Matthieu Josuat-Vergès
email: matthieu.josuat-verges@u-pem.fr

C Y Amy Pang

Subalgebras of Solomon's descent algebra based on alternating runs

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.

Introduction

An alternating run of a permutation σ ∈ S n is a maximal monotone sequence of consecutive elements in σ 1 , . . . , σ n . For example, the alternating runs of σ = 14523687 are 145, 52, 2368, and 87. We denote by run(σ) the number of alternating runs of σ. The enumeration of permutations refined by the number of alternating runs have been the subject of various previous works, beginning with André [3] as early as the late 19th century, then Carlitz in the 70's [START_REF] Carlitz | Enumeration of permutations by sequences[END_REF][START_REF] Carlitz | Enumeration of permutations by sequences[END_REF][START_REF] Carlitz | The number of permutations with a given number of sequences[END_REF][START_REF] Carlitz | Enumeration of permutations by sequences[END_REF]. See also [START_REF] Bóna | A combinatorial proof of the log-concavity of the numbers of permutations with k runs[END_REF][START_REF] Canfield | Counting permutations by their alternating runs[END_REF][START_REF] Chow | Counting signed permutations by their alternating runs[END_REF][START_REF] Fewster | Enumerating permutations by their run structure[END_REF][START_REF] Ma | An explicit formula for the number of permutations with a given number of alternating runs[END_REF][START_REF] Ma | Enumeration of permutations by number of alternating runs[END_REF][START_REF] Zhuang | Counting permutations by runs[END_REF] for more recent references. It is worth mentioning that the work [START_REF] Fewster | Enumerating permutations by their run structure[END_REF] was motivated by computations in quantum field theory. Also, alternating runs play a role in some algorithms such as pattern matching [START_REF] Bruner | A fast algorithm for permutation pattern matching based on alternating runs[END_REF].

However, we take here a different perspective, our goal being to define and study various algebras based on this notion of alternating runs. Such an algebra first arose in Doyle and Rockmore's study of "ruffle" card-shuffling: [14, Section 5.5] shows that the elements

W k = σ∈Sn run(σ)=k σ,
for 1 ≤ k ≤ n -1, linearly span a commutative subalgebra of Z[S n ], called the reduced turning algebra. Doyle and Rockmore also considered the number of alternating runs when we add an initial 0 in front of the permutation and obtain in this way another algebra, called the turning algebra [START_REF] Doyle | Riffles, ruffles, and the turning algebra[END_REF]Section 5.4]. We will in particular obtain new proofs of these results in the present article.

In fact, it is easily seen that run(σ) only depends on the descent set of σ, so that the elements W k lie in the descent algebra D n ⊆ Z[S n ] (see [START_REF] Solomon | A Mackey formula in the group ring of a finite Coxeter group[END_REF] or the next section for a definition). This widely studied algebra provides other examples of combinatorial statistics such that there is an algebra linearly spanned by sums of permutations having the same value. Consider for example the descent statistic, then the n elements

E k = σ∈Sn des(σ)=k σ,
for 0 ≤ k ≤ n -1 linearly span a subalgebra of D n called the Eulerian algebra E n (see [START_REF] Gelfand | Noncommutative symmetric functions[END_REF][START_REF] Loday | Opérations sur l'homologie cyclique des algèbres commutatives[END_REF]). But those being particularly relevant here are the Eulerian peak algebra and its left peak analogue [START_REF] Aguiar | New results on the peak algebra[END_REF][START_REF] Novelli | Representation theory of the higher-order peak algebras[END_REF][START_REF] Petersen | Enriched P -partitions and peak algebras[END_REF][START_REF] Schocker | The peak algebra of the symmetric group revisited[END_REF], because peaks and alternating runs are tightly connected.

Our main goal is to introduce a new algebra based on alternating runs, as the linear span of the 2n -2 elements:

W + k :=
σ∈Sn run(σ)=k, the first run is ascending σ, W - k := σ∈Sn run(σ)=k, the first run is descending σ, for 1 ≤ k ≤ n -1. This algebra is noncommutative, but it contains several commutative subalgebras, in particular the two considered by Doyle and Rockmore and the two Eulerian peak subalgebras.

We give in Section 3 a bijective proof of its existence. Then we show that once we know the existence of this 2n -2 dimensional algebra, the existence of the various subalgebras easily follows.

Then in Section 4, we provide a Hopf algebraic proof of the existence of run algebras, using the formalism of noncommutative symmetric functions [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]. Indeed, whereas each homogeneous component has an internal product which is essentially that of the descent algebra, the external Hopf structure permits to build new bases. Also, this algebraic approach easily gives the commutativity of the subalgebras, a property that seems very difficult to get by bijective proofs.

We go on in Section 5 by computing primitive orthogonal idempotents for each of the various run algebras, as generating functions in noncommutative symmetric functions. Part of these calculations appear to be equivalent to Petersen's computation of peak idempotents via order polynomials of enriched P -partitions [START_REF] Petersen | Enriched P -partitions and peak algebras[END_REF]. We also give some properties of the related characters of the symmetric groups, in the same way as idempotents of the Eulerian subalgebra are related with Foulkes characters [START_REF] Foulkes | Eulerian numbers, Newcomb's problem and representations of symmetric groups[END_REF].

Preliminaries 2.1 Notations

A permutation σ ∈ S n is denoted as the sequence σ 1 , . . . , σ n where σ i = σ(i). We denote by Des(σ) the set of descents of σ, i.e. integers i such that 1 ≤ i ≤ n -1 and σ(i) > σ(i + 1). Let also des(σ) = # Des(σ). Let [n -1] denote {1, . . . , n -1}. The descent class D I for I ⊆ [n -1] is the set of σ ∈ S n with Des(σ) = I.

A partition λ of n is a decreasing sequence of positive integers

λ 1 ≥ λ 2 ≥ • • • ≥ λ k called parts, such that k i=1 λ i = n.
We also denote by |λ| the sum of parts of λ. The integer k is the length of λ, and is denoted (λ). Also, we denote e (λ) (respectively, o (λ)) the even length (respectively, the odd length) of λ, i.e. the number of even parts (respectively, odd parts). Write z λ for 1 i1 i 1 !2 i2 i 2 ! . . . , where i r is the number of parts of size r in λ. We denote λ n to say that λ is a partition of n.

A composition of n is a sequence I = (i 1 , . . . , i r ) of positive integers such that r j=1 i j = n. We also denote by |I| the sum of parts of I. The integer r is called the length of I, and is denoted (I). We write I n to say that I is a composition of n.

In a partition or a composition, we write 1 n to mean that 1 is repeated n times.

The descent set of a composition I = (i 1 , . . . , i r ), denoted Des(I), is

i 1 , i 1 + i 2 , . . . , r-1 j=1 i j .
The map I → Des(I) is a bijection from compositions of n to subsets of [n -1]. When I, J n, we say that J is a refinement of I, denoted I J, if Des(I) ⊆ Des(J). When we write I J, we always understand that |I| = |J|.

The descent algebra

For I n, consider the following sums in the group algebra Z[S n ]:

R I = σ∈Sn Des(σ)=Des(I) σ, S I = σ∈Sn Des(σ)⊆Des(I) σ.
Note that by an immediate inclusion-exclusion, we have:

S I = J I R J , R I = J I (-1) (J)-(I) S J .
The linear span of the elements R I (or equivalently, S I ) for I n is denoted D n and called the descent algebra. It was indeed shown by Solomon [START_REF] Solomon | A Mackey formula in the group ring of a finite Coxeter group[END_REF] that D n is a subalgebra of the group algebra Z[S n ]. However, to facilitate our construction of idempotents of various subalgebras, it will be more convenient to work with rational coefficients, i.e. to consider D n as a subalgebra of

Q[S n ].

The Eulerian peak algebras.

Two previously identified subalgebras of the descent algebra will play a key role in the analysis of our new subalgebras. These are based on peaks: Definition 2.1. A peak of a permutation σ ∈ S n is an integer i with 2 ≤ i ≤ n -1 and σ(i -1) < σ(i) > σ(i + 1). We denote by pk(σ) the number of peaks of σ. Definition 2.2. A left peak of a permutation σ ∈ S n is an integer i with 1 ≤ i ≤ n -1 and σ(i -1) < σ(i) > σ(i + 1) (with the convention σ(0) = 0). We denote by pk • (σ) the number of left peaks of σ.

Proposition 2.3. [25, Sec. 9] The elements

P k = σ∈Sn pk(σ)=k σ for 0 ≤ k ≤ n-1 2
span the Eulerian peak algebra P n , a commutative subalgebra of D n . Note that dim(P n ) = n 2 . The Eulerian peak algebra first appeared in Schocker's analysis [START_REF] Schocker | The peak algebra of the symmetric group revisited[END_REF] of a larger, noncommutative, peak algebra, spanned by sums of permutations with the same peak positions. This work determined the idempotents of P n , which we will recover in the present Theorem 5.3.

Aguiar, Bergeron and Nyman [START_REF] Aguiar | The peak algebra and the descent algebras of types B and D[END_REF] also studied this larger peak algebra, by viewing it as an ideal within the image of projecting the type B descent algebra to D n . This viewpoint uncovered a left peak variant of P n : Proposition 2.4. [1, Sec. 6.3] The elements

P • k = σ∈Sn pk • (σ)=k σ for 0 ≤ k ≤ n 2 span the Eulerian left peak algebra P • n , a commutative subalgebra of D n . Note that dim(P • n ) = n 2 + 1.
Petersen [START_REF] Petersen | Enriched P -partitions and peak algebras[END_REF] computed a complete set of primitive orthogonal idempotents of both P n and P • n , in terms of enriched P -partitions. Our re-computation in the present Theorem 5.3 is uncannily similar, despite being in the different language of noncommutative symmetric functions.

Both P n and P • n are subalgebras of our new algebra W ± n . This idea of adding an initial 0 is also relevant in the case of runs. As mentioned in the introduction, this definition was used by Doyle and Rockmore [START_REF] Doyle | Riffles, ruffles, and the turning algebra[END_REF]. Definition 2.5. With the convention σ 0 = 0, a left run of a permutation σ ∈ S n is a maximal monotone sequence of consecutive elements in σ 0 , . . . , σ n . We denote by run • (σ) the number of left runs of σ.

Note that if the first run of σ is ascending, then run • (σ) = run(σ), and the first left run is obtained from the first run by adding the initial zero. If the first run is descending, then run • (σ) = run(σ) + 1, and the first left run is 0, σ 1 .

Noncommutative symmetric functions

To find and express the idempotents of the run algebras, it is useful to view the descent algebra from another perspective, that contains extra algebraic structure.

Let Sym denote the Hopf algebra of noncommutative symmetric functions, which is the direct sum of Sym n , its degree n homogeneous components (see [START_REF] Gelfand | Noncommutative symmetric functions[END_REF] as a general reference on this subject). Its product and coproduct ∆ are defined below. It will be convenient to also work in Sym, the completion, which allows infinite sums of the elements of Sym, of unbounded degree.

As a vector space, we have the identification Sym n = D n . Under this identification, the image of S I are the complete noncommutative symmetric functions, and the image of R I are the ribbon noncommutative symmetric functions. We write S I , R I for these images also.

The product on D n is extended to Sym by the rule xy = 0 if x ∈ Sym m , y ∈ Sym n , and m = n. We call it the internal product of Sym, to distinguish from the external product. However, it is to be noted that the usual internal product of noncommutative symmetric functions is taken in the opposite order (see [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]).

Sym also admits an external product , for which degree is additive: Sym n Sym n ⊆ Sym n+n . In the complete basis, this is given by concatenation of compositions: S (i1,...,i (I) ) S (j1,...,j (J) ) = S (i1,...,i (I) ,j1,...,j (J) ) .

Hence, for any composition I = (i 1 , . . . , i r ), we have S I = S i1 • • • S ir , so Sym is a free algebra for the external product, generated by the S i , the complete noncommutative symmetric functions indexed by compositions with a single part.

In the ribbon basis, the external product results in "near-concatenation": R (i1,...,i (I) ) R (j1,...,j (J) ) = R (i1,...,i (I) ,j1,...,j (J) ) + R (i1,...,i (I)-1 ,i (I) +j1,j2,...,j (J) ).

For example, S (2,3) S (5,1,1) = S (2,3,5,1,1) and R (2,3) R (5,1,1) = R (2,3,5,1,1) + R (2,8,1,1) . We write F k for powers under the external product, to distinguish from the internal product power F k . The two products are related by the splitting formula of [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]Prop. 5.2]; this involves a third algebraic operation, the coproduct. This is the (external) algebra homomorphism ∆ :

Sym → Sym ⊗ Sym with ∆(S n ) = n i=0 S i ⊗ S n-i . (The convention is that S 0 = 1.) It follows that n≥0 R n and n≥0 R (1 n ) are grouplike -that is, they satisfy ∆(G) = G⊗G.
It is a standard fact of Hopf algebras that (external) products of grouplike elements are also grouplike. For grouplike G, the splitting formula simplifies to:

G(F 1 • • • F k ) = (GF 1 ) • • • (GF k ).
This simplified version is the only case of the splitting formula that we will need.

Remark 2.6. Since we are interested in subalgebras of the descent algebra, it is natural to use a special symbol for the external product of (noncommutative) symmetric functions, the product without a special symbol meaning the internal product. However this differs from the usual notation.

Symmetric functions

Let Sym denote the algebra of symmetric functions and Sym n its degree n homogeneous component. We follow the notation of [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF] so that h λ , e λ , p λ , s λ are respectively the homogeneous, elementary, power sum, and Schur symmetric functions associated with a partition λ. If I = (i 1 , . . . , i (I) ) is a composition, there is a unique ribbon (i.e. a skew shape which is connected and contains no 2 × 2 square) whose row lengths from top to bottom in French notation are i 1 , . . . , i (I) , and the skew Schur function associated with it is denoted r I . It is given by r I = J I (-1) (I)-(J) h J where h J = h j where j runs through the parts of J. To be coherent with our notation on noncommutative symmetric functions, the usual product of Sym is denoted , whereas the internal product on Sym n has no particular symbol.

There is a well-known map Γ : Sym → Sym, defined by the two equivalent formulas:

Γ(R I ) = r I , Γ(S I ) = h I .
In particular, we have:

Γ(S n ) = h n , Γ(R (1 n ) ) = e n .
This map is an algebra morphism for both the internal and external product of symmetric functions:

Γ(xy) = Γ(x)Γ(y), Γ(x y) = Γ(x) Γ(y).
One fact about Sym n that will be particularly useful for us is that the rescaled power sums 1 z λ p λ form a basis of orthogonal idempotents (under the internal product). It follows that, if Λ 1 , Λ 2 , . . . , Λ K are disjoint subsets of partitions of n, then λ∈Λ k 1 z λ p λ form a coarser set of orthogonal idempotents. In fact, every subalgebra of Sym n has a basis of orthogonal idempotents of this form, and identifying the Λ k is particularly interesting if we view Sym n as the character ring of S n [START_REF] Stanley | Enumerative combinatorics[END_REF]Sec. 7.18]. (In this light, Γ is the homomorphism of Solomon from the descent algebra of a finite Coxeter group to its character ring [START_REF] Solomon | A Mackey formula in the group ring of a finite Coxeter group[END_REF].) For example, the idempotents of Γ(E n ), the symmetric function image of the Eulerian algebra, are

(λ)=k 1 z λ p λ [16, Sec. 5.3],
corresponding to the indicator functions on permutations with k cycles. By definition, (I)=k r I is also a basis of Γ(E n ). So, writing χ I for the character corresponding to r I , this slickly recovers a result of Foulkes [START_REF] Foulkes | Eulerian numbers, Newcomb's problem and representations of symmetric groups[END_REF]Th. 3.1], [START_REF] Kerber | Eulerian numbers, Foulkes characters and Lefschetz characters of S n[END_REF]:

   (I)=k χ I : 1 ≤ k ≤ n   
gives a basis for the subspace of functions S n → R depending only on the number of cycles.

Applying this idea to our new commutative run algebras yields similar characters that depend only on the numbers of odd cycles and of even cycles, see Corollary 5.4.

Bijective proof for the existence of run algebras

Recall that run(σ) is defined at the beginning of the introduction, and run • (σ) in Definition 2.5.

Definition 3.1. Let W + k := {σ ∈ S n : run(σ) = k and the first run is ascending}, W - k := {σ ∈ S n : run(σ) = k and the first run is descending}, W k := {σ ∈ S n : run(σ) = k} = W + k ∪ W - k , for 1 ≤ k ≤ n -1, and
W • k := {σ ∈ S n : run • (σ) = k} = W + k ∪ W - k-1
for 1 ≤ k ≤ n. The given range for k indicates when these sets are nonempty. Note that in the definition of W • k we need to specify that W + n = W - 0 = ∅. And let

W - k := σ∈W - k σ, W + k := σ∈W + k σ, W k := σ∈W k σ, W • k := σ∈W • k σ. Note that W k = W - k + W + k and W • k = W - k-1 + W + k .
Then we define subspaces of the descent algebra D n as follows:

W ± := Vect(W + k , W - k : 1 ≤ k ≤ n -1}, W := Vect(W k : 1 ≤ k ≤ n -1}, W • := Vect(W • k : 1 ≤ k ≤ n}, C := Vect(W 2k : 1 ≤ k ≤ n-1 2 } ⊕ Vect(W + 2k-1 , W - 2k-1 : 1 ≤ k ≤ n 2 }.
Note that in each definition, the given generators are linearly independent, since they are sums of permutations over disjoint nonempty subsets. It follows that:

dim(W ± ) = 2n -2, dim(W) = n -1, dim(W • ) = n, dim(C) = 3n-2 2 .
Theorem 3.2. The space W ± is a subalgebra of D n , and the spaces W, W • , C, P and P • are subalgebras of W ± .

Note that W and W • are respectively the reduced turning algebra and the turning algebra of Doyle and Rockmore [START_REF] Doyle | Riffles, ruffles, and the turning algebra[END_REF]. And P and P • are the Eulerian peak and Eulerian left peak algebra, as described in Subsection 2.3.

The rest of this section contains the bijective proof. In Subsections 3.1 to 3.3, we prove the result for W ± . Then, in Subsection 3.4, we show how the other parts of the theorem follow.

The beginning of the proof, in particular using the connectedness in the left weak order, essentially follows Atkinson's proof of the existence of the descent algebra [START_REF] Atkinson | A new proof of a theorem of Solomon[END_REF] (which is also in Tits [30, §6] although in a somewhat different language and very concise style).

Alternating runs and the left weak order

This subsection contains a preliminary result about the sets W + j and W - j . Let s i denote the simple transposition (i, i + 1) ∈ S n . Let us recall that the left weak order on S n is defined by the covering relations σ s i σ if i is to the left of i + 1 in σ, for all σ ∈ S n and 1 ≤ i ≤ n -1. It is known that descent classes are intervals for this order [START_REF] Björner | Generalized quotients in Coxeter groups[END_REF]. In particular, each descent class is a connected subset in S n (seen as an undirected graph whose edges are given by the cover relation of the left weak order). Lemma 3.3. Let I, J ⊆ [n -1] with I = J. The descent classes D I and D J are connected to each other (in the sense that there are u ∈ D I , v ∈ D J such that u v or v u) if and only if one of the following conditions is true:

• I ⊆ J and #I = #J -1, or • J ⊆ I and #J = #I -1.
Proof. We compare the descents sets of σ and s i σ. If i and i + 1 are not adjacent in σ, Des(σ) = Des(s i σ). Otherwise, either Des(σ) = Des(s i σ) ∪ {i} or Des(s i σ) = Des(σ) ∪ {i}. It follows that if D I and D J are connected, I and J satisfy the given conditions.

As for the other implication, suppose I and J satisfy the given conditions, for example I = J ∪ {i}. It is easy to find σ ∈ D I such that σ i = σ i+1 + 1. Then the two permutations σ and s i σ ∈ D J show that D I and D J are connected. Lemma 3.4. Each set W + j or W - j is connected in the left weak order.

Proof. By symmetry, we only need to consider the case of W + j . This set is a union of descent classes. Since each descent class is connected, it remains to show that these descent classes are connected to each other, using the previous lemma.

The descent classes included in W + j are those indexed by subsets X ⊆ {1, . . . , n -1} such that:

• X is a union of j/2 intervals and no less (i.e. the number of descending runs),

• 1 / ∈ X (since the elements in W + j begin with an ascending run),

• n -1 ∈ X if and only if j is even (since the first run is ascending, the last run is descending if and only if j is even).

By adding or removing elements in these subsets as in the previous lemma, (and staying in the same class of subsets) we can reach a particular chosen subset, for example {2, 4, 6, . . . , 2 j/2 } if j is odd and {2, 4, 6, . . . , 2 j/2 -1, n -1} if j is even. This fact is rather straightforward so we omit the full formal proof. Connectedness easily follows.

The scheme of proof

To prove that W ± is an algebra, we show W + j W + k ∈ W ± . The other products, i.e. W + j W - k , W - j W + k , and W - j W - k , can be done similarly (or more formally, one can use symmetry properties described in Subsection 3.4).

For each permutation σ ∈ S n , let

C σ j,k = {(α, β) ∈ W + j × W + k : αβ = σ}. Then c σ j,k = #C σ j,k is the coefficient of σ in the product W + j W + k . Our goal is to prove that c σ j,k = c τ j,k if σ, τ are both in W + m or both in W - m .
We only consider the case where σ and τ are neighbours in the left weak order, i.e., either σ τ or τ σ. Indeed, using the connectedness shown in Lemma 3.4, this implies the general case. So we assume s i σ = τ for some i.

Let us first consider the case where i and i + 1 are not adjacent in σ = σ 1 . . . σ n (and hence in τ = s i σ). It implies that σ and τ are in the same descent class. But since W + j W + k is in the descent algebra (as W + j and W + k both are), the coefficients of σ and τ are equal, i.e. c σ j,k = c τ j,k as we wanted.

It remains to consider the case where i and i + 1 are adjacent in σ (and hence in τ ). We will give a bijection Φ : C σ j,k → C τ j,k . This bijection is rather long to define as we need to consider many cases, and it is described in the next subsection. Note that σ and τ have symmetric roles here, since we did not specify which of σ τ or τ σ is true. So by exchanging the role of σ and τ , the definition in the next subsection also describes another map Ψ : C τ j,k → C σ j,k . At each step of the definition, we can check that Ψ is in the inverse map of Φ, so that these maps are indeed bijections.

The bijection Φ

We are now in the case where i and i + 1 are adjacent in σ. So there is an integer h such that τ = s i σ = σs h , and also σ({h, h + 1}) = τ ({h, h + 1}) = {i, i + 1}.

Note that for any u ∈ S n , u and us 1 cannot be both in W + m or both in W - m . The same is true for u and us n-1 . So we have 2 ≤ h ≤ n -2.

First case

In the first case, we assume that s i α ∈ W + j . Thus we can define Φ((α, β)) = (s i α, β), this pair being clearly in C τ j,k . Note that the pair (s i α, β) ∈ C τ j,k also leads to the first case when we apply the map Ψ, so that Ψ • Φ is the identity when applied on (α, β) falling in the first case.

From now on, assume s i α / ∈ W + j . It follows that s i α is not in the same descent class as α, so i and i + 1 are adjacent in α. So there is an integer g such that s i α = αs g . We also have s g β = βs h since αs g β = s i αβ = αβs h . Also, α({g, g + 1}) = {i, i + 1} and β({h, h + 1}) = {g, g + 1}.

Second case

In the second case, we assume that

s g β ∈ W + k . We define Φ((α, β)) = (α, s g β), which is in C τ j,k (since αs g β = s i αβ = s i σ = τ ).
Note that the pair (α, s g β) ∈ C τ j,k also leads to this second case when we apply the map Ψ, so that Ψ • Φ is the identity when applied on (α, β) falling in the second case.

So from now on, assume that s g β / ∈ W + k . This is our last case.

Third case

We will extensively use the following fact. The proof is clear upon inspection of a few cases. To begin, let us apply this lemma to β ∈ W + k and s g β = βs h / ∈ W + k . Since we already know 2 ≤ h ≤ n -2, we obtain:

• either β(h -1) < g and β(h + 2) > g + 1, (A) • or β(h -1) > g + 1 and β(h + 2) < g. (B)
Note that this implies 2 ≤ g ≤ n -2.

Next, we apply the lemma to α ∈ W + j and s i α = αs g / ∈ W + j . Since we just got 2 ≤ g ≤ n -2, it follows:

• either α(g -1) < i and α(g + 2) > i + 1, (C)

• or α(g -1) > i + 1 and α(g + 2) < i.

(D)
Next, we apply the lemma to σ and s i σ = σs h , which are both in W + m . Since 2 ≤ h ≤ n -1, we get (writing αβ instead of σ):

• either α(β(h -1)) < i and α(β(h + 2)) < i, (E) • or α(β(h -1)) > i + 1 and α(β(h + 2)) > i + 1. (F)
To define Φ, we only need to distinguish between cases C or D, and E or F. Suppose first that we are in case C-E. Lemma 3.6. In case C-E as above, there is a unique (word) factorization α 1 . . . α n = X 1 X 2 X 3 X 4 where the four factors are nonempty and such that:

X 2 = i, i + 1 or X 2 = i + 1, i, X 3 contains only letters > i + 1, the first letter of X 4 is < i.
Proof. To begin, X 2 is defined as the factor containing i and i+1 (we have α({g, g+1}) = {i, i+1}). Since g ≥ 2, we deduce that X 1 is nonempty. Since α(g + 2) > i + 1, i.e. the first letter to the right of X 2 is > i + 1, we can define X 3 as the largest factor following X 2 and containing letters > i + 1.

Then, we have α(β(h -1)) < i and α(β(h + 2)) < i. Since at least one of β(h -1) and β(h + 2) is > g + 1, it follows that there is at least one letter < i to the right of X 2 . So X 4 is nonempty.

Then we define Φ((α, β)) = (α , β ) as follows:

• α = X 1 X 3 X 2 X 4 where X 2 is X 2 in reversed order. • β is such that β (h) = β(h) + |X 3 |, β (h + 1) = β(h + 1) + |X 3 |, if α(β(u)) appears in X 3 then β (u) = β(u) -2, otherwise (i.e. α(β(u)) appears in X 1 or X 4 ) then β (u) = β(u).
Here |X 3 | is the length of X 3 .

Lemma 3.7. We have (α , β ) ∈ C τ j,k . Proof. Since the last letter of X 1 (i.e. α(g -1)) and the first of X 4 are both < i, and X 3 contains only letters > i + 1, we easily get that α ∈ W + j (knowing α ∈ W + j ). Among the two letters α(β(h -1)) and α(β(h + 2)), one appears in X 1 and the other in X 4 . One can deduce that four letters β (h -1), . . . , β (h + 2) are in the same relative order as β(h -1), . . . , β(h + 2). Then, suppose we are in the case

β (u) = β(u) -2, i.e. α(β(u)) appears in X 3 . Let v = u ± 1, then α(β(v)) can appear in X 1 , X 3 , or X 4 .
In each case, we check that β (u) and β (v) are in the same relative order as β(u) and β(v). So β is in the same descent class as β and in paticular β ∈ W + k . By distinguishing which of the factors X 1 , . . . , X 4 contains α(β(u)), from the definition of β we can check that α (β (u)) = s i (α(β(u))). So α β = s i αβ.

Let g = g + |X 3 |. We have s i α = α s g and s g β = β s h by definition of α and β . So (α , β ) falls in the third case when we apply Ψ. Moreover:

• α (g -1) > i + 1 (since this number is the last letter of X 3 ) , so (α , β ) falls in case D.

• α (β (h -1)) = α(β(h -1)) < i (since α β = s i αβ ) , so (α , β ) falls in case E. So (α , β ) falls in case D-E. Consequently, there is only one way to define Φ in case D-E in order to get that Ψ•Φ is the identity. Namely, if (α, β) falls in case D-E, we factor α = X 1 X 2 X 3 X 4 where X 3 is i, i + 1 or i + 1, i, X 2 contains only letters > i + 1, the last letter of X 1 is < i. Once again the factorization exists and is unique. Then we define Φ((α, β)) = (α , β ) as follows:

• α = X 1 X 3 X 2 X 4 . • β is such that β (h) = β(h) -|X 2 |, β (h + 1) = β(h + 1) -|X 2 |, if α(β(u)) appears in X 2
then β (u) = β(u) + 2, otherwise (i.e. α(β(u)) appears in X 1 or X 4 ) then β (u) = β(u).

From the definitions, we check that Ψ • Φ is the identity on all (α, β) that falls in case C-E or D-E. It remains to define Φ and Ψ in a similar manner to exchange the case C-F with the case D-F. This is completely similar except that we exchange the conditions "< i" and "> i + 1".

This ends the definition of the bijection, hence of the proof of W + j W + k ∈ W ± .

The case of the other subalgebras

Knowing that W ± is an algebra, we can finish the proof of Theorem 3.2.

Proposition 3.8. W • is a subalgebra of W ± .
Proof. For each permutation σ, we define:

D σ j,k = {(α, β) ∈ W • j × W • k : αβ = σ},
and d σ j,k = #D σ j,k . If σ ∈ W + m or σ ∈ W - m , then d σ j,k is the coefficient of W + m or W - m , respectively, in W • j W • k . We will show that for each m such that 2 ≤ m ≤ n -1, we have d σ j,k = d τ j,k for some well-chosen σ ∈ W + m and τ ∈ W - m-1 . Indeed this implies W • j W • k ∈ W • . It is easy to find an element σ ∈ W + m such that σ 1 = n -1 and σ 2 = n. We deduce that τ = s n-1 σ = σs 1 is in W - m-1 .
Then, it remains only to find a bijection beween D σ j,k and D τ j,k . This bijection ∆ is defined as follows. Let (α, β) ∈ D σ j,k . In the first case, if

s n-1 α ∈ W • j , we set ∆(α, β) = (s n-1 α, β).
Otherwise, it easily follows s n-1 α = αs n-1 . In the second case, if

s n-1 β ∈ W • k , we set ∆(α, β) = (α, s n-1 β).
There is in fact no other case to consider: if we are not in the first case, we have necessarily

s n-1 β ∈ W • k .
Otherwise, we would have s n-1 β = βs n-1 , and it would follow that s n-1 σ = σs n-1 (this is a contradiction to s n-1 σ = σs 1 ).

As for the other subalgebras, we don't need to do a similar bijection, as it is more convenient to use some symmetry properties. Let ω := W - 1 = n . . . 21. We clearly have run(ωσ) = run(σω) = run(σ) for any σ ∈ S n . Moreover, 1 ∈ Des(σ) iff 1 / ∈ Des(ωσ), so that:

ωW + k = W - k , ωW - k = W + k . Note that if σ ∈ S n
has an odd number of alternating runs, the first and last runs are of similar nature (both ascending or both descending), but if σ has an even number of runs, they are of different nature (one is ascending, the other descending). It is then easy to read the alternating runs of σω = σ n . . . σ 1 from those of σ = σ 1 . . . σ n . It follows:

W + k ω = W - k if k is odd, W + k if k is even, W - k ω = W + k if k is odd, W - k if k is even, (1)
Note that this implies

W - j W + k = ωW + j W + k W + j W - k = W + j W + k if j is even, ωW + j W + k if j is odd, W - j W - k = W + j W + k if j is odd, ωW + j W + k if j is even.
As a consequence, to show that W ± is an algebra it suffices to show that W + j W + k ∈ W ± , as was mentioned earlier.

Proposition 3.9. W is a subalgebra of W ± .

Proof. From the elementary properties of ω = W - 1 (and W + 1 , the unit element), we have

W 1 W + k = W 1 W - k = W k .
It follows that W is the right ideal W 1 W ± and a fortiori a subalgebra. Remark 3.10. The previous proposition should be understood in the sense of nonunital algebras, since W does not contain contain the unit of W ± . Still, W has a unit element as an abstract algebra, namely 1 2 W 1 (at the condition of allowing rational coefficients). Proposition 3.11. C is a subalgebra of W ± .

Proof. From the elementary properties of ω = W - 1 , we have:

ωW + k ω = W + k if k is odd, W - k if k is even, ωW - k ω = W - k if k is odd, W + k if k is even.
Using that x → ωxω is a linear symmetry, it follows that

{x ∈ W ± : ωxω = x} = Vect(W + 2k+1 , W - 2k+1 , W 2k ) = C. If x, y ∈ C, we have ωxyω = ωxω 2 yω = xy, so C is closed under product.
We turn to the case of peak algebras.

Lemma 3.12. We have:

P k = W + 2k + W - 2k+1 + W + 2k+1 + W - 2k+2 , (2)
P • k = W - 2k-1 + W + 2k + W - 2k + W + 2k+1 . (3)
Proof. For the first equality, consider a permutation σ with pk(σ) = k. It has at least 2k runs, since each peak follows an ascending run, and precedes a descending run. Besides these 2k runs, whose first is ascending, there is possibly an additional descending run at the beginning, and possibly an additional ascending run at the end. The four possibilities give the four terms in the sum.

As for the second equality, consider a permutation pk • (σ) = k. It has at least 2k -1 runs, since now the first peak might not follow an ascending run (if it is equal to 1). Besides these 2k -1 runs, whose first is descending, there is possibly an additional ascending run at the beginning, and possibly an additional ascending run at the end. Once again, the four possibilities give the four terms in the sum. Proposition 3.13. P and P • are subalgebras of W ± .

Proof. The previous lemma shows that P and P • are subspaces of W ± , so it remains to show that they are closed under product.

We begin with the case of P • . From the previous lemma, we have

P • k = W • 2k + W • 2k+1 . Now let x ∈ W • , then it is in P • iff the coefficients of W •
2k and W • 2k+1 are the same for all k. This is also equivalent to the equality of the coefficients of W + 2k and W - 2k in x, i.e. to the condition x ∈ C. So we have P • = W • ∩ C, and this is a subalgebra as an intersection of subalgebras.

In the case of P, we first note that it is closed under product iff ωPω is also closed under product. We have: or I = (j 1 , . . . , j (J)-1 , j (J) + j 1 , j 2 , . . . , j (J ) ), of the products of the coefficient of R J in F with the coefficient of R J in G. Hence the coefficient of R I in V - 2k,n is the weighted number of ways to "cut" σ I into k pieces, possibly trivial, so that each piece contains no peak. The weight is 2 k , where k is the number of non-trivial pieces (because of the coefficient 2 in V - 2 = 1 + 2 R I ). Since each piece must be peakless, there must be a cut immediately to the left or immediately to the right of any peak; other than this the positions of cuts are unconstrained. These pair of positions, for any set of peaks, are disjoint, so the number of ways to cut σ I into k non-trivial peakless pieces depends only on the number of peaks in σ I .

ωP k ω = W - 2k + W - 2k+1 + W + 2k+1 + W + 2k+2 = W • 2k+1 + W • 2k+2 , so ωPω ⊆ W • . Now let x ∈ W • ,
If σ I has more than k -1 peaks, then σ I cannot be cut into k peakless pieces, so R I does not appear in V - 2k . If σ I has exactly k -1 peaks, then the ways to cut σ I into k peakless pieces are precisely when there is one cut either immediately to the left or immediately to the right of each peak. There are 2 k-1 such ways, and the weight of this cut is 2 k , since no piece is trivial. Hence the coefficient of

P k-1,n in V - 2k,n is 2 2k-1 .
Next, we apply a similar argument to V + 2k+1,n .

Proposition 4.3. For all k, we have

V + 2k+1,n ∈ P • n , the Eulerian left peak algebra. Moreover, if k ≤ n 2 , then V + 2k+1,n = 2 2k P • k,n + l<k a • k,l,n P • l,n for some constants a • k,l,n . Hence {V + 2k+1,n : 0 ≤ k ≤ n 2 } is a basis for P • n .
Proof. By the same logic as above, the coefficient of R I in V + 2k+1,n is the weighted number of ways to "cut" σ I into k + 1 pieces, possibly trivial, so that the leftmost piece has no descents, and the k other pieces each contain no peak. The weight is 2 k , where k is the number of non-trivial pieces among the k peakless pieces.

Since each piece must be peakless, there must be a cut immediately to the left or immediately to the right of all peaks. Additionally, if σ I has a descent in position 1, then the first cut must be immediately before or after position 1, because the first piece must not contain descents. Hence the two conditions may be summarised as requiring a cut immediately to the left or immediately to the right of all left peaks.

As before, these positions, for any set of left peaks, are disjoint, so the number of ways to cut into σ I into a descentless first piece and k further non-trivial peakless pieces depends only on the number of left peaks in σ I .

If σ I has more than k left peaks, then σ I cannot be cut into a descentless first piece and k further peakless pieces, so R I does not appear in V + 2k+1 . If σ I has exactly k left peaks, then the ways to cut σ I into a descentless first piece and k further peakless pieces are precisely when there is one cut either immediately to the left or immediately to the right of each left peak. There are 2 k such ways, and the weight of this cut is 2 k , since none of the peakless pieces are trivial. Hence the coefficient of

P • k,n in V + 2k+1,n is 2 2k .
From the above two propositions, and Equations ( 2) and (3) relating P k,n and P

• k,n to W + k,n , W - k,n
, we see that:

V - 2k,n = 2 2k-1 W - 2k,n + l<k a k,l,n W 2l+1,n + a k,l,n W + 2l,n + a k,l-1,n W - 2l,n ; V + 2k+1,n = 2 2k W + 2k+1,n + l≤k a • k,l,n W 2l,n + a • k,l-1,n W + 2l-1,n + a • k,l,n W - 2l-1,n .
By exchanging ascending and descending runs, it is also true that

V + 2k,n = 2 2k-1 W + 2k,n + l<k a k,l,n W 2l+1,n + a k,l,n W - 2l,n + a k,l-1,n W + 2l,n ; V - 2k+1,n = 2 2k W - 2k+1,n + l≤k a • k,l,n W 2l,n + a • k,l-1,n W - 2l-1,n + a • k,l,n W + 2l-1,n . Also, because P • k = W • 2k + W • 2k+1
, and

P k = W + 2k + W - 2k+1 + W + 2k+1 + W - 2k+2 , W • 2k+1 + W • 2k+2 = W - 2k + W + 2k+1 + W - 2k+1 + W + 2k+2 ,
we have

V + 2k+1,n = 2 2k W • 2k+1 + 2 2k W • 2k + l<k a • k,l,n (W • 2l+1,n + W • 2l,n ); V + 2k,n = 2 2k-1 W • 2k + 2 2k-1 W • 2k-1 + l<k a • k,l-1,n (W • 2l,n + W • 2l-1,n ).
Combining all this with the multiplicativity of the V k 's in Proposition 4.1, we can conclude:

Theorem 4.4. For each n,

Vect(V + k,n , V - k,n : k ∈ N) = Vect(V + k,n , V - k,n : 1 ≤ k < n) = W ±
n is an algebra under the internal product. Five commutative subalgebras of W ± n are:

Vect(V k,n : k ∈ N) = Vect(V k,n : 1 ≤ k < n) = W n , Vect(V + k,n : k ∈ N) = Vect(V + k,n : 1 ≤ k ≤ n) = W • n , Vect(V - 2k,n : k ∈ N) = Vect(V - 2k,n : 1 ≤ 2k ≤ n + 1) = P n , Vect(V + 2k+1,n : k ∈ N) = Vect(V + 2k+1,n : 1 ≤ 2k + 1 ≤ n + 1) = P • n , and
Vect(V + 2k+1,n , V - 2k+1,n , V 2k,n : k ∈ N) = Vect(V + 2k+1,n , V - 2k+1,n , V 2l,n : 1 ≤ 2k + 1, 2l < n) = C n . Furthermore, P • n is central in W ± n .
Remark 4.5. The quantities V + k,n , V - k,n previously appeared in [START_REF] Petersen | Enriched P -partitions and peak algebras[END_REF], as order polynomials of enriched P -partitions. The translation between his notation and ours is

ρ(x) = V - 2x,n ; ρ(x) = V + 2x,n ; ρ (l) (x) = V + 2x+1,n ; ρ (r) (x) = V - 2x+1,n .
Our Proposition 4.1, the multiplication rule, is his Theorems 3.1, 3.3 and 3.5.

Computation of the orthogonal idempotents

In this section, we compute a complete set of primitive idempotents for W ± n and for its five commutative subalgebras in Theorem 4.4.

Orthogonal idempotents of the commutative subalgebras

The starting point is the orthogonal idempotents of the Eulerian peak and left peak algebras. These were computed in [START_REF] Petersen | Enriched P -partitions and peak algebras[END_REF][START_REF] Schocker | The peak algebra of the symmetric group revisited[END_REF], but we rederive them here so this paper is self-contained. The argument here mirrors the approach of Loday (see [START_REF] Gelfand | Noncommutative symmetric functions[END_REF]Sec. 5.3]) for the idempotents of the Eulerian (descent) algebra. The key is the following observation: Lemma 5.1. For some multiplicatively-closed subset I ⊆ N, suppose {X i : i ∈ I} generate a subalgebra A of Sym with X i X j = X ij (under the internal product). Assume also that there are elements J k in this subalgebra such that, for each i, we have X i = k≥0 i k J k (as an identity of formal power series within Sym). Then the non-zero J k form a basis of orthogonal idempotents for A.

Proof. For all i, j ∈ I, we have

k (ij) k J k = X ij = X i X j = r,s i r j s J r J s .
Equating coefficients of (ij) k then shows that J r J s = 0 if r = s, and J 2 k = J k .

Recall from Theorem 4.4 that the Eulerian peak algebra is generated by {V - i,n : i even} and the Eulerian left peak algebra by {V + i,n : i odd}. By Proposition 4.1, these generating sets satisfy the multiplicative condition of Lemma 5.1. Thus, to calculate the orthogonal idempotents, it suffices to find

I - k,n , J + k,n such that V - 2i,n = k≥0 (2i) k I - k,n and V + 2i+1,n = k≥0 (2i + 1) k J + k,n
. Such expressions come from the formal exponential and logarithm operations in Sym, defined via their familiar power series expansions:

log (1 + F ) : = F - 1 2 F 2 + 1 3 F 3 -. . . , exp F : = 1 + F + 1 2! F 2 + 1 3! F 3 + . . . ,
for F ∈ Sym with no term in degree 0. It can be checked that exp (F + G) = (exp F ) (exp G) whenever F and G commute. Also, (1 + F ) k = exp (k log (1 + F )) for all positive integers k (and all F ∈ Sym with no term in degree 0), and this can be used to define (1 + F ) k when k is not a positive integer. Now we have

V - 2i = (V - 2 ) i = exp (i log V - 2 ) = k≥0 1 k! (i log V - 2 ) k = k≥0 (2i) k 1 2 k k! (log V - 2 ) k , and
V + 2i+1 = V + 1 (V - 2 ) (-1 2 ) (V - 2 ) (i+ 1 2 ) = V + 1 (V - 2 ) (-1 2 ) exp i + 1 2 log V - 2 = k≥0 (2i + 1) k 1 2 k k! V + 1 (V - 2 ) (-1 2 ) (log V - 2 ) k . (4)
This motivates the definitions:

I - 1 := 1 2 log V - 2 , I - k := 1 k! I - 1 k , I + 1 := 1 2 log V + 2 , I + k := 1 k! I + 1 k , J + 0 := V + 1 (V - 2 ) (-1 2 ) , J + k := J + 0 I - k , J - 0 := V - 1 (V + 2 ) (-1 2 ) , J - k := J - 0 I + k .
Also, make the convention I - 0 = I + 0 = 1. As with the V s, write I + k,n , I - k,n , J + k,n , and J - k,n for the degree n part of I + k , I - k , J + k , and J - k . The above calculations show that I - k,n , J + k,n are orthogonal idempotents for the Eulerian peak and left peak algebras respectively. However, we will see that I - k,n is not orthogonal to J + k,n . A first attempt to rectify this might be to calculate products such as I - k,n J + k,n , and guess simple linear combinations of these idempotents that are orthogonal. These product calculations are considerably easier in the symmetric functions -that is, we calculate the images of I + k,n , I - k,n , J + k,n , J - k,n under the homomorphism Γ : Sym → Sym, then find simple linear combinations whose images have the form λ∈Λi 1 z λ p λ over disjoint subsets Λ i of partitions of n (see the last paragraph of Subsection 2.5).

Remark 5.2. Note that J - k,n is not a system of orthogonal idempotents. Indeed, the proof below will show that the symmetric function images Γ(J - k,n ) are not of the form

λ∈Λ k 1 z λ p λ .
The problem is that {V - 2i+1 } is not closed under the internal product, so the analogue of Equation ( 4) cannot be applied in Proposition 5.1.

Theorem 5.3. With notation as above:

i) A basis of orthogonal idempotents of W n is given by

1 2 (I + k,n + I - k,n ), 1 ≤ k ≤ n and k ≡ n mod 2 and 1 2 (J + l,n + J - l,n -I + l,n -I - l,n ), 0 ≤ l ≤ n -4 and l ≡ n mod 2.
Their images under Γ : Sym → Sym are

λ n o (λ)=k, e (λ)=0 1 z λ p λ , 1 ≤ k ≤ n and k ≡ n mod 2 and λ n o(λ)=l, e(λ)>0 even 1 z λ p λ , 0 ≤ l ≤ n -4 and l ≡ n mod 2.
ii) A basis of orthogonal idempotents of W • n is given by

I + k,n , 1 ≤ k ≤ n and k ≡ n mod 2 and 1 2 (J + l,n -I + l,n ), 0 ≤ l ≤ n -2 and l ≡ n mod 2.
Their images under Γ : Sym → Sym are

λ n o (λ)=k, e (λ)=0 1 z λ p λ , 1 ≤ k ≤ n and k ≡ n mod 2 and λ n o (λ)=l, e (λ)>0 1 z λ p λ , 0 ≤ l ≤ n -2 and l ≡ n mod 2.
iii) A basis of orthogonal idempotents of C n is given by

1 2 (I + k,n + I - k,n ), 1 ≤ k ≤ n and k ≡ n mod 2 and 1 2 (J + l,n + J - l,n -I + l,n -I - l,n ), 0 ≤ l ≤ n -4 and l ≡ n mod 2 and 1 2 (J + m,n -J - m,n ), 0 ≤ m ≤ n -2 and m ≡ n mod 2.
Their images under Γ : Sym → Sym are iv) A basis of orthogonal idempotents of P n is given by

λ n o (λ)=k, e(λ)=0 1 z λ p λ , 1 ≤ k ≤ n and k ≡ n mod 2
I - k,n , 1 ≤ k ≤ n and k ≡ n mod 2.
Their images under Γ :

Sym → Sym are λ n o(λ)=k, e(λ)=0 1 z λ p λ , 1 ≤ k ≤ n and k ≡ n mod 2.
v) A basis of orthogonal idempotents of P • n is given by

J + k,n , 0 ≤ k ≤ n and k ≡ n mod 2.
Their images under Γ : Sym → Sym are

λ n o (λ)=k 1 z λ p λ , 0 ≤ k ≤ n and k ≡ n mod 2.
Proof. Observe that each set of claimed images under Γ have cardinality equal to the dimension of the corresponding subalgebra. Thus, if these images are correct, Γ : Sym → Sym must restrict to an algebra isomorphism on each of these subalgebras. Hence the claimed sets are orthogonal and idempotent if and only if their images under Γ are orthogonal and idempotent. Each claimed symmetric function image is a sum of p λ z λ , the orthogonal idempotents of Sym, hence their sums are also idempotent. Furthermore, for each subalgebra, no partition λ appears in more than one sum, so the sums are orthogonal.

So it suffices to check that the claimed idempotents are indeed in the correct subalgebras, and have the claimed images under Γ. Note that I - 1 is a series in V - 2 , and I + 1 is a series in V + 2 , so

I - k,n ∈ Vect(V - 2k,n ) = P n , I + k,n ∈ Vect(V + 2k,n ), J + k,n ∈ Vect(V + 2k+1,n ) = P • n , J - k,n ∈ Vect(V - 2k+1,n ).
By symmetry in the definitions of

I + k , I - k , J + k , J - k , it is clear that I + k,n + I - k,n ∈ Vect(V 2k,n ), J + k,n + J - k,n ∈ Vect(V 2k+1,n ).
Thus all claimed idempotents are in the correct subalgebras.

To calculate the symmetric function images, first recall that Also, As remarked at the end of Section 2.5, the identification of the commutative images of the idempotents leads to analogues of the characters of Foulkes [START_REF] Foulkes | Eulerian numbers, Newcomb's problem and representations of symmetric groups[END_REF], that only depend on certain features of the cycle type:

Γ(V + 1 ) = n≥0 h n = exp   i≥1 p i i   , Γ(V - 1 ) = n≥0 e n = exp
Γ(J + 0 ) = Γ(V + 1 ) Γ(V - 1 ) (-1 2 ) Γ(V + 1 ) (-1 2 ) = Γ(V + 1 ) ( 1 2 ) Γ(V - 1 ) (-1 2 ) = exp   1 2   i≥1 p i i - (-1) i-1 p i i     = exp i>0 even p i i = o (λ)=0 1 z λ p λ , and
Γ(J - 0 ) = Γ(V - 1 ) Γ(V + 1 ) (-1 2 ) Γ(V - 1 ) (-1 2 ) = Γ(V + 1 ) (-1 2 ) Γ(V - 1 ) ( 1 2 ) = exp   1 2   i≥1 - p i i + (-1) i-1 p i i     = exp i>0 even - p i i = o (λ)=0

Lemma 3 . 5 .

 35 Let u ∈ S n and a, b such that s a u = us b (so that u({b, b + 1}) = {a, a + 1}). Then u and s a u are both in W + m or both in W - m for some m, if and only if:• 2 ≤ b ≤ n -2, and• u(b -1) and u(b + 2) are both < a or both > a + 1.

  then it is in ωPω iff the coefficients of W • 2k+1 and W • 2k+2 are the same for all k. This is also equivalent to the equality of the coefficients of W + 2k+1 and W - 2k+1 in x, and also to the condition xω = x. So we have ωPω = {x ∈ W • : xω = x}, which is clearly closed under product.

  )=l, e (λ) even 1 z λ p λ , 0 ≤ l ≤ n -4 and l ≡ n mod 2 and λ n o (λ)=m, e(λ)>0 odd 1 z λ p λ , 0 ≤ m ≤ n -2 and m ≡ n mod 2.

o

  (λ)=k, e(λ)=0 1 z λ p λ .

(- 1 )o

 1 (λ)=k, e (λ)=0 1 z λ p λ   = o (λ)=k, e (λ) odd 1 z λ p λ .

Corollary 5 . 4 .

 54 Write χ I for the symmetric group character corresponding to the ribbon skew shape for I. Let run(I), pk(I) etc. denote run(σ), pk(σ) for any permutation σ with Des(σ) = Des(I). Then i) The set of characters  for functions S n → R depending only on the number of odd cycles and whether there are even cycles, and are zero on permutations with an odd number of even cycles.ii) The set of characters   run • (I)=k χ I : 1 ≤ k ≤ n   is a basis for functions S n → R depending only on the number of odd cycles and whether there are even cycles.iii) The set of characters for functions S n → R depending only on the number of odd cycles and whether the number of even cycles is zero, even positive, or odd.iv) The set of characters   pk(I)=k χ I : 1 ≤ 2k ≤ n + 1   is a basis for functions S n → R depending only on the number of odd cycles and are zero on permutations with an odd number of even cycles.v) The set of characters   pk • (I)=k χ I : 1 ≤ 2k + 1 ≤ n + 1   is a basis for functions S n → R depending only on the number of odd cycles.Proof. If I∈I k R I span a commutative subalgebra of the descent algebra (for some disjoint sets I k of compositions), and the symmetric function image of this subalgebra has a basis of orthogonal idempotents of the formλ∈Λ k p λ z λ , thenI∈I k χ I is a basis for functions S n → R that are constant on permutations with cycle type within each Λ k , and are zero on permutations with cycle type not in any Λ k .

Acknowledgements

We thank Jean-Yves Thibon for suggesting an exploration of algebras based on alternating runs. We thank Kyle Petersen for helpful discussions, and for pointing out to us the reference by Doyle and Rockmore. We are also grateful to François Bergeron, Christophe Hohlweg and Franco Saliola for their help and comments. Sage computer software [29] was very useful throughout this project. We also thank the Laboratoire International Franco-Québécois de Recherche en Combinatoire (LIRCO), who supported the French author in his visits to Montréal. Eventually, we thank the reviewers for their nice comments and suggestions.

* Supported by the Laboratoire International Franco-Québecois de Recherche en Combinatoire (LIRCO), and by ANR CARMA (ANR-12-BS01-0017).

† Supported by a CRM-ISM postdoctoral fellowship, and an NSERC individual research grant of François Bergeron.

Algebraic proof for the existence of run algebras

This section gives a second proof, using noncommutative symmetric functions, that grouping permutations by the number of runs gives an algebra. This will require working in all degrees n at the same time, so from now on add a second index to all previous quantities indicating the degree, if these quantities involve one degree only. For example, we now write W n for the commutative run algebra in degree n, and W k,n for its basis.

First define the following elements in Sym:

The intuition is that V + 1 is an ascending run and V - 1 is a descending run. To obtain multiple runs, one concatenates ascending runs and descending runs alternately. Since external product roughly expresses concatenation, it is no surprise that we define

and

Write V + k,n for the degree n part of V + k , and similarly for V - k,n , V k,n , so

Note that V + k,n , V - k,n are non-zero even when k > n. Recall from Section 2.4 that V + 1 , V - 1 are grouplike. Being a product of grouplike elements, V + k , V - k are also grouplike. Thus the simplified splitting formula in Section 2.4 holds, and this allows a straightforward proof that the V s are "multiplicative" (under the internal product) in the following way: Proposition 4.1. We have:

If k is even, then

Proof. The splitting formula asserts that, for any F 1 , . . . , F l in Sym, it holds that

and similarly for V - k . So in particular:

where the ± is + or -depending on the parity of l. For even k, once we know that

), this simplifies to

For odd k, once we know that

), this simplifies to

The other products are treated similarly, and we are left with the case l = 1. Recall V + 1 = i R i and R i is the identity for the internal product, so

Internal product to the right by ω i sends a permutation σ to its reversal σ i , . . . , σ 1 , which exchanges the ascents and descents and reverses their order. Given a composition I, let Ī denote the result after this exchanging and reversing, e.g.

.

By linearity, it follows that, for any

, since reversing a permutation twice does not change it. As a result, if k is even, then

And if k is odd, then

(Note that the sign rule here is the same as in Equation ( 1). This will be explained by the relation between Vs and Ws given below.)

Now it is clear that Vect(V + k,n , V - k,n : k ∈ N) is closed under the internal product, and it is easy to identify subspaces, such as Vect(V + k,n + V - k,n : k ∈ N), which form subalgebras. It remains to equate these to the run algebras W ± n and W n , and see how the Vs give them new bases. We will do this by relating V - 2k and V + 2k+1 to the Eulerian peak and left peak algebras respectively, via a triangular change of basis. Proposition 4.2. For all k, we have V - 2k,n ∈ P n , the Eulerian peak algebra.

for some constants a k,l,n . Hence {V - 2k,n : 1

} is a basis for P n .

Proof. For the first assertion, it suffices to show that, for each composition I, the coefficient of R I in V - 2k,n depends only on the number of peaks in any permutation with descent composition Ilet σ I denote such a permutation.

First observe that V - 2k = (V - 2 ) k by definition, and that V - 2 = 1 + 2 R I over all compositions I where σ I has no peaks.

Recall that R (i1,...,i (I) ) R (j1,...,j (J) ) = R (i1,...,i (I) ,j1,...,j (J) ) + R (i1,...,i (I)-1 ,i (I) +j1,j2,...,j (J) ).

So the coefficient of R I in any external product F G of noncommutative symmetric functions is the sum, over all J := (j 1 , . . . , j (J) ), J := (j 1 , . . . , j (J ) ) with I = (j 1 , . . . , j (J) , j 1 , . . . , j (J ) )

5.2 Orthogonal idempotents for the noncommutative run algebra W ± n

We now show that the idempotents of C n constructed in Theorem 5.3 are in fact a complete set of primitive orthogonal idempotents for W ± n .

Theorem 5.5. A complete set of primitive orthogonal idempotents for W ± n is given by

Proof. The key is to show that C n is a complement to K n := ker Γ ∩ W ± n . As noted in the previous section, Γ restricted to C n is an isomorphism, so C n and K n have trivial intersection. Since dim C n = n -1 + n 2 , it must be that dim K n ≤ n 2 -1. Now W + 2k,n = ωW - 2k,n ω for 1 ≤ k ≤ n 2 -1, so W + 2k,n -W - 2k,n ∈ K n , and these elements are linearly independent, so dim K n = n 2 -1, and W ± n = C n ⊕ K n . By [26, Cor. 2.2], the Jacobson radical of any subalgebra A of Sym n (under the internal product) is equal to ker Γ ∩ A. Hence K n is the Jacobson radical of W ± n , so any complete set of primitive orthogonal idempotents for the complement C n is a complete set of primitive orthogonal idempotents for W ± n .


Miroslav Iliaš

Hans Jørgen

Aa Jensen

Radovan Bast

Trond Saue
email: trond.saue@irsamc.ups-tlse.fr

Gauge origin independent calculations of molecular magnetizabilities in relativistic four-component theory

Keywords: molecular properties, magnetic properties, molecular magnetizabilities, relativistic effects, London atomic orbitals, property surfaces

The use of magnetic-field dependent London atomic orbitals, also called gauge including atomic orbitals, is known to be an efficient choice for accurate non-relativistic calculations of magnetizabilities. In this work the appropriate formulas were extended and implemented in the framework of the four-component relativistic linear response method at the self-consistent field single reference level. Benefits of employing the London atomic orbitals in relativistic calculations are illustrated with Hartree-Fock wave functions on the XF 3 (X = N, P, As, Sb, Bi) series of molecules. Significantly better convergence of magnetizabilities with respect to the basis set size is observed compared to calculations employing a common gauge origin. In fact, it is mandatory to use London atomic orbitals unless you want to use ridiculously large basis sets. Relativistic effects on magnetizabilities are found to be quite small (< 5%) for this particular set of molecules, but should be investigated on a larger set of molecules. We emphasize the breakdown of the connection between the paramagnetic contribution to magnetizabilities and rotational g tensors in the relativistic domain and discuss its origin. Finally, we visualize the magnetizability density which shows markedly atomic features evocative of Pascal's rules.

Introduction

Presently the preferred way to ensure that calculated magnetic properties of molecules, such as magnetizabilities, are independent of the gauge origin of the external magnetic field, is through the use of the physically motivated magneticfield dependent London atomic orbitals (LAOs) [1][2][3][4][5][6][7], also called gauge including atomic orbitals (GIAOs) [8]. LAO-based approaches for the calculation of magnetizabilities have accordingly been reported with non-relativistic Hamiltonians at the Hartree-Fock (HF) [9], multiconfiguration self-consistent field (MCSCF) [10], density functional theory (DFT) [11][12][13][14], and coupled cluster CCSD(T) [15] levels. Schwerdtfeger and co-workers included relativistic effects with relativistic pseudopotentials, but noted a breakdown of the pseudopotential approximation in LAO-based calculations of magnetizabilities of Au, AuF, and Sn n clusters [16]. Proper correction terms were derived by van Wüllen [17] and successfully applied in a joint publication with the previous authors [18]. Yoshizawa and Hada [19][20][21] have reported HF and MP2 calculations of molecular magnetizabilities based on the second-order Douglas-Kroll-Hess Hamiltonian, using a common gauge origin (CGO) combined with extended basis sets [22].

In the present work we present the theory for an LAO-based calculation of magnetizabilities at the 4-component relativistic HF level, our implementation of the theory, and sample applications. This work is a natural extension of our previous development of a method for gauge origin independent 4-component relativistic calculations of NMR shielding constants [23,24]. As a first application we have chosen to study the magnetizability of the trifluorides of the group 15 elements. Our choice was motivated by the study of Ruud and Helgaker [25] in which they demonstrated the crucial role of LAOs for the proper convergence of the magnetizability of PF 3 . The paper is organized as follows: In Section 2 the theory for 4-component relativistic HF calculations of magnetizabilities is derived. Computational details are given in Section 3, whereas results are presented and discussed in Section 4. Finally we conclude and provide perspectives in Section 5.

Theory

The static 3 × 3 magnetizability tensor may be defined as the second derivative of the molecular energy with respect to the components of the magnetic field:

ζ αβ = - d 2 E dB α dB β B=0 (1)
In the present work we employ SI-based atomic units [26] in which the magnetizability is given in units of e 2 a 2 0 /m e = 7.89104•10 -29 JT -2 = 4.752042•10 -6 erg G -2 mol -1 . We will also make extensive use of the Einstein convention of summation over repeated indices.

Our starting point is the 4-component Dirac-Coulomb (DC) Hamiltonian extended by a uniform external magnetic field B which is represented through the vector potential

A G (r) = 1 2 B × r G ; r G = r -R G , (2)
where the subscript G refers to the arbitrarily chosen gauge origin. The subsequent development, restricted to closed-shell HF theory, follows closely that of the seminal paper by Helgaker and Jørgensen [7] (we note that the implementation can be straightforwardly extended to DFT). We employ an exponential parametrization of a single reference Slater determinant in terms of the orbital rotation operator κ

|0(B) = exp[-κ] |0(0) ; κ = κ ai a † i -κ * ai i † a. (3)
Here and in the following we will use indices i, j, . . . and a, b, . . . for occupied and virtual orbitals, respectively, and reserve p, q, . . . for general orbitals. The molecular orbitals will be expanded in the magnetic-field dependent LAOs

ω µ (r) = exp[-ieA G (R µ ) • r] χ µ (r), (4)
with R µ representing the center of the basis function χ µ . In our case each LAO is formally a 4-component vector function (a spinor), however, only one of the four components is non-zero. An immediate advantage of using such an effectively scalar basis set expansion is that it allows for a straightforward use of non-relativistic integral codes. In the LAO basis the integrals of the one-electron part of the DC Hamiltonian are given by

h µν = χ µ | exp i 2 B • (R µν × r) ( ĥD -B • mν ) | χ ν = h D;µν -B • m ν;µν , (5)
where appear integrals over the Dirac Hamiltonian ĥD in the molecular field, that is, the electrostatic field of fixed nuclei, as well as over the magnetic dipole operator mν with respect to center R ν of the basis function

χ ν mν = - 1 2 (r ν × cα). (6)
One may note that the imaginary phase factor in the above integral vanishes for one-center integrals due to the presence of the vector R µν = R µ -R ν . The corresponding integrals over the two-electron Coulomb term are

g µν,ρσ = χ µ χ ρ | exp i 2 B • (R µν × r 1 + R ρσ × r 2 ) 1 r 12 | χ ν χ σ . (7)
Simplification of the ensuing formalism is obtained by first defining a set of unmodified molecular orbitals (UMOs) in terms of the zero-field molecular coefficients [27,28]

ψ q (r, B) = ω µ (r, B)c µq (0). (8)
The electronic Hamiltonian in second-quantized form will, however, be constructed from a set of orthonormalized molecular orbitals (OMOs) (denoted by tilde) ψp (r, B) = ψ q (r, B) T qp (B);

ψp | ψq = T * rp ψ r | ψ s T sq = δ pq , (9)
which conserves the algebra of creation and annihilation operators at any field strength. Orthonormality of the OMOs at any field strength is assured by the connection matrix T . Throughout this work we will employ the natural connection [10,29,30] which provides maximum similarity in a least-squares sense between the OMOs and the unperturbed UMOs. In the OMO basis the electronic Hamiltonian may be separated into two parts H = H0 -B • mpq p † q; H0 = hD;pq p † q + 1 2 gpq,rs p † r † sq (10) and it should be noted that this separation is different from the common gauge origin (CGO) case where would appear integrals of the magnetic dipole operator mG with respect to the arbitrary gauge origin and not the individual basis function centers as here.

The HF energy now depends explicitly on the magnetic field through the LAOs in addition to the magnetic interaction term of the Hamiltonian, Eq. ( 10), and implicitly through the orbital rotation parameters and the connection matrix. At any field strength we impose that the HF energy is optimized with respect to non-redundant orbital rotation parameters {κ pq }. The components of the magnetizability tensor, Eq. (1), are accordingly given by

ζ αβ = ζ linres αβ + ζ expval αβ . (11)
The linear response part is given by

ζ linres αβ = -0 dκ dB β , d H0 dB α -mα 0 B=0 = -(κ β; * ai F α ai + F α ia κ β ai ) (12)
where appear magnetic derivatives of the OMO Fock matrix

F α pq = d Fpq dB α B=0 = -m α;pq + hα D;pq + Lα pq,jj ; L pq,rs = (pq|rs) -(ps|rq) . ( 13
)
The non-redundant first-order amplitudes κ β are found by solving a response equation involving the generalized property gradient F β as described in Ref. 23. The expectation value part is given by

ζ expval αβ = -0 d 2 H0 dB α dB β - d mα dB β - d mβ dB α 0 B=0 (14)
= -(

hαβ D;ii + 1 2 Lαβ ii,jj ) + P αβ mβ α,ii ,
where P αβ symmetrizes the expression with respect to components α and β of the magnetic field. The first-and second-order magnetic derivatives appearing in the above expressions can be expressed compactly using the brace notation of Helgaker and Jørgensen [7] for the one-index transformation of one-and two-electron matrix elements

{T α , h} pq = T α * p p h p q + h pq T α q q (15) {T α , L} pq,rs = T α * p p L p q,rs + L pq ,rs T α q q + T α * r r L pq,r s + L pq,rs T α s s (16)
For one-electron operators in OMO basis the first and second magnetic derivatives are then given as

1 hα D;mn = h α D;mn + {T α , h D } mn ( 17
)
hαβ D;mn = h αβ D;mn + P αβ {T α , h β D } mn + {T αβ , h D } mn (18) + 1 2 P αβ ({T α , {T β , h D }} mn -{T β T α , h D } mn ).
The corresponding expressions for two-electron operators are equivalent, but involve four indices and the brace notation of Eq. ( 16). The final expressions, Eqs. ( 12) and ( 14), for components of the magnetizability tensor are quite similar to non-relativistic theory [7,9,10], but involve some new integrals such as the first-and second derivatives of the Dirac Hamiltonian in LAO-basis

h α D;µν = i 2 χ µ | (R µν × r) α ĥD | χ ν , (19)
h αβ D;µν = - 1 4 χ µ | (R µν × r) α (R µν × r) β ĥD | χ ν , (20)
which has been implemented in the integral driver HERMIT [START_REF] Hermit | a molecular integral program[END_REF] in a development version of the DIRAC12 code for relativistic molecular calculations [START_REF] Dirac | a relativistic ab initio electronic structure program, Release DIRAC12[END_REF]. An important difference, though, is that the relativistic expression does not contain an explicit diamagnetic term. In a CGO formalism the diamagnetic contribution can be extracted from the part of the linear response function involving rotations between occupied positive-energy and virtual negative-energy orbitals [START_REF] Aucar | [END_REF]34]. In an LAO-based formalism the situation is somewhat more complicated since reorthonormalization of UMOs, Eq. ( 8), involve both positive-and negative-energy orbitals. The separation of para-and diamagnetic contributions in a LAO framework will not be pursued in the present work.

Computational details

Using a development version of the DIRAC12 code [START_REF] Dirac | a relativistic ab initio electronic structure program, Release DIRAC12[END_REF] we have calculated HF magnetizabilities of the series of molecules XF 3 (X = N, P, As, Sb, Bi) in different basis sets of increasing size. We have employed the experimental structures, as given in Table 1, all corresponding to trigonal pyramidal structures. Most calculations were based on the 4-component DC Hamiltonian, but in order to investigate separately scalar and spin-orbit effects on magnetizabilities we have also performed spin-orbit free and non-relativistic calculations based on the modified Dirac [35,36] and the Lévy-Leblond [37] Hamiltonians, respectively. In addition, for non-relativistic reference values we have performed calculations with the Dalton [START_REF] Dalton | a molecular electronic structure program[END_REF] code. In all calculations we have chosen a Gaussian model for the nuclear charge distributions [START_REF] Visscher | [END_REF].

For the lighter atoms (F, N, P and As) we have employed sequences of precc-pVXZ basis sets of Dunning and coworkers [40][41][42][43], where pre may indicate single (aug) and double (d-aug) augmentation. For the heavier atoms Sb and Bi we employed the corresponding basis sets by Dyall [44,45], denoted dyall.vnz (n = 2, 3, 4, dyall.avnz with augmentation) in the DIRAC basis set library. In this work we will specify basis sets with the notation pre-XZ which is to be understood as an alias for both the Dunning basis sets (F, N, P and As) and the Dyall basis sets (Sb and Bi). All basis sets have been employed in uncontracted form. In the DIRAC code the small component basis sets were generated by restricted kinetic balance (RKB) in the SCF step and extended according to the simple magnetic balance (sMB) scheme in the magnetizability calculations [24]. The (SS|SS) class of twoelectron integrals was neglected throughout the four-component DC relativistic calculations. For the LAO calculations we used the natural connection [29] in the reorthonormalization terms. In the CGO calculations the gauge origin was placed at the center of mass of the molecule.

Results and discussion

In Table 2 we report our calculated magnetizabilities for the selected molecules XF 3 (X = N, P, As, Sb, Bi). In addition to parallel (ζ ) and perpendicular (ζ ⊥ ) components of the magnetizability with respect to the C 3 axis, we also report the isotropic and anisotropic components defined as:

ζ iso = 1 3 (ζ + 2ζ ⊥ ) (21)
ζ ani = ζ ⊥ -ζ (22)
In Figure 1 we illustrate the convergence of ζ iso of BiF 3 in the basis sets XZ and aug-XZ. As was already shown by Ruud and Helgaker [25] for the PF 3 molecule, the use of LAOs is crucial for obtaining fast convergence of magnetizabilities with respect to the basis set. Taking the LAO result for ζ iso of BiF 3 in aug-QZ basis as reference, the CGO result in the DZ basis is in error by 152%, whereas the corresponding LAO number is within 1% of the reference value. Augmenting the basis with diffuse functions reduces the CGO error to 42% (aug-DZ), but even in the aug-QZ basis the CGO error with respect to the reference value is more than 6%. The effect of augmentation of the basis was studied in more detail for NF 3 . As seen from Table 2, the calculated CGO value of ζ iso is -11.47 e 2 a 2 0 /m e and -6.46 e 2 a 2 0 /m e in the DZ and aug-DZ bases, respectively. Simply adding the diffuse d functions of fluorine reduces the DZ value to -6.10 e 2 a 2 0 /m e , whereas the effect of adding the other diffuse functions is quite small. However, for consistency we have chosen to employ the same augmentation level for the basis sets of all atoms in this study. The LAO results are in comparison hardly sensitive to augmentation. For all molecules the isotropic magnetizability ζ iso is converged to within 1% using LAOs in conjunction with the aug-TZ basis set. In passing we note that adding the (SS|SS) class of two-electron integrals reduces both the parallel and perpendicular magnetizabilities of BiF 3 , obtained with LAOs in aug-QZ basis, by 0.0025 e 2 a 2 0 /m e , which is outside the precision with which our results are reported in Table 2, thus justifying the neglect of this integral class.

Table 2 shows that relativistic effects on magnetizabilities are rather small for the selected set of molecules; for BiF 3 relativity increases the isotropic magnetizability by about 5%. Under the assumption of additivity of atomic susceptibilities in diamagnetic molecules, as proposed by Pascal [46][47][48], and taking his value of -1.4 e 2 a 2 0 /m e for the atomic susceptibility of fluorine [49], we find that that the atomic susceptibility increases as -2.3 • Z 0.38 e 2 a 2 0 /m e for the group 15 atoms. Furthermore, relativity will increase the atomic susceptibility of the bismuth atom by 7%, from -9.3 to -8.7 e 2 a 2 0 /m e . This result is in line with the rather small relativistic effects on magnetizabilities of noble gas atoms and group 14, 15, 16 and 17 hydrides reported by Yoshizawa and Hada [20,21]. However, the same authors found relativistic effects on the order of 20% for the IF molecule. Qualitatively we can understand these observations in the following way: One would expect large effects of 4-component calculations of magnetizabilities compared to non-relativistic calculations when spin-orbit coupling is important for the valence electrons. As the fluorine atoms are very electronegative, the central atom of the XF 3 (X = N, P, As, Sb, Bi) series of molecules will have an oxidation state of +3, that is, the valence np orbitals are formally not occupied. Indeed, projection analysis [50] of BiF 3 gives a charge of +2.2 of bismuth with a valence configuration of 6s 1. 8 1/2 6p 0.5 1/2 6p 0.6 3/2 in terms of the calculated atomic orbitals of neutral bismuth in its ground state electronic configuration. Further analysis, using the modified Dirac [35,36] and the Lévy-Leblond [37] Hamiltonians, shows that the relativistic effect is almost completely dominated by scalar relativistic effects. On the other hand, for IF the valence np orbitals are formally occupied, so it is perhaps not surprising to observe large relativistic effects on the magnetizability of this molecule. Clearly relativistic effects on magnetizabilities merit further study. Experimental values for our selected set of molecules are scarce, but have been included in Table 2 whenever available. Comparison of our HF results with experiment is meaningful since the effects of electron correlation and vibrational corrections are found to be generally small for magnetizabilities [13,15,51,52] of diamagnetic molecules. For instance, a recent benchmark study by Lutnaes et al. [13] of a set of 28 small molecules shows that the contribution from electron correlation and zero-point vibrational corrections are typically within 2% and 1%, respectively, but with outliers such as paramagnetic O 3 and BH [15] for correlation and N 2 , CO, O 3 , LiH, and PN for vibration. The anisotropic component of the magnetizability can be obtained from microwave spectroscopy of molecules in an external magnetic field [53]. Experimental values of ζ ani obtained in this manner have been reported for NF 3 and PF 3 by Stone et al. [54] and later refined for PF 3 by Hüttner and co-workers [55]. For NF 3 the agreement with our HF/LAO-results is quite good, but less so for PF 3 . For the latter molecule Stone et al. [54] provide an estimate of the isotropic contribution based on Pascal's rules. The discrepancy, on the order of 15%, may in part be due to the possible breakdown of Pascal's rules for fluorine-containing molecules [52]. Experimental data for bulk susceptibilities of SbF 3 (ζ iso =-9.68 e 2 a 2 0 /m e ) and BiF 3 (ζ iso =-12.8 e 2 a 2 0 /m e ) can be found in the compilation by Föex [START_REF] Föex | Constantes Sélectionnées. Diamagnétisme et Paramagnétisme[END_REF], referring to experimental work by Pascal [49] and Chowdhury [START_REF] Chowdhury | [END_REF], respectively, and agree reasonably well with our HF values.

In a non-relativistic framework the paramagnetic component of the magnetizability can be extracted from the rotational g tensor [53] much in the same way as the paramagnetic component of the NMR shielding tensor can be related to the spin-rotation constant [58][59][60], and has accordingly been given for the PF 3 molecule by Stone et al. [54]. It is important, however, to stress that these connections do not hold in the relativistic domain [61][62][63]. This can be understood from the following argument: Component ζ αβ of the magnetizability tensor can be interpreted as component α of the magnetic dipole moment induced to first order by component β of an external magnetic field and has the general form

ζ αβ = 1 2 dr (r G × j (Bβ) (r)) α . (23)
The current density j corresponds to the electron charge times the density associated with the appropriate velocity operator, which is different in the relativistic and non-relativistic domains. Starting from the Legendre transformation

p • v = H (r, p) + L (r, v) (24)
one can extract the relativistic and non-relativistic velocity operators as

v R = ∂H R ∂p = cα; v NR = ∂H NR ∂p = π m ; π = p + eA (25)
The relativistic form, in terms of the Dirac α matrices scaled with the speed of light c, can be interpreted in terms of the Zitterbewegung superimposed on the mean trajectory of the electron [64]. The non-relativistic form of the velocity operator naturally splits into a paramagnetic and diamagnetic part, where the former, inserted into Eq. ( 23), gives rise to an orbital angular momentum operator with respect to the gauge origin G. Setting the gauge origin to the center of mass of the molecule then provides the connection to the rotational g tensor, a connection which is absent in the relativistic case. Starting from Eq. ( 23) one may define a magnetizability density

ζ CGO αβ (r) = 1 2 (r G × j (Bβ) (r)) α . (26)
which integrates to a specific component ζ αβ of the magnetizability tensor. The first-order magnetically induced current density j (Bβ) can be constructed from the perturbed orbital rotation parameters κ β which appear in Eq. ( 12). In the relativistic case the orbital rotation parameters split into two classes, κ ++ ai and κ -+ ai , corresponding to rotations between the occupied positive-energy orbitals and virtual positive-and negative-energy orbitals, respectively. This separation provides a separation of the para-and diamagnetic contributions to the induced current density, as demonstrated in Ref. 34. In Figure 2 the total isotropic average of ζ CGO αβ (r) in BiF 3 is represented using an isosurface plot together with its para-and diamagnetic contributions using the above recipe. For technical reasons the plots are based on calculations using a common gauge origin and a large basis (aug-QZ). We indeed observe that the para-and diamagnetic densities have opposite signs. The diamagnetic contribution resembles a superposition of atomic contributions, evocative of Pascal's rules, and dominates the total isotropic density in BiF 3 . While the density isosurface around Bi atom is negative, the contributions from the F atoms are polarized and exhibit both positive and negative isosurfaces. The densities corresponding to the parallel and one of the perpendicular contributions, ζ CGO (r) and ζ CGO ⊥ (r), are visualized in Figures 3 and4, respectively. Figure 3 representing the parallel component ζ CGO (r) gives direct visual evidence that there is almost no contribution to the paramagnetic part from the Bi atom. The diamagnetic contribution on the other hand features a torus-shaped isosurface around the Bi atom.

Conclusions and Perspectives

We have presented the theory and an implementation of 4-component relativistic LAO-based HF calculations of molecular magnetizabilities. An obvious extension of the present work would be to the Kohn-Sham DFT framework. We have already implemented the DFT extension of the linear response term ζ linres αβ , Eq. ( 12), and have applied it to the calculation of NMR shielding constants [24]. We will next add the missing DFT contributions to the expectation value term ζ expval αβ , Eq. ( 14). It would also be interesting to explore static correlation effects in the framework of 4-component relativistic MCSCF [65]. We furthermore foresee a study of the separation of para-and diamagnetic contributions of the magnetizability in an LAO-based relativistic framework.

As a first application we have calculated magnetizabilities of the XF 3 (X = N, P, As, Sb, Bi) molecules with LAOs. We observe, contrary to CGO-based calculations, fast basis set convergence in 4-component calculations, just as in the non-relativistic domain [9,25]. Our code also allows visualization of the magnetizability density which for BiF 3 shows markedly atomic features evocative of Pascal's rules. Since the effects of electron correlation and vibrational corrections are generally small for magnetizabilities [51], the agreement to experiment, when available, is quite reasonable. For the selected set of molecules relativistic effects are rather small (5%), but a study by Yoshizawa and Hada [21] indicates that relativistic effects on magnetizabilities may reach 20%. We therefore plan further studies of magnetizabilities of molecules containing heavy elements. Another interesting aspect to explore is the breakdown of the connection between rotational g tensor and the paramagnetic contribution to the magnetizabilities in the relativistic domain. Table 2. Magnetizabilities (in e 2 a 2 0 /me) of XF3 (X = N, P, As, Sb, Bi). The relativistic effect ∆ rel is calculated as the difference between Dirac-Coulomb (DC) and non-relativistic (NR) results, the latter obtained with the Dalton code. The basis set notation pre-XZ refers to pre-cc-pVXZ basis sets of Dunning and co-workers [40][41][42][43] for the atoms F, N, P and As and corresponding basis sets by Dyall [44,45]

Figure 1 .

 1 Figure 1. Convergence of ζ iso (in e 2 a 2 0 /me) of BiF 3 in XZ and aug-XZ basis sets using LAOs or CGO.

Figure 2 .

 2 Figure 2. Isotropic average total (left), paramagnetic (middle), and diamagnetic (right) magnetizability density ζ iso (r) in BiF 3 (CGO aug-QZ; DC Hamiltonian; gray/light isosurface: -0.035 e 2 /mea 0 ; red/dark isosurface: +0.035 e 2 /mea 0 ).

Figure 3 .

 3 Figure 3. Parallel paramagnetic (left) and diamagnetic (right) magnetizability density ζ (r) in BiF 3 (CGO aug-QZ; DC Hamiltonian; gray/light isosurface: -0.035 e 2 /mea 0 ; red/dark isosurface: +0.035 e 2 /mea 0 ).

Figure 4 .

 4 Figure 4. Perpendicular paramagnetic (left) and diamagnetic (right) magnetizability density ζ ⊥ (r) in BiF 3 (CGO aug-QZ; DC Hamiltonian; gray/light isosurface: -0.035 e 2 /mea 0 ; red/dark isosurface: +0.035 e 2 /mea 0 ).

Table 1 .

 1 Taylor & Francis and I.T. Consultant Experimental structure parameters of the XF3 (X = N, P, As, Sb, Bi) molecules. Ref. 66; b Ref. 67; c Ref. 68; d Ref. 69.

		NF 3	a	PF 3	b	AsF 3	c	SbF 3	d	BiF 3	d
	R(XF)/ Å	1.3676 1.5644 1.7043 1.880	1.987
	angle(FXF)/deg 101.84	97.8	95.88	94.9		96.1

a

  for Sb and Bi.

					CGO				LAO
	Molecule		Basis	ζ	ζ ⊥	ζiso	ζani	ζ	ζ ⊥	ζiso	ζani
	NF3	DC	DZ	-14.83	-9.78	-11.47	5.04	-4.27	-4.87	-4.67	-0.60
			TZ	-7.88	-6.52	-6.97	1.37	-4.36	-4.96	-4.76	-0.60
			QZ	-5.65	-5.54	-5.58	0.11	-4.43	-5.04	-4.83	-0.61
			aug-DZ	-6.75	-6.31	-6.46	0.44	-4.57	-5.22	-5.00	-0.65
			aug-TZ	-5.01	-5.42	-5.28	-0.41	-4.64	-5.24	-5.04	-0.60
			aug-QZ	-4.70	-5.26	-5.08	-0.56	-4.64	-5.24	-5.04	-0.60
			d-aug-DZ	-6.37	-6.18	-6.24	0.19	-4.57	-5.22	-5.00	-0.65
			d-aug-TZ	-5.01	-5.45	-5.31	-0.44	-4.69	-5.27	-5.08	-0.58
			d-aug-QZ	-4.76	-5.31	-5.13	-0.55	-4.68	-5.28	-5.08	-0.60
		NR	aug-QZ					-4.68	-5.28	-5.08	-0.60
		Exp. a									-0.63 (±0.32)
		∆ rel	aug-QZ					0.04	0.04	0.04	0.00
	PF3	DC	DZ	-19.52	-14.04	-15.87	5.48	-6.68	-6.31	-6.43	0.37
			TZ	-10.39	-9.18	-9.59	1.21	-6.66	-6.27	-6.40	0.39
			QZ	-7.74	-7.71	-7.72	0.02	-6.66	-6.29	-6.41	0.37
			aug-DZ	-9.91	-9.52	-9.65	0.39	-6.84	-6.61	-6.69	0.23
			aug-TZ	-7.29	-7.96	-7.74	-0.67	-6.82	-6.50	-6.61	0.33
			aug-QZ	-6.88	-7.46	-7.27	-0.58	-6.76	-6.43	-6.54	0.33
		NR	aug-QZ					-6.81	-6.47	-6.59	0.34
		Exp. a								(-7.87)	0.278 (±0.04)
		Exp. b									0.286 (±0.04)
		∆ rel	aug-QZ					0.06	0.04	0.05	-0.01
	AsF3	DC	DZ	-23.87	-19.08	-20.68	4.80	-8.30	-8.01	-8.11	0.29
			TZ	-13.16	-12.49	-12.71	0.68	-8.38	-8.13	-8.21	0.25
			QZ	-9.85	-10.39	-10.21	-0.55	-8.39	-8.17	-8.24	0.22
			aug-DZ	-12.15	-12.26	-12.22	-0.10	-8.48	-8.31	-8.37	0.17
			aug-TZ	-9.12	-10.29	-9.90	-1.18	-8.50	-8.30	-8.37	0.20
			aug-QZ	-8.62	-9.77	-9.39	-1.14	-8.49	-8.29	-8.36	0.20
		NR	aug-QZ					-8.59	-8.36	-8.44	0.23
		∆ rel	aug-QZ					0.09	0.07	0.08	-0.03
	SbF3	DC	DZ	-30.68	-25.42	-27.17	5.26	-11.12	-10.73	-10.86	0.38
			TZ	-17.28	-17.18	-17.22	0.10	-11.22	-10.91	-11.01	0.30
			QZ	-12.92	-14.14	-13.74	-1.22	-11.16	-10.89	-10.98	0.26
			aug-DZ	-15.93	-15.72	-15.79	0.20	-11.23	-11.00	-11.08	0.23
			aug-TZ	-12.03	-13.85	-13.24	-1.82	-11.25	-11.00	-11.08	0.26
			aug-QZ	-11.38	-13.14	-12.55	-1.76	-11.23	-10.97	-11.06	0.26
		NR	aug-QZ					-11.42	-11.10	-11.21	0.32
		Exp. c								-9.68
		∆ rel	aug-QZ					0.19	0.13	0.15	-0.06
	BiF3	DC	DZ	-35.97	-30.39	-32.25	5.58	-12.83	-12.59	-12.67	0.24
			TZ	-20.44	-19.08	-19.53	1.36	-12.80	-12.68	-12.72	0.13
			QZ	-15.24	-15.44	-15.37	-0.20	-12.80	-12.70	-12.74	0.10
			aug-DZ	-18.43	-17.99	-18.14	0.44	-12.97	-12.86	-12.90	0.10
			aug-TZ	-13.78	-14.43	-14.22	-0.65	-12.85	-12.75	-12.79	0.10
			aug-QZ	-13.06	-13.92	-13.63	-0.86	-12.86	-12.76	-12.79	0.10
		NR	aug-QZ					-13.59	-13.32	-13.41	0.27
		Exp. d								-12.8
		∆ rel	aug-QZ					0.73	0.56	0.62	-0.17

These expressions may be compared to Eqs.[START_REF] Dirac | a relativistic ab initio electronic structure program, Release DIRAC12[END_REF] and (33) of Ref. 10; it will be seen that for the second magnetic derivative there is a sign difference in the second term which we believe is correct for our expression.

Taylor & Francis and I.T. Consultant

ACKNOWLEDGMENTS

It is a great pleasure to dedicate this paper to a dear and very much respected colleague, Trygve Helgaker, on his 60th birthday. According to Picasso, one starts to get young at the age of sixty and then it is too late. Trygve may prove him wrong. On a more serious note, the scientific impact of Trygve Helgaker is evident in this contribution: he wrote the first version of the integral package HERMIT [31] used for the integrals in this work back in the 1980's, and he has given significant contributions to eight of the central references in this work.

This work was done under Slovak -French Research and Development Cooperation (Grant No. SK-FR-00180-11). MI is grateful for the financial support from the Slovak Research and Development Agency (Grant No. APVV-0059-10).