Algebraic combinatorics related to the free Lie algebra

D. Blessenohl and H. Laue

Mathematisches Seminar der Universität Ludewig-Meyn-Str. 4 DW-2300 Kiel 1

During the past decade numerous fruitful contributions to the theory of the free Lie algebra have been made. Results and methods in this area are characterized by a subtle interplay between algebraic and combinatorial ideas. The quantity and the wealth of the material accumulated in the last years might be accompanied by the certainly undesired side-effect of concealing its own roots and history. We have therefore decided to restrict ourselves to a concise approach to some specially chosen topics. It is essentially self-contained and takes its course starting from a completely elementary source. At the same time the attempt is made to duly provide the reader with appropriate references for the various contributions involved. We take the opportunity to refer to [18] as a useful supplement to this article.

In the first section we describe certain aspects of the theory and a number of known results. In the second section we show that these are different branches of the same key result (Ind) for which we then give a proof, by means of elementary combinatorial reasoning, in the third section. The underlying idea of our approach is to transfer problems on free Lie algebras into the area of group rings of symmetric groups. On the one hand, this provides a powerful tool to solve those problems. On the other hand, the arising questions are a challenging contribution to the classical representation theory of the symmetric group: By passing from the free Lie algebra to group rings, several notions are focused which apparently have not been considered as of central importance before. A most interesting problem in this sense is to analyze the role of the Solomon algebra in the general representation theory of the symmetric group. In our fourth section we introduce this algebra and add some hints in that direction.

1 Free Lie algebras

In the following we write ℕ_{0} for the set of all non-negative integers and set ℕ := ℕ_{0}\\{0\}, \underline{n} := {k | k ∈ ℕ, 1 ≤ k ≤ n} for all n ∈ ℕ_{0}.

Let R be a commutative unitary ring, n ∈ ℕ, F be the monoid generated freely by n letters x_{1}, …, x_{n}. Any element x_{i_{1}} ⋯ x_{i_{m}} ∈ F is called a word (over {x_{1}, …, x_{n}}), the number m its length or degree. If the number of all r ∈ \underline{m} such that x_{i_{r}} = x_{j} is denoted by k_{j} (j ∈ \underline{n} ), then k_{1} + ⋯ + k_{n} = m, and (k_{1}, …, k_{n}) is called the multidegree of x_{i_{1}} ⋯ x_{i_{m}}.

Let A_{R} be the free R-module with basis F. The multiplication in F extends canonically to A_{R} which thus becomes an associative R-algebra generated freely by {x_{1}, …, x_{n}}. For all m ∈ ℕ_{0} let A_{R,m} be the R-submodule of A_{R} generated by all words of length m, and for all (k_{1}, …, k_{n}) ∈ ℕ_{0}^{n} let A_{R}(k_{1}, …, k_{n}) be the R-submodule of A_{R} generated by all words of multidegree (k_{1}, …, k_{n}). Then

(1)  \begin{aligned}A_{R} & = & ⨁_{m∈ℕ_{0}} & A_{R,m} \\ A_{R,m} & = & ⨁_{k_{1}+···+k_{n}=m} & A_{R}(k_{1}, …, k_{n}) (m ∈ ℕ_{0}). \end{aligned}

The standard Lie product,

a ∘ b := ab − ba for all a, b ∈ A_{R},

turns A_{R} into a Lie algebra over R. By [31], [6, II, §3, Theorem 1], {x_{1}, …, x_{n}} generates freely a Lie subalgebra of A_{R} which will be denoted by L_{R}. A Lie monomial in A_{R} is an element of the ∘-closure of {x_{1}, …, x_{n}}. A Lie monomial of the particular form (⋯ (x_{i_{1}} ∘ x_{i_{2}}) ∘ ⋯) ∘ x_{i_{ℓ}} is called left-normed. For simplicity, we shall use for it the bracket-free notation x_{i_{1}} ∘ x_{i_{2}} ∘ ⋯ ∘ x_{i_{ℓ}}. It is easy to see that the R-module L_{R} is generated by the set of all left-normed Lie monomials. Surprisingly, no R-basis of L_{R} consisting of left-normed Lie monomials is known. Set L_{R,m} := L_{R} ∩ A_{R,m} for all m ∈ ℕ_{0}, L_{R}(k_{1}, …, k_{n}) = L_{R} ∩ A_{R}(k_{1}, …, k_{n}) for all (k_{1}, …, k_{n}) ∈ ℕ_{0}^{n}. The multidegree of a Lie monomial a ≠ 0 in A_{R} is the unique n-tuple (k_{1}, …, k_{n}) such that a ∈ L_{R}(k_{1}, …, k_{n}). We have

(2)  \begin{aligned}L_{R} & = & ⨁_{m∈ℕ_{0}} & L_{R,m} \\ L_{R,m} & = & ⨁_{k_{1}+···+k_{n}=m} & L_{R}(k_{1}, …, k_{n}) (m ∈ ℕ_{0}). \end{aligned}

By [6, II, §2.5], the following holds:

1.1 Proposition. The mapping

\begin{matrix}\{x_{1} …, x_{n}\} & → R ⊗_{ℤ}L_{ℤ} \\ x_{j} & ↦ 1 ⊗ x_{j} \end{matrix}

extends uniquely to a Lie R-algebra isomorphism ϕ_{R} of L_{R} onto R ⊗_{ℤ}L_{ℤ}. In particular,

L_{R,m}ϕ_{R} = R ⊗_{ℤ}L_{ℤ,m} for all m ∈ ℕ_{0}, and L_{R}(k_{1}, …, k_{n})ϕ_{R} = R ⊗_{ℤ}L_{ℤ}(k_{1}, …, k_{n}) for all

(k_{1}, …, k_{n}) ∈ ℕ_{0}^{n}.

As an abelian group, L_{ℤ}(k_{1}, …, k_{n}) is torsion-free and finitely generated (for example, by the set of all Lie monomials in A_{ℤ} of multidegree (k_{1}, …, k_{n})). Therefore, L_{ℤ}(k_{1}, …, k_{n}) is a free ℤ-module of finite rank. As a consequence, L_{R}(k_{1}, …, k_{n}) is a free R-module of the same rank. This rank is known to be the so-called necklace number as has been proved by Witt [31]. By 1.1, it is justified to specialize R, and for our purposes it is convenient to put R := ℂ. Subsequently we simply write A (L resp.) for A_{ℂ}(L_{ℂ} resp.), similarly A_{m} (A(k_{1}, …, k_{n}) resp.) for the space A_{ℂ,m} (A_{ℂ}(k_{1}, …, k_{n}) resp.) of all homogeneous elements of degree m (of multidegree (k_{1}, …, k_{n}) resp.), etc. Now Witt’s Dimension Formula reads as follows:

(WDF)  dim L(k_{1}, …, k_{n}) = \frac{1}{m} ∑_{d|k_{1},...,k_{n}} μ(d) \frac{\frac{m}{d}!}{\frac{k_{1}}{d}! ⋯ \frac{k_{n}}{d}!}

where m = k_{1} + ⋯ + k_{n}. The necklace number on the right-hand side of (WDF) is the number of Lyndon words in F of multidegree (k_{1}, …, k_{n}). In group theoretic terms, it is the number of orbits of length m of the subgroup ⟨(1 … m)⟩ of the symmetric group S_{m} with respect to its left action on the set of left cosets of a Young subgroup of S_{m} of isomorphism type S_{k_{1}} × ⋯ × S_{k_{n}}.

Putting (x_{i_{1}} ⋯ x_{i_{ℓ}})∘ := x_{i_{1}} ∘ ⋯ ∘ x_{i_{ℓ}} for all i_{1}, …, i_{ℓ} ∈ \underline{n} , we obtain a vector space epimorphism ∘ : A → L, sometimes called the Dynkin mapping. Grün (see [20, footnote 12]) expressed this mapping by means of the Weyl action of S_{m} on the space A_{m}: By the rule

σx_{i_{1}} ⋯ x_{i_{m}} := x_{i_{1σ}} ⋯ x_{i_{mσ}} (i_{1}, …, i_{m} ∈ \underline{n} , σ ∈ S_{m}),

A_{m} is made into a ℂS_{m}-left module. Obviously, the spaces A(k_{1}, …, k_{n}) where k_{1} + ⋯ + k_{n} = m are ℂS_{m}-submodules of A_{m}. Let

X_{m} := {π | π ∈ S_{m}, 1π > 2π > … > 1 < … < (m − 1)π < mπ}

and

ω_{m} := ∑_{π∈X_{m}}(−1)^{1π^{−1}−1}π ∈ ℂS_{m}.

Then

(3)  a∘ = ω_{m}a for all a ∈ A_{m}.

The important criterion by Dynkin [8], Specht [25], Wever [29] characterizes the Lie elements of A_{m} by means of the Dynkin mapping:

(DSW)  a ∈ L_{m} ⟺ a∘ = ma, for any a ∈ A_{m}.

Putting ν_{m} := \frac{1}{m}ω_{m}, we have, by (3),

(4)  ν_{m}A_{m} = L_{m},

as the Dynkin mapping is onto. Hence the essential content of (DSW) is the following:

(5)  ν_{m}^{2} = ν_{m}.

In the special case of m = n, (4) implies that

(6)  ν_{n}A(1, …_{n}, 1) = L(1, …_{n}, 1).

As an operator, every ϕ ∈ ℂS_{n} such that ϕA(1, …_{n}, 1) = L(1, …_{n}, 1) maps the space of all homogeneous elements of degree n of an arbitrary free associative algebra onto its subspace of homogeneous Lie elements of degree n. This is due to the fact that A is free over {x_{1}, …, x_{n}}.

We now fix n ∈ ℕ and choose a primitive n-th root of unity ε. An element ϕ ∈ ℂS_{n} is called a Lie idempotent if ϕA(1, …_{n}, 1) = L(1, …_{n}, 1) and ϕ^{2} = ϕ. By (5) and (6),

ν_{n} = \frac{1}{n} ∑_{π∈X_{n}}(−1)^{1π^{−1}−1}π

is a Lie idempotent. Using (5), it is easy to check, for an arbitrary element ϕ ∈ ℂS_{n}, that

(7)  ϕ is a Lie idempotent if and only if ϕν_{n} = ν_{n}, ν_{n}ϕ = ϕ.

This is a first example of the general phenomenon that significant notions in the theory of free Lie algebras may be characterized by means of formally simple equations in the group ring ℂS_{n}. Applying (7), we have: (8)

The Lie idempotents in ℂS_{n} are the idempotent generators of the right ideal ν_{n}ℂS_{n}.

A second important example of a Lie idempotent was given by Klyachko in 1974. For every σ ∈ S_{n} set

ind σ := ∑\{j | j ∈ \underline{n − 1} , jσ > (j + 1)σ\}

(called the (major) index of σ, [19, sect. III, VI, 104.]). Then

λ_{n} := \frac{1}{n} ∑_{σ∈S_{n}} ε^{ind σ}σ

is a Lie idempotent ([16], [2,3.4.3]). ^{1})

Starting from any Lie idempotent, an analysis of [3, 1^{st} Theorem 2.3] yields a simple method of constructing a family of related Lie idempotents:

1.2 Proposition. Let K be a subfield of ℂ, ϕ = ∑_{σ∈S_{n}} c_{σ}σ ∈ KS_{n} be a Lie idempotent, B be a ℚ-basis of K such that 1 ∈ B. For all b ∈ B let ϕ_{b} ∈ ℚS_{n} such that ϕ = ∑_{b∈B} bϕ_{b}. (Almost all ϕ_{b} are 0.) Then

ϕ_{1} + ∑_{b∈B\{1}} d_{b}ϕ_{b}

is a Lie idempotent, for every choice of the coefficients d_{b} ∈ ℂ.

Proof. As ν_{n} ∈ ℚS_{n}, (7) implies that ϕ_{1}ν_{n} = ν_{n}, ϕ_{b}ν_{n} = 0 for all b ∈ B\{1}, and ν_{n}ϕ_{b} = ϕ_{b} for all b ∈ B. But these equations imply our claim, again by (7). □

Of course, new Lie idempotents are obtained by means of (7) only if not all the coefficients c_{σ} of ϕ are rational. The case of ϕ := λ_{n}, K := ℚ(ε), B := {1, ε, ε^{2}, ε^{3}, …} has been considered in [3, (2.20)].

In 1986, Reutenauer discovered a further Lie idempotent; it has rational coefficients: For every σ ∈ S_{n} we define the defect set of σ by

D(σ) := \{j | j ∈ \underline{n − 1} , jσ > (j + 1)σ\},

and the defect of σ by

d(σ) := |D(σ)|.
(9)  ∑_{σ∈S_{n}} t^{ind σ}σ = ∏_{j=1}^{n}(id + tτ_{j} + t^{2}τ_{j}^{2} + ⋯ + t^{j−1}τ_{j}^{j−1})

∑_{σ∈S_{n}} t^{ind σ}Φ(σ) = ∏_{j=1}^{n}(Φ(id) + tΦ(τ_{j}) + t^{2}Φ(τ_{j})^{2} + ⋯ + t^{j−1}Φ(τ_{j})^{j−1}).

\begin{aligned}∑_{σ∈S_{n}} t^{ind σ} & = ∏_{j=1}^{n}(1 + t + t^{2} + ⋯ + t^{j−1}) ([27, 4.5.9]), \\ ∑_{σ∈S_{n}} sgn(σ)t^{ind σ} & = ∏_{j=1}^{n}(1 + (−1)^{j+1}t + t^{2} + (−1)^{j+1}t^{3} + ⋯ + (−1)^{j+1}t^{j−1}) resp. \end{aligned}

1For any variable t we have the identity where τ_{j} = (j … 1). This yields a product representation of λ_{n} if we put t := ε. Furthermore, if Φ is a representation of the group ring of S_{n} over the field ℂ(t), then (9) implies that In the special case of the 1-dimensional representations this reduces to the following identities:

Then

ρ_{n} := \frac{1}{n} ∑_{σ∈S_{n}} \frac{(−1)^{d(σ)}}{\binom{n−1}{d(σ)}} σ

is a Lie idempotent ([21, (1.4)]).

We have an easy characterization of the elements π ∈ X_{n} which occur in ν_{n}, in terms of defect sets: Let π ∈ S_{n} and r := d(π). Then the following three statements are equivalent: (10)

π ∈ X_{n}

D(π) = \underline{r}

π = (j_{1} … 1) ⋯ (j_{r} … 1) for some j_{1}, …, j_{r} ∈ \underline{n} such that j_{1} > j_{2} > … > j_{r} > 1.

The equation π = (j_{1} … 1) ⋯ (j_{r} … 1) where j_{1} > j_{2} > … > j_{r} > 1 implies that j_{ℓ} = ℓπ for all ℓ ∈ \underline{r} , hence D(π)π = \{j_{1}, …, j_{r}\}. In particular, we have:

(11)

For every C ⊆ \underline{n} \\{1\} there exists a unique π ∈ X_{n} such that D(π)π = C.
The mapping π ↦ D(π)π is a bijection of X_{n} onto the power set of \underline{n} \\{1\}.

The following three simple properties of the elements π ∈ X_{n} will be useful at a later stage:

(12)  D(π^{−1}) = D(π)π − 1

(In particular, the inverses of any two distinct elements of X_{n} have distinct defect sets.) For k, ℓ ∈ \underline{n} we have

(13)  If k < ℓ and kπ > ℓπ, then k ∈ D(π).
(14)  If k < ℓ and kπ < ℓπ, then ℓ ∉ D(π).

Moreover, by (10),

(15)  ν_{n} = \frac{1}{n} ∑_{π∈X_{n}}(−1)^{d(π)}π = \frac{1}{n}(id − (n … 1))(id − (n − 1 … 1)) ⋯ (id − (21)).

This last description of ν_{n} as a product has first been noted by Magnus [20].

The coefficients of any Lie idempotent have a remarkable property which was discovered by Wever [30, Satz 4] in the case of the particular Lie idempotent ν_{n} (see also [13], [5]). The following general version of this result is due to Garsia [10, Proposition 5.1]:

1.3 Theorem. Let ϕ= ∑_{σ∈S_{n}} c_{σ}σ ∈ℂS_{n} be a Lie idempotent. Then for any conjugacy class C of S_{n}

∑_{σ∈C} c_{σ} = \{ \begin{matrix}\frac{μ(d)}{n} if d | n and (1 … n)^{\frac{n}{d}} ∈ C \\ 0 otherwise \end{matrix} .

By (8), ϕℂS_{n} = ν_{n}ℂS_{n} for all Lie idempotents ϕ ∈ ℂS_{n}. Hence the general statement 1.3 is implied by Wever’s special result by means of the following proposition due to Frobenius [9, §1], [7, §9, exerc. 16]:

(16)

If ϕ = ∑_{σ∈S_{n}} c_{σ}σ, ψ = ∑_{σ∈S_{n}} d_{σ}σ are idempotent elements of ℂS_{n}, then

ϕℂS_{n} ≅_{ℂS_{n}} ψℂS_{n} if and only if ∑_{σ∈C} c_{σ} = ∑_{σ∈C} d_{σ} for all conjugacy classes C of S_{n}.

We now turn to another aspect of the theory which concerns representations of general linear groups. The vector space A_{n} may be identified with the n-fold tensor product V ⊗…_{n} ⊗V where V = A_{1}. Therefore, in a natural way, A_{n} is a GL(V)-(right) module. In his doctoral thesis of 1901 [22] and in a famous paper of 1927 [23], Schur described the decomposition of A_{n} into irreducible GL(V)-modules in terms of irreducible representations of S_{n}: If p is a partition of n (p ⊢ n) and U^{p} is an irreducible ℂS_{n}-module corresponding to a Young diagram of shape p, then U^{p} ⊗_{ℂS_{n}} A_{n} is either 0 or is an irreducible GL(V)-module. A_{n} is a direct sum of modules of this type, and the multiplicity of U^{p} ⊗_{ℂS_{n}}A_{n} in A_{n} is the number of standard Young tableaux of shape p, denoted by st^{p}. That is,

(17)  A_{n} ≅_{GL(V)} ⨁_{p⊢n} st^{p}(U^{p} ⊗_{ℂS_{n}}A_{n}).

Obviously, L_{n} is a GL(V)-submodule of A_{n}. More than fifty years ago, the question was raised as to how the GL(V)-module structure of L_{n} could be described [28]. In the meantime, various contributions to this problem have been achieved, but a satisfactory answer in the spirit of Schur’s result (17) was discovered only recently. Let us first recall a module isomorphism of preliminary character proved by Klyachko in 1974. We write C_{n} for the eigenspace of the cycle (1 … n) in A_{n} with respect to the eigenvalue ε.

1.4 Proposition ([16, Proposition 1]). L_{n} ≅_{GL(V)} C_{n}.

The desired decomposition of L_{n} into GL(V)-irreducible constituents was finally obtained in 1987: For every Young tableau T put

maj T := ∑\{j | j ∈ \underline{n − 1} , j + 1 is in a lower row of T than j\}

(called the major index of T). For p ⊢ n and i ∈ \underline{n}, let st_{i}^{p} be the number of all standard Young tableaux T of shape p such that maj T ≡ i mod n. The main result on the GL(V)-module structure of L_{n} is the following:

1.5 Theorem (see [10, 8.]). L_{n} ≅_{GL(V)} ⨁_{p⊢n} st_{1}^{p}(U^{p} ⊗_{ℂS_{n}}A_{n}).

2 On (Ind), a key result

A proof of 1.5 is obtained by means of two non-trivial results which are interesting in their own right (2.1 and (Ind)).

For every j ∈ \underline{n} ∪ \{0\} let M_{j} be a 1-dimensional ⟨(1 … n)⟩-module over ℂ such that the character of (1 … n) is ε^{j}. The first result to be mentioned here is the following:

2.1 Theorem (Kraskiewicz, Weyman [17], (see also Springer [26, 4.5]))

M_{j}^{S_{n}} ≅_{ℂS_{n}} ⨁_{p⊢n} st_{j}^{p}U^{p} for every j ∈ \underline{n} ∪ \{0\}.

As for the second result, we remark first that the natural action of S_{n} on {x_{1}, …, x_{n}} gives rise to a ℂS_{n}-right module structure on the spaces A(1, …_{n}, 1) and L(1, …_{n} , 1). We observe

(18)  A(1, …_{n}, 1) is a regular ℂS_{n}-right module,

and

(19)  L_{n} ≅_{GL(V)} L(1, …_{n}, 1) ⊗_{ℂS_{n}} A_{n}.

The following statement proves to be a key result for the whole context as the discussion in this section will show. An equivalent form of it is already contained in Wever’s paper [30, Satz 5] and was rediscovered in 1974 by Klyachko [16, Corollary 1]:

(Ind)  L(1, …_{n}, 1) ≅_{ℂS_{n}} M_{1}^{S_{n}}.

The induced ℂS_{n}-module M_{1}^{S_{n}} is obviously isomorphic to the right ideal of ℂS_{n} generated by the following idempotent element:

ζ_{n} := \frac{1}{n} ∑_{i=0}^{n−1} ε^{−i}(1 … n)^{i}.

Hence M_{1}^{S_{n}} ⊗_{ℂS_{n}}A_{n} is GL(V)- isomorphic to the eigenspace C_{n}.

(a) Now 1.4 is a consequence of the isomorphisms

L_{n} ≅_{GL(V)} L(1, …_{n}, 1) ⊗_{ℂS_{n}} A_{n} ≅_{GL(V)} M_{1}^{S_{n}} ⊗_{ℂS_{n}} A_{n}

which follow from (19) and (Ind).

(b) Applying 2.1 (where j = 1), we obtain 1.5.

(c) Theorem 1.3, too, follows easily from (Ind): Let ϕ ∈ ℂS_{n} be any Lie idempotent. By (18), A(1, …_{n}, 1) ≅_{ℂS_{n}} ℂS_{n}, hence

ϕℂS_{n} ≅_{ℂS_{n}} ϕA(1, …_{n}, 1) = L(1, …_{n}, 1) ≅_{ℂS_{n}} M_{1}^{S_{n}} ≅_{ℂS_{n}} ζ_{n}ℂS_{n}.

Therefore, by (16), the property stated in the formula of 1.3 for the coefficients of ϕ follows once it is verified for the coefficients of ζ_{n}. But for ζ_{n} it is an easy consequence of well-known properties of roots of unity.

(d) Finally, we sketch a short proof of (WDF) exploiting (Ind) (see [4,2.] for more details). Let k_{1}, …, k_{n} ∈ ℕ_{0} and m = k_{1} + ⋯ + k_{n}. Let Y be a Young subgroup of type S_{k_{1}} × ⋯ × S_{k_{n}} of S_{m}, χ be the character of the ℂS_{m}-right module L(1, …_{m}, 1), and ψ be a faithful irreducible character of ⟨(1 … m)⟩. Now (Ind) implies that (χ, 1_{Y}^{S_{m}})_{S_{m}} = (ψ^{S_{m}}, 1_{Y}^{S_{m}})_{S_{m}}. But

(χ, 1_{Y}^{S_{m}})_{S_{m}} = (χ |_{Y} , 1_{Y})_{Y} = dim C_{L(1,..._{m},1)}(Y)

which is equal to the dimension of L(k_{1}, …, k_{m}), and

(ψ^{S_{m}}, 1_{Y}^{S_{m}})_{S_{m}} = (ψ, 1_{Y}^{S_{m}}|_{<(1...m)>})_{<(1...m)>}

which is the number of orbits of length m of the subgroup ⟨(1 … m)⟩ of S_{m} with respect to its left action on the set of left cosets of Y.

3 A self-contained approach

We now present an elementary combinatorial approach to the theory by giving selfcontained proofs of (5) and (Ind). For every D ⊆ \underline{n − 1} we call

S_{n}(D) := {σ | σ ∈ S_{n}, D(σ) = D}

the defect class of D in S_{n} and put

δ_{D} := ∑_{σ∈S_{n}(D)} σ ∈ ℂS_{n}.

Any defect class contains exactly one of the inverses of the elements of X_{n} (cf. (12)).

The following basic lemma by F. Bergeron, N. Bergeron, and Garsia [3,(1.11)] reveals a surprising connection between the concept of the Lie multiplication and that of the defect of permutations:

3.1 Lemma. δ_{D}ν_{n} = (−1)^{|D|}ν_{n} for all D ⊆ \underline{n − 1} .

A direct simple proof of 3.1 would be of interest, as has been remarked already in [3]. We propose to proceed as follows: For every σ ∈ S_{n} we put D_{0}(σ) := D(σ) ∪ {0},

\begin{matrix}P_{σ} := D(σ)\(1 + D_{0}(σ)), \\ T_{σ} := (1 + D_{0}(σ))\D(σ) (= (1 + D_{0}(σ))\D_{0}(σ)), \end{matrix}

and call the elements of P_{σ} the peaks, the elements of T_{σ} the troughs of σ. Let j ∈ \underline{n}. Then

j ∈ P_{σ} if and only if j ≠ 1, j ≠ n, and (j − 1)σ, (j + 1)σ < jσ

and

j ∈ T_{σ} if and only if j = 1 and 2σ > 1σ, or j = n and (n − 1)σ > nσ,
or 1 < j < n and (j − 1)σ, (j + 1)σ > jσ.

By (10), we have

(20) σ ∈ X_{n} ⟺ P_{σ} = ∅ ⟺ T_{σ} = {1σ^{−1}}. ^{2})

2As an easy consequence, we mention a further characterization: σ ∈ X_{n} if and only if \underline{k}σ^{−1} is an interval, for every k ∈ \underline{n}.

3.2 Proposition. Let σ ∈ S_{n}, D ⊆ \underline{n − 1} , L := (1 + (D\D(σ))) ∪ (D(σ)\D). (Then T_{σ} ∩ L = ∅.)

a) There is an element π ∈ X_{n} such that D(σπ^{−1}) = D if and only if P_{σ} ⊆ (D\(1 + D)) ∪ ((1 + D)\D).

b) Suppose that there is an element ψ ∈ X_{n} such that D(σψ^{−1}) = D. Let π ∈ X_{n}. Then D(σπ^{−1}) = D if and only if Lσ ⊆ D(π)π ⊆ (T_{σ} ∪ L)σ.

Proof. For all A, B ⊆ ℤ it is straightforward to verify that

(21) ((1 + (A\B)) ∪ (B\A))\((1 + (A ∪ B))\(B ∩ A)) = (B\(1 + B))\((A\(1 + A)) ∪ ((1 + A)\A)),

(22) ((1 + (A ∪ B))\(A ∩ B))\((1 + (A\B)) ∪ (B\A)) = (1 + B)\B.

Set A := D ∪ {0}, B := D_{0}(σ), R := (1 + (A ∪ B))\(A ∩ B). Obviously, L = (1 + (A\B)) ∪ (B\A). Now (21) and (22) easily imply that L\R = P_{σ}\((D\(1 + D)) ∪ ((1 + D)\D)), R\L = T_{σ}. Hence

(23)  R = T_{σ} ∪ L ⟺ L ⊆ R ⟺ P_{σ} ⊆ (D\(1 + D)) ∪ ((1 + D)\D).

The main step of our proof is to show the following, for all π ∈ X_{n}:

(24)  D(σπ^{−1}) = D ⟺ L ⊆ D(π)πσ^{−1} ⊆ R.

Suppose first that D(σπ^{−1}) = D. For every i ∈ L, one of the following two statements holds:

⟨ \begin{matrix}i ≠ 1, & iσπ^{−1} & < & (i − 1)σπ^{−1}, & and & (i − 1)σ & < & iσ \\ i ≠ n, & iσπ^{−1} & < & (i + 1)σπ^{−1}, & and & (i + 1)σ & < & iσ \end{matrix} .

By (13), iσπ^{−1} ∈ D(π). Hence L ⊆ D(π)πσ^{−1}. Furthermore, for every i ∈ \underline{n}\R, one of the following two statements holds:

⟨ \begin{matrix}i ≠ 1, & iσπ^{−1} & > & (i − 1)σπ^{−1}, & and & (i − 1)σ & < & iσ \\ i ≠ n, & iσπ^{−1} & > & (i + 1)σπ^{−1}, & and & (i + 1)σ & < & iσ \end{matrix} .

By (14), iσπ^{−1} ∉ D(π). Hence D(π)πσ^{−1} ⊆ R.

Conversely, suppose that L ⊆ D(π)πσ^{−1} ⊆ R. We show, for all i ∈ \underline{n − 1} , that

(25)  i ∈ D(σπ^{−1}) ⟺ i ∈ D.

Suppose first that i ∈ D(σ). Then (i + 1)σ < iσ. By hypothesis, D(σ)\D ⊆ D(π)πσ^{−1} ⊆ \underline{n}\(D(σ) ∩ D), and therefore

i ∈ D ⟺ i ∉ D(π)πσ^{−1} ⟺ iσπ^{−1} ∉ D(π) ⟺ iσπ^{−1} > (i + 1)σπ^{−1},

by (13) and (14). Similarly, if i ∉ D(σ), then iσ < (i+1)σ. By hypothesis, 1+(D\D(σ)) ⊆ D(π)πσ^{−1} ⊆ 1 + (D ∪ D_{0}(σ)), and therefore

i ∈ D ⟺ i + 1 ∈ D(π)πσ^{−1} ⟺ (i + 1)σπ^{−1} ∈ D(π) ⟺ iσπ^{−1} > (i + 1)σπ^{−1},

by (14) and (13). Thus in both cases (25) holds. The proof of (24) is complete. As L_{σ} ⊆ \underline{n}\\{1\}, the statements (24) and (11) imply that

(26)  (∃π ∈ X_{n} D(σπ^{−1}) = D) ⟺ L ⊆ R.

By (23), this implies a). Under the hypothesis of b), (26) implies that L ⊆ R, hence R = T_{σ} ∪ L, by (23). By means of (24), we obtain b). □

3.3 Corollary. Let σ ∈ S_{n}, D ⊆ \underline{n − 1} , X (σ, D) := \{π|π ∈ X_{n}, D(σπ^{−1}) = D\}.

a) If σ ∈ X_{n}, then X (σ, D) contains exactly one element π, and we have (−1)^{d(π)} = (−1)^{|D|+d(σ)}.

b) If σ ∉ X_{n}, then ∑_{π∈X (σ,D)}(−1)^{d(π)} = 0.

Proof. a) If σ ∈ X_{n}, then (20) implies that P_{σ} = ∅ and T_{σ} = {1σ^{−1}}. By 3.2a), X (σ, D) ≠ ∅. If π ∈ X (σ, D), then Lσ ⊆ D(π)π ⊆ {1} ∪ Lσ by 3.2b), hence D(π)π = Lσ. By (11), X (σ, D) = \{π\}. Furthermore, D(σ) = \underline{d(σ)}, and therefore (1+(D\D(σ)))∩(D(σ)\D) = ∅. Hence

d(π) = |L| = |D\D(σ)| + |D(σ)\D| ≡ |D| + |D(σ)| mod 2.

b) If σ ∉ X_{n}, then |T_{σ}| ≥ 2 by (20). By 3.2b) and (11), there is a 1-1 correspondence between X (σ, D) and the power set of T_{σ}\{1σ^{−1}}. Hence

∑_{π∈X (σ,D)}(−1)^{|D(π)|} = (−1)^{|L|} · ∑_{S⊆T_{σ}\{1σ^{−1}}}(−1)^{|S|} = 0. □

Proof of 3.1. For all D ⊆ \underline{n − 1} we have, by 3.3,

δ_{D}ω_{n} = ∑_{σ∈S_{n}(D)} ∑_{π∈X_{n}}(−1)^{d(π)}σπ = ∑_{ρ∈S_{n}} ∑_{{π∈Xn \atop D(ρπ^{−1})=D}}(−1)^{d(π)}ρ = ∑_{ρ∈X_{n}}(−1)^{|D|+d(ρ)}ρ = (−1)^{|D|}ω_{n}.

□

As a first application of 3.1, we obtain a simple proof of (5): For every π ∈ X_{n} we have d(π) = 1π^{−1} − 1, and therefore

(27)  ω_{n} = ∑_{d=0}^{n−1} (−1)^{d}δ\underline{d}

(cf. [3, Theorem 1.1]). Hence ν_{n}^{2} = \frac{1}{n} ∑_{d=0}^{n−1} (−1)^{d}δ\underline{d}ν_{n} = \frac{1}{n} ∑_{d=0}^{n−1} (−1)^{2d}ν_{n} = ν_{n}. □

A further immediate consequence of 3.1 is the following:

(28)  ∑_{D⊆\underline{n−1}} t^{ΣD}δ_{d}ν_{n} = ∏_{j=1}^{n−1} (1 − t^{j})ν_{n}  (t a variable),

where ∑ D := ∑_{i∈D} i ([3, Theorem 2.1], [4, (9)]). Putting t := ε we obtain

(29)  λ_{n}ν_{n} = ν_{n}.

This equation leads to a short proof of (Ind) and, simultaneously, of the fact that λ_{n} is a Lie idempotent:

By a direct calculation one has the equation λ_{n}ζ_{n} = λ_{n} ([16, Lemma 2, 1)], [2,3.4.3]). Hence, by (29), ν_{n}ℂS_{n} = λ_{n}ζ_{n}ν_{n}ℂS_{n} ⊆ λ_{n}ζ_{n}ℂS_{n}. Now dim λ_{n}ζ_{n}ℂS_{n} ≤ dim ζ_{n}ℂS_{n} =

dim M_{1}^{S_{n}} = (n − 1)!, and dim ν_{n}ℂS_{n} = dim ν_{n}A(1, …_{n}, 1) = dim L(1, …_{n}, 1) by (18) and

(6). It is well known that the Lie monomials x_{1} ∘ x_{2σ} ∘ ⋯ ∘ x_{nσ} (σ ∈ Stab_{S_{n}}(1)) form a basis of L(1, …_{n}, 1) (cf., e.g., [2, 4.8.1]). Hence dim ν_{n}ℂS_{n} = |Stab_{S_{n}}(1)| = (n − 1)!.^{3}) We conclude that

(30)  ν_{n}ℂS_{n} = λ_{n}ζ_{n}ℂS_{n} = λ_{n}ℂS_{n},

and the left multiplication by λ_{n} induces a ℂS_{n}-right module isomorphism of ζ_{n}ℂS_{n} onto ν_{n}ℂS_{n}. This yields (Ind). □

As ν_{n} is an idempotent, (30) implies that

(31)  ν_{n}λ_{n} = λ_{n}.

Now (29) and (31) show that λ_{n} is a Lie idempotent, by (7). □

We conclude this section by a further application of 3.2:

3.4 Corollary. Let D ⊆ \underline{n − 1} , 0 ≤ k < n. For all S ⊆ \underline{n − 1} set b_{S} := \binom{|S\(1+S_{0})|}{k−|(1+(D\S))∪(S\D)|}

where S_{0} := S ∪ {0}. Then

δ_{D}δ\underline{k} = ∑_{S} b_{S}δ_{S}

where the sum ranges over all S ⊆ \underline{n − 1} such that S\(1+S_{0}) ⊆ ((1+D)\D)∪(D\(1+D)).

3Alternatively, we may use the fact that ν_{n} is an idempotent to derive this formula for dim ν_{n}ℂS_{n} from a result due to Frobenius [9]: The coefficient of id in ν_{n} is \frac{1}{n}. Hence, if χ is the character of the ℂS_{n}-module ν_{n}ℂS_{n}, then ∑_{σ∈S_{n}} \frac{1}{n} = χ(id) = dim ν_{n}ℂS_{n}.

Proof. For every σ ∈ S_{n} put X^{k}(σ, D) := {π|π ∈ X_{n}, D(σπ^{−1}) = D and d(π) = k}. We show: (32)

If σ ∈ S_{n} such that S\(1 + S_{0}) ⊆ ((1 + D)\D) ∪ (D\(1 + D)) (where S := D(σ)), then |X^{k}(σ, D)| = b_{S}.

By 3.2a), the hypothesis of (32) implies that there exists an element π ∈ X_{n} such that D(σπ^{−1}) = D. By 3.2b) there is then a 1-1 correspondence between X^{k}(σ, D) and the set of all subsets of order k of (L ∪ T_{σ})\{1σ^{−1}} containing L. Therefore

|X^{k}(σ, D)| = \binom{|T_{σ}| − 1}{k − |L|} = b_{S},

as |S\(1 + S_{0})| = |(1 + S_{0})\S| − 1. This shows (32). We conclude that

δ_{D}δ\underline{k} = ∑_{ρ∈S_{n}(D)} ∑_{{π∈Xn \atop d(π)=k}} ρπ = ∑_{σ∈S_{n}} |X^{k}(σ, D)|σ = ∑ b_{D(σ)}σ,

where the last sum ranges over all σ ∈ S_{n} such that P_{σ} ⊆ ((1 + D)\D) ∪ (D\(1 + D)).□

We summarize the logical structure of the principal parts of the preceding sections by means of the following diagram:

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(WDF)  1.3  1.4  1.5 (Ind) (29)  (5) (∼ (DSW)) 3.1 3.2  (via 2.1)

4. Some remarks about Solomon’s descent algebra

Let Δ_{n} be the subspace of ℂS_{n} generated by all elements δ_{S} (S ⊆ \underline{n − 1} ). A particular aspect of 3.4 is that for every D ⊆ \underline{n − 1} , 0 ≤ k < n, the product δ_{D}δ\underline{k} is contained in Δ_{n}. This is a special case of the following result due to Solomon [24]:

4.1 Theorem. Δ_{n} is multiplicatively closed.

Hence Δ_{n} is a subalgebra of ℂS_{n}, called the Solomon algebra (with respect to \underline{n} ). It should be noted that all Lie idempotents mentioned before (ν_{n}, λ_{n}, ρ_{n}) are elements of Δ_{n}. The equations in (5), 3.1, (29), (31), 3.4 may be viewed as details of the multiplicative structure of Δ_{n}. Garsia and Reutenauer proved a remarkable characterization of the Solomon algebra: By means of certain Lie terms, they defined a set of subspaces of the free associative algebra which are normalized by an element γ ∈ ℂS_{n} if and only if γ ∈ Δ_{n} ([12, Theorem 4.5]).

In the following we give two rather different but equally simple proofs of 4.1. In the first one we introduce a graph structure on the set of points S_{n}. The second one will consist in showing 4.3.

We define a lexicographic ordering on S_{n} by putting π <_{lex} ρ if π ≠ ρ and iπ < iρ for the smallest i ∈ \underline{n} such that iπ ≠ iρ (π, ρ ∈ S_{n}).

An element σ^{∗} ∈ S_{n} is called a neighbour of σ ∈ S_{n} if there is a number k ∈ \underline{n − 1} such that

σ^{∗} = σ(k, k + 1) and |kσ^{−1} − (k + 1)σ^{−1}| ≠ 1.

The relation on S_{n} defined in this manner is obviously symmetric and hence yields a nonoriented graph structure on the set of vertices S_{n}. We denote by [σ] the component of σ. Then we have D(ρ) = D(σ) for every ρ ∈ [σ]. This observation is the trivial part of the following result:

4.2 Proposition. Let D ⊆ \underline{n − 1} , σ ∈ S_{n}(D). Then [σ] = S_{n}(D).

Proof. For all ℓ ∈ ℕ_{0} we put M_{ℓ} := {μ|μ ∈ S_{ℓ}, (i + 1)μ = iμ − 1 for all i ∈ D(μ)}. The following statement is easily seen:

(33)

Let λ ∈ S_{n}, k := nλ. Then λ ∈ M_{n} if and only if (k + j)λ = n − j for all

j ∈ \underline{n − k} ∪ \{0\}

and λ|\underline{k−1} ∈ M_{k−1}.

As a consequence, we show

(34) For every T ⊆ \underline{n − 1} there exists a unique element μ_{n}^{T} ∈ M_{n} ∩ S_{n}(T),

in other words, M_{n} is a set of representatives for the defect classes in S_{n}. In order to prove (34) by induction on n, we put k := max((\underline{n − 1} ∪ \{0\})\T) + 1. Then we may assume that M_{k−1} ∩ S_{k−1}(T ∩ \underline{k − 1} ) contains a unique element μ. By (33), M_{n} ∩ S_{n}(T) contains the permutation

λ := ( \begin{matrix}1 & … & k − 1 & k & k + 1 & … & n − 1 & n \\ 1μ & … & (k − 1)μ & n & n − 1 & … & k + 1 & k \end{matrix} )

as its only element.

Our next step is to prove

(35) If ρ ∈ S_{n}\M_{n}, then there exists a neighbour ρ^{∗} of ρ such that ρ^{∗} <_{lex} ρ.

We have to show that there is a number k ∈ \underline{n − 1} such that kρ^{−1} − (k + 1)ρ^{−1} > 1 as then the element ρ^{∗} := ρ(k, k + 1) has the required properties. By hypothesis we have iρ − (i + 1)ρ ≥ 2 for some i ∈ \underline{n − 1} . Now it suffices to put k := min\{j|(i + 1)ρ ≤ j ≤ iρ, jρ^{−1} ≥ i} − 1.

By (34), M_{n} ∩ [ρ] ⊆ M_{n} ∩ S_{n}(D(ρ)) = {μ_{n}^{D(ρ)}} for all ρ ∈ S_{n}. The assumption that μ_{n}^{D(ρ)} ∉ [ρ] leads, by (35), to the contradiction that [ρ] is infinite. Hence μ_{n}^{D(ρ)} ∈ [ρ] for all ρ ∈ S_{n}. In particular, μ_{n}^{D} ∈ [ρ] ∩ [σ] for all ρ ∈ S_{n}(D). □

Proof of 4.1. For any σ, σ^{∗} ∈ S_{n} which are neighbours of each other we set

N_{σ,σ^{∗}}^{1} := {(ξ, ρ)|ξ, ρ ∈ S_{n}, ξρ = σ, (σ^{∗}σ^{−1})ξ is a neighbour of ξ},
N_{σ,σ^{∗}}^{2} := {(ξ, ρ)|ξ, ρ ∈ S_{n}, ξρ = σ, ρ(σ^{−1}σ^{∗}) is a neighbour of ρ}.

Let k ∈ \underline{n − 1} such that σ^{−1}σ^{∗} = (k, k + 1). If ξ, ρ ∈ S_{n} such that ξρ = σ, then ξ(kρ^{−1}, (k + 1)ρ^{−1}) = ξρ(k, k + 1)ρ^{−1} = σ(k, k + 1)ρ^{−1} = σ^{∗}σ^{−1}ξ. Hence σ^{∗}σ^{−1}ξ is a neighbour of ξ if and only if |kρ^{−1} − (k + 1)ρ^{−1}| = 1, i.e., if and only if ρσ^{−1}σ^{∗} is not a neighbour of ρ. We write Π_{σ} for the set of all pairs (ξ, ρ) ∈ S_{n} × S_{n} such that ξρ = σ. Then it follows that

(36)  Π_{σ} is the disjoint union of N_{σ,σ^{∗}}^{1} and N_{σ,σ^{∗}}^{2}.

It is straightforward to verify the following, for any ξ, ρ ∈ S_{n}:

(37)  (ξ, ρ) ∈ N_{σ,σ^{∗}}^{1} ⟹ (σ^{∗}σ^{−1}ξ, ρ) ∈ N_{σ^{∗},σ}^{1},
(38)  (ξ, ρ) ∈ N_{σ,σ^{∗}}^{2} ⟹ (ξ, ρσ^{−1}σ^{∗}) ∈ N_{σ^{∗},σ}^{2}.

By symmetry, we conclude that, in particular, |N_{σ,σ^{∗}}^{j}| = |N_{σ^{∗},σ}^{j}| (j = 1, 2). We observe that the mapping

(ξ, ρ) ↦ \{ \begin{matrix}(σ^{∗}σ^{−1}ξ, ρ) & if (ξ, ρ) ∈ N_{σ,σ^{∗}}^{1} \\ (ξ, ρσ^{−1}σ^{∗}) & if (ξ, ρ) ∈ N_{σ,σ^{∗}}^{2} \end{matrix}

is a bijection of Π_{σ} onto Π_{σ^{∗}}. In (37), we have D(ξ) = D(σ^{∗}σ^{−1}ξ), and in (38), similarly, D(ρ) = D(ρσ^{−1}σ^{∗}). As a consequence, we obtain

(39) |Π_{σ} ∩ (S_{n}(D) × S_{n}(D′))| = |Π_{σ^{∗}} ∩ (S_{n}(D) × S_{n}(D′))| for all D, D′ ⊆ \underline{n − 1} .

Up to this point our hypothesis was that σ, σ^{∗} were neighbours of each other. But 4.2 shows now that (39) holds, in fact, for any σ, σ^{∗} ∈ S_{n} such that D(σ) = D(σ^{∗}). Hence

δ_{D} · δ_{D′} = ∑_{σ∈S_{n}} |Π_{σ} ∩ (S_{n}(D) × S_{n}(D′))|σ = ∑_{T ⊆\underline{n−1}} |Π_{σ_{T}} ∩ (S_{n}(D) × S_{n}(D′))|δ_{T}

where, for T ⊆ \underline{n − 1} , the element σ_{T} is an arbitrary representative of S_{n}(T). □

It is obvious that the coefficient of the basis element δ_{T} in the representation of δ_{D} ·δ_{D′} must necessarily be the one given in the last formula of the proof once it is known that δ_{D} · δ_{D′} is contained in Δ_{n}. Therefore, the essential statement which yields the claim is not that formula but the foregoing assertion (39) and its subsequent comment.

We now describe another basis of Δ_{n} and prove a formula for the product of any two of its elements as a linear combination of basis elements whose coefficients, by contrast, turn out to be combinatorially interesting and not obvious at all (4.3). This formula, in a more general context, is essentially due to Solomon ([24, Theorem 1]). A simple application of Coleman’s lemma [15, 4.3.7] suffices to gather from Solomon’s result the special version which is of interest here. It has been stated explicitly by Garsia and Reutenauer in [12, Proposition 1.1] who refer to [11]. We give an independent elementary proof.

If q_{1}, …, q_{ℓ} ∈ ℕ such that q_{1} + ⋯ + q_{ℓ} = n, the ℓ-tuple q = (q_{1}, …, q_{ℓ}) is called a decomposition of n, and we write q ⊧ n. We define the length of q by |q| := ℓ. Then D(q) := \{q_{1}, q_{1} + q_{2}, …, q_{1} + ⋯ + q_{ℓ−1}\} is a subset of \underline{n − 1} . The mapping q ↦ D(q) is a bijection from the set of all decompositions of n onto the power set of \underline{n − 1} . We put

Ξ^{q} := ∑_{T ⊆D(q)} δ_{T} .

Then {Ξ^{q}|q ⊧ n} is a basis of Δ_{n}. For r = (r_{1}, …, r_{k}), q = (q_{1}, …, q_{ℓ}) ⊧ n let M_{r,q} be the set of all (k × ℓ)-matrices M = (m_{ij}) over ℕ_{0} such that

∑_{j∈\underline{ℓ}} m_{ij} = r_{i} for all i ∈ \underline{k} ,
∑_{i∈\underline{k}} m_{ij} = q_{j} for all j ∈ \underline{ℓ} .

In short, summing up the rows (columns resp.) of M gives r (q resp.). Furthermore, let w(M) be the sequence of all non-zero entries of M written according to the natural order of its rows. More formally, if t is the number of all non-zero entries of M and s ∈ \underline{t} , we set w_{s} := m_{ij} if m_{ij} ≠ 0 and if there exist exactly s − 1 entries m_{x,y} ≠ 0 such that (x, y) is lexicographically smaller than (i, j). Then w(M) = (w_{1}, …, w_{t}).

4.3 Proposition. Ξ^{r} · Ξ^{q} = ∑_{M∈M_{r,q}} Ξ^{w(M)} for all r, q ⊧ n.

The sum on the right is equal to ∑_{s⊧n} c_{r,q,s}Ξ^{s} where c_{r,q,s} is the number of all M ∈ M_{r,q} such

that w(M) = s. Obviously, 4.1 is an immediate consequence of 4.3. We will derive 4.3 from our next lemma for which we need some more preparations. For every q = (q_{1}, …, q_{ℓ}) ⊧ n, the standard partition relative to q is defined to be the ℓ-tuple P^{q} := (P_{1}^{q}, …, P_{ℓ}^{q}) where

P_{j}^{q} := (q_{1} + ⋯ + q_{j−1}) + \underline{q_{j}} (j ∈ \underline{ℓ} ).

18 The stabilizer of P^{q} in S_{n} is a Young subgroup Y^{q} of S_{n} of type S_{q_{1}} ×⋯×S_{q_{ℓ}}. Every coset Y^{q}σ (σ ∈ S_{n}) contains a lexicographically smallest element. The set S^{q} of these elements is called the Solomon system of Y^{q} in S_{n}. We have the following obvious characterization:

(40)  σ ∈ S^{q} if and only if σ|_{P_{j}^{q}} is increasing, for all j ∈ \underline{ℓ} .

This implies that

Ξ^{q} = ∑_{σ∈S^{q}} σ for all q ⊧ n.

For every r, q ⊧ n and M = (m_{ij}) ∈ M_{r,q} we put (following the main idea of Coleman’s lemma [15, 4.3.7])

S^{r}(M) := \{ρ|ρ ∈ S^{r}, |P_{i}^{r} ∩ P_{j}^{q}ρ^{−1}| = m_{ij} for all i ∈ \underline{k} , j ∈ \underline{ℓ} \}

where k = |r|, ℓ = |q|. We have the following remark:

(41)  P_{i}^{r} ∩ P_{j}^{q}ρ^{−1} is an interval, for all i ∈ \underline{k} , j ∈ \underline{ℓ} , ρ ∈ S^{r}.

For, if x, y ∈ P_{i}^{r} ∩ P_{j}^{q}ρ^{−1} and z ∈ ℕ such that x < z < y, then z ∈ P_{i}^{r} and, by (40), xρ < zρ < yρ. Now xρ, yρ ∈ P_{j}^{q}, hence zρ ∈ P_{j}^{q}, i.e., z ∈ P_{j}^{q}ρ^{−1}. ^{4})

4.4 Lemma.^{5}) Let r, q ⊧ n and M ∈ M_{r,q}. Then the product mapping (ρ, σ) ↦ ρσ (ρ, σ ∈ S_{n}) induces a bijection of S^{r}(M) × S^{q} onto S^{w(M)}.

Proof. Let M = (m_{ij}), k := |r|, ℓ := |q|. For all (i, j) ∈ \underline{k} × \underline{ℓ} we put

R_{ij} := ( ∑_{(x,y)∈\underline{i−1} ×\underline{ℓ}} m_{x,y} + ∑_{y∈\underline{j−1}} m_{i,y}) + \underline{m_{ij}} .

Up to empty sets (which arise if m_{ij} = 0), the sequence (R_{11}, R_{12}, …, R_{kℓ}) is the standard partition of \underline{n} relative to w(M). As |P_{i}^{r}| = ∑_{j} m_{ij} = ∑_{j} |R_{ij}|, we have

(42)  P_{i}^{r} = ⋃_{j} R_{ij} for all i ∈ \underline{k} .

Next we prove

(43)  P_{i}^{r} ∩ P_{j}^{q}ρ^{−1} = R_{ij} for all i ∈ \underline{k} , j ∈ \underline{ℓ} , ρ ∈ S^{r}(M).

4The proof shows that (41) holds for arbitrary intervals I, J instead of P_{i}^{r}, P_{j}^{q} whenever ρ|_{I} is increasing or decreasing.

5D. B.

We have P_{i}^{r} = ⋃_{j}(P_{i}^{r} ∩ P_{j}^{q}ρ^{−1}) and, by definition of S^{r}(M), |P_{i}^{r} ∩ P_{j}^{q}ρ^{−1}| = m_{ij} = |R_{ij}|.

Hence it suffices to show that for any j_{1}, j_{2} ∈ \underline{ℓ} such that j_{1} < j_{2} every element x_{1} ∈ P_{i}^{r} ∩ P_{j_{1}}^{q}ρ^{−1} is smaller than every element x_{2} ∈ P_{i}^{r} ∩ P_{j_{2}}^{q}ρ^{−1}. But for such elements x_{1}, x_{2} we have x_{1}ρ ∈ P_{j_{1}}^{q}, x_{2}ρ ∈ P_{j_{2}}^{q}, hence x_{1}ρ < x_{2}ρ as P^{q} is a standard partition of \underline{n} . This implies, by (40), that x_{1} < x_{2}.

In particular, (43) implies that

(44)  P_{j}^{q} = ⋃_{i} R_{ij}ρ for all j ∈ \underline{ℓ} , ρ ∈ S^{r}(M).

Now we show that

(45)  ρσ ∈ S^{w(M)} for all ρ ∈ S^{r}(M), σ ∈ S^{q}.

By (43) and (40), ρσ|_{R_{ij}} is the composition of two increasing functions, hence is increasing. Therefore, ρσ induces an increasing function on every part of the standard partition of \underline{n} relative to w(M). Now (45) follows from (40).

Let τ ∈ S^{w(M)}. If ρ ∈ S^{r}(M), σ ∈ S^{q} such that ρσ = τ, then, by (44), we have

(46)  P_{j}^{q}σ = (⋃_{i} R_{ij}) τ for all j ∈ \underline{ℓ} .

Hence σ is the uniquely determined permutation of \underline{n} which maps P_{j}^{q} increasingly onto

(⋃_{i} R_{ij}) τ, for all j ∈ \underline{ℓ} . The uniqueness of σ implies that of ρ as ρ = τσ^{−1}. On the other hand, to prove existence, we note first that

|P_{j}^{q}| = ∑_{i} m_{ij} = | ⋃_{i} R_{ij}| = |(⋃_{i} R_{ij})τ| for all j ∈ \underline{ℓ} .

Hence there exists an element σ ∈ S^{q} with the property (46). Let ρ := τσ^{−1}. We claim

(47)  ρ ∈ S^{r}.

Let i ∈ \underline{k} and x_{1}, x_{2} ∈ P_{i}^{r} such that x_{1} < x_{2}. By (42), there exist j_{1}, j_{2} ∈ \underline{ℓ} such that j_{1} ≤ j_{2} and x_{1} ∈ R_{i,j_{1}}, x_{2} ∈ R_{i,j_{2}}. By definition of σ we have (46), hence x_{h}τ ∈ R_{i,j_{h}}τ ⊆ P_{j_{h}}^{q}σ (h = 1, 2). If j_{1} = j_{2}, then x_{1}τ < x_{2}τ as τ ∈ S^{w(M)}, hence

(48)  x_{1}ρ = x_{1}τσ^{−1} < x_{2}τσ^{−1} = x_{2}ρ,

since σ ∈ S^{q}. Finally, if j_{1} < j_{2}, then again (48) holds, because x_{h}ρ = x_{h}τσ^{−1} ∈ P_{j_{h}}^{q} (h = 1, 2). This proves (47).

Furthermore, (46) and (42) imply that P_{i}^{r} ∩ P_{j}^{q}ρ^{−1} = P_{i}^{r} ∩ P_{j}^{q}στ^{−1} = ⋃_{g} R_{ig} ∩ ⋃_{h} R_{h,j} =

R_{ij}. This and (47) show that ρ ∈ S^{r}(M). □

Proof of 4.3. Using 4.4, we obtain that for all r, q ⊧ n,

\begin{aligned}Ξ^{r} · Ξ^{q} & = ∑_{ρ∈S^{r}} ρ · ∑_{σ∈S^{q}} σ = ( ∑_{M∈M_{r,q}} ∑_{ρ∈S^{r}(M)} ρ) · ∑_{σ∈S^{q}} σ \\ & = ∑_{M∈M_{r,q}} ( ∑_{ρ∈S^{r}(M)} ρ · ∑_{σ∈S^{q}} σ) = ∑_{M∈M_{r,q}} ∑_{τ∈S^{w(M)}} τ = ∑_{M∈M_{r,q}} Ξ^{w(M)}. □ \end{aligned}

There is a strong connection between the Solomon algebra and the character theory of S_{n}. For every q ⊧ n we write ξ^{q} for the Young character with respect to Y^{q}, that is, ξ^{q} = (1_{Y^{q}})^{S_{n}}. If q, r ⊧ n such that ξ^{q} = ξ^{r} (or, equivalently, Y^{q} is conjugate to Y^{r} in S_{n}), we write q ≈ r.

It is well known that the same rule as in 4.3 holds for the (tensor) product of two Young characters ξ^{r}, ξ^{q} if the Ξ’s are replaced by ξ’s ([14, 2.9.16]). Hence

Ξ^{q} ↦ ξ^{q} (q ⊧ n)

extends linearly to an algebra epimorphism c of Δ_{n} onto the character ring Cl(S_{n}) of S_{n} over ℂ. Solomon [24] showed that its kernel is the Jacobson radical of Δ_{n}:

(49)  ker c = J(Δ_{n}).

This may be seen as follows: By the semisimplicity of Cl(S_{n}) we know that J(Δ_{n}) ⊆ ker c. On the other hand, ker c is the linear span of the elements Ξ^{q}−Ξ^{r} where q ≈ r. Exploiting the multiplication rule 4.3, the nilpotency of ker c is readily seen (see the proof of Theorem 3.4 in [1]). Hence ker c is a nilpotent ideal, and (49) follows.

Moreover, Atkinson has shown that the nilpotency index of J(Δ_{n}) is n − 1 ([1, 3.5]). If ξ ∈ Cl(S_{n}) and C is a conjugacy class of S_{n}, we write ξ(C) for the unique value

of ξ on C. The conjugacy classes of S_{n} may be indexed by the partitions of n. More precisely, we define C^{p} (where p ⊢ n) to be the set of all elements of S_{n} which have a cycle decomposition of type p. We note:

(50) The linear mappings c^{p} (where p ⊢ n) such that

c^{p}(Ξ^{q}) = ξ^{q}(C^{p}) for all q ⊧ n

are a full set of irreducible representations for Δ_{n}.

(For a different description see [1,3.].) To prove (50) it suffices to define d^{p}(ξ) := ξ(C^{p}) for all ξ ∈ Cl(S_{n}), p ⊢ n, and to observe that the mappings d^{p} are a full set of irreducible representations of Cl(S_{n}). Since c^{p} is the composition of c and d^{p}, this yields (50), in view of (49).

Furthermore, it should be mentioned here (without proof) that c(δ_{D}) is the character of a certain skew representation of S_{n}, for every D ⊆ \underline{n − 1} .

We conclude this exposition by some remarks about the ideal of Δ_{n} generated by all Lie idempotents in Δ_{n}. Using (7) and 3.1, this is easily seen to be ν_{n}Δ_{n}. Moreover, the ideal ν_{n}Δ_{n} ∩ J(Δ_{n}) has codimension 1 in ν_{n}Δ_{n}, and the coset ν_{n} + (ν_{n}Δ_{n} ∩ J(Δ_{n})) is the set of all Lie idempotents in Δ_{n}. We state, without giving details here:

4.5 Proposition. The dimension of ν_{n}Δ_{n} is the number of all decompositions of n which are Lyndon words over the alphabet ℕ.

(A proof of this result and related topics will be given in a forthcoming paper.)

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