On the Automorphism Group of Solomon’s Descent Algebra

Thorsten Bauer

Mathematisches Seminar Christian–Albrechts–Universität zu Kiel D-24118 Kiel, Germany

By Δ_{n} we denote the subalgebra of ℚS_{n} which is known as Solomon’s Descent Algebra [7, 6, 1]. This article gives a sketch of a proof for the following theorem about the automorphisms of Δ_{n}. (Details and related developments will be given in a forthcoming publication.)

Main Theorem For the automorphism group of Solomon’s Descent Algebra Δ_{n} the following holds:

Aut(Δ_{n}) = \{ \begin{matrix}Inn(Δ_{n}) & if n odd, \\ Inn(Δ_{n}) × C_{2} & if n even. \end{matrix}

The proof of this theorem splits into four main steps:

• the reduction of the problem to show that a stabilizer of a certain set is generated by inner automorphisms,

• the definition of a graph Γ_{n},

• the mentioned stabilizer is determined by its action on Γ_{n},

• the action of the stabilizer on Γ_{n} is induced by conjugation by invertible elements of Δ_{n}, i.e. by inner automorphisms, and possibly a central involutionary outer automorphism.

1 Notation

We write ℕ for the set of positive integers {1, 2, … }, ℕ^{∗} for the free monoid generated by ℕ and ℚℕ^{∗} for the free algebra generated by ℕ over the field of the rational numbers ℚ. The elements of ℕ^{∗} are written as words in the alphabet ℕ, e.g. 1 3 2 ∈ ℕ^{∗} is the product of 1, 3 and 2. If w = w_{1} … w_{k} is a word, we set |w| := k, the length of w.

Let n ∈ ℕ and p = p_{1} … p_{k} ∈ ℕ^{∗} such that p_{1}, …, p_{k} ∈ ℕ. The word p is called a partition of n (p ⊢ n) if p_{1} + ⋯ + p_{k} = n and p_{1} ≥ p_{2} ≥ ⋯ ≥ p_{k}. We write p(n) for the number of all partitions of n. For each letter c we set

a_{c}(p) := |{i | p_{i} = c}|,

the number of occurences of the letter c in p. Then we may write

p = n^{a_{n}(p)}(n − 1)^{a_{n−1}(p)} … 1^{a_{1}(p)}.

2 Reduction of the problem

By a theorem of Malcev [4], [5, 11.6] we know that the set 1 + Rad(Δ_{n}) of invertible elements acts transitively by conjugation on the set of complements of Rad(Δ_{n}).

By Frattini’s lemma [3, 3.3]^{1}, for each complement H of Rad(Δ_{n}) we have:

Aut(Δ_{n}) = Stab_{Aut(Δ_{n})}(H) \overline{(1 + Rad(Δ_{n}))},  (1)

where \overline{(1 + Rad(Δ_{n}))} denotes the group of automorphisms induced by conjugation by the elements of 1 + Rad(Δ_{n}).

Now we construct a complement H of Rad(Δ_{n}) to which the above mentioned theorem and lemma will be applied.

In [2], D. Blessenohl and H. Laue define elements^{2} ν^{p}, p ⊢ n, with the following properties [2, 1.2 Proposition]:

(a) ν^{p} is an idempotent for each p ⊢ n.

(b) ν^{p}ν^{r} = 0 for all partitions p, r such that p ≠ r.

(c) ∑_{p⊢n} ν^{p} = 1.

1Though the lemma is only stated for finite groups, it holds for infinte groups, too.

2In [2] these elements are not indexed by the partitions themselves but by listing them in the lexicographically decreasing order: ν^{(j)} instead of ν^{p} if p is the j-th partition.

(d) Δ_{n} = ⟨ ν^{p} | p ⊢ n ⟩_{ℚ} ⊕ Rad(Δ_{n}).

The set H := ⟨ ν^{p} | p ⊢ n ⟩_{ℚ} is a subalgebra-complement of Rad(Δ_{n}) in Δ_{n}. We observe that

H ≅ \underbrace{ℚ ⊕ ⋯ ⊕ ℚ}_{p(n)},

and that {ν^{p} | p ⊢ n} is the unique set of p(n) mutually orthogonal idempotents ≠ 0 of H.

By Equation (1) it suffices to show that Stab_{Aut(Δ_{n})}(H) is generated by inner automorphisms in the case that n is even and by inner automorphisms and an involutionary outer automorphism in the case that n is odd.

3 Directed graph of partitions

Now we define a directed graph Γ_{n}, the nodes of which are the partitions of n. The node r is called connected with p (r |∼ ·_{pf} p) if |r| = |p| + 1 and there exist letters c, d ∈ ℕ such that c ≠ d, a_{c}(r) = a_{c}(p) + 1, a_{d}(r) = a_{d}(p) + 1 and a_{c+d}(r) + 1 = a_{c+d}(p), i.e. p can be obtained from r by coalescing two different letters of r and reordering the letters to obtain a partition.

The shape of the graph Γ_{7} is shown in Figure 1.

Obviously Γ_{n} has two (three resp.) connected subgraphs in the case that n is odd (even resp.). More precisely, the partition 1 … 1 if n is odd and 1 … 1 and 2 … 2 if n is even are not connected with any other node. We observe further that for each k < n the subgraph of Γ_{n} induced by all partitions which include the letter k is isomorphic to Γ_{n−k}. It is not at all trivial to see that, if φ is an automorphism of Γ_{n}, for each k ∈ ℕ this subgraph is invariant under φ. We get by an inductive argument the following lemma about Γ_{n}:

Lemma 1 For the automorphism group of Γ_{n} holds:

Aut(Γ_{n}) = \begin{cases}\{id\} & if n odd, \\ \{id, τ\} & if n even, \end{cases}

where τ is the automorphism of Γ_{n} that exchanges the nodes 1 … 1 and 2 … 2 and fixes the other nodes.

4 The action of Stab_{Aut(Δ_{n})}(H) on Γ_{n}

Now let φ be an automorphism of Δ_{n} such that H^{φ} ⊆ H.

[figure: 7; 6 1; 5 2; 4 3; 5 1 1; 4 2 1; 3 3 1; 3 2 2; 4 1 1 1; 3 2 1 1; 2 2 2 1; 3 1 1 1 1; 2 2 1 1 1; 2 1 1 1 1 1; 1 1 1 1 1 1 1]

Figure 1: The graph Γ_{7}

Since the elements ν^{p}, p ⊢ n, are mutually orthogonal idempotents, φ acts on the set {ν^{p} | p ⊢ n}. Therefore φ induces an action on the set of partitions of n.

It turns out that the action of φ on the set of partitions of n is compatible with the relation |∼ ·_{pf}, i.e. for all partitions r, p of n we have:

r |∼ ·_{pf} p ⟺ r^{φ} |∼ ·_{pf} p^{φ}.

Therefore φ induces an automorphism of Γ_{n}.

By Lemma 1 we see that the spaces ν^{p}Δ_{n}ν^{r} are φ-invariant for all partitions p, r such that r |∼ ·_{pf} p.

By [1, 2.2 Corollary] ^{3} we get then

Lemma 2 Let p, r partitions of n such that r |∼ ·_{pf} p. Then ν^{p}Δ_{n}ν^{r} is a one-dimensional φ-invariant space, and φ|_{ν^{p}}_{Δ_{n}}_{ν^{r}} has an eigenvalue ≠ 0.

By [1, 2.4 Corollary] we can deduce

Rad(Δ_{n}) = ⟨ ν^{p}Δ_{n}ν^{r} | r |∼ ·_{pf} p ⟩_{(+,·)}

from which we see that the action of φ on Rad(Δ_{n}) is determined by the action on the subspaces ν^{p}Δ_{n}ν^{r}, r |∼ ·_{pf} p.

Now we define a certain subgraph Γ_{n}^{∗} of Γ_{n}, the nodes of which are the partitions of n, too. At first, we set

P_{n}^{∗} := {p ∈ P_{n} | p = p_{1} … p_{k}, p_{1} ≠ p_{k}},

i.e. the set of all partitions of n that have at least two different letters. For each partition p = p_{1} … p_{k} ∈ P_{n}^{∗} let

ι(p) := min{i | i ∈ {1, …, k}, p_{i} ≠ p_{1}}

and

ζ(p) := (p_{1} + p_{ι(p)})p_{2} … p_{ι(p)−1}p_{ι(p)+1} … p_{k},

i.e. we form the sum of the two largest different letters that occur in p, delete these two letters from p and add the sum as a new letter. If p ∈ P_{n}^{∗} then ζ(p) is a partition of n and it holds p |∼ ·_{pf} ζ(p).

The partitions p, r of n are called connected in Γ_{n}^{∗} (p ≻ r), if p ∈ P_{n}^{∗} and r = ζ(p) or if r = d^{k} for some d, k ∈ ℕ and p = d^{k−1}(d − 1)1.

The relation ≻ is coarser than |∼ ·_{pf} and defines a spanning tree for the “big” connected component of Γ_{n}, seen as an undirected graph, that contains the node n. Figure 2 shows Γ_{7}^{∗}.

We obtain

3In [1], the spaces ω_{p}Λ^{r} are treated. But these are isomorphic to ν^{p}Δ_{n}ν^{r}.

[figure: 7; 6 1; 5 2; 4 3; 5 1 1; 4 2 1; 3 3 1; 3 2 2; 4 1 1 1; 3 2 1 1; 2 2 2 1; 3 1 1 1 1; 2 2 1 1 1; 2 1 1 1 1 1; 1 1 1 1 1 1 1]

Figure 2: The edges defined by ≻ Main Lemma Let φ ∈ Stab_{Aut(Δ_{n})}(H) such that φ|_{ν^{p}}_{Δ_{n}}_{ν^{r}} = id for all partitions p, r such that r ≻ p. Then φ|_{Rad(Δ_{n})} = id.

The proof of the Main Lemma needs hard conclusions. For this reason we give only some instructive examples in Section 5 that demonstrate the important ideas.

But now we may easily construct for each φ ∈ Stab_{Aut(Δ_{n})}(H) an invertible element h ∈ H such that the conjugation by h coincides with the action of φ on the subspaces ν^{p}Δ_{n}ν^{r}, r ≻ p. By the Main Lemma, we see that the action of h and φ coincide on Rad(Δ_{n}).

Therefore the automorphism ψ : Δ_{n} → Δ_{n}, x ↦ (x^{φ})^{h^{−1}} centralizes Rad(Δ_{n}) and stabilizes the set {ν^{p} | p ⊢ n}. By what we have observed above, ψ induces an automorphism of Γ_{n}. Now Lemma 1 implies, that ψ is the identity or an involution. Hence, it follows

Lemma 3

Stab_{Aut(Δ_{n})}(H) = \begin{cases}\{inner automorphisms induced by H\} & if n odd, \\ \{inner automorphisms induced by H\} × C_{2} & if n even. \end{cases}

5 Example for n = 7

We illustrate the proof of the Main Lemma by discussing the example of n = 7 which provides a spectrum of all three typical cases which may occur in general. This discussion therefore does not only give a flavour of the general proof but presents all its basic elements in a concrete form.

In [1, p. 718] a basis of Δ_{n} consisting of idempotents ν_{q}, q ⊧ n, is given. In the following, we consider the linear extension ν : ⟨q | q ⊧ n⟩_{ℚ} → Δ_{n} of the mapping {q | q ⊧ n} → Δ_{n}, q ↦ ν_{q}.

We use the Lie product ∘ on ℚℕ^{∗} defined by a ∘ b := ab − ba for all a, b ∈ ℚℕ^{∗}.

E.g. we write

ν_{1∘2} = ν_{12−21} = ν_{12} − ν_{21}.

In order to illustrate the proof of the Main Lemma we may assume φ fixes elementwise the subspaces ν^{7}Δ_{n}ν^{61}, ν^{7}Δ_{n}ν^{52}, ν^{7}Δ_{n}ν^{43}, ν^{61}Δ_{n}ν^{511}, ν^{61}Δ_{n}ν^{421}, ν^{52}Δ_{n}ν^{322}, ν^{43}Δ_{n}ν^{331}, ν^{511}Δ_{n}ν^{4111}, ν^{511}Δ_{n}ν^{3211}, ν^{322}Δ_{n}ν^{2221}, ν^{4111}Δ_{n}ν^{31111}, ν^{3211}Δ_{n}ν^{22111}, ν^{31111}Δ_{n}ν^{211111}.

We have to show that the subspaces ν^{52}Δ_{n}ν^{421}, ν^{43}Δ_{n}ν^{421}, ν^{421}Δ_{n}ν^{3211}, ν^{331}Δ_{n}ν^{3211} are fixed elementwise by φ, too.

Figure 3 shows this situation. The thick edges represent the subspaces on which the action of φ is assumed as identity. The thin edges represent the subspaces on which the action of φ is not known. The eigenvalues of φ on these eigenspaces are denoted by a, b, c and d. We have to show that a = b = c = d = 1.

In the following considerations we use rules described in [1, 1.5 Theorem, 2.1 Proposition].

Case 1: The calculation of a and b. There are the one-dimensional spaces

(ν^{7}Δ_{n}ν^{61})(ν^{61}Δ_{n}ν^{421})

= ⟨ ν^{7}ν_{7}ν_{61}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{6∘1}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{(4∘2)∘1} ⟩_{ℚ}

with eigenvalue 1,

(ν^{7}Δ_{n}ν^{52})(ν^{52}Δ_{n}ν^{421})

= ⟨ ν^{7}ν_{7}ν_{52}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{5∘2}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{(4∘1)∘2} ⟩_{ℚ}
= ⟨ ν^{7}ν_{(1∘4)∘2} ⟩_{ℚ}

with eigenvalue a,

(ν^{7}Δ_{n}ν^{71})(ν^{43}Δ_{n}ν^{421})

= ⟨ ν^{7}ν_{7}ν_{43}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{4∘3}ν_{421} ⟩_{ℚ}
= ⟨ ν^{7}ν_{4∘(2∘1)} ⟩_{ℚ}
= ⟨ ν^{7}ν_{(2∘1)∘4} ⟩_{ℚ}

with eigenvalue b.

Applying the Jacobi identity ((x ∘ y) ∘ z + (y ∘ z) ∘ x + (z ∘ x) ∘ y = 0) we get

0 = 0^{φ}
= (ν^{7})^{φ}(\overbrace{ν_{(4∘2)∘1} + ν_{(1∘4)∘2} + ν_{(2∘1)∘4}}^{=0})^{φ}
= (ν^{7})^{φ}(ν_{(4∘2)∘1} + aν_{(1∘4)∘2} + bν_{(2∘1)∘4})
= (ν^{7})^{φ}((−ν_{(1∘4)∘2} − ν_{(2∘1)∘4}) + aν_{(1∘4)∘2} + bν_{(2∘1)∘4})
= (ν^{7})^{φ}((a − 1)ν_{(1∘4)∘2} + (b − 1)ν_{(2∘1)∘4}).

The summands in the last equation are linearly independent. It follows that a = 1 and b = 1.

[figure: 7; 6 1; 5 2; 4 3; 5 1 1; 4 2 1; 3 3 1; 3 2 2; 4 1 1 1; 3 2 1 1; 2 2 2 1; 3 1 1 1 1; 2 2 1 1 1; 2 1 1 1 1 1; 1 1 1 1 1 1 1; a; b; c; d]

Figure 3: Γ_{7} with the eigenvalues as weights In a similar manner we treat the other two cases. Case 2: The calculation of c. There are the one-dimensional spaces

(ν^{61}Δ_{n}ν^{511})(ν^{511}Δ_{n}ν^{3211})(ν^{3211}Δ_{n}ν^{22111}) = ⟨ ν^{61}ν_{(((2∘1)∘2)∘1)1)} ⟩_{ℚ}

with eigenvalue 1,

(ν^{61}Δ_{n}ν^{421})(ν^{421}Δ_{n}ν^{3211})(ν^{3211}Δ_{n}ν^{22111}) = ⟨ ν^{61}ν_{(((2∘1)∘1)∘2)1)} ⟩_{ℚ}

with eigenvalue c.

The anticommutative law and the Jacobi identity imply that both spaces are generated by the same element. It follows that c = 1.

Case 3: The calculation of d. There are the one-dimensional spaces

(ν^{43}Δ_{n}ν^{421})(ν^{421}Δ_{n}ν^{3211}) = ⟨ ν^{43}ν_{(3∘1)(2∘1)} ⟩_{ℚ}

with eigenvalue b = 1,

(ν^{43}Δ_{n}ν^{331})(ν^{331}Δ_{n}ν^{3211}) = ⟨ ν^{43}ν_{(3∘1)(2∘1)} ⟩_{ℚ}

with eigenvalue d. Both spaces are generated by the same element. It follows that d = b = 1.

References

[1] D. Blessenohl, H. Laue: On the Descending Loewy Series of Solomon’s Descent Algebra. J. Algebra 180 (1996), 698-724.

[2] D. Blessenohl, H. Laue: The Module Structure of Solomon’s Descent Algebra. Berichtsreihe des Mathematischen Seminars Kiel, 96-6 (1996).

[3] H. Kurzweil: Endliche Gruppen. Springer Verlag, Berlin 1977.

[4] A. Malcev: On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra. C. R. (Doklady) Acad. Sci. URSS (N.S.) 36 (1942), 42-45.

[5] R. Pierce: Associative Algebras. Springer Verlag, New York 1982.

[6] C. Reutenauer: Free Lie algebras. Oxford Univ. Press, London 1993.

[7] L. Solomon: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41 (1976), 255-268.