Maxime Pelletier

Nicolas Ressayre

Some unexpected properties of Littlewood-Richardson coefficients

We are interested in identities between Littlewood-Richardson coefficients, and hence in comparing different tensor product decompositions of the irreducible modules of the linear group GL n (C). A family of partitions -called near-rectangular -is defined, and we prove a stability result which basically asserts that the decomposition of the tensor product of two representations associated to near-rectangular partitions does not depend on n. Given a partition λ, of length at most n, denote by V n (λ) the associated simple GL n (C)-module. We conjecture that, if λ is near-rectangular and µ any partition, the decompositions of V n (λ) ⊗ V n (µ) and V n (λ) * ⊗ V n (µ) coincide modulo a mysterious bijection. We prove this conjecture if µ is also near-rectangular and report several computer-assisted computations which reinforce our conjecture.

Introduction

In this paper we study some properties of Littlewood-Richardson coefficients when some partitions are near-rectangular (see below for a precise definition). Let n ≥ 2 be an integer. The irreducible representations of GL n (C) are parametrized by all non-increasing sequences of n integers. As is often the case, we focus on the polynomial representations among those, which correspond to the sequences containing only non-negative integers, also called partitions with at most n parts. Denote by V n (λ) the representation of GL n (C) associated to such a partition λ. The Littlewood-Richardson coefficient c ν λµ appears in the tensor product decomposition

V n (λ) ⊗ V n (µ) = ν∈Λn c ν λµ V n (ν),
where Λ n denotes the set of partitions of length at most n. Denote moreover by V n (λ) * the GL n (C)-module which is dual to V n (λ).

Obviously, given λ and µ two partitions with at most n parts, V n (λ) ⊗ V n (µ) and V n (λ) * ⊗ V n (µ) are not isomorphic as GL n (C)-modules. Nevertheless, we may want to compare their decompositions in irreducible modules.

Problem 1. Compare the two GL n (C)-modules, V n (λ) ⊗ V n (µ) and V n (λ) * ⊗ V n (µ).

For example, Coquereaux-Zuber [START_REF] Coquereaux | On sums of tensor and fusion multiplicities[END_REF] showed that the sums of the multiplicities of these two representations coincide. For λ ∈ Λ n , set λ * = λ 1 -λ n ≥ λ 1 -λ n-1 ≥ • • • ≥ λ 1 -λ 2 ≥ 0. Then V n (λ) * and V n (λ * ) only differ by a tensor power of the determinant. More precisely, V n (λ) * is the irreducible representation of GL n (C) corresponding to the sequence -λ n ≥ -λ n-1 ≥ • • • ≥ -λ 1 . As a consequence V n (λ * ) is isomorphic to V n (λ) * tensored by the λ 1 -th power of the determinant representation of GL n (C).

Following these observations, the Coquereaux-Zuber's result can be written as

ν∈Λn c ν λµ = ν∈Λn c ν λ * µ . (1)
More generally, we may want to compare {c ν λµ : ν ∈ Λ n } and {c ν λ * µ : ν ∈ Λ n } as multisets. For instance, for n = 3, λ = (5, 3) and µ = (6, 3), V 3 (λ) ⊗ V 3 (µ) decomposes as V 3 (7, 5, 5) + V 3 (7, 7, 3) + V 3 (8, 8, 1) + V 3 (9, 4, 4) + V 3 (9, 8) + V 3 (10, 7) +V 3 (11, 6) + V 3 (11, 3, 3) + V 3 (11, 4, 2) + V 3 (11, 5, 1) + V 3 (6, 6, 5) +2V 3 (7, 6, 4) + 2V 3 (8, 5, 4) + 2V 3 (8, 7, 2) + 2V 3 (9, 7, 1) + 2V 3 (10, 4, 3) +2V 3 (10, 5, 2) + 2V 3 (10, 6, 1) + 3V 3 (8, 6, 3) + 3V 3 (9, 5, 3) + 3V 3 (9, 6, 2), while V 3 (λ * ) ⊗ V 3 (µ) expands as V 3 (7, 7, 2) + V 3 (8, 4, 4) + V 3 (10, 3, 3) + V 3 (8, 8) + V 3 (9, 7) + V 3 (10, 6) +V 3 (11, 3, 2) + V 3 (11, 4, 1) + V 3 (6, 5, 5) + V 3 (6, 6, 4) + 2V 3 (7, 5, 4) +2V 3 (7, 6, 3) + 2V 3 (8, 7, 1) + 2V 3 (9, 4, 3) + 2V 3 (9, 6, 1) + 2V 3 (10, 4, 2) +2V 3 (10, 5, 1) + 3V 3 (8, 5, 3) + 3V 3 (8, 6, 2) + 3V 3 (9, 5, 2) + V 3 (11, 5).

One can then notice that the multiplicities in the two expansions are the same: 11 occurrences of "1", 7 occurrences of "2" and 3 occurrences of "3" in both cases. A natural question is then: is this always true?

We check in this article that it is the case for GL 3 (C) in general (note that this was already proven by Coquereaux and Zuber in [START_REF] Coquereaux | Conjugation properties of tensor product multiplicities[END_REF]; they even give several proofs of this fact in this paper from 2014): using a computer, we are able to compute explicitly, for n = 3, the function

(Λ 2 n ) × N -→ N (λ, µ, c) -→ Nb n (c • λµ > c) := #{ν ∈ Λ n : c ν λµ > c}.
See Section 4 for details. In this introduction we report on this computation as follows:

Proposition 1. The function Nb 3 (c • λµ > c) : Λ 3 × Λ 3 × N -→ N (λ, µ, c) -→ #{ν ∈ Λ n : c ν λµ > c} is piecewise polynomial of degree 2 with 7 cones 1 . Moreover, Nb 3 (c • λµ > c) = Nb 3 (c • λ * µ > c). (2)
We then aim to suggest a generalization to GL n (C) for any n: we will say that a partition

λ ∈ Λ n is near-rectangular if λ = λ 1 λ n-2 2 λ n for some integers λ 1 ≥ λ 2 ≥ λ n ; that is, if λ 2 = • • • = λ n-1 . We formulate the following Conjecture 1. Let λ and µ in Λ n . If λ is near-rectangular then ∀c ∈ N #{ν ∈ Λ n : c ν λµ = c} = #{ν ∈ Λ n : c ν λ * µ = c}.
Equivalently, we conjecture that there exists a bijection ϕ : Λ n -→Λ n , depending on λ and µ such that

∀ν ∈ Λ n c ν λµ = c ϕ(ν) λ * µ , if λ is near-rectangular. In other words, the multisets {c ν λµ : ν ∈ Λ n } and {c ν λ * µ : ν ∈ Λ n } are expected to be equal, if λ is near-rectangular.
Note that any partition of size ≤ 3 is near-rectangular. Then, both the last assertion of Proposition 1, and the result of [START_REF] Coquereaux | Conjugation properties of tensor product multiplicities[END_REF] are equivalent to Corollary 2. Conjecture 1 holds for n = 3.

In literature, there are a lot of equalities between Littlewood-Richardson coefficients. There are symmetries (see [START_REF] Briand | The 144 symmetries of the Littlewood-Richardson coefficients of SL 3[END_REF] and references therein), stabilities (see [START_REF] Briand | Rectangular symmetries for coefficients of symmetric functions[END_REF]), reductions (see [START_REF] Cho | Reduction formulae of Littlewood-Richardson coefficients[END_REF]). None of these numerous results really seems to explain whether or why Conjecture 1 should hold. Moreover, there are various combinatorial models for the Littlewood-Richardson coefficients (see [START_REF] Fulton | Young tableaux[END_REF][START_REF] Littelmann | Paths and root operators in representation theory[END_REF][START_REF] Zelevinsky | A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence[END_REF][START_REF] Knutson | The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture[END_REF][START_REF] Vakil | A geometric Littlewood-Richardson rule[END_REF][START_REF] Coskun | A Littlewood-Richardson rule for two-step flag varieties[END_REF]). None of these models seems to prove Conjecture 1 easily.

What made this Conjecture 1 even more unexpected for us is that it seems that, should the aforementioned bijection ϕ : Λ n -→Λ n exist, it does not seem to be expressible simply in terms of λ and µ. Indeed, even if n = 3, we remark in Section 4 that (λ, µ, ν)-→(λ * , µ, ϕ(ν)) cannot be linear. We checked, using Sagemath, Conjecture 1 on a few million examples for GL 4 (C), GL 5 (C), GL 6 (C), and GL 10 (C): see Section 7.2 for details. Moreover, we also give in Section 7.2 an example showing that the assumption on λ is truly necessary.

We now consider the tensor products of representations associated to two near-rectangular partitions. Observe that a partition λ of length l parametrizes representations V n (λ) of GL n (C) for any n ≥ l. A priori, c ν λµ could depend on n. It is a classical result (see e.g. [START_REF] Fulton | Young tableaux[END_REF]) of stability that it actually does not. Our second result is a similar stability result but for partitions of arbitrarily large length. Indeed, fix two near-rectangular partitions λ = λ 1 λ n-2 2 λ n and µ = µ 1 µ n-2 2 µ n . We prove that the decomposition of V n (λ) ⊗ V n (µ) does not depend on n ≥ 4 but only on the six integers λ 1 , λ 2 , λ n , µ 1 , µ 2 and µ n . More precisely, we get a simple expression of the Littlewood-Richardson coefficients that appear in this tensor product, expression independent of n. Proposition 3. Let n ≥ 4. Let λ = λ 1 λ n-2 2 and µ = µ 1 µ n-2 2 be two near-rectangular partitions. Let ν be a partition with at most n parts. Then c ν λµ = 0 unless ν = ν 1 ν 2 (λ 2 + µ 2 ) n-4 ν n-1 ν n for four integers ν 1 , ν 2 , ν n-1 and ν n such that

ν 1 ≥ ν 2 ≥ λ 2 + µ 2 ≥ ν n-1 ≥ ν n . In this case c ν λµ = # M, m , where M = max(0, λ 2 + µ 1 -ν 1 , -µ 2 + ν n ) and m = min(λ 1 + µ 1 -ν 1 , λ 2 + µ 1 -ν 2 , -λ 2 -µ 2 + ν n-1 + ν n , -µ 2 + ν n-1 ).
In particular, this value does not depend on n ≥ 4.

In Proposition 3, for simplicity we assumed that λ n = µ n = 0. Yet nothing is lost with such an assumption, since

V n (λ 1 λ n-2 2 λ n ) ≃ det λn ⊗V n ((λ 1 -λ n )(λ 2 -λ n ) n-2 ).
Proposition 3 positively answers [PW20, Question 2] by giving a much stronger result. It also allows to check a particular case of Conjecture 1. Indeed, with the help of a computer we computed Nb 4 (c • λµ > c) for λ and µ near-rectangular. Let Λ nr n = {λ 1 λ n-2 2 λ n : λ 1 ≥ λ 2 ≥ λ n } be the set of near-rectangular partitions of length at most n.

Proposition 4. The function

Nb 4 (c • λµ > c) : Λ nr 4 × Λ nr 4 × N -→ N (λ, µ, c) -→ #{ν ∈ Λ n : c ν λµ > c}
is piecewise polynomial of degree 3 with 36 cones. Moreover,

Nb 4 (c • λµ > c) = Nb 4 (c • λ * µ > c). (3)
The 36 polynomial functions and cones are given in Section 6.3. As a consequence of Propositions 3 and 4, we get Corollary 5. Let n ≥ 4. Conjecture 1 holds for GL n (C) assuming in addition that µ is near-rectangular.

A much weaker version of Conjecture 1 is

Conjecture 2. If λ ∈ Λ n is near-rectangular then #{ν ∈ Λ n : c ν λµ = 0} = #{ν ∈ Λ n : c ν λ * µ = 0}. Equivalently, we ask whether, for λ ∈ Λ nr n , ∀µ ∈ Λ n Nb n (c • λµ > 0) = Nb n (c • λ * µ > 0).
For n = 4 and λ near-rectangular, we computed Nb 4 (c • λµ > 0) and checked Conjecture 2. Here we report on this computation as follows (see Section 6.4 for details). Proposition 6. The function

Nb 4 (c • λµ > 0) : Λ nr 4 × Λ 4 -→ N (λ, µ) -→ #{ν ∈ Λ n : c ν λµ > 0}
is piecewise quasi-polynomial2 of degree 3 with 205 cones. The only congruence occurring is the parity of λ 1 + |µ|. Moreover,

Nb 4 (c • λµ > 0) = Nb 4 (c • λ * µ > 0). (4)
This symmetry with the complete duality (λ, µ) -→ (λ * , µ * ) gives an action of (Z/2Z) 2 . This group acts on the 205 pairs (cone,quasi-polynomial) with 83 orbits.

This work is based on numerous computer aided computations with Barvinok [VSB + 07], Normaliz [BIS] and SageMath [START_REF]SageMath, the Sage Mathematics Software System[END_REF]. Details on these computations can be found on the webpage of the second author [Res20, Supplementary material].

Remark. After a version of this work was posted on ArXiv, Darij Grinberg offered a solution of our main conjecture in [START_REF] Grinberg | The Pelletier-Ressayre hidden symmetry for Littlewood-Richardson coefficients[END_REF]. Therein he defines a piecewise linear involution ϕ from

Z n to Z n satisfying ∀ν ∈ Λ n c ν λµ = c ϕ(ν)
λ * µ , if λ is near-rectangular, thus solving our conjecture. An amazing fact is that this bijection does not necessarily map a partition to a partition: if ϕ(ν) is not a partition then c ν λµ simply vanishes, allowing ϕ to work.

Generalities on the function Nb

n (c • λµ > c)
Recall that for λ, µ ∈ Λ n and c ∈ N we set

Nb n (c • λµ > c) = #{ν ∈ Λ n : c ν λµ > c}. Since V n (λ) ⊗ V n (µ) ≃ V n (µ) ⊗ V n (λ) ≃ (V n (λ * ) ⊗ V n (µ * )) * as SL n (C)-modules, the function Nb n (c • λµ > c) satisfies Nb n (c • λµ > c) = Nb n (c • µλ > c) = Nb n (c • λ * µ * > c) = Nb n (c • µ * λ * > c). (5)
Let 1 n ∈ Λ n denote the partition with n parts equal to 1. Then V n (1 n ) is the one dimensional representation of GL n (C) given by the determinant. In particular, as an

SL n (C)- module V n (λ) ≃ V n (λ -k n ) for any k. Set Λ 0 n = {λ ∈ Λ n : λ n = 0}.
It can be seen as the set of dominant weights for the group SL n (C). For λ ∈ Λ n , set λ = λ -λ n n , the partition obtained by substracting λ n to each part of λ. Then

Nb n (c • λµ > c) = Nb n (c • λμ > c),
and even more

c ν λµ = c ν-(λn+µn) n λμ . ( 6
) Set Horn n = {(λ, µ, ν) ∈ (Λ n ) 3 : c ν λµ = 0}
. By a Brion-Knop's result (see [É92]), Horn n is a finitely generated semigroup. The Knutson-Tao saturation theorem [START_REF] Knutson | The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture[END_REF] shows that Horn n is the set of integer points in a convex cone, the Horn cone. The Horn cone is polyhedral and the minimal list of inequalities defining it is known (see e.g. [START_REF] Fulton | Eigenvalues, invariant factors, highest weights, and Schubert calculus[END_REF][START_REF] Belkale | Local systems on P 1 -S for S a finite set[END_REF][START_REF] Knutson | The honeycomb model of GL n (C) tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone[END_REF]). These inequalities contain the Weyl inequalities

ν i+j-1 ≤ λ i + µ j whenever i + j -1 ≤ n; (7)
and are all of the form

k∈K ν k ≤ i∈I λ i + j∈J µ j , (8)
for some triples (I, J, K) of three subsets of {1, . . . , n} of the same cardinality.

Proposition 7. Fix n ≥ 0. The function

Nb n (c • λµ > 0) : Λ n × Λ n -→ N (λ, µ) -→ #{ν ∈ Λ n : c ν λµ > 0}. is piecewise quasi-polynomial.
This means that there exist a sub-lattice Λ of Z 2n ⊃ Λ n × Λ n of finite index and a collection of piecewise polynomial functions (see Footnote 1) parametrized by the classes in Z 2n /Λ. Applying to (λ, µ) the piecewise polynomial function corresponding to its class, one then gets Nb n (c • λµ > 0). Proof. We have

Nb n (c • λµ > 0) = # Horn n ∩ ({(λ, µ)} × Λ n ) . (9)
Consider the Horn cone Horn Q n generated by Horn n . By the discussion above this proposition, it is defined as a subset of Q 3n by an explicit list of linear inequalities, namely the Horn inequalities (8). Knutson-Tao's saturation Theorem [START_REF] Knutson | The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture[END_REF] asserts that Horn n is precisely the set of integer points (that is, belonging to

(Λ n ) 3 ) in Horn Q n . Now, equality (9) describes Nb n (c • λµ > 0)
as the number of integer points in the affine section of the Horn cone obtained by fixing (λ, µ).

Moreover each inequality 8 of the Horn cone depends linearly on (λ, µ). Now the conclusion is a consequence of the general theory of multivariate Ehrhart polynomials (see e.g. vertices of the triangles on Figure 1 with integers, with the labels on the boundaries determined by λ, µ and ν as drawn on the left.

[BBDL + 19, Theorem 1.1] or [Stu95]). • • • • • • • • • • • • • • • 0 λ 1 λ 1 + λ 2 • • • |λ| |λ| + µ 1 |λ| + µ 1 + µ 2 • • • |λ| + |µ| |ν| ν 1 ν 1 + ν 2 • • • • • • • • • • • • • • • • • • λ 1 λ 2 • • • λ n µ 1 µ 2 • • • µ n ν n • • • ν 1 ν 2 Figure 1: Hives with boundary conditions R 1 R 2 R 3 • c • • a • d b γ α β δ •c • b • a • d γ β δ α • a • b • d •c β α γ δ
Then, neighbouring vertices define three distinct types of rhombus (see Figure 2), each coming with its own constraint condition on the integers labelling the vertices. For each extracted rhombus we impose the following constraint (using the notation of Figure 2

): b + c ≥ a + d (10)
Alternatively, one can label the edges of the drawing rather than the vertices: each edge is labelled by the difference between the values on its vertices, with an orientation as shown on the right of Figure 1. Then each rhombus gives also a constraint on the labels of the edges, equivalent to the previous one on the labels of the vertices (still using the notation of Figure 2):

β ≥ δ or equivalently α ≥ γ. ( 11
)
The equivalence between these two inequalities comes from the fact that α

+ δ = β + γ.
By definition a hive is a labelling in Z (n+1)(n+2) 2

satisfying Inequalities (10) for each one of the 3 n(n-1) 2 rhombi. The Knutson-Tao's result is Theorem 8. (see [START_REF] Knutson | The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture[END_REF]Appendix]) Let λ, µ and ν in Λ n . Then c ν λµ is the number of hives with boundary conditions determined by λ, µ and ν as on the left of Figure 1.

4 The case of GL 3 It is known that the function (Λ n ) 3 -→N, (λ, µ, ν) -→ c ν λµ is piecewise polynomial (see [Ras04]) of degree n 2 -3n+2

2

. For n = 3, we get precisely the following.

Proposition 9. Let λ = (λ 1 , λ 2 , 0), µ = (µ 1 , µ 2 , 0), and ν = (ν 1 , ν 2 , ν 3 ) in Λ 3 be such that |ν| = |λ| + |µ|. Then c ν
λµ is the number of integer points in the interval

max(µ 1 -λ 2 , µ 2 , ν 1 -λ 1 , µ 1 -ν 3 , ν 2 -λ 2 , µ 1 + µ 2 -ν 2 ), min(µ 1 , ν 1 -λ 2 , µ 1 + µ 2 -ν 3 ) .
This statement is well known and can easily be checked using the hive model. Indeed, once λ, µ and ν are fixed, there is only one interior entry x to choose in order to determine the hive. The solution set to the system of the 9 rhombus inequalities is an interval from which the interval of the statement is obtained by translation by λ 1 + λ 2 . Proposition 9 implies that, for any nonnegative integer c, c ν λµ > c if and only if, for any linear form ϕ and ψ appearing in the min and max respectively, we have ϕ -ψ ≥ c. Namely c ν λµ > c if and only if

λ 1 -λ 2 -c ≥ 0 λ 2 -c ≥ 0 µ 1 -µ 2 -c ≥ 0 µ 2 -c ≥ 0 ν 1 -ν 2 -c ≥ 0 ν 2 -ν 3 -c ≥ 0 λ 1 + µ 1 -ν 1 -c ≥ 0 λ 1 + µ 1 -ν 2 -ν 3 -c ≥ 0 λ 1 + µ 2 -ν 2 -c ≥ 0 λ 1 + λ 2 + µ 1 -ν 1 -ν 3 -c ≥ 0 λ 1 -ν 3 -c ≥ 0 λ 1 + λ 2 + µ 2 -ν 2 -ν 3 -c ≥ 0 λ 2 + µ 1 -ν 2 -c ≥ 0 λ 1 + µ 1 + µ 2 -ν 1 -ν 3 -c ≥ 0 µ 1 -ν 3 -c ≥ 0 λ 2 + µ 1 + µ 2 -ν 2 -ν 3 -c ≥ 0 λ 2 + µ 2 -ν 3 -c ≥ 0 λ 1 + λ 2 + µ 1 + µ 2 -ν 1 -ν 2 -c ≥ 0 (12)
and

|ν| = |λ| + |µ|. ( 13
)
Note that, for c = 0, we recover the 6 inequalities saying that λ, µ and ν are dominant, the 6 Weyl inequalities and the 6 others inequalities of the Horn cone (see e.g. [START_REF] Fulton | Eigenvalues, invariant factors, highest weights, and Schubert calculus[END_REF]).

We now want to compute the function mapping (λ, µ, c) to the number of solutions of this system of inequalities in ν. The method consists in restating this problem in the langage of vector partition functions as in [START_REF] Sturmfels | On vector partition functions[END_REF].

Start with the 18×8 matrix H whose rows give the coefficients of the 18 inequalities (12). Set

Λ = {(λ 1 , λ 2 , µ 1 , µ 2 , ν 1 , ν 2 , ν 3 , c) ∈ Z 8 : |ν| = |λ| + |µ|} and Λ + = {(λ 1 , λ 2 , µ 1 , µ 2 , ν 1 , ν 2 , ν 3 , c) ∈ Λ : λ, µ, ν dominant and c ≥ 0}.
Let Horn 3 denote the set of points in Λ + that satisfy the inequalities (12).

To get nonnegative variables, we make the following change of coordinates

a 1 = λ 1 -λ 2 -c b 1 = µ 1 -µ 2 -c c 1 = ν 1 -ν 2 -c a 2 = λ 2 -c b 2 = µ 2 -c c 2 = ν 2 -c
Then Horn 3 identifies with Horn

′ 3 = {X ∈ N 7 | NX ≥ 0}, where N =                       -1 -2 -1 -2 1 3 -3 1 1 1 1 -1 -1 1 1 1 0 1 0 -1 1 0 -1 -1 -2 1 2 -2 0 1 1 1 0 -1 1 -1 -2 0 -1 1 2 -2 -1 -1 -1 -1 1 2 -2 0 -1 0 -1 1 1 -1 0 0 0 -1 0 1 -1 0 0 -1 -1 1 1 -1 0 -1 0 0 0 1 -1 -1 -1 0 0 1 1 -1 1 2 1 2 -1 -2 2                      
.

Set Ñ = (N| -I 13 ) in such a way that

Horn ′ 3 ≃ {(X, Y ) ∈ N 7 × N 13 | NX = Y } ≃ {X ∈ N 20 | ÑX = 0}.
We now proceed to the affine section mentioned in the proof of Proposition 7. Thus, up to our changes of variables, the function

(λ, µ, c) → Nb 3 (c • λµ > c) is the map N 5 -→ N Y -→ #{X ∈ N 15 : MX = -BY },
where M = (M | -I 13 ), M is the matrix formed by columns 5 and 6 of the matrix N and B is the matrix formed by the other columns of N. Note that M is not unimodular: the lowest common multiple of the maximal minors is not 1, but 6. There are 83 such nonzero minors. Then [START_REF] Sturmfels | On vector partition functions[END_REF] implies that (λ, µ, c) → Nb(c • λµ > c) is piecewise quasi-polynomial with chambers obtained by intersecting some 83 explicit simplicial cones. We used [VSB + 07], an implementation of Barvinok algorithm [START_REF] Barvinok | A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed[END_REF] to compute this function. The surprise was that we got only polynomial functions and only 7 cones. Actually, the software gave 36 cones that can be glued to give those 7 described in Proposition 10.

Proposition 10. We use the basis of fundamental weights to set

λ = k 1 ̟ 1 + k 2 ̟ 2 = (k 1 + k 2 , k 2 ) and µ = l 1 ̟ 1 + l 2 ̟ 2 = (l 1 + l 2 , l 2 ). Then Nb(c • λµ > c) = 0 unless c ≤ min(k 1 , k 2 , l 1 , l 2 ).
Moreover, this cone decomposes into 7 cones C 1 , . . . , C 7 on which Nb(c • λµ > c) is given by polynomial functions P 1 , . . . , P 7 . Five of these seven pairs (C i , P i ) are kept unchanged by switching k 1 and k 2 . The two others are swapped by this operation.

In particular, Conjecture 1 holds for GL 3 .

In the basis of fundamental weights, we are interested in the function

ψ : N 5 -→ N (k 1 , k 2 , l 1 , l 2 , c) -→ #{ν ∈ Λ 3 | c ν k 1 ̟ 1 +k 2 ̟ 2 ,l 1 ̟ 1 +l 2 ̟ 2 > c} .
Notice moreover that switching k 1 and k 2 corresponds then to taking λ * . Define now the following seven polynomials in k 1 , k 2 , l 1 , l 2 , c:

P 1 = 2 c 2 -c(k 1 + k 2 + l 1 + l 2 + 2) - 1 2 (k 1 + k 2 -l 1 -l 2 ) 2 + k 1 k 2 + l 1 l 2 + 1 2 (k 1 + k 2 + l 1 + l 2 ) + 1 P 2 = 3c 2 -3c(k 1 + k 2 + 1) + 1 2 (k 1 + k 2 ) 2 + k 1 k 2 + 3 2 (k 1 + k 2 ) + 1 P 3 = 3c 2 -3c(l 1 + l 2 + 1) + 1 2 (l 1 + l 2 ) 2 + l 1 l 2 + 3 2 (l 1 + l 2 ) + 1 P 4 = 5 2 c 2 -c 2 k 1 + 2 k 2 + l 1 + 5 2 + k 1 k 2 + (k 1 + k 2 )(l 1 + 1) - l 1 2 (l 1 -1) + 1 P 5 = 5 2 c 2 -c 2 k 1 + 2 k 2 + l 2 + 5 2 + k 1 k 2 + (k 1 + k 2 )(l 2 + 1) - l 2 2 (l 2 -1) + 1 P 6 = 5 2 c 2 -c k 1 + 2l 1 + 2l 2 + 5 2 + l 1 l 2 + (l 1 + l 2 )(k 1 + 1) - k 1 2 (k 1 -1) + 1 P 7 = 5 2 c 2 -c k 2 + 2l 1 + 2l 2 + 5 2 + l 1 l 2 + (l 1 + l 2 )(k 2 + 1) - k 2 2 (k 2 -1) + 1
Notice that P 1 , . . . , P 5 are symmetric in k 1 , k 2 , whereas P 6 and P 7 are swapped when one swaps k 1 and k 2 . One might also add that P 3 is the image of P 2 under the involution corresponding to swapping λ and µ -i.e. swapping (k 1 , k 2 ) and (l 1 , l 2 ) -, as P 6 is the image of P 4 and P 7 the one of P 5 under this same involution.

Then, for c ≥ 0, k 1 ≥ c, k 2 ≥ c, l 1 ≥ c, and l 2 ≥ c, the function ψ is given by the following piecewise polynomial:

Cones of polynomiality Polynomial giving ψ C

1 : k 1 + k 2 ≥ max(l 1 , l 2 ) + c, l 1 + l 2 ≥ max(k 1 , k 2 ) + c P 1 C 2 : k 1 + k 2 ≤ min(l 1 , l 2 ) + c P 2 C 3 : l 1 + l 2 ≤ min(k 1 , k 2 ) + c P 3 C 4 : l 1 + c ≤ k 1 + k 2 ≤ l 2 + c P 4 C 5 : l 2 + c ≤ k 1 + k 2 ≤ l 1 + c P 5 C 6 : k 1 + c ≤ l 1 + l 2 ≤ k 2 + c P 6 C 7 : k 2 + c ≤ l 1 + l 2 ≤ k 1 + c P 7
One can then see that the cones C 1 to C 5 are stable under permutation of k 1 and k 2 whereas the cones C 6 and C 7 are swapped when k 1 and k 2 are. Thus, for all k 1 , k 2 , l 1 , l 2 , c ≥ 0,

ψ(k 1 , k 2 , l 1 , l 2 , c) = ψ(k 2 , k 1 , l 1 , l 2 , c),
which proves Proposition 10.

Remark. The last part of Proposition 10 asserts that there exists a bijection

(Λ 0 3 ) 2 × Λ 3 -→(Λ 0 3 ) 2 × Λ 3 , (λ, µ, ν) -→ (λ * , µ, ν) such that c ν λµ = c ν λ * µ .
One could hope for such a bijection to be linear. Unfortunately, it CANNOT.

One can check this claim as follows. The matrix of such a linear bijection ϕ : (Λ 0 3 ) 2 × Λ 3 -→(Λ 0 3 ) 2 × Λ 3 , (λ, µ, ν) -→ (λ * , µ, ν) would only depend on 7 vectors ν 1 , . . . , ν 7 in Z 3 . Assume that this matrix does exist.

For example, the image of (λ 1 , λ 2 , µ 1 , µ 2 , ν 1 , ν 2 , ν 3 ) = (1, 0, 0, 0, 1, 0, 0) is then (1, 1, 0, 0, ν 1 + ν5 ). Moreover this image has to correspond to some nonzero Littlewood-Richardson coefficient. Hence it is necessary that ν 1 + ν 5 = (1, 1, 0).

Similarly, considering the image of (0, 0, 1, 1, 1, 1, 0), one gets that it must be (0, 0, 1, 1, ν 3 + ν 4 + ν 5 + ν 6 ) = (0, 0, 1, 1, 1, 1, 0), and then ν 3 + ν 4 + ν 5 + ν 6 = (1, 1, 0). Now the image of (1, 0, 1, 1, 1, 1, 1) has to be a ray of the Horn cone. We deduce that this image is (1, 1, 1, 1, 2, 2, 0), and thus ν 1 + ν 3 + ν 4 + ν 5 + ν 6 + ν 7 = (2, 2, 0).

Combining these three constraints one gets ν 5 = ν 7 , which contradicts the invertibility of ϕ.

Note also that the linear automorphisms of (Λ 3 ) 3 preserving the Littlewood-Richardson coefficients are proved to form a group of cardinality 288 (so big !) in [START_REF] Briand | The 144 symmetries of the Littlewood-Richardson coefficients of SL 3[END_REF].

introduction.

Proof of Proposition 3. We know that the Littlewood-Richardson coefficient c ν λµ is equal to the number of hives with exterior edges labelled by λ, µ, and ν. Let us assume that c ν λµ > 0, meaning that such a hive exists, and consider any hive like this. In this hive, a certain number of edges have a label that is already fixed by these "boundary conditions". These edges and values are shown in the following picture (made for the sake of example with a hive of size 6, but the picture is strictly the same for all sizes at least 4):

ν 1 ν 2 ν 3 ν n-2 ν n-1 ν n λ 1 0 λ 2 λ 2 λ 2 λ 2 0 µ 1 µ 2 µ 2 µ 2 µ 2 ν 1 -λ 1 νn µ 1
Then, using a number of hive conditions (including the fact that, in any triangle, the values on two of the edges determine the value on the third one), we obtain the following values for some interior edges: all the edges parallel to the north-west side of the triangle, with no endpoint on the other two sides, correspond to the value λ 2 ; the ones parallel to the north-east side, with no endpoint on the other two sides, have the value µ 2 ; finally the edges parallel to the south side, strictly inside the triangle and with no endpoint on the other two sides, must have the value λ 2 + µ 23 . Thus the edges on the south side corresponding to the parts ν 3 to ν n-2 must in fact have the value λ 2 +µ 2 . That is, ν must be of the aforementioned form: ν = ν 1 ν 2 (λ 2 +µ 2 ) n-4 ν n-1 ν n . Notice moreover that, even if n = 4, two rhombi inequalities show immediately that one must still have

ν 2 ≥ λ 2 + µ 2 ≥ ν n-1 .
From now on we assume that ν has this particular form. Consider once again a hive with exterior edges labelled by λ, µ, and ν. The same conditions as before apply, and the values of all the remaining edges have then to be chosen in order to determine completely the hive. Let us name these values a 0 , a 1 , . . . , a 7 as shown on the following picture:

a 0 a 0 a 0 a 0 a 1 a 1 a 1 a 1 a 2 a 2 a 2 a 2 a 3 a 3 a 3 a 3 a 4 a 5 a 6 a 6 a 6 a 7 a 7 a 7 ν 1 ν 2 ν n-1 ν n λ 2 + µ 2 λ 2 + µ 2 λ 1 0 λ 2 λ 2 λ 2 λ 2 0 µ 1 µ 2 µ 2 µ 2 µ 2 ν 1 -λ 1 νn µ 1
The fact that many of these edges must be given the same value comes everytime from the fact that, in any rhombus inside a hive, if two opposite edges have the same value, then it must also be the case for the other pair of opposite edges.

Using now once again the fact that, in any triangle, the values corresponding to two edges determine the value on the third one, these 8 integers a 0 , . . . , a 7 are related by the following equations:

                       a 0 + a 1 = µ 1 a 0 + µ 2 = a 3 λ 2 + a 1 = a 2 ν 1 -λ 1 + a 5 = a 2 a 4 + ν n = a 3 a 4 + a 7 = ν n-1 a 5 + a 6 = ν 2 a 6 + a 7 = λ 2 + µ 2 ⇐⇒                    a 1 = µ 1 -a 0 a 2 = λ 2 + µ 1 -a 0 a 3 = µ 2 + a 0 a 4 = µ 2 -ν n + a 0 a 5 = λ 1 + λ 2 + µ 1 -ν 1 -a 0 a 6 = λ 2 + 2µ 2 -ν n-1 -ν n + a 0 a 7 = -µ 2 + ν n-1 + ν n -a 0 (let us recall that the equality |λ|+|µ| = |ν| means that λ 1 +2λ 2 +µ 1 +2µ 2 = ν 1 +ν 2 +ν n-1 +ν n ).
In particular, this means that the hive is for instance entirely determined by the value of a 0 .

We can now look at all the hive inequalities that must be satisfied by these a i 's:

a 0 ≥ 0 λ 2 ≥ a 0 a 1 ≥ µ 2 ν 1 ≥ a 2 a 1 ≥ ν 1 -λ 1 a 2 ≥ ν 2 µ 2 ≥ a 6 a 3 ≥ ν n ν n ≥ a 0 ν n-1 ≥ a 3
The inequalities of the first column can be obtained from the hive inequalities in the north corner of the hive, those of the second from the south-west corner, and those of the third from the south-east corner (keep in mind that some of them can of course be obtained in several ways). Thanks to the previous relations between the a i 's, all these inequalities can be expressed in terms of a 0 only, giving in the end exactly the following necessary and sufficient conditions on a 0 to obtain a hive:

a 0 ≥ max(0, λ 2 + µ 1 -ν 1 , ν n -µ 2 ) a 0 ≤ min(λ 2 , µ 1 -µ 2 , λ 1 + µ 1 -ν 1 , λ 2 + µ 1 -ν 2 , -λ 2 -µ 2 + ν n-1 + ν n , ν n , ν n-1 -µ 2 )
As a consequence, c ν λµ is the number of integer points in the interval [M; m] where

M = max(0, λ 2 + µ 1 -ν 1 , ν n -µ 2 ), and m = min(λ 2 , µ 1 -µ 2 , λ 1 + µ 1 -ν 1 , λ 2 + µ 1 -ν 2 , -λ 2 -µ 2 + ν n-1 + ν n , ν n , ν n-1 -µ 2 ). Observe finally that ν n-1 ≤ λ 2 + µ 2 and ν 2 ≥ λ 2 + µ 2 give ν n-1 -µ 2 ≤ λ 2 , λ 2 + µ 1 -ν 2 ≤ µ 1 -µ 2 , -λ 2 -µ 2 + ν n-1 + ν n ≤ ν n .
Hence c ν λµ is the cardinality of

max(0, λ 2 +µ 1 -ν 1 , -µ 2 +ν n ), min(λ 1 +µ 1 -ν 1 , λ 2 +µ 1 -ν 2 , -λ 2 -µ 2 +ν n-1 +ν n , -µ 2 +ν n-1 ) .
This stability can be interpreted as a proof of the existence of a bijection between sets of hives. Such a bijection can for instance be obtained as follows.

Starting from a hive of size n (n ≥ 4), consider the three areas coloured in the picture above (on the left): the four triangles in the north corner, the four in the south-east one, and the seven in the south-west one. Then send this hive to the one of size 4 obtained by keeping these three coloured-areas (picture on the right). The rhombus inequalities in the hive show that the values on the edges in these three particular areas determine indeed completely the hive. This means that this map is well defined and that it is truly a bijection.

Remark. Let α, β, γ be three partitions such that c γ α,β = 1. By the Fulton's conjecture (see [START_REF] Knutson | The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture[END_REF] or [START_REF] Belkale | Geometric proof of a conjecture of Fulton[END_REF][START_REF] Prakash Belkale | A generalization of Fulton's conjecture for arbitrary groups[END_REF][START_REF] Ressayre | A short geometric proof of a conjecture of Fulton[END_REF]), we have c kγ kα,kβ = 1, for any k ≥ 0. Let ( α, β, γ) be a second triple of partitions. The stability result of [START_REF] Sam | Proof of Stembridge's conjecture on stability of Kronecker coefficients[END_REF] (see also [START_REF] Paradan | Stability property of multiplicities of group representations[END_REF][START_REF] Pelletier | A geometric approach to the stabilisation of certain sequences of Kronecker coefficients[END_REF]) asserts that c γ+kγ α+kα, β+kβ does not depend on the integer k big enough. Returning to the setting of Proposition 3, consider α = 1 λ 2 , β = 1 µ 2 and γ = 1 λ 2 +µ 2 . Set also α = (λ 1 λ 2 λ 2 ) ′ , β = (µ 1 µ 2 ) ′ and γ = (ν 1 ν 2 ν 3 ν 4 ) ′ , where ′ denote the conjugated partition. Using the invariance of the Littlewood-Richardson coefficient by simultaneous conjugation we get c ν λµ = c γ+kγ α+kα, β+kβ with k = n -2. Thus, the mentioned stability result asserts that c ν λµ does not depend on n big anough. Proposition 3 asserts that this sequence is in fact constant for n ≥ 2.

The case of GL 4 (C)

This section is about GL 4 (C). But Proposition 3 allows to extend several results to any GL n (C) for n ≥ 4.

The Horn cone

The set of points in Horn n with λ and/or µ near-rectangular is the set of integer points on a face of this cone. Proposition 3 implies that the geometry of this face and the Littlewood-Richardson coefficients on it do not depend on n ≥ 4. We denote by Horn n the set of points in Horn n with the first two partitions λ and µ in Λ 0 n . Then Horn n ≃ Z 2 × Horn n . Set

Horn nr 2 4 = {(λ, µ, ν) ∈ Horn 4 : λ and µ are near-rectangular} and Horn nr 4 = {(λ, µ, ν) ∈ Horn 4 : λ is near-rectangular}. The inequalities defining the Horn cone Horn Q n are well known (see Section 2). By convex geometry and explicit computation, one can deduce the minimal lists of inequalities for Horn nr 2 4 and Horn nr 4 . Softwares like Normaliz [BIS] allow to make the computation.

Proof. By Proposition 3, one may assume that n = 4. Then, the set of self-dual isotypic components of V 4 ((2k)k 2 ) ⊗ V 4 ((2l)l 2 ) are obtained by adding the condition

ν 1 + ν 4 = ν 2 + ν 3
to the conditions (14). The corollary follows by explicit computations that can be performed with [VSB + 07].

6.3 Computation of Nb(c • λµ > c) for λ and µ near-rectangular

In this subsection, we report on the computation of the function

Nb 4 (c • λµ > c) : (Λ nr 4 ) 2 × N -→ N (λ, µ, c) -→ #{ν ∈ Λ 4 : c ν λµ > c}.
By Proposition 3, this function determines Nb n (c • λµ > c) for any near-rectangular partitions λ and µ of length n ≥ 4.

Since Propositions 9 and 3 give similar expressions for the Littlewood-Richardson coefficient, we can apply the strategy of Section 4.

We get that Nb 4 (c

• λµ > c) is the number of points ν ∈ Λ 4 such that λ 1 + 2λ 2 + µ 1 + 2µ 2 = ν 1 + ν 2 + ν 3 + ν 4 and -λ 2 -µ 2 + ν 2 ≥ 0 λ 2 + µ 2 -ν 3 ≥ 0 λ 1 -λ 2 ≥ c -λ 2 + ν 3 ≥ c ν 1 -ν 2 ≥ c ν 3 -ν 4 ≥ c λ 1 + µ 1 -ν 1 ≥ c λ 2 + µ 1 -ν 2 ≥ c -λ 2 -µ 2 + ν 3 + ν 4 ≥ c -µ 2 + ν 3 ≥ c λ 1 + µ 2 -ν 2 ≥ c λ 1 + λ 2 + µ 2 -ν 2 -ν 4 ≥ c λ 1 + µ 1 + µ 2 -ν 1 -ν 4 ≥ c λ 2 + µ 1 + µ 2 -ν 2 -ν 4 ≥ c
In particular, it is the number of integer points in some polytope depending linearly on the data (λ, µ, c). Then Nb 4 (c • λµ > c) is piecewise quasi-polynomial, and can be computed using Barvinok's algorithm. Surprisingly, here Λ = (Λ nr 4 ) 2 × Z and Nb 4 (c • λµ > c) is piecewise polynomial.

As in Section 4, from this point on we use the basis of fundamental weights to write

λ = k 1 ̟ 1 + k 2 ̟ * 1 and µ = l 1 ̟ 1 + l 2 ̟ * 1 .
Thus the symmetry we want to observe is once again with respect to swapping k 1 and k 2 . Consider the function

ψ : N 5 -→ N (k 1 , k 2 , l 1 , l 2 , c) -→ #{ν ∈ Λ 4 | c ν k 1 ̟ 1 +k 2 ̟ * 1 ,l 1 ̟ 1 +l 2 ̟ * 1 > c} .
We now give details about the results in Proposition 4:

Proposition 17. We have ψ(k 1 , k 2 , l 1 , l 2 , c) = 0 unless c ≤ min(k 1 , k 2 , l 1 , l 2 ).
Moreover, this cone decomposes into 36 cones C 1 , . . . , C 36 on which ψ is given by polynomial functions P 1 , . . . , P 36 . 12 of these pairs (C i , P i ) are kept unchanged by swapping k 1 and k 2 (namely (C 1 , P 1 ) to (C 12 , P 12 )). The 24 other such pairs are pairwise swapped by this operation (for all i ∈ {7, . . . , 18}, (C 2i-1 , P 2i-1 ) and (C 2i , P 2i ) are swapped).

In particular, Conjecture 1 holds for GL 4 and λ, µ near-rectangular.

Now to present as clearly as possible these cones and polynomials without writing all of them, let us use the two following involutions: s 1 corresponding to the exchange of k 1 and k 2 , and s 2 corresponding to swapping (k 1 , k 2 ) and (l 1 , l 2 ). Then s 1 , s 2 acts on the set of all pairs (C i , P i ) with 8 orbits. Let us give below one representative for each one of these. The labelling is the one of the complete list [Res20, pol_and_cones_SL4nr2.txt], chosen so that the stability when swapping k 1 and k 2 is easier to see:

C 1 : l 1 + l 2 ≤ k 1 + c, l 1 + l 2 ≤ k 2 + c, P 1 = - 1 2 • (-l 2 + c -1) • (-l 1 + c -1) • (-l 1 -l 2 + 2c -2)
has a s 1 , s 2 -orbit of size 2;

C 16 : l 1 + l 2 ≤ k 1 + c, l 1 + l 2 ≥ k 2 + c, k 2 ≥ l 1 , k 2 ≥ l 2 , P 16 = P 1 - -k 2 + l 1 + l 2 -c + 2 3
has an orbit of size 4;

C 2 : l 1 + l 2 ≥ k 1 + c, l 1 + l 2 ≥ k 2 + c, k 1 ≥ l 1 , k 1 ≥ l 2 , k 2 ≥ l 1 , k 2 ≥ l 2 , P 2 = P 16 - -k 1 + l 1 + l 2 -c + 2 3
has an orbit of size 2;

C 19 : l 1 + l 2 ≥ k 1 + c, k 1 ≥ l 1 , k 2 ≤ l 1 , k 2 ≥ l 2 , P 19 = P 2 + -k 2 + l 1 + 1 3
has an orbit of size 8;

C 21 : l 1 + l 2 ≤ k 1 + c, k 2 ≤ l 1 , k 2 ≥ l 2 , P 21 = P 16 + -k 2 + l 1 + 1 3
has an orbit of size 8;

C 29 : k 1 + k 2 ≥ l 1 + l 2 , l 1 + l 2 ≥ k 1 + c, k 2 ≤ l 1 , k 2 ≤ l 2 , P 29 = P 19 + -k 2 + l 2 + 1 3
has an orbit of size 4;

C 27 : k 1 + k 2 ≤ l 1 + l 2 , k 1 ≥ l 1 , k 1 ≥ l 2 , P 27 = P 29 + -k 1 -k 2 + l 1 + l 2 + 1 3
has an orbit of size 4; finally,

C 36 : l 1 + l 2 ≤ k 1 + c, k 2 ≤ l 1 , k 2 ≤ l 2 , P 36 = P 21 + -k 2 + l 2 + 1 3
also has an orbit of size 4.

Remark. One can observe that the polynomials P i are expressed using each other. We exploit here the fact that the difference between two polynomials associated to two adjacent cones has a simple expression theoretically given by the Paradan formula [Par04, BV09]. As we recalled in Proposition 7, Nb 4 (c • λµ > 0) is the number of integer points in an affine section of the Horn cone. The inequalities defining this cone are explicitly given in Proposition 13. Then, one can compute explicitly the quasi-polynomial function with the program [VSB + 07]. The output is too big (even using symmetries) to be collected there. The interested reader can get details from [Res20, Supplementary material].

Computation of

Proposition 18. The cone Λ nr 4 × Λ 4 decomposes into 205 cones of non empty interior. On 151 of them Nb 4 (c • λµ > 0) is polynomial of degree 3, and on the other 54 it is quasipolynomial. The only congruence occurring is the parity of λ 1 + |µ|.

Moreover, for any pair (C, P ) where C is one of the 205 cones and P the corresponding function, one can see that in this list there is also a pair (C ′ , P ′ ) obtained by replacing λ by λ * (in 57 cases, (C ′ , P ′ ) = (C, P )). In particular, Conjecture 1 holds.

Under the action of Z/2Z × Z/2Z there are 61 orbits of actual polynomials and 22 orbits of quasi-polynomials.

Here we give three examples illustrating some of the variety of cases that one can observe. The function Nb 4 (c • λµ > 0) for λ = k 1 ̟ 1 + k 2 ̟ * 1 ∈ Λ nr 4 ∩ Λ 0 4 and µ = µ 1 µ 2 µ 3 ∈ Λ 0 4 is given:

partitions is then to consider the dominant weights in N̟ 4 ⊕ N̟ 5 . A natural generalization of Conjecture 2 would be: for λ = a̟ 4 + b̟ 5 ∈ N̟ 4 ⊕ N̟ 5 and µ a dominant weight of D 5 , do the two tensor products V D 5 (a̟ 4 + b̟ 5 ) ⊗ V D 5 (µ) and V D 5 (b̟ 4 + a̟ 5 ) ⊗ V D 5 (µ)

contain the same number of isotypic components?

The answer is NO, even assuming that µ ∈ N̟ 4 ⊕ N̟ 5 too. An example is λ = 2̟ 4 + ̟ 5 and µ = ̟ 4 +2̟ 5 . The two tensor products have respectively 31 and 30 isotypic components as checked using SageMath [The20]: sage: D5=WeylCharacterRing("D5",style="coroots") sage: len(D5(0,0,0,2,1)*D5(0,0,0,1,2)) 31 sage: len(D5(0,0,0,1,2)*D5(0,0,0,1,2)) 30

In type A n

The representations of SL n (C) corresponding to near-rectangular partitions are of the form V (a̟ 1 + b̟ n-1 ). Observe that (̟ 1 , ̟ n-1 ) is a pair of mutually dual fundamental weights. One could hope that Conjecture 1 or 2 hold for any linear combination of a given pair of mutually dual fundamental weights. This is not true even for (̟ 2 , ̟ 3 ) and n = 5. Indeed, for λ = ̟ 2 + 2̟ 3 and µ = 3̟ 2 + ̟ 3 , the numbers of isotypic components in V (λ) ⊗ V (µ) and V (λ) * ⊗ V (µ) differ: sage: len(lrcalc.mult([3,3,2],[4,4,1],5)) 34 sage: len(lrcalc.mult([3,3,1],[4,4,1],5)) 33

Mention finally that we checked Conjecture 1 on examples, using SageMath. See [Res20, test_Conj1.sage]:

• Conjecture 1 holds for GL 4 if max(λ 1 -λ 2 , λ 2 ) ≤ 20 and |µ| ≤ 40.

• Conjecture 1 holds for GL 5 if max(λ 1 -λ 2 , λ 2 ) ≤ 20 and |µ| ≤ 30.

• Conjecture 1 holds for GL 6 if max(λ 1 -λ 2 , λ 2 ) ≤ 10 and |µ| ≤ 30.

• Conjecture 1 holds for GL 10 if max(λ 1 -λ 2 , λ 2 ) ≤ 10 and |µ| ≤ 15.

Figure 2 :

 2 Figure 2: Rhombi

  Nb 4 (c • λµ > 0) for λ near-rectangularIn this section, we report on the computation of the functionNb 4 (c • λµ > 0) : Λ nr 4 × Λ 4 -→ N (λ, µ) -→ #{ν ∈ Λ 4 : c ν λµ > 0}.

Namely, the cone Λ 3 × Λ 3 × N is the support of a fan with 7 maximal cones. And, on each one of these 7 cones, the function Nb 3 (c • λµ > c) is given by a polynomial in (λ, µ, c) of degree

See Section 2 right after Proposition 7 for a definition.

A stability resultIn this section, we are interested in the Littlewood-Richardson coefficients c ν λµ with λ and µ near-rectangular. Using the hive model, we give a proof of Proposition 3 stated in the

These three kinds of edges are coloured on both pictures in this proof: the first kind (parallel to the north-west) in red, the second kind (north-east) in blue, and the third one (south) in green.

Acknowledgements. We are very grateful to Vincent Loechner, who helped us in the use of ISCC implementation of Barvinok's algorithm. We also want to thank Dipendra Prasad for useful discussions on [PW20], which motivated this work. Finally, we are very grateful to the anonymous referee for his numerous advice which helped improving the article.

The authors are partially supported by the French National Agency (Project GeoLie ANR-15-CE40-0012).

Proposition 11. Let λ, µ in Λ 0 4 and ν in Λ 4 such that λ and µ are near-rectangular. Then c ν λµ = 0 if and only if

Remark. Proposition 3 also implies that ν 1 + ν 4 ≥ λ 2 + µ 1 , which is a consequence of these 11 inequalities.

Proposition 12. The cone generated by Horn nr 2 4 ∩ Horn 4 has 8 extremal rays generated by the triples (λ, µ, ν) associated to the following inclusions

Each triple (λ, µ, ν) on one of these extremal rays indexes a Littlewood-Richardson coefficient with value 1. The Hilbert basis of Horn nr 2 4 ∩ Horn 4 consists in these 8 generators. We get similar descriptions for Horn nr 4 .

Proposition 13. Let λ, µ in Λ 0 4 and ν in Λ 4 such that λ is near-rectangular. Then c ν λµ = 0 if and only if all of the following inequalities hold:

We have 32 facets.

Proposition 14. The cone generated by Horn nr 4 ∩ Horn 4 has 12 extremal rays generated by the triples (λ, µ, ν) associated to the following inclusions

Each triple (λ, µ, ν) on one of these extremal rays indexes a Littlewood-Richardson coefficient with value 1. The Hilbert basis of Horn nr 4 ∩ Horn 4 consists in these 12 generators.

Special case of self-dual representations

Let k and l be two nonnegative integers and n ≥ 4.

In [PW20, Section 8], conjectural values (for n = 6) are given for the numbers of isotypic components in V n ((2k)k n-2 ) ⊗ V n ((2l)l n-2 ) and for the numbers of self-dual isotypic components. Here, we prove and extend these formulas.

Corollary 15. Assume up to symmetry that l ≤ k. The number of distinct isotypic components in V n ((2k)k n-2 ) ⊗ V n ((2l)l n-2 ) is given by

By Proposition 3, one may assume that n = 4. Then, Proposition 11 shows that ν ∈ Z 4 is the highest weight of an isotypic component of V 4 ((2k)k 2 ) ⊗ V 4 ((2l)l 2 ) if and only if (recall that l ≤ k) all of the following conditions hold:

The corollary follows by explicit computations that can be performed with [VSB + 07].

Similarly, one gets the number of self-dual representations.

Corollary 16. Assume up to symmetry that l ≤ k. The SL n (C)-module V n ((2k)k n-2 ) ⊗ V n ((2l)l n-2 ) contains (l + 1) 2 distinct self-dual isotypic components.

• on the cone defined by

µ 3 ≥ 0, by the polynomial

• on the cone defined by

• on the cone defined by µ Let (̟ 1 , . . . , ̟ 5 ) be the list of fundamental weights. Then V (̟ 4 ) * ≃ V (̟ 5 ) whereas V (̟ 1 ), V (̟ 2 ) and V (̟ 3 ) are self-dual. The natural generalization of near-rectangular


Dr Dejan Segan

Dr George Stanley

Dr Pierluca Messina

Dr Jean-Marie Marie Swiecicki

Dr Kieu Ngo

Dr Vincent Vivier

Dr Olivier Buriez

Eric Labbe

Pr Eric Labbé

Interaction of redox probes and ferrocene-labelled peptides with lipid bilayers observed at lipid bilayer-modified electrodes

Keywords: redox probes, peptides, Supported Bilayer Lipid Membrane, Electrochemical Impedance Spectroscopy, cyclic voltammetry

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Plasma membranes are the interfaces which regulate the flux of ions and molecules getting in and out of living cells. Transmembrane trafficking refers not only to the regular metabolism of cells, but also to the delivery of drugs or pathogens [1][START_REF] Yang | Site-Specific Protein Labeling[END_REF] , hence rising up research topics focusing on the mechanisms of membrane crossing. Molecular transport across membranes is generally classified into passive modes (i.e. energy-independent processes following the concentration and/or electrical field gradients) and active modes (which require energy and specific regulation) [START_REF] Shinoda | [END_REF] . The amphiphile nature of the phospholipids composing the membranes increases the complexity of the trafficking pathways, especially when it concerns the translocation of ionic species through the hydrophobic region of lipid bilayers. The specific example of the internalization of cell penetrating peptides (CPPs) in purely lipidic liposomes, which matches the features and definition of a passive transport, has been the subject of a number of explorations and interpretations in the last two decades, as reported in early [4] , recent [5][6][7] or very recent [8] reviews.

The experimental strategies aimed at exploring and quantifying the passage of peptides across lipid membranes rely on several techniques. These methods may be classified according to the possibility to use label-free peptides (e.g. X-ray [9] or neutron diffraction [10] , Differential Scanning calorimetry [11] ,

Infrared spectroscopy MALDI-ToF MS [12][13] ), and also according to the dynamic information they provide (e.g. NMR [14] ) . Fluorescence-based methods [15][16][17] , which require the functionalization by fluorophores display a unique sensitivity in the micromolar concentration range. The reader is encouraged to consult the paper by Avci et al. for a comprehensive review [18] . However, some methods either require expert knowledge or setups (NMR) or present some drawbacks associated to the chemical quenching of non-internalized peptides [19][20] . Electrochemical techniques provide a variety of time-resolved approaches namely through impedance spectroscopy, amperometry or voltammetry. In 2014, we developed an electrochemical method to monitor and quantify the passage of small molecules across suspended BLMs formed by patch-clamp techniques [21] , a strategy allowing the characterization of fluxes across membranes in contact with solutions at both sides. Since CPP concentrations lie in the 10-100 µM range in vivo, the concentration gradient across a suspended membrane is too low to generate a significant (electrochemically detectable) efflux of CPP, considering a fraction of the CPPs may remain bound to the membrane. With this respect, we have implemented electrochemical methods to explore the permeation and fluxes of model redox probes at bilipidic membranes (BLMs) deposited at the surface of glassy carbon electrodes (GCE). The interactions of CPPs with supported membranes have already been investigated by Quartz Crystal Microbalance with Dissipation monitoring (QCM-D) [22] or by Scanning Electrochemical Microscopy (SECM) [23] . Supported BLMs may afford an adequate electrochemical interface to explore the passage of redox species from the solution to the electrode surface (and vice versa) across the membrane. In 2012, we established that a cationic CPP could be modified with a redox probe such as ferrocene so as to allow its electrochemical detection [24] . In this work, ferrocene-labelled peptides could be detected at a micromolar concentration, exhibiting the characteristic voltammetric wave of the ferrocene probe. In these conditions, the faradaic signal displayed a poor resolution due to the small diffusion rate of the peptide. Two other strategies have been investigated to improve the faradaic signal resolution. The first one used avidin/biotin recognition to preconcentrate the CPP at the electrode surface, but the ferrocene probe showed irreversible voltammograms under these conditions. A second strategy consisted in modifying the electrode with a cation-exchange polymer [25] , which allowed a rapid preconcentration of the cationic CPP at the electrode surface. This led to a very good signal-to-noise ratio and ensured stability of the redox wave. In 2015, the preparation as well as the electrochemical detection of a CPP composed of nine arginines and tagged with a ferrocenyl unit (noted as Fc-R9 in Figure 1) at a carbon fiber electrode immersed in a micro-volume suspended water droplet was successfully investigated. Accordingly, a well-defined sigmoidal voltammogram was obtained at slow scan rate in the presence of micromolar concentration of the Fc-R9 peptide [26] .

Herein, we have first investigated the permeation of classical redox probes (hexaamineruthenium(III) and ferrocene methanol) by electrochemical impedance spectroscopy and cyclic voltammetry at glassy carbon electrodes modified with two phospholipids of interest (diphytanoyl phosphatidylcholine (DPhPC, zwitterionic and in gel ordered phase at all temperatures [27] ) and dioleyl phosphatidyl glycerol (DOPG, anionic and in liquid disordered phase above -18°C [START_REF] Koynova | Encyclopedia of Biophysics[END_REF] )) for the study of the interactions between membranes and peptides. Then, we have explored the behavior of ferrocene-labelled peptides classified as penetrating and non-penetrating in former studies at DOPG-modified GCE. The lipids, probes and ferrocene-tagged peptides explored in this study are represented in Figure 1.

Figure 1

Structure of the lipids and probes used in this work

DPhPC = 1,2-diphytanoyl-sn-glycero-3-phosphocholine O O N+ H P O - O O O O O N H N H NH NH NH N H N H N H N H N H O N H NH 2 O O O O O O O NH NH N H 2 Fe O N H N H N H 2 N H N H N H 2 N H N H N H 2 N H N H N H 2 NH 2 O N H N H N H 2 N H N H NH 2 N H N H N H 2

Experimental

a. Chemicals

Diphytanoyl phosphatidylcholine (DPhPC) and dioleyl phosphatidyl glycerol (DOPG) were purchased from Avanti Lipids, dissolved in chloroform to obtain 2 mL solutions at 2.5 mg/mL and stored under argon in a freezer at -20°C. These solutions were directly used for the formation of BLMs on GCE (see below). Potassium hexacynaoferrate K3Fe(CN)6; ferrocene methanol Fc-MeOH and hexaamineruthenium(III) chloride Ru(NH3)6Cl3 were purchased from Sigma Aldrich and used as received. PKCi, R9 and RW9 were synthesized on an MBHA resin (0.54 mmol.g -1 ) via standard Boc solid phase peptide synthesis [START_REF] Swiecicki | [END_REF] . Once the peptide synthesis was complete, the N-terminal Boc protecting group was removed with Trifluoroacetic acid (TFA). The resin was washed with dichloromethane, Nmethyl-2-pyrrolidone (NMP) and neutralized with 10 % (v/v) N,N-diisopropylethylamine (DIPEA, AnalaR NORMAPUR, Carlo Erba) in NMP and finally washed with NMP. The peptide still on resin was reacted for 30 min with a solution of ferrocene carboxylic acid (5 eq., Sigma-Aldrich) in the presence of Hexafluorophosphate Benzotriazole Tetramethyl Uronium (HBTU, 4.5 eq., Iris Biotech GmbH) and DIPEA ( 10 eq.) in NMP. The peptidyl-resin was finally washed with NMP, methanol and dried overnight under vacuum before HF cleavage. Crude peptides were purified by RP-HPLC and their identity was confirmed by MALDI-ToF MS.

b. Electrode and film procedures

The cyclic voltammograms and electrochemical impedance spectra were recorded using an Autolab (PGSTAT 20) potentiostat (Ferrocene methanol and hexaamineruthenium probes) or an Ametek -Princeton Applied Research 2273 Potentiostat (ferrocene-labelled peptides).

GCE preparation and lipid film formation: The GC Electrode was thoroughly polished with 2400 & 4000

Mecaprex polishing paper using alumina slurry (on 4000), rinsed with ethanol and water, then ultrasonicated. This electrode was left in buffer solution overnight. GCE was polarized at 1.5 V vs Ag/AgCl in a 0.1 M KCl solution for 180 seconds, according to a carbon surface conditioning procedure inspired from the work by Wang et al. [30][31] . The oxidation of GC surface by electrochemical polarization and its characterization have been described in aqueous solutions of variable pH and produce a pHdependent distribution of surface groups [32][33][34] . Although originally described in alkaline solutions, the polarization of the electrode in aqueous KCl afforded a GC oxidized surface prone to immobilize lipid layers, as already observed [35] and described hereafter. The electrode was removed, rinsed with H2O and dried under argon. Lipid solutions (6µL at 2.5 mg/mL in chloroform) were deposited onto the electrode from a 10 μL glass syringe. Due to the hydrophilic polarized surface of the electrode and the amphiphilic character of the phospholipids, the molecules spontaneously assemble into a lipid film structured as a bilayer covering the electrode surface [35] . The electrode was dried under argon and then added to the electrochemical cell containing 1.85 mL of aqueous solution "S" composed of D-(+)glucose (0.32 M), NaCl (0.15 M) and NaN3 (4.6 mM) -that composition of the solution was originally envisioned to form lipid films via LUV deposition / opening on GCE, method finally discarded because the lipid coverage appeared too heterogeneous and poorly reproducible-. The probes and peptides were introduced as 0.15 mL of 1 mM aqueous solution in the cell already containing 1.85 mL of solution "S", resulting in 75 µM solutions in contact with the BLM-modified GCE.

Results and discussion

Lipid film characterization

We first used a qualitative approach to assess the formation of a lipid thin-film on the electrode, and checked the electrochemical response of hexacyanoferrate solutions before and after formation of the lipid film. Figure 2 shows the cyclic voltammograms (CVs) of Fe(CN)6 3-either at neutral (DPhPC), anionic (DOPG) or mixed (DPhPC:DOPG 70:30) lipidic films. The quasi reversible monoelectronic reduction of Fe(CN)6 3-is observed at 0.05 V vs. Ag/AgCl at unmodified GCE electrode. For BLM-modified electrodes, the absence of faradaic response revealed a near-complete blocking of the electrode surface, as previously described for glassy carbon electrodes modified with lipid bilayers [30,35] . For DPhPC-modified GCE (Fig 2A ), a low amplitude S-shape CV still appeared, mostly accounting for micro-defects in the GCE coverage which may be interpreted as the faradaic signal of Fe(CN)6 3-at an array of independent micrometer-size electrodes.

Electrochemical Impedance Spectroscopy (EIS) was also performed on BLM-modified GCE. After recording the impedance of a bare GCE electrode in aqueous 0.1 M NaCl (without redox probe), the BLM modified GCE were examined in the same medium. The Nyquist representations of DPhPC-and DOPG-modified GCE are shown in Figure 3, the Bode plots are in the SI. For both bare and modified electrodes, a non-ideal capacitive behavior is observed in the whole frequency domain. In the case of BLM modified GCE, the electrode-solution interface can be represented by a Constant Phase Element (CPE) accounting for the electrical properties of the membrane in series with the double layer capacitance and the electrolyte resistance. In the case of bare GCE, the electrochemical response consisted in the double layer contribution (which also shows a non-ideal behavior) and the electrolyte resistance, only. Assuming a surface distribution of the interface properties (due to current and potential distribution of the electrified interface), the capacity accounting for this non-ideal behavior can be estimated from Brug's formula [36] :

𝐶 𝐵 = 𝑄 1/𝛼 ( 𝑅 𝑒 + 𝑅 𝑡 𝑅 𝑒 𝑅 𝑡 ) 𝛼-1 𝛼
where Re and Rt are electrolyte resistance and transfer resistance, respectively.

For the experiments presented in Fig. 3, it is obvious that the 𝑅 𝑡 values are several orders of magnitude larger than the electrolyte resistance, thus leading to the simplified formula

𝐶 𝐵 = 𝑄 1/𝛼 (𝑅 𝑒 ) 1-𝛼 𝛼
For the bare GCE (Fig. 3.A),  = 0.82, Q = 1.11 10 -6   s 0.82 , leading to a capacitance value of 2.2 µF cm -2 , which is in fair agreement with the double layer value expected at a CGE.

For modified GCE,  = 0.73 and Q = 8.8 10 -7   s 0.73 was obtained for DPhPC and  = 0.72, Q = 9.3 10 - 7   s 0.72 for DOPG. In both cases, the evaluation of the capacitance leads to CB = 0.46 µF cm -2 .

A B C

Assuming a model of a plane capacitor, CB may also be expressed as CB = °/d where ° is the vacuum permittivity (° = 8.85 10 -14 F cm -1 ) ,  the dielectric constant of the medium (= 2.05 for apolar hydrocarbon chains [37] ) and d the capacitor thickness. With this model, d is estimated at 3.9 nm, a value consistent with the thickness of a bilayer of lipid molecules [35] . It is also interesting to note that this value is obtained by making the assumption that the capacity of the double layer which forms is negligible compared to the capacity of the bilayer. If we assume that the double layer is of the order of 5 µF cm -2 on the bilayer, then the error made is an underestimation of about 7.5% of the thickness (3.6 nm). The cationic Ru(NH3)6 3+ probe reduces at -0.22 V/Ag/AgCl at bare GC electrode (Fig. 4A). The formation of a lipid film dramatically affects the CVs: at DPhPC-GCE interface, the reduction of the ruthenium probe is no longer observed, a behavior that can be ascribed to the blocking of the electrode surface by the DPhPC BLM. Upon introduction of an increasing amount of DOPG the lipid film, a reduction peak of Ru(NH3)6 3+ is recovered, but at more negative potential values. This observation indicates that i) DOPG-enriched lipid films appear permeable to cationic probes and ii) Ru(NH3)6 3+ is stabilized by the DOPG environment (it is reduced at more negative potential values), most likely through electrostatic association between the cationic probe and the anionic phospholipid. If successive scans are performed with the DOPG-GCE in the same conditions, the CVs show a dramatic current increase and both reduction and backward oxidation peaks of the ruthenium complex adopt symmetrical shapes, together with a decrease of the anodic to cathodic peak potential separation (Fig. 4B). For pure DOPG-modified electrodes, Ru(NH3)6 3+ behaves like a redox species that accumulates in a thin layer grafted to the electrode. Actually, the integration of the reduction peak displayed at the scan obtained after 5 min equilibrium in Fig. 4B yields an apparent electrode surface coverage of 3.3 10 -9 mol.cm -2 , a value six times larger than the one observed for packed ferrocene functionalized selfassembled monolayers at gold electrodes ( 5.10 -10 mol.cm -2 ) [38] , the probe no longer acting as a dilute species in a solvent/phase. Such an accumulation of the cationic probe within the anionic DOPG under our conditions progressively leads to the swelling of the lipid membrane, as observed in the inset of Fig. 4B where the CV (in red) obtained after 15 min features both a decreased peak current and a positive potential shift for the Ru III /Ru II redox process. Finally, the evolution of the CVs is consistent with an initial strong interaction between the ruthenium complex and the DOPG film, leading to a progressive destructuration of the lipid film after 5 minutes.

The electrochemical response of Fc-MeOH probed at DOPG-and DPhPC-modified GC electrodes is shown on Figures 4C-E The electrochemical behavior of Fe(CN)6 3-, Ru(NH3)6 3+ and Fc-MeOH at BLM-modified electrodes finally reveals that DPhPC remains impermeable to ions, whatever the sign of their charge. Actually, a cationic species generated at the GC-DPhPC interface is expelled from the film, while cations and anions in solution do not produce any faradaic signal, a proof of the blocking character of the DPhPC film. DOPG films strongly associate with cationic species, that interaction even leading to a swelling of the lipid film. The apparent thickness of the BLM-modified electrodes has been estimated from the capacitance obtained from EIS experiments for Ru(NH3)6 3+ and Fc-MeOH probes as a function of the potential applied to GCE (Figure 5). It shows that most lipid-probe combinations afford dimensionally stable lipid films, except the DOPG-Ru(NH3)6 3+ couple, which exhibits a dramatic increase of the thickness upon reaching potential values that allow reduction of the ruthenium precursor. The polarization of the DOPG-modified GCE at potential values allowing the reduction of Ru(NH3)6 3+ for 70-90 seconds (duration required for the recording of EIS data) appears to considerably affect the DOPG film structure and apparent thickness (Fig. 5). Since the positive charge borne by the ruthenium complex may only decrease upon reduction, the destructuration of the DOPG film may result from the large partition constant of Ru(NH3)6 3+ between the solution and the DOPG film that could form bulky Ru/DOPG aggregates, leaving the GCE coated with a fraction of disorganized and multilayered DOPG molecules.

Actually, the swelling and denaturation of BLMs upon incorporation of small molecules like chloroform [39] or benzene [40] has been described and even explains the mode of action of several anesthetics [41] . A recent work by Kim et al explored the interaction of the Green Malachite cation with anionic DPPG films at the air-water interface and the authors described the adsorption of the cation forming a highly packed condensed phase near the lipid headgroup region [42] . Therefore, an adsorption of Ru(NH3)6 3+ to the anionic heads of DOPG could take place as a first step, followed by a reorganization of the ruthenium-DOPG adduct, to the detriment of the lipid film.

Figure 5

Apparent thickness of DPhPC-GCE and DOPG-GCE reconstructed from EIS experiments as a function

of the probe and the potential applied to GCE.

Electrochemical behavior of ferrocene-labelled peptides at DOPG-modified electrodes

Considering the electrochemical behavior of electrogenerated Fc + -MeOH at GCE coated with neutral gel phase lipid bilayer (Fig 4C ), polycationic peptides are expected to be expelled from the lipid film, a feature also observed with ferrocene and ferrociphenols at a similar phosphatidycholine BLMs (1,2dimyristoyl-sn-glycero-3-phosphocholine DMPC) [35] . That is why we focused on the interactions of peptides with the negatively charged DOPG films. We have explored the electrochemical response of bare-GCE and DOPG-modified GCE towards ferrocene-functionalized peptides, introduced at biologically-relevant concentrations, i.e. in the micromolar range. Our objective was the characterization of interactions between the negatively charged heads of the phospholipids and the positively charged peptides. Fc-pKCi (pKCi = protein kinase C inhibitor) represents a non-penetrating peptide, Fc-R9 is "non-amphipathic" (random coil secondary structure) and Fc-RW9 is "secondary amphipathic" (alpha helix conformation with both cationic and hydrophobic residues [7] ). Their primary structure is displayed in Fig. 1. The cyclic voltammograms of Fc-pKCi , Fc-RW9 and Fc-R9 are presented in Figure 6. and Fc-R9 peptides exhibit a similar electrochemical behavior at bare-GCE and DOPG-GCE, although less pronounced than in the case of Fc-pKCi (Figs. 6B and6C).

The peak current increase at DOPG-GCE (compared to bare GCE), mostly observed for Fc-pKCi and Fc-RW9, shows that the positively charged peptides associate with the DOPG film. A strictly electrostatic association between the polycationic peptides and the phosphate-based anionic heads at the DOPG/solution interface would not give rise to a faradaic response, as commented above for the redox probes that cannot penetrate the lipid layer. The shape of the CVs in Fig. 6A resemble those of free diffusing electroactive species and display an anionic-to-cathodic peak separation of 70 to 90 mV (cycles 1 and 5 respectively). The motion of electroactive probes like Ru(NH3)6 3+ in polymer films attached to the electrode has been described namely in the case of ionomeric polymers [43][44][45] . In such polymer films, two transport regimes are observed : at high scan rates (short time scales), the peaks on the CVs exhibit a dependence on v 1/2 (square root of the scan rate), accounting for a diffusioncontrolled motion within the film, and a dependence on v at lower scan rates, ascribed to the total reduction or oxidation of the species in a thin layer attached to the electrode. The junction between the two regimes allows an estimation of the apparent diffusion coefficients Dapp within the film from an expression derived from the mean square motion of species by diffusion,  = (2Dappt)½ [START_REF] Girault | Analytical and Physical Electrochemistry[END_REF] . This method is well adapted to the description of the motion of species essentially located in the film (i.e.

with high partition coefficients between the film and the solution). The fluid/dynamic character of DOPG at room temperature (DOPG is in fluid disordered phase) also makes it distinct from structurally "frozen" polymer films. In the present study, the attempts to establish a clear dependence of the peak currents with the scan rate have failed. Considering that very low scan rates in cyclic voltammetry allow a partial re-equilibration of the peptide concentration in the DOPG phase, the peak currents tend to represent a mixed contribution of the peptides already bound to DOPG and peptides present in solution. Instead, we envision in our future prospects to use other electrochemical techniques to estimate both the partition coefficient and Dapp, namely voltammetry at rotating disk electrodes as described by Compton et al. for thin films [START_REF] Batchelor-Mcauley | [END_REF] . In the lipid film, one must consider that the cationic peptides most likely recruit DOPG molecules to allow a globally neutral entity to reach the GCE surface.

Finally, the CVs in Figure 6 recorded at DOPG-GCE do not represent the oxidation of a species strictly immobilized within the 4 nanometer-thick lipid bilayer (the anodic and cathodic peaks exhibit a 70-90 mV peak-to-peak separation). If the CVs in Figure 6 clearly underline the interaction between the cationic peptide and the anionic lipid film, their diffusional reversible shape excludes the occurrence of an oxidation restricted to the fraction of peptide already associated to the lipid film.

One could expect cationic peptides to exhibit an electrochemical behavior at DOPG-GCE similar (and even amplified with respect to the multiple cationic sites) to the one observed for Ru(NH3)6 3+ . Several recent studies have focused on the association of model cationic peptides, mostly polyarginine (Arg8 and Arg9) and polylysine (Lys8 and Lys9) and bilayers of anionic phosphoglycerol (PG) lipids (DMPG and POPG) through the combination of multiple experimental techniques (fluorescence, QCM-D, second harmonic generation spectroscopy, nanoplasmonic sensing) and molecular dynamics simulation [48][49] .

Although possessing the same cationic charge at physiological pH, Arg9 and Lys9 show different associative behaviors with PG lipids [48] . Actually, Lys9 peptides underwent electrostatic repulsion towards each other, resulting in an "anticooperative" association with anionic POPG, whereas Arg9 chains could aggregate at the POPG surface, a property ascribed to the better association of the guanidinium groups to the PG heads and to like-charge pairing made possible in guadinidium moieties compared to the ammonium side chains of Lys9. Moreover, computer simulations carried out from Arg8 and Lys8 [49] suggest that Lys8 may adopt a conformation prone to only associate with the PG headgroups, whereas Arg8 displays a "buried" conformation, also interacting with several side chains.

Then, these works underline the complexity of the cationic peptide-anionic lipid association, which are not reducible to the electrostatic attraction, a feature illustrated by the similar electrochemical properties of the ferrocene-labelled peptides used in our study. The integrity of the DOPG bilayer remains unaffected by the association with the peptides (the CVs are recovered after several minutes of equilibration between the film and the solution), thus highlighting the very specific interaction of cationic peptides with DOPG compared to Ru(NH3)6 3+ .

Finally, the electrochemical features of ferrocene-tagged CPPs at DOPG modified electrodes appear to be independent on the "penetrating" nature of the peptides. Actually, the protein kinase C inhibitor pKCi is commonly coined as non-cell penetrating peptide and is often used as a control-peptide in studies devoted to the internalization of CPPs. Unexpectedly, Fc-pKCi displays the most intense faradaic signature at DOPG-GCE, suggesting it may possess the most favorable partition coefficient between the DOPG film and the solution. The penetrating character of a peptide may in that case not be related to the magnitude of its association with the lipid. The electrochemical properties of the Fc-labelled peptides used in this work indifferently correspond to species both partitioning between the solution and the BLM and diffusing across the lipid film, preventing a discrimination between penetrating and non-penetrating peptides.

Concluding remarks

The behavior of small redox probes at DPhPC modified glassy carbon electrodes features a clear discrimination between ionic forms and neutral ones. Only neutral probes are allowed to penetrate the gel ordered bilayer of DPhPC, whereas charged probes are expelled from the lipid film. With anionic DOPG all probes display clear electrochemical responses in cyclic voltammetry. The cationic Ru(NH3)6 3+

probe undergoes a strong electrostatic interaction with the anionic lipids which leads to the swelling and destructuration of the DOPG film. Apart from this specific electrostatic association, the DOPG film seems to allow access to the electrode surface for all probes, a property to be compared to the liquid disordered state of DOPG at ambient temperature. The CVs at DOPG-GCE are significantly different from the one observed at DPhPC and DMPC [35] modified electrodes, which display unambiguous behavior towards the probes : neutral species accumulate in the lipid bilayer and exhibit symmetrical "thin layer" CVs while charged ions are blocked or expelled from gel phase phosphatidylcholine (PC) lipid films. The anionic and fluid phase DOPG shows more complex CVs with ferrocene-labelled peptides exhibiting a diffusional shape and an instability upon multiple cycling. The shape and magnitude of the peak currents at DOPG-GCE may therefore be linked to the weak and slow partition equilibrium of the peptides between the solution and the DOPG film.

FcDOPG = 1 , 2 -

 12 = Fc-RFARKGALRQKNV-CONH 2 dioleoyl-sn-glycero-3-phospho-(1'-rac-glycerol)

Figure 2

 2 Figure 2 CVs of K3Fe(CN)6 5 mM in solution "S" at bare GCE (solid line) and BLM-modified GCE (dashed line) at 40 mV.s -1 . A: BLM = pure DPhPC ; B: BLM = pure DOPG ; C: BLM = DPhPC:DOPG 70:30 mass ratio. Scan rate: 0.1 V.s -1 for all CVs

  vs. Ag/AgCl) E (V vs. Ag/AgCl)

Figure 3

 3 Figure 3 Nyquist plots of bare GCE (A), and DPhPC (B) and DOPG (C) modified GCE in aqueous 0.1 M NaCl. Applied potential 0.25 V vs. Ag/AgCl (perturbation 10 mVrms)

Figure 4 A

 4 Figure 4

  . Two very different voltammograms are obtained on pure DPhPC-GCE and pure DOPG-GCE. The oxidation of Fc-MeOH on DPhPC-GCE (Fig 4C) displays a mixed diffusional-adsorbed shape peak during the forward scan and a much smaller reduction peak during the backward scan. The neutral Fc-MeOH probe oxidizes to a cationic ferricinium which is expelled from the neutral DPhPC film, resulting in an asymmetric CV. Thus, only a small fraction of Fc + -MeOH electrogenerated during the forward scan remains in the vicinity of the GCE surface to be reduced during the subsequent cathodic scan. At DOPG-GCE, Fc-MeOH exhibits symmetrical CVs, with peak currents increasing with the equilibrium time (Fig 4E). This observation suggests that i) Fc-MeOH progressively accumulates in the DOPG film and ii) Fc + -MeOH produced during the forward scan is not expelled from the lipid film and reduces along a mostly diffusional process. At mixed DPhPC:DOPG 50:50 films (Fig. 4D), the initial voltammograms of Fc-MeOH essentially resemble those obtained at pure DOPG-GCE. Increasing the equilibrium time results in a progressive evolution of the CVs towards pure "DPhPC-like" CVs, with asymmetric shapes. The voltammograms in Fig. 4D indicate that Fc-MeOH initially associates to DOPG but the lipid film most likely undergoes slow demixing and the final CVs feature the oxidation of Fc-MeOH at DPhPC-enriched phases within the lipid film.

Figure 6 Multi-

 6 Figure 6

Acknowledgements

The authors would like to thank Pr. Solange Lavielle and Dr. Gérard Chassaing for their constant support and encouragement along this work. We also wish to acknowledge the CNRS, Sorbonne-Université, École Normale Supérieure-PSL and French Agence Nationale de la Recherche (grant number: ANR-12-BS08-0002-01 "ELIPTIC") for their support.

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