J.-C Aval

IDEALS AND QUOTIENTS OF B-QUASISYMMETRIC POLYNOMIALS

Dédié à Xavier Viennot

Abstract. The space QSym n (B) of B-quasisymmetric polynomials in 2 sets of n variables was recently studied by Baumann and Hohlweg [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF]. The aim of this work is a study of the ideal QSym n (B) + generated by B-quasisymmetric polynomials without constant term. In the case of the space QSym n of quasisymmetric polynomials in 1 set of n variables, Aval, Bergeron and Bergeron [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF] proved that the dimension of the quotient of the space of polynomials by the ideal QSym + n is given by Catalan numbers C n = 1 n+1 2n n . In the case of B-quasisymmetric polynomials, our main result is that the dimension of the analogous quotient is equal to 1 2n+1 3n n , the numbers of ternary trees with n nodes. The construction of a Gröbner basis for the ideal, as well as of a linear basis for the quotient are interpreted by a bijection with lattice paths. These results are finally extended to p sets of variables, and the dimension is in this case 1 pn+1 (p+1)n n

, the numbers of p-ary trees with n nodes.

Résumé. L'espace QSym n (B) des polynômes B-quasisymétriques en deux ensembles de n variables a été récemment étudié par Baumann et Hohlweg [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF]. Nous considérons ici l'idéal QSym n (B) + engendré par les polynômes B-quasisymétriques sans terme constant. Dans le cas de l'espace QSym n des polynômes quasisymétriques en 1 ensemble de n variables, Aval, Bergeron et Bergeron [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF] ont montré que la dimension du quotient de l'espace des polynômes par l'idéal QSym + n est donnée par les nombres de Catalan C n = 1 n+1 2n n . Dans le cas des polynômes B-quasisymétriques, notre principal résultat est que la dimension du quotient analogue est ici 1 2n+1 3n n , à savoir le nombre d'arbres ternaires à n noeuds. Nous construisons une base de Gröbner pour l'idéal, de même qu'une base du quotient, toutes deux explicites et en bijection avec des chemins. Nous étendons enfin ces résultats à p ensembles de variables, et montrons que dans ce cas la dimension est

Introduction

To start with, we recall (a part of) the story of the study of ideals and quotients related to symmetric or quasisymmetric polynomials. The root of this work is a result of Artin [1]. Let us consider the set of variables X n = x 1 , x 2 , . . . , x n . The space of polynomials in the variables X n with rational coefficients is denoted by Q[X n ]. The subspace of symmetric polynomials is denoted by Sym n . Symmetric polynomials may be seen (cf. [START_REF] Macdonald | Symmetric Functions and Hall Polynomials[END_REF]) as invariants of the symmetric group S n under the action defined as follows:

for σ ∈ S n anf P ∈ Q[X n ], σ • P (X n ) = P (x σ(1) , x σ(2) , . . . , x σ(n) ).
Let V be a subset of the polynomial ring. We denote by V + the ideal generated by elements of a V with no constant term. Artin's result is given by:

(1.1) dim Q[X n ]/ Sym + n = n! .
Another, more recent, part of the story deals with quasisymmetric polynomials. The space QSym n ⊂ Q[X n ] of quasisymmetric polynomials was introduced by Gessel [START_REF] Gessel | Multipartite P -partitions and products of skew Schur functions[END_REF] as generating functions for Stanley's P -partitions [START_REF] Stanley | Enumerative Combinatorics[END_REF]. This is the starting point of many recent works in several areas of combinatorics [START_REF] Bergeron | Pieri Operations on Posets[END_REF][START_REF] Malvenuto | Duality between quasi-symmetric functions and the Solomon descent algebra[END_REF][START_REF] Gelfand | Noncommutative symmetric functions[END_REF][START_REF] Stanley | [END_REF]. Quasisymmetric polynomials may also be seen as S n -invariants under Hivert's quasisymmetrizing action ( [START_REF] Hivert | Hecke algebras, difference operators, and quasi-symmetric functions[END_REF]), defined as follows.

Let I = {i 1 , . . . , i k } be a subset of {1, . . . , n} and a = (a 1 , . . . , a k ) a sequence of positive (> 0) integers, of the same cardinality. We define

X a I = x a 1 i 1 • • • x a k i k
, where the elements of I are listed in increasing order. Hivert's action is then defined on monomials by σ * X a I = X a σ(I)

where σ(I) is the set {σ(i 1 ), . . . , σ(i k )} arranged in increasing order. In [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF], Aval et. al. study the problem analogous to Artin's work in the case of quasisymmetric polynomials. Their main result is that the dimension of the quotient is given by Catalan numbers:

(1.2) dim Q[X n ]/ QSym + n = C n = 1 n + 1 2n n .
An interesting axis of research is the extension of these results to 2 (or p ≥ 2) sets of variables. In the case of two sets of variables, let A n = A 2 n denote the alphabet

A n = x 1 , y 1 , x 2 , y 2 , . . . , x n , y n .
The diagonal action of S n on Q[A n ] is defined as simultaneous permutation of variables x's and y's:

σ • P (A n ) = P (x σ(1) , y σ(1) , . . . , x σ(n) , y σ(n) ).
Invariants associated to this action are called diagonally symmetric polynomials. Their set is denoted by DSym n .The diagonal coinvariant space Q[A n ]/ DSym + n has been studied extensively in the last 15 years by several authors [START_REF] Bergeron | Lattice Diagram Polynomials and Extended Pieri Rules[END_REF][START_REF] Bergeron | Identities and Positivity Conjectures for Some Remarkable Operators in the Theory of Symmetric Functions[END_REF][START_REF] Garsia | A graded representation model for Macdonald's polynomials[END_REF][START_REF] Garsia | Orbit Harmonics and Graded Representations[END_REF][START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF]. A great achievment in this area is Haiman's proof of the following equality (cf. [START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF]):

dim Q[A n ]/ DSym + n = (n + 1) n-1 .
In [START_REF] Aval | Diagonal Temperley-Lieb invariants and harmonics[END_REF], the space DQSym n of diagonally quasisymmetric polynomials is defined as the invariant space of the diagonal extension of Hivert's action. This space was originally introduced by Poirier [START_REF] Poirier | Cycle type and descent set in wreath products[END_REF], and, with generalizations, has been recently studied in [START_REF] Novelli | Free quasi-symmetric functions of arbitrary level[END_REF] and [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF].

The coinvariant space Q[A n ]/ DQSym + n is investigated in [START_REF] Aval | Diagonal Temperley-Lieb invariants and harmonics[END_REF], and conjectures are stated. In particular, a conjectural basis for this quotient is presented.

To end this presentation, we introduce the space QSym n (B) of B-quasisymmetric polynomials, which is the focus of this article. This space, whose definition appears implicitly in [START_REF] Poirier | Cycle type and descent set in wreath products[END_REF], is studied with more details in [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF]. A precise definition will be given in the next section, and we only mention here that QSym n (B) is a subspace (and in fact a subalgebra, cf. [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF]) of DQSym n .

We now state the main result of this work, which appears as a generalization of equation (1.2).

Theorem 1.1.

(1.3) dim Q[A n ]/ QSym n (B) + = 1 2n + 1 3n n .
Observe that in equations (1.2) and (1.3), the dimensions

1 n+1 2n n and 1 2n+1 3n n
are respectively the numbers of binary and ternary trees (cf. [START_REF] Sloane | A Handbook of Integer Sequences[END_REF]). This will be generalized in the last section of this paper.

The content of this paper is divided into 5 main sections. After this introduction, the Section 2 defines the central objects of this work, the B-quasisymmetric polynomials. Sections 3 and 4 are the proof of the Theorem 1.1. In Section 3 is introduced a set G of polynomials, which is proved in Section 4 to be a Gröbner basis for QSym n (B) + . The Gröbner basis G, as well as the basis of the quotient "deduced" from it, are interpreted in terms of plane paths. Finally, Section 5 gives a generalization of this work for p sets of variables, where the equation analogous to

(1.3) replaces 1 2n+1 3n n by 1 pn+1 (p+1)n n
, the number of p-ary trees.

QSym(B): definitions and notations

For these definitions, we follow [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF], with some minor differences, for the sake of simplicity of the computations we will have to make.

Let N and N denote two occurrrences of the set of nonnegative integers. We shall write N = { 0, 1, 2, . . . } and make no difference between the elements of N and N in any arithmetical expression. We distinguish N and N for the ease of reading.

A bivector is a vector

v = (v 1 , v 2 , . . . , v 2k-1 , v 2k
) such that the odd entries {v 2i-1 , i = 1..k} are in N, and the even entries {v 2i , i = 1..k} are in N.

A bicomposition is a bivector in which there is no consecutive zeros, ie. no pattern 0 0 or 00.

The integer k is called the size of v. The weight of v is by definition the couple

(|v| N , |v|N) = ( k i=1 v 2i-1 , k i=1 v 2i ).
We also set |v| = |v| N + |v|N. For example (1, 0, 2, 1, 0, 2, 3, 0) is a bicomposition of size 4, and of weight [START_REF]Multivariate Fuss-Catalan numbers[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF].

To make notations lighter, we shall sometimes write bivectors or bicomposition as words, for example 1 02 10 23 0 stands for (1, 0, 2, 1, 0, 2, 3, 0) (see also the following definition).

The fundamental B-quasisymmetric polynomials, indexed by bicompositions, are defined as follows

F c 1 c 2 ...c 2k-1 c 2k (A n ) = x i 1 • • • x i |c| N y j 1 • • • y j |c| N ∈ Q[A n ]
where the sum is taken over indices i's and j's such that

i 1 ≤ • • • i c 1 ≤ j 1 ≤ • • • j c 2 < i c 1 +1 ≤ • • • i c 1 +c 3 ≤ j c 2 +1 ≤ • • • ≤ j c 2 +c 4 < i c 1 +c 3 +1 ≤ • • •
We give some examples:

F 1 2 = i≤j≤k x i y j y k , F 0 21 0 = i≤j<k y i y j x k .
It is clear from the definition that the bidegree (ie. the couple (degree in x, degree in y)

) of F c in Q[A n ] is the weight of c. If the size of c is greater than n, we shall set F c (A n ) = 0.
The space of B-quasisymmetric polynomials, denoted by QSym n (B) is the vector subspace of Q[A n ] generated by the F c (A n ), for all bicompositions c.

Let us denote by I 2 n the ideal QSym n (B) + generated by B-quasisymmetric polynomials with zero constant term.

Paths and G-set

The aim of this section is to construct a set G of polynomials, which will be proved in the next section to be a Gröbner basis of I 2 n . These two sections are greatly inspired from [START_REF] Aval | Catalan Paths and Quasi-Symmetric Functions[END_REF][START_REF] Aval | Ideals of Quasi-Symmetric Functions and Super-Coinvariant Polynomials for S n[END_REF].

Let v = (v 1 , v 2 , . . . , v 2k-1 , v 2k ) be a bivector of size n. We associate to v a path π(v) in the plane N × N, with steps (0,1) or (2,0). We start from (0,0) and add for each entry v i (read from left to right): v i steps (2,0), followed by one step (0,1).

As an example, the path associated to (1, 0, 1, 2, 0, 0, 1, 1) is

We have two kinds of path, regarding their position to the diagonal x = y. If a path always remains above this line, we call it a 2-Dyck path, and say that the corresponding vector is 2-Dyck. Conversely, if the path enters the region x < y, we call both the path and the vector transdiagonal. For example, v = (0, 0, 1, 0, 0, 1, 1, 0) is 2-Dyck, whereas w = (0, 0, 1, 1, 1, 0, 0, 0) is transdiagonal.

(w) (v) π π A simple but important observation is that a vector v = (v 1 , v 2 , . . . , v 2k-1 , v 2k ) is transdiagonal if and only if there exists 1 ≤ l ≤ k such that (3.1) v 1 + v 2 + • • • + v 2l-1 + v 2l ≥ l.
Our next task is to construct a set G of polynomials, mentionned above. From now on, unless otherwise indicated, vectors are of size n. For w a vector of size k < n, w0 * denotes the vector (of size n) obtained by adding the desired number of 0 0 patterns. We shall define the length ℓ(v) of a vector v as the integer k such that v = v 1 v 2 . . . v 2k-1 v 2k 0 * with v 2k-1 v 2k = 0 0. In the case of bicompositions, the notions of size and length coincide.

For v a vector (of length n), we denote by A v n the monomial

A v n = x v 1 1 y v 2 1 • • • x v 2n-1 n y v 2n n .
To deal with leading terms of polynomials, we will use the lexicographic order induced by the ordering of the variables:

x 1 > y 1 > x 2 > y 2 > • • • > x n > y n .
The lexicographic order is defined on monomials as follows: A v n > lex A w n if and only if the first non-zero entry of v -w (componentwise) is positive.

The set

G = {G v } ⊂ I 2 n
is indexed by transdiagonal vectors. Let v be a transdiagonal vector. For v = c0 * with c a non-zero bicomposition of length ≥ n (which implies that v is transdiagonal), we define

G v = F c .
If v cannot be written as c0 * , the polynomial G v is defined recursively. We look at the rightmost occurrence of two consecutive zeros (on the left of a non-zero entry: we do not consider the subword 0 * ). Two cases are to be distinguished according to the parity of the position of this pattern:

• if v = w0 0αβc0 * , with w a vector of size k -1, α ∈ N (by definition non-zero), β ∈ N, c a bicomposition, we define

(3.2) G w0 0αβc0 * = G wαβc0 * -x k G w(α-1)βc0 * ;
• if v = wα 00βc0 * , with w a vector of size k -1, α ∈ N, β ∈ N (by definition non-zero), c a bicomposition, we define

(3.3) G wα 00βc0 * = G wαβc0 * -y k G wα(β-1)c0 * .
We easily check that both terms on the right of (3.2) and (3.3) are indexed by vectors that are transdiagonal as soon as v is transdiagonal. We do it for (3.2) : let us denote v ′ = wαβc0 * and v ′′ = w(α -1)βc0 * . Let l be the smallest integer such that (3.1) holds for v. If l ≥ k -1 then w is transdiagonal thus so are v ′ and v ′′ , and if not:

v ′ 1 + v ′ 2 + • • • + v ′ 2l-3 + v ′ 2l-2 ≥ l and v ′′ 1 + v ′′ 2 + • • • + v ′′ 2l-3 + v ′′ 2l-2 ≥ l -1. Since v ′ and v ′′ are of length equal to ℓ(v)-1, this defines any G v for v transdiagonal by induction on ℓ(v).
It is interesting to develop an example, where we take n = 3.

G 0 01 00 2 = G 0 01 20 0 -y 2 G 0 01 10 0 = (G 1 20 00 0 -x 1 G 0 20 00 0) -y 2 (G 1 10 00 0 -x 1 G 0 10 00 0) = (F 1 2 -x 1 F 0 2) -y 2 (F 1 1 -x 1 F 0 1) = (x 1 y 2 1 + x 1 y 1 y 2 + x 1 y 1 y 3 + x 1 y 2 2 + x 1 y 2 y 3 + x 1 y 2 3 + x 2 y 2 2 + x 2 y 2 y 3 +x 2 y 2 3 + x 3 y 2 3 -x 1 (y 2 1 + y 1 y 2 + y 1 y 3 + y 2 2 + y 2 y 3 + y 2 3 )) -y 2 (x 1 y 1 + x 1 y 2 + x 1 y 3 + x 2 y 2 + x 2 y 3 + x 3 y 3 -x 1 (y 1 + y 2 + y 3 )) = x 2 y 2 3 -y 2 x 3 y 3 + x 3 y 2 3
The monomials of the result are ordered with respect to the lexicographic order and we observe that the leading monomial (denoted LM) of G 0 01 00 2 is A 0 01 00 2 3

. The following proposition shows that this fact holds in general for the family G. Proposition 3.1. Let v be a transdiagonal vector. The leading monomial of G v is

(3.4) LM(G v ) = A v n .
The proof of this proposition is, as the definition of the G v polynomials, inductive on the length of v. First observe that the definitions of the F c and of the lexicographic order imply (3.4) when v = c0 * with c a bicomposition. Now Proposition 3.1 is a consequence of the following lemma.

We shall write A n\k = x k+1 , y k+1 , . . . , x n , y n .

Lemma 3.2. Let w be a vector of size k, and c a bicomposition, then we have

(3.5) G wc0 * (A n ) = A w k F c (A n\k ) + (terms < A w k ). Proof. If w is a bicomposition, then (3.5
) is a consequence of the definition of the polynomials F 's. If not this is readily done by induction on ℓ(w), by using the recursive definition of the G's.

We suppose that we are in the case of recursion (3.2), ie. w can be written w = u0 0αβd with u a bivector of size l, α ∈ N, α > 0, β ∈ N, and d a bicomposition.

We first observe that the F polynomials obey to recursive relations. We suppose we have a bicomposition γδg, with γ > 0. Then the definition of the fundamental quasisymmetric polynomials implies:

(3.6) F γδg (A n ) = F γδg (A n\1 ) + x 1 F (γ-1)δg (A n )
To reduce (4.8), we use the following relations, where w denotes a bivector, d a bicomposition, α and β elements of N and N, not simultaneously zero: The reduction of the product (4.8) is made possible because of the order of the multiplications: the successive "shifts" are processed from left to right.

Our final task is to show that all vectors u generated in this process are transdiagonal and that their length never exceeds n.

Let us first check that the generated vectors are all transdiagonal. In the case of relations (4.10) and (4.12), this is obvious. Now, let us consider, for example relation (4.9). Let us denote u = wαβd0 * , u ′ = w(α + 1)βd0 * and u ′′ = w0 0(α + 1)βd0 * . Since u is transdiagonal, there exists 1 ≥ l ≥ ℓ(u) such that

u 1 + u 2 + • • • + u 2l-1 + u 2l ≥ l.
If l > ℓ(w), u ′ and u ′′ are clearly transdiagonal, and if not:

u ′ 1 + u ′ 2 + • • • + u ′ 2l-1 + u ′ 2l ≥ l + 1 ≥ l and u ′′ 1 + u ′′ 2 + • • • + u ′′ 2l+1 + u ′′ 2l+2 ≥ l + 1 whence u ′ and u ′′ are transdiagonal.
Let us now check that the length of the generated vectors never exceeds n. We keep track of the couple e = u 2ℓ(u)-1 , u 2ℓ(u) . We distinguish two cases.

(1) e comes from c 2ℓ(c)-1 , c 2ℓ(c) that is shifted to the right by relations (4.9) and/or (4.11). It may be shifted at most |v| steps to the right, thus:

ℓ(u) ≤ ℓ(c) + |v| ≤ |c| + |v| ≤ n.
(2) e comes from a 1 0 or 0 1 generated by relation (4.10) or (4.12), then shifted to the right (by any relations). We suppose it is created by a multiplication by x k or y k , and we consider the vector

t = v 2n v 2n-1 . . . v 2k v 2k-1 0 * . Since A v n is in C n , the word t is 2-Dyck. Thus: |t| < ℓ(t) = n -k + 1.
This implies that the term 1 0 or 0 1 can be shifted at most to position

k + |t| ≤ k + n -k = n.
Now if we denote by D n,k,l the set of 2-Dyck paths with k horizontal steps at even height (corresponding to x terms) and l horizontal steps at odd height (corresponding to y terms), then the following equality is proven in [START_REF]Multivariate Fuss-Catalan numbers[END_REF]:

# D n,k,l = n + k -1 k n + l -1 l n -k -l n .
5. Quotient of polynomials by QSym(G p )

Every results of this paper can be extended to p sets of variables. Because of the great similarity to the previous sections, we shall only present here the result and a short sketch of the proof.

We denote by A p n the alphabet of p × n variables:

A p n = x (1) 1 x
(2)

1 . . . x (p) 1 x (1)
2 . . . x (p) 2 . . . x (1) n . . . x (p) n . We define p-vectors of size k as vectors of p × k integers. For the ease of reading, we can write for example when p = 3: v = 0 12 1 00 2 01 . A p-composition is a p-vector avoiding 3 consecutive zeros.

The set QSym(G p ) of G p -quasisymmetric polynomials is the vector subspace of Q[A p n ] spanned by fundamental G p -quasisymmetric polynomials, defined for a pcomposition c by:

F c = x (1) i (1) • • • x (p) i (p) with i (1) 1 ≤ • • • ≤ i (1) c 1 ≤ i (2) c 1 +1 ≤ • • • ≤ i (2) c 1 +c 2 ≤ • • • ≤ • • • ≤ i (p) c 1 +•••+cp < i (1)
c 1 +•••+cp+1 ≤ • • • .
We give an example (here p = 3 and we use letters x, y, z for the alphabets x (1) , x (2) , x (3) ): Proof. We shall only give a brief description of the proof, which is very similar to the one of Theorem 4.4.

We first associate to any monomial a plane path, as in Section 3, with the difference that horizontal steps are of length p. Paths (and associated monomials) are said to be p-Dyck if they stay above the diagonal, and transdiagonal if not.

The construction of the set G indexed by transdiagonal path is the same, with p cases of recurrence. We prove that G is a Gröbner basis of I p n as in Section 4, and conclude that a basis of the quotient Q p n is given by the monomials associated to p-Dyck paths.

To conclude, we observe that the depth-first search [START_REF] Cormen | Introduction to Algorithms[END_REF] gives a bijection between p-ary trees, enumerated by the right-hand side of (5.1), and p-Dyck paths.

(4. 9 )

 9 x k G wαβd0 * = G w(α+1)βd0 * -G w0 0(α+1)βd0 * or (4.10) x k G w0 * 00 * = G w0 * 10 * -G w0 * 0 010 * for the x k factors and: (4.11) y k G wαβd0 * = G wα(β+1)d0 * -G w0 0α(β+1)d0 * or (4.12) y k G w0 * 0 0 * = G w0 * 1 0 * -G w0 * 0 0 10 * for the y k factors. All these equations are direct consequences of the recursive definition of the G polynomials.

F 0 10 2

 2 01 = i<j≤k≤l y i x j x k z l . We define the ideal I p n = QSym(G p ) + and the quotient Q p n = Q[A p n ]/I p n . The result which generalizes Theorem 4.4 is Theorem 5.1. For p ≥ 1,

Acknowledgement. The author thanks C. Hohlweg for introducing him to B-quasisymmetric polynomials, and for valuable comments and explanations.

if γδ = 1 0, and

The same kind of equalities holds when γ = 0, with recursive terms multiple of y 1 . We now prove (3.5). We have to distinguish two cases. We first suppose αβ = 1 0, and use (3.2) and (3.6) to write:

All this process can be done in the case of recurrence (3.3), and this completes the proof.

Proof of the main theorem

The aim of this section is to prove Theorem 1.1, by showing that the set G constructed in the previous section is a Gröbner basis for I 2 n . This will be achieved in several steps.

We introduce the notation

Proof. It clearly suffices to show that (4.1) holds for any monomial A v n , with v transdiagonal. We assume that there exists A v n not reducible of the form (4.1) and we choose A w n to be the smallest amongst them with respect to the lexicographic order. Let us write

n -G w ) are lexicographically smaller than A w n , thus they are reducible. This contradicts our assumption and completes our proof. This lemma implies that B n spans the quotient Q n . We will now prove its linear independence. The next lemma is a crucial step.

We have already obtained the following reduction for any monomial A w n in

which is equivalent to

We then apply the reduction (4.1) to each monomial of the Q c 's. Now we use the algebra structure of QSym(B) (cf. Proposition 37 of [START_REF] Baumann | A Solomon theory for the wreath product G ≀ S n[END_REF]) to reduce products of fundamental B-quasisymmetric polynomials as linear combinations of F c 's. We obtain (4.2) in a finite number of operations since degrees strictly decrease at each operation, because |c| > 0 implies deg Q c < |w|.

Now we come to the final step in the proof. Before stating this lemma, we introduce some notations, and make an observation.

For

). In the same way, let R(A n ) denote the reverse alphabet of A n : R(A n ) = y n , x n , . . . , y 1 , x 1 . Then one has for any bicomposition c:

n ∈ B n }. Now we reduce the problem, using (4.6) and the algebra structure of QSym n (B) to write:

. Now we have to prove that for any monomial A v n ∈ C n and any non-zero bicomposition c:

Thanks to Lemma 4.1, any monomial of degree at least equal to n is in I 2 n , thus we can restrict to |v| + |c| < n.

We consider the product (4.8)

To illustrate the recursive reduction of a product of the form (4.8), we give the following example, where n = 5:

= y 2 (G 2 00 10 00 00 0 -G 0 02 00 10 00 0) = y 2 G 2 00 10 00 00 0 -y 2 G 0 02 00 10 00 0 = G 2 00 20 00 00 0 -G 2 00 00 2 -G 0 02 10 10 00 0 + G 0 02 00 10 10 0.

Now we are able to complete the proof of Theorem 1.1. We can even state a more precise result.

Theorem 4.4. A basis of the quotient Q n is given by the set

Since I 2 n is bihomogeneous, the quotient Q n is bigraded and we can consider H k,l (Q n ) the subspace of Q n consisting of polynomials of bidegree (k, l), then

Proof. By Lemma 4.1, the set B n spans Q n . Assume we have a linear dependence:

By Lemma 4.3, the set G spans I 2 n , thus

This implies LM(P ) = A u n , with u transdiagonal, which is absurd. Hence B n is a basis of the quotient Q n .

For the combinatorial part, we refer to [START_REF]Multivariate Fuss-Catalan numbers[END_REF], but we give a short proof of (4.13).

A ternary tree is a tree in which every internal node has exactly 3 sons. Ternary trees are known [START_REF] Sloane | A Handbook of Integer Sequences[END_REF] to be enumerated by

To conclude we observe that the depth-first search [START_REF] Cormen | Introduction to Algorithms[END_REF] of a tree gives a bijection between ternary trees and 2-Dyck paths ; we recall that we search recursively the left son, the middle son, the right son, and finally the root, and associate to each external node (except the leftmost one) a (0, 1) step, and to each internal node a (2, 0) step. Below is given an illustration of this bijection, where we put in dashed lines the final horizontal sequence, to coincide with our definition of 2-Dyck paths.


Slavkov Dragan

Hajdukovic
email: dragan.hajdukovic@cern.chdragan.hajdukovic@physac.me

Quantum Vacuum and Solar System Ephemerides

Quantum vacuum might be a source of gravity, and if it is so, it would be a third revolution (Newton, Einstein, Quantum vacuum) in our understanding of gravitation. It was argued recently that, if quantum vacuum fluctuations are virtual gravitational dipoles, dark matter and dark energy can be considered as local and global effects of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter. The major question is, if quantum vacuum, as an additional source of gravity, can be compatible with known orbits of planets in Solar System. In order to answer this question, it is necessary to create ephemerides including quantum vacuum in dynamical equations of motion which describe gravitational physics of Solar system. We suggest a simple way how to include quantum vacuum in the ephemerides creation process and suggest a careful revision of recent work done by different authors.

Quantum vacuum [1] is an essential part of the Standard Model of Particles and Fields. If you are not familiar with quantum vacuum, just consider it as a new state of matter-energy, radically different from more familiar states (solid, fluid, gas, plasma…) but as real as they are. You may imagine quantum vacuum as an omnipresent fluid (superfluid) composed of quantum vacuum fluctuations, or, in a more popular wording, composed of virtual particle-antiparticle pairs with an extremely short lifetime (for instance the lifetime of a virtual electron-positron pair is of the order of 21 10  seconds).

In principle quantum vacuum can be a source of gravity. In a recent series of papers [2][3][4][5][6][7] it was argued that phenomena , usually attributed to dark matter and dark energy, can be considered as local and global effects, of the gravitational polarization of the quantum vacuum by the immersed Standard Model matter (i.e. matter composed from quarks and leptons, interacting through exchange of gauge bosons). More precisely, what we call a halo of dark matter might be a halo of the gravitationally polarized quantum vacuum; all halos together may be considered as a cosmological fluid which, in certain epochs, can cause the accelerated expansion of the Universe. Within the Solar system, there is an excellent agreement between astronomical observations and predictions of the Newtonian theory of gravity, based on the existing quantity of the Standard Model matter. Hence, the eventual gravitational field caused by the quantum vacuum must be only a tiny fraction of the total gravitational field. In conclusion, if quantum vacuum is the true nature of what we call dark matter and dark energy it must be dominant source of gravity in larger systems as galaxies but negligible in small systems as Solar system. In a way it is close to our intuition; while the quantity of the Standard Model matter remains constant, the quantity of quantum vacuum increases with volume.

A major support to the emerging model of the gravitational properties of the quantum vacuum would be to show that including quantum vacuum as an additional source of gravity doesn't produce effects which are in conflict with the existing empirical evidence for Solar system. Consequently, we propose to create new ephemerides (with quantum vacuum included as a source of gravity) and to show that new ephemerides are not inferior to the old once.

Ephemerides are result of a numerical integration of the dynamical equations of motion which describe the gravitational physics of the Solar system. Historically, the first ephemerides were created using only Newtonian gravity with the basic result that the acceleration, caused by a pointlike body, of mass M at a distance r is equal to Within the emerging model [2][3][4][5][6][7] of the gravitational properties of the quantum vacuum, each point-like body is surrounded by a halo of the polarized quantum vacuum and, in addition to Newtonian or General relativistic effects, there is an additional acceleration   r g qv towards the body, which is caused by the quantum vacuum.

    r M f GP r g g gv , 4 max   (1)
Here, max g P denotes a universal constant related to quantum vacuum; in principle this constant must be measured but it shouldn't be too different from a rough theoretical estimate

2 max 06 . 0 m kg P g  which is used in this paper. Function   r M f
, is not known exactly but a reasonably good approximation is

  max 4 , tanh , g sat sat P M R r R r m f          (2)
Within the Solar system, a significant simplification is that for Sun and planets r R sat  ; consequently, according to Eq. ( 2),  

1 ,  r M f
. An oversimplification is to neglect contribution of planets to the gravitational polarization of the quantum vacuum. If the Sun is considered as the only source of gravitational polarization, the unique effect is an additional constant acceleration towards the Sun ( This acceleration is about 20 times smaller than acceleration in the Pioneer anomaly.

Let us note that this acceleration causes a retrograde precession of planets, with the following precession per orbit

  m M G a e g Sun qv qv      2 2 max 1 2  (3)
where a and m denote respectively semi-major axes and mass of the planet. For instance, in the case of Mercury we have

century arc qv sec 68 . 0    .
The observed perihelion precession of Mercury is known with high accuracy; we quote the value from Ref. [8] which is   0015 . 0 31 . 575  arc-seconds per century. Note that uncertainty is a few milli-arc-seconds per century what is two orders of magnitude smaller than the presumed value qv   . However, observations tell us only the total precession rate but not separate contributions of different sources of precession. Sun, planets and smaller bodies in Solar system, are sources of Newtonian precession (estimated to be about 532"/century), while the Schwarzschild metric of the Sun which cannot be neglected at the distance of Mercury should be source of an additional precession of 43"/century. Apparently, it is possible that quantum vacuum as one more of many sources of precession is "hidden" between them and cannot be easily distinguished from other sources. Hence, it seems questionable if (as it was done in a series of papers [START_REF] Fienga | Proceedings IAU Symposium[END_REF][START_REF] Iorio | [END_REF][11]) a hypothetical source of precession can be disproved only because it causes precession which is bigger than the uncertainty of the measurement of the total precession rate.

In addition, let us underline that the value determined by Eq. ( 3) must be considered only as an upper bound. Namely, the halo of the Sun is just the biggest halo, perturbed by sub-halos, i.e. halos of planets and smaller bodies. An outer planet with the halo of the polarized quantum vacuum acts on Mercury as if it has a slightly bigger mass; result is a tiny perihelion precession. The key feature is that precession caused by halo of the Sun and halo of a planet have opposite sign. Namely, the acceleration of Mercury caused by an external planet is directed outward, opposite to the acceleration caused by the Sun. Approximately, the magnitude of acceleration caused by 7 outer planets [12] is

    8 2 2 2 2 i i i i p R a a R GM g (4)
Here, a denotes semi-major axis of Mercury, while i M and i R are mass of the i-th planet and the radius of its orbit (orbits are assumed to be circular). The corresponding perihelion precession per orbit is

  m M G a e g Sun p p    2 2 1 2  (5)
Consequently the total perihelion precession caused by quantum vacuum is a sum of precessions of the opposite sign determined by equations ( 3) and (5).

All this suggests that there is need for a careful revision of the previous work done by different authors [START_REF] Fienga | Proceedings IAU Symposium[END_REF][START_REF] Iorio | [END_REF]11]. Apparently an essential minimum is to include halos for Sun and eighth planets.

At galactic and larger scales, the emerging theory [2][3][4][5][6][7] of the gravitational properties of the quantum vacuum predicts the right size of the effects, showing potential to explain phenomena usually attributed to dark matter and dark energy. As a major task, it remains to check if the theory predicts the right size of effects at small scales as Solar system.

  ephemerides include General relativistic effects. What we propose now is to include quantum vacuum as well.


Armel Pitelet

Emilien Mallet

Rabih Ajib

Caroline Lemaître

Emmanuel Centeno

Antoine Moreau
email: antoine.moreau@uca.fr

Plasmonic enhancement of spatial dispersion effects in prism coupler experiments

Spatial dispersion, i.e. the dependency of the permittivity on the wavevector, arises in metals because of the repulsive interaction between the free electrons inside the jellium. This phenomenon puts a limit to the validity of Drude's model 1 , which has proven extremely accurate in plasmonics for more than a century now. This success can be related to the fact that Drude's model is the zeroth order term of all the more advanced descriptions of the free electron gas 2-6 . The first-order correction to the model, however, is intrinsically nonlocal: because of the repulsion between free electrons, the metal can still be described using an effective polarization, but the polarization in a given point depends on the electric field in the surroundings -at a typical distance close to the free mean path of the electrons.

The idea that spatial dispersion could have an impact on the optical response of metals and on surface plasmons dates back to the 1960s. Very advanced models 3,7 , including the hydrodynamic model 2,4 , have been proposed at the time to tackle the problem. However, it became clear in the 1980s that no optical experiment with noble metals could show an impact of spatial dispersion large enough to threaten the predominance of Drude's model in plasmonics with an optical excitation. While spatial dispersion and more advanced models for surface plasmons [8][9][10] continued to be thoroughly studied in surface science especially using EELS [11][12][13] , the consensus seemed to be that Drude's model is largely sufficient in plasmonics [START_REF] Boardman | Electromagnetic surface modes[END_REF][START_REF] Frostmann | Metal optics near the plasma frequency[END_REF] .

For many in this community, the experiments showing Drude's model failure with large enough nanoparticles excited optically [START_REF] Ciracì | [END_REF]16 thus came as a surprise. These experiments renewed the interest paid to the hydrodynamic model 17 , which proved accurate enough, and to the enhancement of spatial dispersion brought by small gaps between metals instead of relying on tiny particles 18,19 . However, the sub-nanometer gaps required to observe the impact of spatial dispersion raised some skepticism 20 , all the more so that in that case the effects studied in surface science, like the spill-out, are likely to intervene strongly [21][22][23] . A lot of work has subsequently been devoted to (i) study theoretically in which situations spatial dispersion is likely to have an impact [24][25][26][27] (ii) develop numerical tools based on the hydrodynamical model [28][29][30] to accurately predict these effects in complex geometries and finally (iii) to better understand the fundamental reasons why the hydrodynamic model, while it presents well documented deficiencies 31 , seems accurate enough for noble metals in plasmonics 32,33 .

Here we propose to use the classical prism-coupler configuration 10,34 to excite high wavevector plasmonic modes, in order to enhance the effects of spatial dispersion alone. We predict, relying on the hydrodynamic model and accurate material parameters, that the impact of spatial dispersion will be two orders of magnitude larger than what can be reached with simple surface plasmons. The plasmonic resonances excited can be linked to the excitation of Gap-Plasmons 26,35 (GP), Long-Range Surface Plasmons (LRSP) and Short-Range Surface Plasmons 19,[36][37][38] (SRSP) supported by multilayered structures with dimensions that are large enough to exclude other phenomenon like the spill-out 23 . Were such experiments to be conducted, they should allow to measure accurately crucial parameters of the hydrodynamic model.

Fig. 1 shows the prism-coupler structures for which the coupling condition can be written as a relation between the effective index n ef f = k x /k 0 of a guided mode (characterized by a wavevector k x at a frequency ω = k 0 c) to the prism index n p and angle of incidence at the prism bottom interface θ i

n ef f = n p sin(θ i ) (1)
showing it is a way to measure the wavevector directly. Any change in k x induced by spatial dispersion leads to a discrepancy between the coupling angle predicted using Drude's model and taking nonlocality into account. We underline however, as will be clear in the following, that the change in the coupling angle is not the only change brought by nonlocality.

In the hydrodynamic model [START_REF] Boardman | Electromagnetic surface modes[END_REF]17,26,32,33 , the link the electric field E in the metal to the induced electronic current J is written as interactions between free electrons in metal. A special attention must be paid to (i) the material parameters -since our version of the hydrodynamic model makes an explicit distinction between the response of free electrons, considered as nonlocal, and the response of core electrons, considered as purely local -and (ii) the necessary additional boundary condition -here we impose that no electron is allowed to leave the metal. For the material parameters, we rely on a Brendel-Bormann model that has been proven to be particularly accurate 39 . Finally, regarding parameters, several theoretical expressions exist for the nonlocal parameter β. We use a value of β = 1.35 × 10 6 m.s -1 coming from the best available experimental data 16 . We use this value to assess whether nonlocality can have an impact on the optical response, but we stress that the setups we proposes actually constitute a way to measure this crucial parameter.

-β 2 ∇(∇.J) + J + γ J = 0 ω 2 p Ė (
The reflectance of the structures is computed using Moosh, an open code 40 that is able to take spatial dispersion into account in the framework of the hydrodynamic model through the use of a specifically designed scattering matrix algorithm 29 . The situations described below result from a choice of the geometrical parameters which maximizes the impact of spatial dispersion -but the phenomenon is always easy to spot. We first study the optical excitation of a Gap-Plasmon resonance in a gap formed by an insulator layer sandwiched between two metallic slabs (see Fig. 1 (a), MIM structure). For the GP propagating in a dielectric layer with a permittivity d between two metals with a permittivity m = 1 + χ f + χ b where χ f and χ b are the susceptibilities linked to the free and bound electrons respectively, the dispersion relation for a mode at frequency and with the hydrodynamic model (dashed red line) for the MIM structure with air. The dashed black line corresponds to the coupler without the second metallic interface. The vertical dotted line shows where the undisturbed surface plasmon is expected. For large gaps only the surface plasmon on the lower interface is excited. For the thinner gap the symmetric and antisymmetric modes can clearly be seen. The angle of excitation for the symmetric mode is higher than for the SP signaling the gap-plasmon regime. (b) Same situation with a dielectric loaded gap and a lower frequency. The impact of nonlocality begins to be noticeable because the resonance is at larger incidence angle.

ω and with a wavevector k x can be written 19,26

κ d d tanh κ d h 2 + κ m m = Ω (3)
where

κ i = k 2 x -i k 2 0 with i = d, m, k 0 = ω/c and Ω = k 2 x κ l 1 m -1 1+χ b with κ 2 l = k 2 x + ω 2 p β 2 1 χ f + 1 χ b .
The parameter Ω vanishes when there is no spatial dispersion, but increases with k x , which leads to expect a larger impact for higher wavevector. The wavevector increases when h decreases, so that decreasing h leads to a higher discrepancy between Drude's model and the hydrodynamic model.

The effective index of the GP is controlled by the thickness of the dielectric h gap supporting it 35 . As h gap decreases, the wavevector of the GP increases and even diverges when h gap → 0. For an ultra-thin h gap , the GP wavevector is large enough to make the GP sensitive to nonlocality and thus to deviate from Drude's model prediction regarding the coupling angle 26 . Defining the coupling angle as the angle for which reflectivity reaches its lowest value, we define ∆θ i as the difference between the local and nonlocal coupling angle -the nonlocal angle being always the smallest. In order to reach a meaningful difference we have to push the GP wavevector as high as possible by using the smallest dielectric thickness. We are however limited by the prism refractive index which determines the maximum reachable effective index. That is the reason why we consider T iO 2 41 prisms: they present the highest possible refractive index in the visible range and are commercially available. We use Au as metal and air as dielectric (a setup for which the gap can thus be changed progressively). Fig. 2 shows how, when the gap is decreased, the two interfaces become coupled, and the entrance in the gap-plasmon regime when the effective index of the symmetric mode, and thus the angle of excitation, increase clearly compared to the surface plasmons. We stress that, when the interfaces are decoupled, the discrepancy between the local and the nonlocal descriptions is below 0.01 • , a result coherent with previous studies of the phenomenon 10 .

Fig 3 the reflectivity of our structure as a function of the incident angle for a wavelength λ of 600 nm. With h gap = 13.75 nm ( see left part of Fig 3), non-locality causes a shift of the coupling angle of more than 1 • , which is a two orders of magnitude increase compared to the surface plasmon. Here, the position of the reflectivity minimum is close to the coupling angle that can be determined from the dispersion relation -a regime that we thus call the plain coupling regime and for which the effective index of the mode is lower than the prism index.

For a slightly thinner gap (h gap = 11.50 nm), the GP wavevector is beyond the maximum incident wavevector reachable but near enough to still impact the reflectivity. The reflectivity still presents a minimum, but its position cannot be predicted using the effective index of the mode any more. The presence of the guided mode, even if it can be only imperfectly coupled is still responsible for the minimum. We attribute the difference between Drude's model prediction and the predictions of the hydrodynamic model to the fact that the effective index of the gap-plasmon is always lower when spatial dispersion is taken into account. The resonance being further off with a local description, the minimum is higher. While the presence of a closer resonance in the nonlocal case makes the minimum lower. This situation, that we call the near coupling regime, gives rise to a discrepancy ∆R for the minimum of the reflectivity, shown Fig 3 . Such a difference appears at grazing incidence, but it may eventually be easier to measure than the angular shift in the plain coupling regime.

For both regimes, the only condition we have to satisfy to study non-locality is to reach a high enough wavevector for the gap-plasmon. We have done this so far by using an extremely thin gap filled with air, but we now show it is possible to reach similar sensitivity to nonlocality relying on much larger thicknesses provided the refractive index of the dielectric is significantly higher than 1. The prism is made of T iO2 with a permittivity p = 6.41, the metal is Au with a permittivity m = -18.50 + 1.50i and the dielectric is LASF9 with a permittivity d = 3.37 + 1.10e -07 i. The thickness of the upper metallic layer is hc = 18 nm.

For a fixed wavelength, using such materials actually leads plasmonic guided modes to present higher wavevectors than using air as dielectric for the same gap width. Very similar results to the above ones are obtained for much large thicknesses of more than 60 nm. We have studied the case of a dielectric waveguide filled with LASF9 at a wavelength of λ = 750 nm. Exactly as previously, the impact of spatial dispersion is observed in the plain coupling regime, producing a shift of the coupling angle (see Fig. 4(a)), and in the near coupling regime (see Fig. 4(b)), producing a very large discrepancy in the reflectivity of 0.17. This setup presents several advantages as (i) it is probably easier to control the gap over macroscopic distances horizontally if it is filled with a dielectric (ii) using a dielectric allows to reach higher effective index for quite large wavelengths (iii) such a large thickness allows to completely neglect other phenomena like the spill-out, which intervene when extremely small gaps are involved.

Finally, we show that it is possible to excite the guided modes supported by the complementary structure: a metallic slab 19,38 buried in a dielectric, a structure we will call IMI shown on Fig. 1(b). We underline that the modes of a single metallic slab have been envisaged a long time ago in the framework of EELS 42 . Compared to the MIM structure, the IMI is able to support not only one but two guided modes, the LRSP and the SRSP. These two modes have two distinct wavevectors which behave differently when the thickness of the metallic slab supporting them varies. The dispersion relation for the symmetric SRSP mode can be written

κ m m tanh κ m h 2 + κ d d = Ω (4)
and for the antisymmetric LRSP

κ m m coth κ m h 2 + κ d d = Ω. ( 5
)
As a consequence, while the SRSP tends to behave as the GP, i.e to present a diverging wavevector when h m tends to zero, the LRSP has a lower wavevector which stays much more stable when h m decreases. The reflectivity of our IMI structure excited through the prism thus presents two dips for two different angles corresponding respectively to the LRSP and SRSP modes. Here too the parameter Ω appears in the dispersion relation. Since it is larger when the wavevector k x is larger, the impact of spatial dispersion on the can be expected to be much lower for the LRSP than for the SRSP 19,38 . Fig 5 shows that while spatial dispersion has a small impact on the LRSP, there is a clear shift of 0.72 • for the higher effective index SRSP when nonlocality in taken into account. We underline that for such a setup the LRSP can thus be used to retrieve the material and geometrical parameters while the SRSP allows for a measurement of β.

In conclusion, we have shown how feasible multilayered structures could be used to enhance the impact of spatial dispersion in metals, through the excitation of three of the most emblematic guided mode in plasmonics: the gap-plasmon, and the short-and long-range surface plasmons. Physically, this enhancement by two orders of magnitude can be directly linked to the high wavevectors these plasmonic guided modes present. It is obtained here for dielectric gaps whose typical thicknesses, ranging from 10 nm to 70 nm are one to two orders of magnitude larger than in previous experiments [START_REF] Ciracì | [END_REF]16 . This allows to rule out any role of more complex phenomena like the spill-out of the electron gas outside of the metal [21][22][23]33 because of the finite extraction work.

The geometrical parameters for which so strong nonlocal effects can be found suggest spatial dispersion has probably to be taken into account in plasmonics for much larger structures than previously thought. We underline that the present setups do not involve a large number of chemically synthesized nanoparticles [START_REF] Ciracì | [END_REF]16,18,43 and, furthermore, that the working wavelengths are in the red part of the optical spectrum -which shows that these effects manifest themselves even far below the plasma frequency of metals.

We hope our work will pave the way for well controlled optical experiments allowing to assess whether or not the hydrodynamic model is an accurate replacement for Drude's in plasmonics. Given the increasing number of devices relying on gap-plasmon excitation in dielectricfilled gaps [44][45][46][47] such an evolution may be a necessity soon.

1 FIG. 1 .

 11 FIG. 1. Schematic representation of the prism couplers:(a) resembles the Otto configuration, i.e a prism on top of a metallic slab which ensures coupling to the gap-plasmon. while(b) resembles the Kretschmann-Raether configuration, i.e a prism on top of a dielectric layer through which the guided modes of an IMI structure are coupled.

  FIG. 2. (a): Reflectivity in the local approximation (solid black line) and with the hydrodynamic model (dashed red line) for the MIM structure with air. The dashed black line corresponds to the coupler without the second metallic interface. The vertical dotted line shows where the undisturbed surface plasmon is expected. For large gaps only the surface plasmon on the lower interface is excited. For the thinner gap the symmetric and antisymmetric modes can clearly be seen. The angle of excitation for the symmetric mode is higher than for the SP signaling the gap-plasmon regime. (b) Same situation with a dielectric loaded gap and a lower frequency. The impact of nonlocality begins to be noticeable because the resonance is at larger incidence angle.

FIG. 3 .

 3 FIG.3. Reflectivity of the structure shown on Fig.1(a) as a function of the incident angle θi for two value of hgap. The wavelength of the incident light is λ = 600 nm. The prism is made of T iO2 with a permittivity p = 6.78, the metal is Au with a permittivity m = -8.44 + 1.41i and the dielectric is air. The upper metallic layer thickness is hc = 18 nm. The resonance corresponds to the excitation of a gap-plasmon.

FIG. 4 .

 4 FIG. 4. Reflectivity of the structure shown on Fig.1(a) as a function of the incident angle θi for two values of hgap. The wavelength of the incident light is λ = 750 nm. The prism is made of T iO2 with a permittivity p = 6.41, the metal is Au with a permittivity m = -18.50 + 1.50i and the dielectric is LASF9 with a permittivity d = 3.37 + 1.10e -07 i. The thickness of the upper metallic layer is hc = 18 nm.

1 FIG. 5 .

 15 FIG. 5. Reflectivity of the structure shown on Fig. 1(b) as a function of the incident angle θi for hm = 44 nm. The wavelength of the incident light is λ = 700. Prism is T iO2 with permittivity p = 6.50 metal is Au with permittivity m = -15.04+1.31i and dielectric is LASF9 with permittivity d = 3.28 + 1.15 × 10 -07 i (a permittivity very close to the one of other high index dielectrics like Al2O3). The thickness of the layer under the prism is hc = 65 nm. The two dips are due to the excitation of LRSP (left) in the plain regime and SRSP (right) in the near coupling regime.

FUNDING INFORMATION

This work has been supported by the French National Research Agency, Project Physics of Gap-Plasmons No. ANR-13-JS10-0003.