℂ(a) is embedded in the endomorphism algebra of H_{∞}(T). This implies that the dimension of H_{∞}(T) is at least as large as the dimension of ℂ(a) over ℂ, but the latter is continuum. This contradiction completes the proof.

4 The structure of the algebra A_{0}

In the last two sections we aim to describe the structure of the algebras A_{m}. Here we consider the commutative algebra A_{0}; see Proposition 3.9. We construct generators of A_{0} and show that it is isomorphic to the algebra of shifted symmetric functions.

4.1 Generators of A_{0}

Let Z(S(n)) denote the center of the algebra ℂ[S(n)]. For 0 ≤ M ≤ n denote by Z^{M}(S(n)) the M-th term of the filtration on Z(S(n)) inherited from the algebra A(n); see (3.20). Note that

Z^{0}(S(n)) = Z^{1}(S(n)) = ℂ 1  (4.1)

because, for a permutation s ∈ S(n) the inequality deg s ≤ 1 implies s = 1. For any partition M = (M_{1}, …, M_{r}) with

|M| = M_{1} + … + M_{r} ≤ n  (4.2)

introduce the element c_{n}^{M} of the group algebra ℂ[S(n)] as follows

c_{n}^{M} = ∑(i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) ⋯ (k_{1}, …, k_{M_{r}})  (4.3)

where the sum is taken over the sequences i_{1}, …, i_{M_{1}}; j_{1}, …, j_{M_{2}}; …; k_{1}, …, k_{M_{r}} of |M| pairwise distinct indices taken from ℕ_{n}. By (i_{1}, …, i_{M_{1}}) etc. in (4.3) we denote cycles in the symmetric group. For the empty partition ∅ we set c_{n}^{∅} = 1. Note that c_{n}^{(1)} = ∑_{i=1}^{n}(i) = n 1. Given two partitions M and L we denote by M ∪ L the partition whose parts are those of M and L rewritten in the decreasing order. We have

c_{n}^{M∪1∪...∪1} = (n − |M|) ⋯ (n − |M| − p + 1) c_{n}^{M}  (4.4)

where p stands for the number of 1’s in the left hand side of the relation. By definition (3.16) of the degree of an element of Γ(n) we have

deg c_{n}^{(1)} = 0,  and  deg c_{n}^{(M)} = M for M ≥ 2.  (4.5)

More generally,

deg c_{n}^{(M_{1},... ,M_{r})} = ∑_{i, M_{i}≥2} M_{i}.  (4.6)

Proposition 4.1 Each of the families

c_{n}^{M},  |M| = n,  (4.7)

and

c_{n}^{M},  |M| ≤ n and M has no part equal to 1,  (4.8)

forms a basis of Z(S(n)). Moreover, the elements of degree ≤ M of each family form a basis of Z^{M}(S(n)).

Proof. The elements (4.7) are proportional to the characteristic functions of the conjugacy classes of the group S(n) and so, they form a basis of Z(S(n)). By (4.4) the elements of type (4.8) are proportional to those of type (4.7).

Proposition 4.2 For any two partitions M = (M_{1}, …, M_{r}) and L = (L_{1}, …, L_{t}) with |M| + |L| ≤ n we have

c_{n}^{M}c_{n}^{L} = c_{n}^{M∪L} + (…),  (4.9)

where (…) stands for a linear combination of the elements c_{n}^{K} with |K| < |M| + |L|.

Proof. For a permutation s ∈ S(n) or s ∈ S(∞) define its support as

supp s = {i ∈ ℕ_{n} | s(i) ≠ i} or supp s = {i ∈ ℕ | s(i) ≠ i},  (4.10)

respectively. (The degree of a permutation is then given by deg s = | supp s|; cf. (3.16)).

Let s ∈ S(n) be a permutation which occurs in the expansion of c_{n}^{M}, that is, s is of cycle type M ∪ 1 ∪ … ∪ 1 (with n − |M| units). Similarly, let s′ be a permutation occurring in c_{n}^{L}. If the supports supp s and supp s′ are disjoint then s and s′ commute, and the product ss′ occurs in the expansion of c_{n}^{M∪L}. In particular, deg ss′ = |M| + |L|. If supp s and supp s′ have a non-empty intersection then the degree of ss′ is strictly less than |M| + |L|.

Remark. A detailed investigation of the structure constants for the products of type (4.9) have been recently given by Ivanov and Kerov [7].

Corollary 4.3 Let k = (k_{1}, …, k_{n}) run over the n-tuples of non-negative integers such that 2k_{2} + ⋯ + nk_{n} ≤ n. Then the monomials

(c_{n}^{(2)})^{k_{2}} ⋯ (c_{n}^{(n)})^{k_{n}}  (4.11)

form a basis of Z(S(n)). Moreover, for any M ≥ 0, the elements (4.11) with 2k_{2} + ⋯ + nk_{n} ≤ M form a basis of Z^{M}(S(n)).

Proof. It suffices to prove that

(c_{n}^{(2)})^{k_{2}} ⋯ (c_{n}^{(n)})^{k_{n}} = c_{n}^{M} + (…)  (4.12)

where M = 2^{k_{2}} ⋯ n^{k_{n}} and (…) stands for a certain linear combination of the elements c_{n}^{M′} with |M′| < |M| = 2k_{2} + ⋯ + nk_{n}. But this follows from Proposition 4.2.

Now we will define analogs of the elements c_{n}^{M} for the algebra A_{0}(n). Namely, for any partition M = (M_{1}, …, M_{r}) with |M| ≤ n set

Δ_{n}^{M} = ∑(i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) ⋯ (k_{1}, …, k_{M_{r}})(1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}}) (4.13)

where, as in (4.3), the sum is taken over all sequences of |M| pairwise distinct indices taken from ℕ_{n}. In particular,

Δ_{n}^{(1)} = ∑_{i=1}^{n}(1 − ε_{i}).  (4.14)

For the empty partition ∅ we set Δ_{n}^{∅} = 1. By (3.16), we have

deg Δ_{n}^{M} = |M| for any partition M,  (4.15)

cf. (4.6). Note that Δ_{n}^{M} can also be written as

Δ_{n}^{M} = ∑(1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}})(i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) ⋯ (k_{1}, …, k_{M_{r}}), (4.16)

and as

\begin{matrix}Δ_{n}^{M} = ∑(1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}}) & (i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) \\ ⋯ & (k_{1}, …, k_{M_{r}})(1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}}). \end{matrix}  (4.17)

Indeed, (1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}}) is invariant under the conjugation by the permutation (i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) ⋯ (k_{1}, …, k_{M_{r}}) which implies (4.16). To derive (4.17) it suffices to note that (1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{r}}) is an idempotent.

Proposition 4.4 The element Δ_{n}^{M} belongs to A_{0}(n) for any M.

Proof. Since Γ(n) is generated by the group S(n) and the pairwise commuting idempotents ε_{1}, …, ε_{n}, it suffices to show that Δ_{n}^{M} commutes both with S(n) and with the ε_{i}. The first claim is clear since Δ_{n}^{M} is invariant under the conjugation by the elements of S(n). To prove the second claim, we observe that any ε_{l}, 1 ≤ l ≤ n, commutes with any term

σ = (i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) ⋯ (k_{1}, …, k_{M_{r}})(1 − ε_{i_{1}}) ⋯ (1 − ε_{k_{Mr}})  (4.18)

in (4.13). Indeed, this is clear if l does not occur in the set of indices in (4.18) because ε_{l} commutes with the corresponding cycle. But if l coincides with one of the indices i_{1}, …, k_{M_{r}}, then ε_{l} σ = σ ε_{l} = 0. This follows from (4.17) and the relation (1 − ε_{l})ε_{l} = 0.

Proposition 4.5 We have

θ_{n}(Δ_{n}^{M}) = Δ_{n−1}^{M}  (4.19)

where we adopt the convention that

Δ_{k}^{M} = 0 if |M| > k.  (4.20)

Proof. By the definition of the projection θ_{n} (see Section 3) we need to calculate Δ_{n}^{M} ε_{n}. However, as it follows from the proof of Proposition 4.4, the effect of multiplying Δ_{n}^{M} by ε_{n} reduces to striking from (4.13) all terms (4.18) such that n occurs among the corresponding indices. If |M| = n, then all the terms are vanished, so that the result of the multiplication is 0. If |M| < n, then the terms that survive are just the terms of the sum defining Δ_{n−1}^{M}.

Corollary 4.6 For any partition M, there exists an element Δ^{M} ∈ A_{0} such that

θ^{(n)}(Δ^{M}) = Δ_{n}^{M} for any n ≥ 1  (4.21)

with the convention (4.20).

Proof. By Proposition 4.4, Δ_{n}^{M} ∈ A_{0}(n). Now we apply Proposition 4.5 and note that the degrees of the elements Δ_{n}^{M} are uniformly bounded by (4.15).

We now aim to prove an analog of Proposition 4.1 for the algebra A_{0}(n); see Proposition 4.10 below. For this we need the following three lemmas.

Let I(n) = A(n)(1 − ε_{n}) denote the left ideal of the algebra A(n) generated by the element 1 − ε_{n}.

Lemma 4.7 For any n,

I(n) ∩ A_{0}(n) = Z(S(n))(1 − ε_{1}) ⋯ (1 − ε_{n}).  (4.22)

Proof. First suppose that x ∈ A(n) can be written as y (1 − ε_{1}) ⋯ (1 − ε_{n}) where y ∈ Z(S(n)). The argument of the proof of Proposition 4.4 shows that x ∈ A_{0}(n). Moreover, we obviously have x ∈ I(n).

Conversely, suppose x ∈ I(n) ∩ A_{0}(n). Then x ε_{n} = 0. Using the invariance of x under the conjugation by elements of S(n) we also obtain x ε_{i} = 0 for i = 1, …, n. Therefore x is invariant under the right multiplication by (1−ε_{1}) ⋯ (1−ε_{n}). Further, we may write

x = y + ∑_{r=1}^{n} ∑_{1≤i_{1}<···<i_{r}≤n} y_{i_{1}...i_{r}} ε_{i_{1}} ⋯ ε_{i_{r}},  (4.23)

where y and all the y_{i_{1}...i_{r}} are elements of ℂ[S(n)]; see Proposition 3.1. Multiplying this relation by (1 − ε_{1}) ⋯ (1 − ε_{n}) on the right we obtain

x = y (1 − ε_{1}) ⋯ (1 − ε_{n}).  (4.24)

Finally, for any s ∈ S(n) we may write

x = sxs^{−1} = sys^{−1}(1 − ε_{1}) ⋯ (1 − ε_{n}).  (4.25)

Averaging over s ∈ S(n) turns y into an element of Z(S(n)).

For a subset I = {i_{1}, …, i_{k}} in ℕ_{n} we put ε_{I} = ε_{i_{1}} ⋯ ε_{i_{k}}, and for s ∈ S(n) set

Q(s) = {i ∈ ℕ_{n} | s_{ii} = 1} = ℕ_{n} \ supp s.  (4.26)

Lemma 4.8 The mapping

s ↦ γ,  γ = s ε_{Q(s)} = ε_{Q(s)}s,  (4.27)

defines a bijection of S(n) onto the set of all γ ∈ Γ(n) satisfying the conditions

dom γ = range γ,  (4.28)
deg γ = n.  (4.29)

Proof. The effect of the multiplication of s by ε_{Q(s)} from the left or from the right consists of replacing all the 1’s on the diagonal by zeros. This implies (4.28), and (4.29) is obvious.

Conversely, let γ ∈ Γ(n) satisfy (4.28) and (4.29). Relation (4.28) means that for any i ∈ ℕ_{n} the i-th row and the i-th column are zero or non-zero at the same time, whereas (4.29) means that all the diagonal entries of γ are zero. Now, let the matrix σ be defined as follows. Set σ_{ii} = 1 if the i-th row (and the i-th column) of γ is zero, and set σ_{ij} = γ_{ij} for i ≠ j. It is easy to see that σ ∈ S(n) and that γ is the image of σ under the mapping (4.27).

Lemma 4.9 The restriction of the projection θ_{n} : A_{0}(n) → A_{0}(n−1) to the subspace A_{0}^{n−1}(n) is injective.

Proof. Let x ∈ A_{0}(n) and θ_{n}(x) = 0. We will show that deg x = n unless x = 0. By Lemma 4.7, x can be written as a linear combination of the elements

s (1 − ε_{1}) ⋯ (1 − ε_{n}) = ∑_{I⊆ ℕ_{n}}(−1)^{|I|}s ε_{I},  s ∈ S(n).  (4.30)

Rewrite this as

s (1 − ε_{1}) ⋯ (1 − ε_{n}) = ∑_{I⊇Q(s)}(−1)^{|I|}s ε_{I} + ∑_{I⊉Q(s)}(−1)^{|I|}s ε_{I}.  (4.31)

Then the terms of the first sum are of degree n whereas those of the second sum are of degree strictly less than n. So it suffices to prove that the elements

∑_{I⊇Q(s)}(−1)^{|I|}s ε_{I},  s ∈ S(n),  (4.32)

are linearly independent. Note that

rank s ε_{I} = n − |Q(s)| if I = Q(s),  (4.33)
rank s ε_{I} < n − |Q(s)| if I ⊃ Q(s);  (4.34)

see (3.10). Write S(n) as the disjoint union of n + 1 subsets:

S(n) = ⋃_{k=0}^{n}{s ∈ S(n) | n − |Q(s)| = k}.  (4.35)

If s belongs to the k-th subset then

rank ( ∑_{I⊇Q(s)}(−1)^{|I|}s ε_{I} ) = k.  (4.36)

Moreover, only one term of the sum in (4.36) has rank k, namely that with I = Q(s). Finally, it remains to note that by Lemma 4.8 all the elements sε_{Q(s)} with s ∈ S(n) are pairwise distinct elements of Γ(n).

Proposition 4.10 For any n the elements Δ_{n}^{M}, where M is any partition with |M| ≤ n, form a basis of A_{0}(n). Furthermore, for any M such that 0 ≤ M ≤ n these elements with |M| ≤ M form a basis of A_{0}^{M}(n).

Proof. The first claim of the proposition will follow from the second one. We will prove the second claim using induction on n. The claim is obviously true for n = 1. Assume that n ≥ 2 and M ≤ n − 1. By the induction hypothesis the elements Δ_{n−1}^{M} with |M| ≤ M form a basis of A_{0}^{M}(n − 1). By Proposition 4.5 the image of Δ_{n}^{M} under θ_{n} is Δ_{n−1}^{M}. By Lemma 4.9 the restriction θ_{n} ↓ A_{0}^{M}(n) is injective. Therefore, the elements Δ_{n}^{M} with |M| ≤ M form a basis in A_{0}^{M}(n).

Further, let us show that the elements Δ_{n}^{M} with |M| = n form a basis of I(n) ∩ A_{0}(n). Note that

Δ_{n}^{M} = c_{n}^{M} (1 − ε_{1}) ⋯ (1 − ε_{n}).  (4.37)

By Proposition 4.1 the elements c_{n}^{M}, where M runs over the set of partitions of n, form a basis of Z(S(n)). Due to Lemma 4.7 it now remains to check that the elements c_{n}^{M}, being multiplied by (1 − ε_{1}) ⋯ (1 − ε_{n}), remain linearly independent. However, this follows from the fact that the composite map

ℂ[S(n)] → A(n) → ℂ[S(n)]  (4.38)

is the identity map; here the first arrow is the multiplication by (1 − ε_{1}) ⋯ (1 − ε_{n}), and the second arrow is the retraction homomorphism (3.9).

Finally, let us show that

A_{0}(n) = A_{0}^{n−1}(n) ⊕ (I(n) ∩ A_{0}(n)).  (4.39)

Indeed, as it was shown above, θ_{n} maps A_{0}^{n−1}(n) onto A_{0}^{n−1}(n − 1) = A_{0}(n − 1) . Since I(n) ∩ A_{0}(n) is the kernel of the restriction θ_{n} ↓ A_{0}(n) and since θ_{n}(A_{0}(n)) is contained in A_{0}(n − 1), we obtain the decomposition

A_{0}(n) = A_{0}^{n−1}(n) + (I(n) ∩ A(n)).  (4.40)

Lemma 4.9 implies that

A_{0}^{n−1}(n) ∩ I(n) = {0}  (4.41)

and (4.39) follows.

To complete the proof we need to show that the elements Δ_{n}^{M} with 0 ≤ |M| ≤ n form a basis of A_{0}(n). However, the elements with |M| < n form a basis of the first component of the decomposition (4.39), whereas the elements with |M| = n form a basis in the second component of this decomposition.

The following is an analog of Corollary 4.3.

Corollary 4.11 Let k = (k_{1}, …, k_{n}) run over the n-tuples of non-negative integers such that k_{1} + 2k_{2} + ⋯ + nk_{n} ≤ n. Then the monomials

(Δ_{1}^{(1)})^{k_{1}} ⋯ (Δ_{n}^{(n)})^{k_{n}}  (4.42)

form a basis of A_{0}(n). Moreover, for any M ≥ 0, the monomials (4.42) with k_{1} + 2k_{2} + ⋯ + nk_{n} ≤ M form a basis of A_{0}^{M}(n).

Proof. It suffices to prove that

(Δ_{1}^{(1)})^{k_{1}} ⋯ (Δ_{n}^{(n)})^{k_{n}} = Δ_{n}^{M} + (…)  (4.43)

where M = 1^{k_{1}}2^{k_{2}} ⋯ n^{k_{n}} and (…) stands for a linear combination of the elements Δ_{n}^{M′} with |M′| < |M|. Then our claim will follow from Proposition 4.10. To prove (4.43) we verify that for any partitions M = (M_{1}, …, M_{r}) and L = (L_{1}, …, L_{t}) with |M| + |L| ≤ n

Δ_{n}^{M}Δ_{n}^{L} = Δ_{n}^{M∪L} + (…),  (4.44)

where the rest term (…) has degree strictly less than |M| + |L| and so, it is a linear combination of elements Δ_{n}^{K} with |K| < |M| + |L|. Write

Δ_{n}^{M} = ∑ δ_{I},  Δ_{n}^{L} = ∑ δ′_{J}.  (4.45)

Here I is a sequence i_{1}, …, i_{|M|} of pairwise distinct indices taken from ℕ_{n} and

δ_{I} = (i_{1}, …, i_{M_{1}}) … (i_{M_{1}+...+M_{r−1}+1}, …, i_{|M|}) ∏_{p=1}^{|M|}(1 − ε_{i_{p}});  (4.46)

the δ′_{J} are the corresponding elements for the partition L. Then

Δ_{n}^{M}Δ_{n}^{L} = ∑_{I, J} δ_{I}δ′_{J} = ∑_{I∩J=∅} δ_{I}δ′_{J} + ∑_{I∩J≠∅} δ_{I}δ′_{J}.  (4.47)

The first sum on the right hand side of (4.47) is Δ_{n}^{M∪L} whereas the second sum is of degree strictly less than |M| + |L|.

Consider the elements Δ^{M} ∈ A_{0} introduced in Corollary 4.6. We shall denote by ℙ the set of all partitions.

Theorem 4.12 The elements Δ^{M}, M ∈ ℙ form a basis of the algebra A_{0}. Moreover, for any M ≥ 0, the elements Δ^{M} with |M| ≤ M form a basis of the M-th subspace A_{0}^{M} in A_{0}.

Proof. The first claim follows from the second one. The second claim follows from Proposition 4.10 and the definition of A_{0}^{M} as the projective limit of the spaces A_{0}^{M}(n).

Corollary 4.13 For n > M, the mapping

θ_{n} : A_{0}^{M}(n) → A_{0}^{M}(n − 1)  (4.48)

is an isomorphism of vector spaces and so is the mapping

θ^{(n)} : A_{0}^{M} → A_{0}^{M}(n),  n ≥ M.  (4.49)

In particular, dim A_{0}^{M} < ∞.

Theorem 4.14 The monomials

(Δ^{(1)})^{k_{1}}(Δ^{(2)})^{k_{2}} ⋯  (4.50)

with k_{1}, k_{2}, … ∈ ℤ_{+} and k_{1} +2k_{2} +… < ∞ form a basis of the algebra A_{0}. Moreover, for any M ≥ 0 the monomials (4.50) with k_{1} + 2k_{2} + … ≤ M form a basis of the subspace A_{0}^{M}.

Proof. It suffices to check that

(Δ^{(1)})^{k_{1}}(Δ^{(2)})^{k_{2}} … ≡ Δ^{M}  mod A_{0}^{M−1}  (4.51)

where M = 1^{k_{1}}2^{k_{2}} ⋯ and M = |M|. However, this follows from the relation (4.43).

Corollary 4.15 The elements Δ^{(1)}, Δ^{(2)}, … are algebraically independent and generate the algebra A_{0}.

4.2 The algebra Λ^{∗} of shifted symmetric functions, and the isomorphism A_{0} ≃ Λ^{∗}

Let Λ^{∗}(n) ⊆ ℂ[x_{1}, …, x_{n}] denote the subalgebra of polynomials in n variables x_{1}, …, x_{n} which are symmetric in the new variables

y_{1} = x_{1} − 1, y_{2} = x_{2} − 2, …, y_{n} = x_{n} − n.  (4.52)

Following [24], we refer to Λ^{∗}(n) as the algebra of shifted symmetric polynomials in n variables. We equip Λ^{∗}(n) with the filtration with respect to the usual degree of polynomials. Set Λ^{∗}(0) = ℂ and for n ≥ 1 define the projection Λ^{∗}(n) → Λ^{∗}(n − 1) by specializing x_{n} = 0. Note that this projection preserves the filtration.

Definition 4.16 The algebra Λ^{∗} of shifted symmetric functions is the projective limit of the filtered algebras Λ^{∗}(n) as n → ∞.

In other words, an element f ∈ Λ^{∗} is a sequence (f_{n} | n ≥ 0) such that

(i) f_{n} ∈ Λ^{∗}(n) for any n;

(ii) for any n ≥ 1, f_{n} ↦ f_{n−1} under the projection Λ^{∗}(n) → Λ^{∗}(n − 1); (iii) deg f_{n} remains bounded as n → ∞. For an element f = (f_{n}) ∈ Λ^{∗}, we define its degree by

deg f = sup_{n} deg f_{n},  (4.53)

and for M = 0, 1, … we denote by (Λ^{∗})^{M} the subspace in Λ^{∗} consisting of the elements of degree ≤ M. The algebra Λ^{∗} was first introduced in [26]. A detailed study of Λ^{∗} is contained in [24].

Note an evident similarity between the shifted symmetric functions and the symmetric functions. Recall (see [13]) that the algebra Λ of symmetric functions is defined as the projective limit as n → ∞ of the graded algebras Λ(n) ⊆ ℂ[x_{1}, …, x_{n}] of symmetric polynomials in n variables. A difference between Λ^{∗} and Λ consists in a shift of variables and the replacement of the gradation by a filtration. The algebra Λ^{∗} may be viewed as a deformation of the algebra Λ. Indeed, let h be a numerical parameter, and let Λ_{h}^{∗} be defined similarly to Λ^{∗} but with y_{i} = x_{i} − ih instead of (4.52). Then the algebras Λ_{h}^{∗} with h ≠ 0, are naturally isomorphic to each other. Moreover, Λ_{1}^{∗} coincides with Λ^{∗} while Λ_{0}^{∗} coincides with Λ. Another relation between Λ^{∗} and Λ is given by

Proposition 4.17 The graded algebra

gr Λ^{∗} = ℂ ⊕ ⨁_{M=1}^{∞} ((Λ^{∗})^{M}/(Λ^{∗})^{M−1})  (4.54)

is isomorphic to the algebra Λ.

Proof. For any M ≥ 1 and any n, Λ^{∗}(n)^{M}/Λ^{∗}(n)^{M−1} is naturally isomorphic to the M-th homogeneous component of the algebra Λ(n) ⊆ ℂ[x_{1}, …, x_{n}]. Moreover, this isomorphism is compatible with the projections Λ^{∗}(n) → Λ^{∗}(n − 1) and Λ(n) → Λ(n − 1). This yields an isomorphism gr Λ^{∗} → Λ.

In the following example we give some families of generators of the algebra Λ^{∗}. Note that there also exist other important families analogous to the basic symmetric functions; see [24].

Example 4.18 For M = 1, 2, …, elements e_{M}, h_{M}, and p_{M} defined by the formulas below, are shifted symmetric functions:

E(t) = 1 + ∑_{M=1}^{∞} e_{M}t^{M} = ∏_{k=1}^{∞} \frac{1 + (x_{k} − k)t}{1 − kt},
H(t) = 1 + ∑_{M=1}^{∞} h_{M}t^{M} = ∏_{k=1}^{∞} \frac{1 + kt}{1 − (x_{k} − k)t},
p_{M} = ∑_{k=1}^{∞} ((x_{k} − k)^{M} − (−k)^{M}) .

The generating functions satisfy the following relations; cf. [13]:

E(t)H(−t) = 1,  ∑_{k=1}^{∞} p_{M}t^{M} = t\frac{d}{dt} log H(t).  (4.55)

Proposition 4.19 The algebra Λ^{∗} is isomorphic to the algebra of polynomials in countably many generators. Furthermore, we have

Λ^{∗} = ℂ[e_{1}, e_{2}, …] = ℂ[h_{1}, h_{2}, …] = ℂ[p_{1}, p_{2}, …].  (4.56)

Proof. The corresponding statement for the algebra Λ of symmetric functions is well known, see [13, Ch. 1, Section 2]. Now, we apply Proposition 4.17 and note that the image of the shifted symmetric function e_{M}, h_{M} or p_{M} in the space (Λ^{∗})^{M}/(Λ^{∗})^{M−1} ≃ Λ^{M} is the corresponding M-th symmetric function (elementary, complete or power sum). This implies that each of the three families is algebraically independent and generates the algebra Λ^{∗}.

Let Fun ℙ denote the algebra of complex functions on the set of partitions ℙ. By Propositions 3.10 and 3.11, there is an algebra homomorphism

A_{0} → Fun ℙ,  a ↦ â,  (4.57)

such that for a ∈ A_{0} and λ ∈ ℙ, the element a acts in H_{∞}(T_{λ}) as the scalar operator â(λ) · 1. On the other hand, any λ ∈ ℙ can be viewed as a sequence (λ_{1}, λ_{2}, …, 0, 0, …) with finitely many non-zero coordinates, and so, any element of Λ^{∗} may be viewed as a function on ℙ. Thus we obtain an algebra homomorphism Λ^{∗} → Fun ℙ which is clearly an embedding.

Let λ be a partition with m = |λ| ≤ n. Consider the corresponding irreducible representation π_{λ} of S(m), and the representation T_{n}(λ) of the semigroup Γ(n); see Section 2.2.

Proposition 4.20 The eigenvalue of the central element Δ_{n}^{(r)} in T_{n}(λ) is 0 if r > m. If r ≤ m then the eigenvalue coincides with that of the element c_{m}^{(r)} in the representation π_{λ} of S(m).

Proof. Recall the construction of T_{n}(λ) given in Section 2.2. Let ω be an injective map from {1, …, m} to {1, …, n}. Regarding ω as an m-tuple ω = (ω_{1}, …, ω_{m}) we have

ε_{a} f(ω) = \begin{cases}0 & if a ∈ ω, \\ f(ω) & if a ∉ ω. \end{cases}  (4.58)

Therefore, the product (1−ε_{i_{1}}) ⋯ (1−ε_{i_{r}}) is a projection to the subspace of functions f such that the indices i_{1}, …, i_{r} belong to any ω ∈ supp f. This implies the first statement. The second follows from the obvious embedding H(π_{λ}) ⊆ H(T_{n}(λ)) whose image consists of the functions supported by the maps ω such that {ω_{1}, …, ω_{m}} = {1, …, m}.

It was proved in [10] (see also [24]) that the eigenvalue of c_{n}^{(r)} in the irreducible representation π_{λ} of S(n) is a shifted symmetric function whose highest homogeneous component is the power sum symmetric function p_{r}.

Theorem 4.21 Let Λ^{∗} be identified with its image in Fun ℙ. Then the mapping (4.57) is an isomorphism A_{0} → Λ^{∗} of filtered algebras.

Proof. By Proposition 4.20 the images of the generators Δ^{(r)} ∈ A_{0} with respect to the homomorphism (4.57) are shifted symmetric functions which are algebraically independent generators of the algebra Λ^{∗}. The map obviously respects the filtrations.

Recall that by Propositions 3.10 and 3.11, elements of the center A_{0} act in irreducible tame representations of S(∞) by scalar operators. Hence, any such representation determines a homomorphism A_{0} → ℂ.

Corollary 4.22 The center A_{0} separates irreducible tame representations of S(∞). That is, non-equivalent irreducible tame representations give rise to distinct homomorphisms A_{0} → ℂ.

Proof. By Theorem 2.10, the irreducible tame representations are precisely the representations T_{λ}. Hence, our claim is equivalent to the fact that the map Λ^{∗} → Fun ℙ defined above is an embedding.

5 The structure of the algebra A_{m}, m > 0

Here we generalize the results of Section 4 to the algebra A_{m}, where m = 1, 2, . . .. Throughout the section we assume 0 ≤ m ≤ n and use the notation

ℕ_{mn} = {m + 1, …, n}.  (5.1)

For γ ∈ Γ(n), set

\begin{aligned}J_{m}(γ) & = \{i | i ∈ ℕ_{mn}, γ_{ii} = 0\}, \\ deg_{m} γ & = | J_{m}(γ)|. \end{aligned}  (5.2)

We shall call deg_{m} γ the m-degree of γ.

Proposition 5.1 For γ, δ ∈ Γ(n),

deg_{m} γδ ≤ deg_{m} γ + deg_{m} δ.  (5.3)

Proof. For any i ∈ ℕ_{mn} we have

(γδ)_{ii} = 0 ⇒ γ_{ij}δ_{ji} = 0 for all j = 1, …, n.  (5.4)

In particular, (γδ)_{ii} = 0 implies γ_{ii}δ_{ii} = 0, i.e.,

J_{m}(γδ) ⊆ J_{m}(γ) ∪ J_{m}(δ),  (5.5)

and (5.3) follows.

Definition 5.2 Using the m-degree we define a new filtration in A(n), called the m-filtration, by

A(m) = F_{m}^{0}(A(n)) ⊆ F_{m}^{1}(A(n)) ⊆ … ⊆ F_{m}^{n−m}(A(n)) = A(n).  (5.6)

Here F_{m}^{M}(A(n)), the M-th term of the filtration, is formed by the elements a ∈ A(n) which are linear combinations of the elements of Γ(n) of m-degree ≤ M. For any subspace S of A(n) we will use the symbol F_{m}^{M}(S) to indicate the M-th term of the induced filtration.

By Proposition 5.1, the m-filtration is compatible with the algebra structure of A(n), so the corresponding graded algebra exists. But contrary to the case m = 0, this graded algebra is not commutative for m ≥ 1 since it contains, as the 0-component, the non-commutative algebra A(m).

Let D be a multiplicative semigroup with unity 1. Consider the union D ∪ {0}, where 0 is an extra symbol, and adopt the convention that

d 0 = 0 d = 0, d + 0 = 0 + d = d for any d ∈ D.  (5.7)

Definition 5.3 (i) The semigroup S(m, D) consists of the m × m matrices α = [α_{ij}] with entries in D ∪ {0} such that any row and column contains exactly one non-zero entry. The product is the matrix multiplication with the conventions (5.7).

(ii) The semigroup Γ(n, D) is defined as in (i) by allowing any row and column contain at most one non-zero entry.

Note that if D = {1}, then S(n, D) and Γ(n, D) coincide with S(n) and Γ(n), respectively. If D is a group, then S(n, D) is the wreath product of S(n) and D.

We shall be assuming now that D is the free abelian semigroup {1, z, z^{2}, … } with unity 1 and one generator z. This semigroup is isomorphic to the additive semigroup ℤ_{+}. We denote the corresponding semigroups introduced in Definition 5.3 by S(m, ℤ_{+}) and Γ(m, ℤ_{+}).

Set ord z^{k} = k for k = 0, 1, …, and for α ∈ Γ(m, ℤ_{+}), set

ord α = ∑_{i,j; α_{ij}≠0} ord α_{ij}.  (5.8)

Definition 5.4 (i) Set

Γ(m, n) = {σ ∈ Γ(n) | dom σ and range σ contain ℕ_{mn}}.  (5.9)

(ii) Consider the linear span of Γ(m, n) and let Z_{m}(n) ⊂ A(n) denote the subspace in this span formed by the elements invariant under the conjugation by the elements of the group S_{m}(n).

In particular, Γ(0, n) = S(n) and Z_{0}(n) = Z(S(n)) is the center of ℂ[S(n)]. The role of Γ(m, n) and Z_{m}(n) will be similar to that of S(n) and Z(S(n)) in Section 4. Note also that Z_{m}(n) contains ℂ[S(n)]^{S_{m}(n)}, the centralizer of S_{m}(n) in the group algebra ℂ[S(n)].

Now our purpose is to construct a convenient basis in Z_{m}(n). To do this, we need to classify the S_{m}(n)-orbits in Γ(m, n) where the elements of S_{m}(n) act by conjugations.

Proposition 5.5 There is a natural parameterization of the S_{m}(n)-orbits in Γ(m, n) by the couples (α, M), where α ∈ Γ(m, ℤ_{+}) and M is a partition such that

ord α + |M| = n − m.  (5.10)

Proof. Fix an arbitrary element σ ∈ Γ(m, n) and assign to it an m × m-matrix α = α(σ) as follows. For i, j ∉ ℕ_{mn} set

α_{ij} = 0  if j ∉ dom σ,  (5.11)
α_{ij} = 1  if j ∈ dom σ and σ(j) = i,  (5.12)
α_{ij} = z^{k}  if j ∈ dom σ,  (5.13)

and there exist k points p_{1}, …, p_{k} ∈ ℕ_{mn} such that σ(j) = p_{1}, σ(p_{1}) = p_{2}, …, σ(p_{k−1}) = p_{k}, σ(p_{k}) = i. Thus, to any j ∈ dom σ with σ(j) ∈ ℕ_{mn} we have assigned a subset {p_{1}, …, p_{k}} ⊆ ℕ_{mn}. It is clear that these subsets are pairwise disjoint. Let P = P (σ) denote their union. Then ord α = | P | ≤ n − m. It is also clear that α ∈ Γ(m, ℤ_{+}).

Further, let P^{∗} = P^{∗}(σ) be the complement of P in ℕ_{mn}. Then P^{∗} is contained in the domain of σ, and P^{∗} is σ-invariant. Therefore, the restriction of σ to P^{∗} defines a permutation of P^{∗}. Let M = M(σ) be the partition of the number | P^{∗}| which is defined by the lengths of the cycles of this permutation. Then the couple

(α, M) = (α(σ), M(σ))  (5.14)

satisfies (5.10). It is clear that the couple (5.14) remains unchanged if σ is replaced by sσs^{−1} with s ∈ S_{m}(n). Moreover, it is also clear that if the couples (5.14) corresponding to two elements of Γ(m, n) are the same, then these elements belong to the same orbit. Finally, any couple satisfying (5.10) can be obtained from an element of Γ(m, n).

Remark. A couple (5.14) corresponds to an element of S(n) ⊆ Γ(m, n) if and only if α ∈ S(m, ℤ_{+}).

We shall now define analogs of the elements c_{n}^{M}. First, for any subset Q ⊆ ℕ_{mn} and any partition M = (M_{1}, …, M_{r}) such that |M| = | Q| we set

c_{Q}^{M} = ∑(i_{1}, …, i_{M_{1}})(j_{1}, …, j_{M_{2}}) … (k_{1}, …, k_{M_{r}}),  (5.15)

where (i_{1}, …, i_{M_{1}}) etc. are cyclic permutations of the corresponding indices and the summation is taken over all the orderings (i_{1}, …, i_{M_{1}}; j_{1}, …, j_{M_{2}}; …; k_{1}, …, k_{M_{r}}) of the elements of Q. We shall suppose that c_{∅}^{∅} = 1.

Second, for any α ∈ Γ(m, ℤ_{+}) and any subset P ⊆ ℕ_{mn} such that ord α = | P |, we set

Γ(α, P) = {σ ∈ Γ(m, n) | α(σ) = α, P (σ) = P, M(σ) = (1^{n−m−| P |})}, (5.16)

i.e., σ has to fix all the points in ℕ_{mn} \ P.

Definition 5.6 For any couple (α, M), where α ∈ Γ(m, ℤ_{+}) and M is a partition such that ord α + |M| ≤ n − m we set

c_{n}^{α,M} = ∑_{P, Q} ∑_{σ∈Γ(α,P)} σ c_{Q}^{M},  (5.17)

where P, Q are disjoint subsets in ℕ_{mn} such that

| P | = ord α,  | Q| = |M|.  (5.18)

Proposition 5.7 Each of the families

c_{n}^{α,M}  with ord α + |M| = n − m,  (5.19)

and

c_{n}^{α,M}  with ord α + |M| ≤ n − m, and M has no part equal to 1, (5.20)

forms a basis of Z_{m}(n).

Proof. Note that c_{n}^{α,M∪1∪...∪1} is proportional to c_{n}^{α,M}. Therefore, it suffices to consider the family (5.19). By Proposition 5.5 the elements c_{n}^{α,M} with ord α + |M| = n − m are proportional to characteristic functions of the S_{m}(n)-orbits in Γ(m, n).

Now we introduce analogs of the elements Δ_{n}^{M}.

Definition 5.8 For any couple (α, M), where α ∈ Γ(m, ℤ_{+}) and M is a partition such that ord α + |M| ≤ n − m we set

Δ_{n}^{α,M} = ∑_{P, Q} ∑_{σ∈Γ(α,P)} ε(P) σ c_{Q}^{M} ε(Q) ε(P),  (5.21)

where ε(I) := (1 − ε_{i_{1}}) ⋯ (1 − ε_{i_{k}}) for I = {i_{1}, …, i_{k}}. Here P, Q are disjoint subsets in ℕ_{mn} satisfying (5.18). We set Δ_{n}^{1,∅} = 1, where ∅ stands for the empty partition.

Note that (5.21) can be written in an equivalent form where the term ε(Q) takes the leftmost position; cf. (4.13) and (4.16).

Proposition 5.9 The elements Δ_{n}^{α,M} belong to the algebra A_{m}(n).

Proof. The semigroup Γ_{m}(n) is generated by the subgroup S_{m}(n) and the idempotents ε_{m+1}, …, ε_{n}. Therefore, it suffices to check that Δ_{n}^{α,M} is stable under the conjugation by the elements of S_{m}(n) and commutes with the idempotents. The first claim is immediate from (5.21). The second claim is verified exactly as its counterpart for the elements Δ_{n}^{M}; see the proof of Proposition 4.4.

The following is an analog of Proposition 4.5 and it is proved by the same argument.

Proposition 5.10 We have

θ_{n}(Δ_{n}^{α,M}) = Δ_{n−1}^{α,M},  (5.22)

where we adopt the convention that

Δ_{k}^{α,M} = 0 if ord α + |M| > k − m.  (5.23)

Our aim now is to prove an analog of Propositions 4.1 and 4.10; see Proposition 5.14 below. We need the following three lemmas.

Lemma 5.11 For m < n

I(n) ∩ A_{m}(n) = (1 − ε_{m+1}) ⋯ (1 − ε_{n}) Z_{m}(n) (1 − ε_{m+1}) ⋯ (1 − ε_{n}).  (5.24)

Proof. Suppose that x ∈ A(n) can be written as

x = (1 − ε_{m+1}) ⋯ (1 − ε_{n}) y (1 − ε_{m+1}) ⋯ (1 − ε_{n}),  (5.25)

where y ∈ Z_{m}(n). Then x ∈ A_{m}(n) since x is invariant under the conjugation by the elements of S_{m}(n) and is annihilated when multiplied (from the left or from the right) by any idempotent ε_{m+1}, …, ε_{n}. Moreover, this also implies that x ∈ I(n).

Conversely, suppose x ∈ I(n)∩A_{m}(n). Then x ε_{n} = ε_{n} x = 0. Using the invariance of x under the conjugation by the elements of S_{m}(n) we obtain x ε_{i} = ε_{i} x = 0 for i = m+1, …, n. Thus x is invariant under the multiplication by (1−ε_{m+1}) ⋯ (1−ε_{n}) both from the left and from the right.

Further, we can write x = y + y′ where y and y′ are spanned by elements of Γ(m, n) and Γ(n) \ Γ(m, n), respectively. However,

(1 − ε_{m+1}) ⋯ (1 − ε_{n}) y′ (1 − ε_{m+1}) ⋯ (1 − ε_{n}) = 0  (5.26)

since for each element γ ∈ Γ(n) \ Γ(m, n) there exists i > m such that γ ε_{i} = γ or ε_{i} γ = γ. This implies

x = (1 − ε_{m+1}) ⋯ (1 − ε_{n}) y (1 − ε_{m+1}) ⋯ (1 − ε_{n}).  (5.27)

Finally, averaging over the group S_{m}(n) transforms y into an element of Z_{m}(n); cf. the proof of Lemma 4.7.

For σ ∈ Γ(n), set

Q(σ) = {i ∈ ℕ_{mn} | σ_{ii} = 1}.  (5.28)

Lemma 5.12 The mapping

σ ↦ γ,  γ = σ ε_{Q(σ)} = ε_{Q(σ)} σ  (5.29)

defines a bijection of Γ(m, n) onto the set of all γ ∈ Γ(n) satisfying the conditions

dom γ ∩ ℕ_{mn} = range γ ∩ ℕ_{mn},  (5.30)
deg_{m} γ = n − m.  (5.31)

Proof. The effect of the multiplication of σ by ε_{Q(σ)} from the left or from the right consists of replacing all the diagonal entries σ_{ii} = 1 with i > m by zeros. Therefore γ satisfies (5.30). Relation (5.31) follows from this observation and the fact that both dom σ and range σ contain ℕ_{mn}.

Conversely, let γ ∈ Γ(n) satisfy (5.30) and (5.31). Note that (5.30) can be reformulated as follows: for any i = m + 1, …, n the i-th row and the i-th column are zero or non zero at the same time, whereas (5.31) means that all the diagonal entries γ_{ii} with i > m + 1 vanish. Now, let σ be defined by

σ_{ij} = γ_{ij}  if either i ≠ j or min{i, j} ≤ m,
σ_{ii} = 1  if i ∈ ℕ_{mn}

and the i-th row (or the i-th column) of γ is zero. Then it is easy to see that σ ∈ Γ(m, n) and that γ is the image of σ under the mapping (5.29).

Lemma 5.13 For m < n the restriction of the projection θ_{n} : A_{m}(n) → A_{m}(n − 1) to the subspace F_{m}^{n−m−1}(A_{m}(n)) is injective.

Proof. Suppose that x ∈ A_{m}(n) and θ_{n}(x) = 0. We will show that then deg_{m} x = n − m unless x = 0.

By Lemma 5.11, x can be written as a linear combination of the elements of type

(1 − ε_{m+1}) ⋯ (1 − ε_{n}) σ (1 − ε_{m+1}) ⋯ (1 − ε_{n})

= ∑_{R,S⊆N_{mn}}(−1)^{| R|+| S|}ε_{R} σ ε_{S}, σ ∈ Γ(m, n). (5.32)

Let us divide the terms in the sum (5.32) into two groups depending on whether R ∪ S contains Q(σ) or not. Then the terms of the first group are of m-degree n − m whereas those of the second group are of m-degree < n − m. So, it suffices to prove that the elements

∑_{R∪S⊇ Q(σ)}(−1)^{| R|+| S|}ε_{R} σ ε_{S},  σ ∈ Γ(m, n),  (5.33)

are linearly independent. Note that, in the case R ∪ S = Q(σ),

ε_{R} σ ε_{S} = σ ε_{Q(σ)} and rank σ ε_{Q(σ)} = rank σ − | Q(σ)|,  (5.34)

whereas, in the case R ∪ S strictly contains Q(σ),

rank σ ε_{Q(σ)} < rank σ − | Q(σ)|.  (5.35)

Therefore, we now need to show that for any fixed k the elements

∑_{R∪S=Q(σ)}(−1)^{| R|+| S|}σ ε_{Q(σ)},  (5.36)

where σ runs over the subset of the elements in Γ(m, n) with rank σ − | Q(σ)| = k, are linearly independent.

Lemma 5.12 implies that the elements σ ε_{Q(σ)} ∈ Γ(n) are pairwise distinct. Hence it remains to prove that all the coefficients in (5.36) are non-vanishing. This is implied by the following general fact: if Q is an arbitrary finite set, then

∑_{R,S⊆Q, R∪S=Q}(−1)^{| R|+| S|} ≠ 0.  (5.37)

We will prove that the sum in (5.37) equals (−1)^{q} where q = | Q|. Indeed, for any

r = 0, 1, …, q, there are \binom{n}{r} subsets R ⊆ Q with | R| = r. Given R, for any

t = 0, 1, …, r, there are \binom{r}{t} subsets S ⊆ Q such that R ∪ S = Q and | R ∩ S| = t. Since

| R| + | S| = r + t + (q − r) = q + t,  (5.38)

the sum in (5.37) equals

(−1)^{q} ∑_{r=0}^{q} \binom{n}{r} ∑_{t=0}^{r}(−1)^{t}\binom{r}{t}.

If r = 0 then the interior sum is equal to 1, otherwise it is zero. Therefore the entire sum is (−1)^{q}.

Proposition 5.14 The elements Δ_{n}^{α,M} with

ord α + |M| ≤ n − m  (5.39)

form a basis of A_{m}(n). Moreover, for any M with 0 ≤ M ≤ n − m the elements Δ_{n}^{α,M} satisfying

ord α + |M| ≤ M  (5.40)

form a basis of F_{m}^{M}(A_{m}(n)).

Proof. It suffices to prove the second claim. We use induction on n and follow the argument of the proof of Proposition 4.10. The claim is obviously true for n = m. Assume that n ≥ m + 1 and M ≤ n − m − 1. Lemma 5.13 implies that the elements Δ_{n}^{α,M} with ord α + |M| ≤ M form a basis of F_{m}^{M}(A_{m}(n)).

To show that the elements Δ_{n}^{α,M} with ord α + |M| = n − m form a basis of I(n) ∩ A_{m}(n) note that

Δ_{n}^{α,M} = (1 − ε_{m+1}) ⋯ (1 − ε_{n}) c_{n}^{α,M} (1 − ε_{m+1}) ⋯ (1 − ε_{n});  (5.41)

see (5.21). Now the claim follows from Proposition 5.7 and the fact that the elements c_{n}^{α,M}, being multiplied by (1 − ε_{m+1}) ⋯ (1 − ε_{n}), remain linearly independent; cf. (4.38).

Using Proposition 5.10 we can introduce the elements Δ^{α,M} ∈ A_{m} as sequences Δ^{α,M} = (Δ_{n}^{α,M} | n ≥ m). Remark. We can regard Δ^{α,M} as a formal series given by (5.21) where the sum is taken over all disjoint subsets P and Q in {m + 1, m + 2, … } satisfying (5.18).

Theorem 5.15 The elements Δ^{α,M} with α ∈ Γ(m, ℤ_{+}) and M ∈ ℙ form a basis of the algebra A_{m}. Moreover, for any M ≥ 0, the elements Δ^{α,M} with ord α+|M| ≤ M form a basis of the M-th subspace F_{m}^{M}(A_{m}) in A_{m}.

Proof. The first claim follows from the second one. The second claim follows from Proposition 5.14 and the definition of F_{m}^{M}(A_{m}) as the projective limit of the spaces F_{m}^{M}(A_{m}(n)).

Corollary 5.16 For n > M, the mapping

θ_{n} : F_{m}^{M}(A_{m}(n)) → F_{m}^{M}(A_{m}(n − 1))  (5.42)

is an isomorphism of vector spaces and so is the mapping

θ^{(n)} : F_{m}^{M}(A_{m}) → F_{m}^{M}(A_{m}(n)),  n ≥ M.  (5.43)

In particular, dim F_{m}^{M}(A_{m}) < ∞.

For each k = 1, …, m consider the following elements of A_{m}(n)

u_{k|n} = ∑_{i=k+1}^{n}(ki)(1 − ε_{k})(1 − ε_{i}) = ∑_{i=k+1}^{n}(1 − ε_{i})(ki)(1 − ε_{i}).  (5.44)

The image of u_{k|n} under the retraction homomorphism (3.9) is the Jucys–Murphy element for S(n); cf. [9], [19]. We obviously have θ_{n}(u_{k|n}) = u_{k|n−1} and so, for each k the element u_{k} ∈ A_{m} can be defined as the sequence u_{k} = (u_{k|n} | n ≥ m). Recall that the algebra A(m) is naturally embedded in A_{m}; see Proposition 3.7.

Proposition 5.17 The following relations hold in the algebra A_{m}:

s_{k} u_{k} = u_{k+1} s_{k} + (1 − ε_{k})(1 − ε_{k+1}),  s_{k} u_{l} = u_{l} s_{k}, l ≠ k, k + 1; (5.45)
u_{k} u_{l} = u_{l} u_{k},  ε_{k} u_{k} = u_{k} ε_{k} = 0,  ε_{i} u_{k} = u_{k} ε_{i}, i ≠ k;  (5.46)

where s_{k} = (k, k + 1).

Proof. For n > m we have u_{1|n} = Δ_{n}^{(2)}−Δ_{n−1}^{′(2)}, where Δ_{n−1}^{′(2)} is the element of the center of ℂ[Γ_{1}(n)] given by (4.13) with the sum taken over the indices from {2, …, n}. Now an easy induction proves that the elements u_{1|n}, …, u_{m|n} pairwise commute, and so do the elements u_{1}, …, u_{m}. The remaining relations easily follow from (5.44) and the relations in the algebra A(n).

We shall denote by H̃_{m} the subalgebra of A_{m} generated by A(m) and the elements u_{1}, …, u_{m}. The following is our main result. The theorem describes the structure of the algebra A_{m}.

Theorem 5.18 We have an algebra isomorphism

A_{m} ≃ A_{0} ⊗ H̃_{m}.  (5.47)

Moreover, the algebra H̃_{m} is isomorphic to an abstract algebra with generators s_{1}, …, s_{m−1}, ε_{1}, …, ε_{m}, u_{1}, …, u_{m} and the defining relations given by (3.2)–(3.4) and (5.45)–(5.46).

Recall that by Theorem 4.21 A_{0} is isomorphic to the algebra of shifted symmetric functions Λ^{∗}. Proof. For any α ∈ Γ(m, ℤ_{+}) and M ∈ ℙ such that ord α + |M| ≤ n − m we have the equality in the algebra A_{m}(n),

Δ_{n}^{α,∅}Δ_{n}^{1,M} = Δ_{n}^{α,M} + lower m-degree terms,  (5.48)

where ∅ stands for the empty partition while 1 ∈ Γ(m, ℤ_{+}) is the m × m identity matrix; cf. the proofs of Proposition 4.2 and Corollary 4.11. On the other hand, we have

deg_{m}(Δ_{n}^{M} − Δ_{n}^{1,M}) < |M|;  (5.49)

see (4.13) for the definition of Δ_{n}^{M}. Now (5.48) and Proposition 5.14 imply that the elements Δ_{n}^{α,∅}Δ_{n}^{M} with ord α + |M| ≤ n − m form a basis of A_{m}(n). Hence the elements Δ^{α,∅}Δ^{M} with α ∈ Γ(m, ℤ_{+}) and M ∈ ℙ form a basis of the algebra A_{m}. In other words, A_{m} is a free A_{0}-module with the basis {Δ^{α,∅} | α ∈ Γ(m, ℤ_{+})}.

Further, if α ∈ Γ(m, ℤ_{+}) has zero rows i_{1}, …, i_{r} then

Δ_{n}^{α,∅} = ε_{i_{1}} ⋯ ε_{i_{r}} Δ_{n}^{α′,∅}  (5.50)

for some element α′ ∈ S(m, ℤ_{+}). Observe now that every element α′ ∈ S(m, ℤ_{+}) can be written as a product of the form

α′ = σ α_{1}^{k_{1}} ⋯ α_{m}^{k_{m}},  k_{i} ≥ 0,  (5.51)

where σ ∈ S(m) and α_{i} ∈ Γ(m, ℤ_{+}) is the diagonal matrix whose ii-th entry is z and all other diagonal entries are equal to 1. This implies that modulo lower m-degree terms, the element Δ_{n}^{α′,∅} coincides with the product

Δ_{n}^{σ,∅}(Δ_{n}^{α_{1},∅})^{k_{1}} ⋯ (Δ_{n}^{α_{m},∅})^{k_{m}};  (5.52)

cf. the proof of (4.44). The claim remains valid if we replace each Δ_{n}^{α_{k},∅} with the element u_{k|n}. Indeed, this follows from the equality

Δ_{n}^{α_{k},∅} = u_{k|n} + elements of m-degree zero.  (5.53)

Note also that the element Δ_{n}^{σ,∅} can be identified with σ. Thus, modulo lower mdegree terms, the element (5.52) coincides with the product σ u_{1|n}^{k_{1}} ⋯ u_{m|n}^{k_{m}}. Using an obvious induction on the m-degree we may conclude that the A_{0}-module A_{m} is generated by the subspace H̃_{m}.

To prove that H̃_{m} generates the A_{0}-module A_{m} freely, we check that for any M > 0 the dimension of the subspace F_{m}^{M}(H̃_{m}) is less or equal to the number of elements α ∈ Γ(m, ℤ_{+}) with ord α ≤ M. Indeed, by Proposition 5.17 the subalgebra H̃_{m} is spanned by the elements of the form γ u_{1}^{k_{1}} ⋯ u_{m}^{k_{m}} with γ ∈ Γ(m). The relation ε_{k} u_{k} = 0 ensures that such a product is zero unless k_{j} = 0 for each zero column j in γ. To each of the nonzero products associate the element α ∈ Γ(m, ℤ_{+}) which has the ij-entry z^{k_{j}} where the j-th column of γ is nonzero with γ_{ij} = 1. This shows that the cardinality of a basis of F_{m}^{M}(H̃_{m}) can be at most the number of elements α ∈ Γ(m, ℤ_{+}) with ord α ≤ M, proving (5.47).

To prove the second claim of the theorem note that by Proposition 5.17 there is an algebra epimorphism from the abstract algebra in question to H̃_{m}. The above argument implies that the nonzero products γ u_{1}^{k_{1}} ⋯ u_{m}^{k_{m}} with γ ∈ Γ(m) form a basis of H̃_{m}.

Corollary 5.19 The mapping

s_{k} ↦ s_{k},  ε_{k} ↦ ε_{k},  u_{k} ↦ u_{k|n}  (5.54)

defines an algebra homomorphism ψ : H̃_{m} → A_{m}(n). The algebra A_{m}(n) is generated by A_{0}(n) and the image of ψ.

The degenerate affine Hecke algebra H_{m} (see [6], [12]) is defined to be generated by elements s_{1}, …, s_{m−1} and u_{1}, …, u_{m} with the defining relations (3.2) and

s_{k} u_{k} = u_{k+1} s_{k} + 1,  s_{k} u_{l} = u_{l} s_{k}, l ≠ k, k + 1;  (5.55)
u_{k} u_{l} = u_{l} u_{k}.  (5.56)

As a linear space, H_{m} is isomorphic to the tensor product ℂ[S(m)] ⊗ ℂ[u_{1}, …, u_{m}]. The following corollary is implied by Theorem 5.18 and provides an analog of the retraction homomorphism (3.9).

Corollary 5.20 The mapping

s_{k} ↦ s_{k},  u_{k} ↦ u_{k},  ε_{k} → 0  (5.57)

defines an algebra epimorphism H̃_{m} → H_{m}.

It can be seen from the proof of Theorem 5.18 that the retraction homomorphisms (3.9) and (5.57) “respect” the homomorphism ψ : H̃_{m} → A_{m}(n) defined in Corollary 5.19. More precisely, the following result takes place. It was announced in [29, Theorem 11], and a proof was given in [25]. We denote by B_{m}(n) the centralizer of S_{m}(n) in the group algebra ℂ[S(n)]; see Introduction.

Corollary 5.21 The mapping

s_{k} ↦ s_{k},  u_{k} ↦ ∑_{i=k+1}^{n}(ki)  (5.58)

defines an algebra homomorphism φ : H_{m} → B_{m}(n). The algebra B_{m}(n) is generated by B_{0}(n) and the image of φ.

References

[1] I. V. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra, Funct. Anal. Appl. 20 (1986), 76–78.

[2] I. V. Cherednik, A new interpretation of Gelfand–Tzetlin bases, Duke Math. J. 54 (1987), 563–577.

[3] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, Vol. 1, AMS Providence, RI, 1964.

[4] J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad. Brasil. Ci., 35 (1963), 491–519.

[5] V. G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.

[6] V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 62–64.

[7] V. N. Ivanov and S. V. Kerov, The algebra of conjugacy classes in symmetric groups, and partial permutations, in: ‘Representation Theory, Dynamical Systems, Combinatorial and Algorithmical Methods III’ (A. M. Vershik, ed.), Zapiski Nauchnyh Seminarov POMI Vol. 256, 1999, pp. 95-120 (Russian).

[8] G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.

[9] A. Jucys, Symmetric polynomials and the centre of the symmetric group ring, Rep. Math. Phys. 5 (1974), 107–112.

[10] S. Kerov and G. Olshanski, Polynomial functions on the set of Young diagrams, Comptes Rendus Acad. Sci. Paris, Sér. I, 319 (1994), 121–126.

[11] A. Lieberman, The structure of certain unitary representations of infinite symmetric groups, Trans. AMS 164 (1972), 189–198.

[12] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635.

[13] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995.

[14] A. I. Molev, A basis for representations of symplectic Lie algebras, Comm. Math. Phys. 201 (1999), 591–618.

[15] A. I. Molev, A weight basis for representations of even orthogonal Lie algebras, in: ‘Combinatorial Methods in Representation Theory’, Proceedings, Kyoto, October–November 1998, to appear; math.RT/9902060.

[16] A. I. Molev, Weight bases of Gelfand–Tsetlin type for representations of classical Lie algebras, Preprint SMS 99–21, University of Sydney; math.QA/9909034.

[17] A. I. Molev, Yangians and transvector algebras, Preprint SMS 98–30, University

of Sydney; math.QA/9811115.

[18] A. Molev and G. Olshanski, Centralizer construction for twisted Yangians, Selecta Math., to appear; q-alg/9712050.

[19] G. E. Murphy, A new construction of Young’s seminormal representation of the symmetric group, J. Algebra 69 (1981), 287–291.

[20] M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996), 664–693.

[21] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), 190–257.

[22] M. Nazarov and V. Tarasov, Representations of Yangians with Gelfand–Zetlin bases, J. Reine Angew. Math. 496 (1998), 181–212.

[23] Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, Oxford Univ. Press, Oxford 1996.

[24] A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math. J. 9 (1998), 239–300.

[25] A. Okounkov, A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), 581–605.

[26] G. I. Olshanski, Unitary representations of the infinite-dimensional classical groups U(p, ∞), SO(p, ∞), Sp(p, ∞) and the corresponding motion groups, Funct. Anal. Appl. 12 (1979), 185–195.

[27] G. I. Olshanski, New “large” groups of type one, J. Soviet Math. 18 (1982), 22–39.

[28] G. I. Olshanski, Unitary representations of the infinite symmetric group: a semigroup approach, in ‘Representations of Lie groups and Lie algebras’ (A. A. Kirillov, Ed.), Academiai Kiadó, Budapest, 1985, pp. 181–197.

[29] G. I. Olshanski, Extension of the algebra U(g) for infinite-dimensional classical Lie algebras g, and the Yangians Y (gl(m)), Soviet Math. Dokl. 36 (1988), no. 3, 569–573.

[30] G. I. Olshanski, Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians, in: ‘Topics in Representation Theory’ (A. A. Kirillov, Ed.), Advances in Soviet Math. 2, AMS, Providence RI, 1991, pp. 1–66.

[31] G. I. Olshanski, On semigroups related to infinite-dimensional groups, in: ‘Topics in Representation Theory’ (A. A. Kirillov, Ed.), Advances in Soviet Math. 2, AMS, Providence RI, 1991, pp. 67–101.

[32] G. I. Olshanski, Twisted Yangians and infinite-dimensional classical Lie algebras, in: ‘Quantum Groups (P. P. Kulish, Ed.)’, Lecture Notes in Math. 1510, Springer, Berlin-Heidelberg, 1992, pp. 103–120.