Introduction

Among the many beautiful connections between statistical physics and combinatorics, those involving integrable lattice models are probably the deepest ones. A major advance on the enumeration of various symmetry classes of alternating sign matrices (ASM) was accomplished by Kuperberg [START_REF] Kuperberg | Symmetry classes of alternating sign matrices under one roof[END_REF] by relating these notoriously difficult combinatorial problems to partition functions of the six-vertex model on a square grid with various boundary conditions. Alternatively, Okada [START_REF] Okada | Enumeration of symmetry classes of alternating sign matrices and characters of classical groups[END_REF] reinterpreted these partition functions as characters of various classical Lie groups.

In this note, we address a related combinatorial mystery, the Razumov-Stroganov conjecture [START_REF] Razumov | Combinatorial nature of ground state vector of O(1) loop model[END_REF], that identifies the suitably normalized groundstate vector entries of the O(1) loop model on a semi-infinite cylinder of perimeter 2n to numbers of configurations of the fully-packed loop model on an n×n square grid. A weaker version of this conjecture, the Razumov-Stroganov sum rule [START_REF] Batchelor | The quantum symmetric XXZ chain at ∆ = -1/2, alternating sign matrices and plane partitions[END_REF] identifies the sum of these entries with the number A n of ASMs of size n × n, and has been proved in [START_REF] Francesco | Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule[END_REF], in an inhomogeneous version (incorporating position-dependent spectral parameters z i ) making extensive use of the integrability of the loop model. It allows for the identification between the sum of entries of the fully inhomogeneous O(1) loop model and the very same partition function as that used by Kuperberg for the enumeration of ASMs, the so-called Izergin-Korepin (IK) determinant [START_REF] Izergin | Partition function of the six-vertex model in a finite volume[END_REF]. This proof was adapted to the case of the inhomogeneous O(1) crossing loop model as well [START_REF] Francesco | Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties[END_REF], and allowed to identify the entries of the groundstate vector with the multidegrees of some matrix varieties [START_REF] Knutson | A scheme related to the Brauer loop model[END_REF]. The case of open boundaries for the loop models was also treated in [START_REF] Francesco | Inhomogeneous loop models with open boundaries[END_REF] both for non-crossing and crossing loop models respectively, giving rise to other sum rules. In particular, this established the open Razumov-Stroganov sum rule, identifying the sum of suitably normalized groundstate vector entries of the O [START_REF] Kuperberg | Symmetry classes of alternating sign matrices under one roof[END_REF] loop model on a semi-infinite strip of width 2n with the number A V (2n + 1) of vertically symmetric alternating sign matrices (VSASM) of size (2n + 1) × (2n + 1).

In the context of q-deformations of quantum hall effect wave functions [START_REF] Pasquier | Quantum incompressibility and Razumov Stroganov type conjectures[END_REF], Pasquier recovered the Razumov-Stroganov sum rule (for q = -e iπ 3 ) by finding the minimal polynomial solution of the quantum Knizhnik-Zamolodchikov (qKZ) equation [START_REF] Frenkel | Quantum affine Algebras and Holonomic Difference Equations[END_REF] for U q (sl(2)) at level 1. In [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], solving qKZ equations for higher rank algebras U q (sl(k)) at level 1 led to a nice generalization of the Razumov-Stroganov sum rule for q = -e iπ k+1 , identifying a weighted sum of entries of the groundstate vector of the inhomogeneous A k-1 interaction-round-a-face (IRF) model on a semi-infinite cylinder of perimeter kn with particular Schur functions generalizing the IK determinant and Okada's result. In particular, DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ in the homogeneous limit where all z i → 1, the sum rule was found to be proportional to the number:

A (k) n = n-1 j=0 j! k-1 i=1 (k + 1)j + i ! k-1 i=0 (kj + i)! (1.1)
reducing to the number of ASMs for k = 2.

In the present paper, we extend the results of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF] to the case of A k-1 IRF models with open boundaries. To this purpose, we introduce in Sect.2 a boundary version of the level 1 qKZ equation which incorporates a parameter r, playing the role of a boundary magnetic field, and such that the ordinary qKZ equation is recovered when r → 0. This extends the boundary qKZ equation of [START_REF] Jimbo | Difference equations in spin chains with a boundary[END_REF] and [START_REF] Kojima | Difference equations for the higher rank XXZ model with a boundary[END_REF], defined respectively for U q (sl(2)) and

U q (sl(k > 2)
), both at level 0. We then use the path representation of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF] for the quotients of the Hecke algebra associated to the A k-1 models to find minimal polynomial solutions to these boundary qKZ equations in Sect.3. At the generalized Razumov-Stroganov point

q = -e iπ k+1
, we compute a weighted sum of entries of this polynomial solution, identified with the groundstate entries of the corresponding inhomogeneous A k-1 IRF model on a semi-infinite strip of width kn, with open boundaries (Sect.4.1). This sum rule produces a simple generalization of the Schur functions of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], in a form very similar to the character formulas of Okada for symplectic groups. In the homogeneous limit where all z i → 1, all these polynomials collapse to polynomials of r with integer coefficients, and lead at r = 1 to a natural generalization of the number of VSASMs:

A (k) V (n) = n-1 j=1 (2j)!(2nk + 2j -1)! ((k + 1)j)!((k + 1)(j + n) -1)! (1.2)
which reduces to A V (2n + 1) for k = 2.

We also investigate the other "combinatorial point" q = -1, corresponding to the rational limit of the underlying integrable model (Sect.4.2). There, in the homogeneous limit, we find that all entries of our solution tend to integers, and discuss their possible interpretation as degrees of nilpotent matrix varieties.

2. Quantum Knizhnik-Zamolodchikov equations without and with a boundary

R matrix and path representation

Throughout this note, we shall use the standard abstract trigonometric solution of the Yang-Baxter equation which reads with the parametrization

Ři,i+1 (z, w) = q -1 z -qw q -1 w -qz I + z -w q -1 w -qz e i (2.
τ = -(q + q -1 ) (2.3) A standard pictorial representation for Ř reads Ři,i+1 (z, w) = z w i+1 i+1 i i (2.4)
where we have also indicated the action on the spaces i and i + 1.

More precisely, we let Ř act on the so-called path representation of the A k-1 IRF models. To this purpose, the e i are further constrained to generate the U q (sl(k)) (also

called A k-1 or SU (k)) quotient H (k) N (τ ) of H N (τ )
obtained by imposing the extra vanishing conditions Y k (e i , e i+1 , . . . , e i+k-1 ) = 0, i = 1, 2, ..., N -k, where the Young q-symmetrizers Y k are defined inductively by Y 1 (e i ) = e i and Y m+1 (e i , . . . , e i+m ) = Y m (e i , . . . , e i+m-1 )(e i+m -µ m )Y m (e i , . . . , e i+m-1 ), where µ m = U m-1 (τ )/U m (τ ), U the Chebyshev polynomials subject to U m (2 cos x) = sin(m + 1)x/ sin x. Note that k = 2 corresponds to the Temperley-Lieb algebra T L N (τ ). The path representation is only briefly sketched here and will be described in detail in [15]. The representation basis is indexed by closed paths of length N = nk, from and to the origin, on the oriented-link Weyl chamber of SU (k), generated by the k vectors

u 1 = ω 1 , u 2 = ω 2 -ω 1 , ..., u k-1 = ω k-1 -ω k-2 , and
u k = -ω k-1
, in terms of the fundamental weights ω i . The paths are allowed to visit only points λ = λ i ω i with all λ i ≥ 0, and may be indexed by their sequence of steps, and by a slight abuse of notation we write π = (i 1 i 2 . . . i N ) for the sequence u i 1 , u i 2 , . . . , u i N ;

we also note π m = i m , the m-th step of the path π. For instance, the path of length nk closest to the origin is (12 . . . k) n , namely n repetitions of the sequence 1, 2, . . . , k, and we denote it by π f . Likewise, the path farthest from the origin is (1

) n (2) n . . . (k) n ,
and we denote it by π 0 . A useful notation consists in representing each step j by a unit segment forming an angle of π(k+2-2j)

2(k+1)
with the horizontal direction: each π becomes a broken line touching the x axis at its ends and staying above it. There are exactly

c (k) nk = (kn)! 0≤j≤k-1 j!/(n + j)! such paths. The number c (k)
nk is the dimension of the path representation and generalizes the Catalan numbers c n = c http://www-spht.cea.fr/articles/T05/135/ in decomposing each path in the broken line representation into "tiles" made of (possibly glued) lozenges L i,j with edges corresponding to pairs 1 ≤ i < j ≤ k of steps. To each tile we then associate a specific polynomial of the e's. For instance, each single lozenge with steps at positions i, i + 1 along the paths corresponds to e i . The tile decomposition of each path is then unique up to elementary moves between tiles allowed by the algebra relations, and we may therefore associate a unique element of H (k) N to each path π, thus forming a representation basis |π . Actually the map is onto the left ideal

H (k)
N Ω, Ω = Y k-1 (e 1 , ..., e k-1 )Y k-1 (e k+1 , ..., e 2k-1 )...Y k-1 (e (n-1)k+1 , ..., e nk-1 ), isomorphic to H (k) N . For illustration, let us detail the case k = 3, n = 2. There are five paths, respectively decomposed as follows: defining Y i,i+1 ≡ Y 2 (e i , e i+1 ) = e i e i+1 e i -e i , Z i,j = e i e j -1, and Ω = Y 1,2 Y 4,5 , we have

|π f = = Ω |π 3 = = e 3 Ω |π 2 = = Z 2,3 Ω |π 1 = = Z 4,3 Ω |π 0 = = e 4 Z 2,3 Ω (2.5)
and the e i read in this basis:

e 1 =      τ 0 0 0 1 0 τ 1 0 0 0 0 0 0 0 0 0 0 τ 1 0 0 0 0 0      e 2 =      τ 1 0 0 0 0 0 0 0 0 0 1 τ 0 0 0 0 0 0 0 0 0 0 1 τ      e 3 =      0 0 0 0 0 1 τ 0 0 0 0 0 τ 0 1 0 0 0 τ 1 0 0 0 0 0      e 4 =      τ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 τ 0 0 0 1 0 τ      e 5 =      τ 0 0 0 1 0 τ 0 1 0 0 0 τ 0 1 0 0 0 0 0 0 0 0 0 0      (2.6)
Let us now simply list a few basic properties of this representation, of crucial importance for the following:

(P1) e i |π = τ |π if π i < π i+1 (π
π i < π i+1 (P4) If C i,π,π = 1 then either (π i , π i+1 ) = (π i+1 , π i ) and π m = π m for all m = i, i + 1, i.e.
π exceeds π by the unit lozenge L π i+1 ,π i in the broken line representation, or π ⊂ π, namely the broken line representation of π lies below that of π (or equivalently π may be completed into π upon adding a number of lozenges).

We conclude with another important property of this representation. There is a duality between the SU (k) and SU (n) representations for N = kn, confirmed by the identity of dimensions c

(k) nk = c (n)
kn . Indeed, there exists an involution ϕ from A k-1 to A n-1 paths of same length kn, defined as follows. First of all, ϕ takes π

0 of A k-1 to π f of A n-1 .
The bijection is then constructed iteratively by simultaneously subtracting (resp. adding) lozenges at the same positions from the A k-1 (resp. to the A n-1 ) paths, so that π is locally convex iff ϕ(π) is locally concave. A simpler way of picturing this involution uses yet another expression of A k-1 paths, as rectangular standard Young tableaux with k row of n boxes, in which successive steps along the path are recorded by their positions in the corresponding row of the tableau, i.e. if the m-th step is j, then we write the number m in the first available box from the left in the j-th row. In this language, ϕ is simply the transposition of the tableaux. Moreover, this duality extends to the path representations, namely

C (n) i,ϕ(π ),ϕ(π) = C (k) i,π,π (2.7)
In practice, this relation is extremely useful: it allows for instance to express the representation (2.5)-(2.6) of H Each intersection between two such lines corresponds to a Ř operator. On the second line, we have represented the diagonal operators of respectively multiplication of the spectral parameter by s (shift), inversion of the spectral parameter with weight r (K (r) (z)) and weight rs (K (rs) (z)), repectively pictured as a crossed dot, empty dot, filled dot. The former is nothing but the composition of the two latter.

qKZ equations

The qKZ equation [START_REF] Frenkel | Quantum affine Algebras and Holonomic Difference Equations[END_REF] reads

Ψ(z 1 , ..., z j-1 , sz j , z j+1 , ..., z N ) = QΨ(z 1 , ..., z j-1 , z j , z j+1 , ..., z N ) Q = Řj+1,j (z j+1 , sz j ) . . . ŘN,j (z N , sz j )S j (z)σ Ř1,j (z 1 , z j ) . . . Řj-1,j (z j-1 , z j ) (2.8)
with the clear pictorial representation of Fig. 1-(a). Here the operator σ rotates cyclically the spectral parameters and the "shift" operator S j (z) acts as the identity on the space j while it multiplies the spectral parameter z by some scalar s (it is represented by a crossed dot in Fig. 1-(a)). In the pictorial representation of Fig. 1-(a), each intersection between two oriented lines stands for an Ř operator (2.4). Note that the "return" z j line (from left to right) passes below the other (vertical) lines, hence there are no intersections, i.e. no Ř matrices involved: the only effect is the operator σ, which rotates the spaces and their spectral parameters according to (z j , z 1 , ..., z j-1 , z j+1 , ..., z N ) → (z 1 , ..., z j-1 , z j+1 , ..., z N , z j ).

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In the presence of boundaries, the qKZ equation is modified as depicted in Fig. 1-(b).

The "return line" now crosses the vertical ones, and the shift of spectral parameter is now replaced by two boundary operators at the far left and right of the return line. The corresponding qKZ equation now reads

Ψ(z 1 , ..., z j-1 , sz j , z j+1 , ..., z N ) = QΨ(z 1 , ..., z j-1 , z j , z j+1 , ..., z N ) Q = Řj+1,j (z j+1 , sz j ) . . . ŘN,j (z N , sz j )K (rs) j ( r z j ) Řj,N ( r z j , z N ) . . . Řj,j+1 ( r z j , z j+1 ) × Řj,j-1 ( r z j , z j-1 ) . . . Řj,1 ( r z j , z 1 )K (r) j (z j ) Ř1,j (z 1 , z j ) . . . Řj-1,j (z j-1 , z j ) (2.9)
The boundary operator K (r) j (z) acts as the identity on the space j, and changes the spectral parameter z into r/z. Together with Ř, it satisfies the boundary Yang-Baxter equation of Ref. [START_REF] Sklyanin | Boundary conditions for integrable quantum systems[END_REF]. In Fig. 1b, we have represented respectively K (r) and K (rs) by an empty and a filled dot. Here r is an additional free parameter, which allows to recover in the limit r → 0 the qKZ equation without boundary (2.8) from that with boundaries (2.9). Indeed, in this limit, the spectral parameter along the return line tends to 0: consequently, all Ř matrices along the return line degenerate into pure undercrossings, and we recover the picture of Fig. 1-(a), upon noting that S(z) = K (rs) (r/z)K (r) (z) is nothing but the shift operator that multiplies the spectral parameter z by s.

The equation (2.8) is equivalently standardly reduced to a system of equations for Ψ, namely

τ i Ψ(z 1 , . . . , z N ) = Ři,i+1 (z i+1 , z i )Ψ(z 1 , . . . , z N ), i = 1, 2, . . . , N -1 σΨ(z 2 , ..., z N , sz 1 ) = cΨ(z 1 , ..., z N ) (2.10)
where τ i acts on functions of the z's by interchanging z i and z i+1 , c is an irrelevant constant, and σ now acts as a rotation of states in the representation picked for the Ř's. Similarly, we may reduce (2.9) to the following system:

τ i Ψ(z 1 , . . . , z N ) = Ři,i+1 (z i+1 , z i )Ψ(z 1 , . . . , z N ), i = 1, 2, . . . , N -1 (2.11a) Ψ( r z 1 , z 2 , ..., z N -1 , z N ) = c 1 (z 1 )Ψ(z 1 , ..., z N ) (2.11b) Ψ(z 1 , z 2 , ..., z N -1 , rs z N ) = c N (z N )Ψ(z 1 , ..., z N ) (2.11c)
for some functions of a single spectral parameter c 1 and c N . Note that Eqs.(2.11b, c) express the covariance of Ψ under the action of K

(q -1 z i+1 -qz i )∂ i Ψ π (z 1 , . . . , z N ) = π C i,π ,π Ψ π (z 1 , . . . , z N ), 1 ≤ i ≤ N -1 (2.12a) Ψ π ( r z 1 , z 2 , ..., z N -1 , z N ) = c 1 (z 1 )Ψ π (z 1 , ..., z N ) (2.12b) Ψ π (z 1 , z 2 , ..., z N -1 , rs z N ) = c N (z N )Ψ π (z 1 , ..., z N ) (2.12c)
where the divided difference operator ∂ i acts on functions of the z's as

∂ i f = τ i f -f z i -z i+1 (2.13)
We also denote by t i = (q -1 z i+1 -qz i )∂ i . The equations (2.12a) express nothing but

t i Ψ = (e i -τ )Ψ.
By analogy with the ordinary qKZ case, when dealing with the A k-1 path representation above, we write

s = q 2(k+l) (2.14)
and call l the level of the (boundary) qKZ equation (2.11). The parameter r is free, and has the dimension of the square of a spectral parameter. In the next section, we construct polynomial solutions of the system (2.11) for level l = 1.

Polynomial solutions for the level 1 boundary qKZ equation

We now look for polynomial solutions of level 1 boundary qKZ equations (2.11) with minimal degree.

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Solutions

We have the remarkable property for all k, that the equations (2.12a) may all be solved for Ψ π in terms solely of Ψ π 0 . This relies on the property (P4) above of the path representation. Indeed, we may order the paths by inclusion. Then each path π locally convex at i, i + 1 may be viewed as the addition of a lozenge L π i ,π i+1 to a smaller path π -, such that π - m = π m for m = i, i + 1 and (π - i , π - i+1 ) = (π i+1 , π i ). Using property (P4), we see that e i π -= π + π , where the sum runs over paths π ⊂ π -. Turning now to Eq.(2.12a), we see that, as all the π involved in the sum on the r.h.s. are such that C i,π ,π = 1, only one of them is strictly included in π, namely π -, while all others contain π. Stripping iteratively lozenges from paths, we may therefore express each Ψ π - as t i Ψ ππ Ψ π , with already known π containing π. This allows for expressing iteratively every Ψ π in terms of Ψ π 0 , in a triangular manner w.r.t. inclusion of paths.

The "fundamental" component Ψ π 0 is fixed by (i) some highest weight conditions (t i + τ )Ψ π 0 = 0 for all i not multiple of n (expressing that π 0 is locally flat except at the convex points multiple of n, see below), (ii) the reflective boundary conditions (2.12b, c), and finally the requirement that it be of smallest possible degree.

As a direct consequence of Eq.(2.12a), Ψ π vanishes whenever z i+1 = q 2 z i and π is concave or flat at i, i + 1, namely π i ≥ π i+1 . Indeed, from property (P3), the vector |π cannot be the image of anything under e i ; but precisely when z i+1 = q 2 z i , the matrix Ř becomes proportional to e i (c.f. the definition (2.1)), hence Ψ π must vanish. This is straightforwardly extended to any portion of π flat or concave say between positions i and j, upon considering the appropriate product or Ř matrices: if z l = q 2 z k for some i ≤ k < l ≤ j, then Ψ π vanishes. Applying this to π 0 which is flat everywhere except at points multiple of n where it is convex, we find that Ψ π 0 must factor out the terms (m-1)n+1≤i<j≤mn (q 2 z i -z j ) for all m = 1, 2, ..., k. Alternatively, wherever π 0 is locally flat, i.e. for all i not multiple of n, we have (t i + τ )Ψ π 0 = 0, as |π 0 has no antecedent under e i : this "highest weight condition" is analogous to that of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF]. It implies that once the above factors are taken out of Ψ, the remaining is a polynomial symmetric in the variables {z (m-1)n+1 , z (m-1)n+2 , ..., z mn } separately for all m = 1, 2, ..., k. Finally, by an inductive argument similar to that used in Ref. [START_REF] Francesco | Inhomogeneous loop models with open boundaries[END_REF], we can show that Ψ π 0 factors out terms (q 2m r -z k z l ) for all pairs k, l such that (m -1)n + 1 ≤ k ≤ l ≤ mn, m = 1, 2, ..., k. Indeed, the left boundary condition (2.12b) allows to show that 2≤j≤n (q 2 r -z 1 z j ) divides Ψ π 0 .

Next, we let z 1 → ∞, namely pick in Ψ the coefficient of top degree in z 1 : in this case the DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ space 1 decouples from the picture of Fig. 1(a), as its crossings with the j line are undone and replaced by undercrossings, and we are left with the same boundary problem, but with line 1 erased, and now a reflective boundary condition with parameter r in terms of z 2 .

Repeating iteratively this process, namely taking successively z 2 , z 3 , ..., z n-1 → ∞ exhausts all the announced extra factors for m = 1. Assuming each variable occurs in Ψ with maximal degree 2(n -1), we find that the leftover dependence on z n must be proportional to z n-1 n , and we may write Ψ π 0 (∞, ..., ∞, z n , z n+1 , ..., z N ) = Cz n-1 n F (z n+1 , ..., z N ). The full vector Ψ(∞, ..., ∞, z n , z n+1 , ..., z N ) now obeys a left reflective boundary condition of the form

Ψ(∞, ..., ∞, r z n , z n+1 , ..., z N ) = c 1 (z n )Ψ(∞, ..., ∞, z n , z n+1 , ..., z N ) (3.1)
We now note that π 0 = e n π -, for the unique π -such that π - i = π i for i = n, n + 1, and (π - n , π - n+1 ) = (π n+1 , π n ). According to (2.12a), we have t n Ψ π 0 = Ψ π -. Expressing that the component Ψ π -obeys the new left reflection condition (3.1) solely in terms of Ψ π 0 yields the following relation for F :

c 1 (z n ) q -1 z n+1 -qz n z n -z n+1 (z n+1 ) n-1 F (z n , ...) -(z n ) n-1 F (z n+1 , ...) = q -1 z n z n+1 -qr qr -z n z n+1 (z n+1 ) n-1 F ( r z n , ...) - r z n n-1 F (z n+1 , ...) (3.2)
Picking z n = q -2 z n+1 finally yields F ( q 2 r z n+1 , ...) = (q 2 r/z 2 n+1 ) n-1 F (z n+1 , ...) and therefore Ψ π 0 now has all the announced extra factors for m = 2 (everything works now as if the left boundary parameter had been rescaled as r → rq 2 ). Proceeding iteratively, we finally get all the announced factors for m = 1, 2, ..., k. We claim that the solution of minimal degree has no extra factors in its Ψ π 0 , namely that

Ψ π 0 = k m=1 (m-1)n+1≤i<j≤mn (q 2 z i -z j )(rq 2m -z i z j ) (3.3)
and that Eqs.(2.12) are satisfied for s = q 2(k+1) and

c 1 (z) = r z 2 n-1 , c N (z) = rq 2(k+1) z 2 n-1 (3.4)
This fixes the solution entirely.

DSM/SPhT-T05/135

http://www-spht.cea.fr/articles/T05/135/ As in the case without boundaries of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], the solution may alternatively be fixed by some global requirement that it vanishes for some specific choices of the spectral parameters. It was found indeed that in the case without boundary Ψ must vanish for any ordered sequence of k + 1 spectral parameters of the form {z, q 2 z, ..., q 2k z}, the so-called "quantum incompressibility" condition, first noticed by Pasquier [START_REF] Pasquier | Quantum incompressibility and Razumov Stroganov type conjectures[END_REF] in the case k = 2, and allowing for interpreting Ψ as q-deformed wave function of fractional quantum Hall effect. In the present case with boundaries, we find that Ψ must vanish for any ordered k + 1-tuple of spectral parameters of the form {z, q 2 z, ..., q 2(m-1) z, rq 2 z , rq 4 z , ...,

rq 2(k-m) z } (3.5)
for m = 2, 3, ..., k + 1. This generalizes the quantum incompressibility condition, in that the wave function must vanish whenever k +1 electrons come into contact with one another or now with their reflections.

As in the case of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], we may also derive recursion relations for the components, relating size N = kn to size N = k(n -1). This is done by simply taking z 1 = z, z 2 = q 2 z, ..., z k = q 2(k-1) z, in which case only paths with π i = i for i = 1, 2, ..., k have nonvanishing components in Ψ. The rest of the path, π = (π k+1 , ..., π N ) is an A k-1 path of length N = N -k, and we have Ψ π (z 1 , ..., z N )

z 1 =q -2 z 2 =...=q -2(k-1) z k =z =   N j=k+1 (q 2k z -z j )(rq 2 -zz j )   Ψ π (z k+1 , ..., z N ) (3.6)
where the prefactor is fixed by the degree and vanishing requirements.

Other sizes

As already touched-upon in the previous section, there is an easy way of extending our polynomial solutions to all sizes N , not necessarily a multiple of k. We proceed as in [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF]. We introduce a covector v such that

e i v = τ v, i = 1, 2, ..., N -1 (4.1)
This covector is unique under the condition that all its entries be strictly positive and that v π f = 1, and exists only if q = -e iπ k+1 or its conjugate, namely τ = 2 cos π k+1 . The entries of v are directly constructed from the path representation and may be expressed as products of ratios of the Chebyshev polynomials U m (τ ) with m < k, associated to the tiles in the decomposition of the paths, and all of which are positive for the above choice of τ . For illustration, for k = 3, n = 2, we take τ = √ 2, and associate the factor

U 1 (τ ) = √ 2
to every lozenge and the factor U 2 (τ ) = 1 to every tile made of two lozenges in the path decomposition (2.5), corresponding respectively to the left eigenvalues of v on e i and on

Z i,j , resulting in v = {1, √ 2, 1, 1, √ 2} in the path basis.
The condition (4.1) immediately implies that v Ři,i+1 (z, w) = v, readily checked from the definition (2.1). Henceforth, the combination v • Ψ is a symmetric polynomial of the z's, as a direct consequence of Eq.(2.11a). Introducing

I (k) kn (z 1 , z 2 , ..., z kn |r) ≡ (-i/q) kn(n-1)/2 v • Ψ(z 1 , ..., z kn ) (4.2) with i = √ -1, we find that I (k)
kn is a polynomial of degree kn(n-1)/2 of r, whose coefficients are all symmetric polynomials of the z's, and the top degree and constant coefficients are respectively

I (k) kn r kn(n-1)/2 = s Y (z 1 , ..., z kn ) I (k) kn r 0 = (z 1 z 2 ...z kn ) n-1 s Y (z 1 , ..., z kn ) (4.3)
where s Y is the Schur function associated to the Young diagram Y with k rows of n -1 boxes, k rows of n -2 boxes, ..., k rows of 1 box. The second line of (4.3) is nothing but the generalized Razumov-Stroganov sum rule of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], recovered in the limit r → 0 as explained above. The polynomial

I (k)
kn (z 1 , ..., z kn |r) has only non-negative integer coefficients in r and the z's. Note also the covariance under z i → r/z i for each i, inherited from the boundary reflections (2.11b, c) and the overall symmetry, namely

I (k) kn (z 1 , ..., z j-1 , r z j , z j+1 , ..., z kn |r) = r z 2 j n-1 I (k)
kn (z 1 , ..., z j-1 , z j , z j+1 , ..., z kn |r) (4.4)

DSM/SPhT-T05/135

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I (k)
kn for different powers of r. In particular, taking simultaneously all z i → r/z i relates the top and constant coefficients, and the first line of (4.3) follows from the second. The polynomial I (k) kn (4.2) is entirely determined up to global normalization by the following properties:

(i) it is a symmetric polynomial of the z's quasi-homogeneous under z i → λz i for all i and r → λ 2 r, with total degree 3kn(n -1)/2 and partial degree 2(n -1) in each of the z's and kn(n -1)/2 in r (ii) it vanishes for all the choices of ordered k-tuples of spectral parameters (3.5), with

q = -e iπ k+1
and the normalization is fixed by (4.3).

Let us first discuss the Temperley-Lieb case k = 2 and n arbitrary, where drastic simplifications occur. At the Razumov-Stroganov point q = -e iπ 3 , τ = 1, we have v π = 1 for all π, hence I

2n is simply the sum of components of Ψ. The quantity I

(2) 2n reduces to the result of [START_REF] Francesco | Inhomogeneous loop models with open boundaries[END_REF] at r = 1. For generic r, the determinantal formulas of [START_REF] Francesco | Inhomogeneous loop models with open boundaries[END_REF] are straightforwardly generalized to

I (2) 2n (z 1 , ..., z 2n |r) = n i,j=1 (z 2 i + z i z j+n + z 2 j+n )(r 2 + rz i z j+n + z 2 i z 2 j+n ) 1≤i<j≤n (z i -z j )(r -z i z j )(z i+n -z j+n )(r -z i+n z j+n ) × det 1≤i,j≤n 1 z 2 i + z i z j+n + z 2 j+n 1 r 2 + rz i z j+n + z 2 i z 2 j+n (4.5)
which is an open version of the Izergin-Korepin determinant [START_REF] Izergin | Partition function of the six-vertex model in a finite volume[END_REF], and the Pfaffian formula becomes

I (2) 2n (z 1 , ..., z 2n |r) 2 = n 1≤i<j≤2n (z 2 i + z i z j + z 2 j )(r 2 + rz i z j + z 2 i z 2 j ) (z i -z j )(r -z i z j ) × Pf 1≤i<j≤2n (z i -z j )(r -z i z j ) (z 2 i + z i z j + z 2 j )(r 2 + rz i z j + z 2 i z 2 j ) (4.6)
These are proved for instance by using the symmetry and the recursion relations obeyed by v • Ψ as a consequence of Eq.(3.6), and showing that the r.h.s. is the unique symmetric polynomial obeying the same constraints. In the homogeneous case where all z i = 1, I

2n reduces to a reciprocal polynomial of r with non-negative integer coefficients, equal respectively to 3 n(n-1)/2 times the number A n of alternating sign matrices (ASM) of size n × n at r = 0 (a result proved in [START_REF] Francesco | Around the Razumov-Stroganov conjecture: proof of a multi-parameter sum rule[END_REF]), and to 3 n(n-1) times the number A V (2n + 1) of DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ vertically symmetric alternating sign matrices (VSASM) of size (2n + 1) × (2n + 1) at r = 1 (a result proved in [START_REF] Francesco | Inhomogeneous loop models with open boundaries[END_REF]), with respective values

A n = n-1 j=0 (3j + 1)!j! (2j + 1)!(2j)! , and A V (2n + 1) = n-1 j=0 (6j + 4)!(2j + 2)! (4j + 4)!(4j + 2)! (4.7)
For general k, we have the following expression for the sum rule I ) is proved as usual by checking that the r.h.s. fulfills all requirements of degree and vanishings above, and is therefore fixed by uniqueness. This formula is a nice multiparameter generalization of that written by Okada [START_REF] Okada | Enumeration of symmetry classes of alternating sign matrices and characters of classical groups[END_REF] for VSASMs, in terms of specialized characters of the symplectic group.

In the particular case n = 2 and k arbitrary, It is easy to show that (4.8) reduces to

I (k) 2k (1, 1, ..., 1|r) = k m=0 r m k + 1 2k + 1 2k + 1 m 2k + 1 k -m (4.11)
and we get

A (k) 2 = 1 k + 1 I (k) 2k (1, 1, ..., 1|0) = c k (4.12)
the k-th Catalan number, while at r = 1 we obtain

A (k) V (2) = 1 (k + 1) 2 I (k) 2k (1, 1, ..., 1|1) = 2 (4k + 1)! (3k + 2)!(k + 1)! (4.13)
DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ expressed in the standard Dyck path basis of T L 6 (τ ), ordered by inclusion. The components of (4.18) sum respectively to 1, 7, 307, the degrees of the Brauer scheme for matrices of size 2, 4, 6 respectively. For completeness, the corresponding r = 0 solutions of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF] yield the following limits ψ (0) 2n when q → -1 of Ψ(z i = 1|r = 0)/(q 2 -1) n(n-1) :

ψ (0) 2 = {1}, ψ (0) 4 = {2, 1}, ψ (0) 6 = {10, 4, 4, 4, 1} (4.19)
The above observation is particularly intriguing, as we already know that another particular point (q = -e iπ 3 ) of the same solution Ψ also displays entries leading to non-negative integers for both r = 0 and r = 1, as mentioned in the previous section. These entries Ψ π were actually identified via the full Razumov-Stroganov conjecture [START_REF] Razumov | Combinatorial nature of ground state vector of O(1) loop model[END_REF] to fully-packed loop configurations on a n × n square grid (at r = 0) or to vertically symmetric fully-packed loop configurations on a (2n + 1) × (2n + 1) square grid (at r = 1) connected in both cases according to the same link pattern as that encoded in the Dyck path π.

The case k > 2 also seems to lead to non-negative integer entries in the rational homogeneous limit, provided we take r = 1 and all z i = 1, and then consider lim q→-1

1 (q 2 -1) kn(n-1) π Ψ π (1, 1, ..., 1) (4.20)
These are presumably related to the degrees of higher order generalizations of the Brauer loop scheme [15]. For illustration, we have for k = 3 and n = 2, in the basis (2.5):

ψ

6 = {60, 28, 13, 15, 6} (4.21) whereas the corresponding r = 0 limit (with (q 2 -1) kn(n-1) replaced by (q 2 -1) kn(n-1)/2 in the limit of (4.20)) yields in the basis (2.5):

ψ (0) 6 = {6, 3, 2, 2, 1} (4.22)
As mentioned in Sect.3.2, we have access to analogous integer Ψ's and sum rules for arbitrary sizes N not necessarily a multiple of k, presumably relating to generalized Brauer schemes of matrices of corresponding sizes.

DSM/SPhT-T05/135

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Conclusion

In this note, we have constructed the minimal polynomial solutions to the level 1 U q (sl(k)) boundary qKZ equation, and investigated its "combinatorial points". We have derived some In view of trying to cook up a full conjecture à la Razumov-Stroganov, we may try to consider the vector w with entries w π = (-i/q) kn(n-1)/2 v π Ψ π : unfortunately, these entries are not real for generic r. For illustration, for k = 3, n = 2, the corresponding entries read in the basis (2.5):

w f = 4(1+9r + 9r 2 + r 3 ), w 3 = 4(1 + 7r + 7r 2 + r 3 ), w 2 = 4(1 + (2 + i)r + (2 -i)r 2 + r 3 ), w 1 = 4(1 + (2 -i)r + (2 + i)r 2 + r 3 ), w 0 = 4(1 + r + r 2 + r 3 ) (5.1)
and sum to the abovementioned value of I

6 (1, 1, ..., 1|r). In this particular case however, the entries of w reduce to non-negative integers in both cases r = 0, 1, with w (0) = 4{1, 1, 1, 1, 1} and w (1) = 8{10, 8, 3, 3, 2}. In general, it is tempting to conjecture that w has integer entries, and we have indeed observed this fact for r = 0 and the cases n = 2, k = 3, 4, 5, and n = 3, k = 3. For r = 1 however we have found a counterexample for n = 2, k = 4, in which the 14 entries of w, although positive and summing to 4 2 A (4) V (2) = 16 × 68, are not all integers. So if a full Razumov-Stroganov conjecture exists it has to be more subtle in the r = 1 case.

Note that, as we have q 2(k+1) = 1 at the generalized Razumov-Stroganov point, the left and right boundary conditions in (2.11) coincide. When r = 0, we are precisely in the case where Ψ is the groundstate vector of the inhomogeneous A k-1 IRF model on a semiinfinite cylinder of perimeter N as pointed out in [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], and governed by the cyclic Hamiltonian H = 1≤i≤N (τ -e i ), e N = σe N -1 σ -1 an additional generator for the cyclic Hecke algebra. For r = 1, Ψ is the groundstate vector of the inhomogeneous A k-1 IRF model on a semi-infinite strip of width N , with reflecting boundary conditions, and governed by the Hamiltonian H = 1≤i≤N -1 (τ -e i ). In both cases, the entries (-i/q) kn(n-1)/2 Ψ π DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ and v π are manifestly positive, as these are respectively the entries of the right and left Perron-Frobenius eigenvectors of the corresponding Hamiltonians. Alternatively, we may view the r = 1 model as describing a stochastically "growing interface" like in [START_REF] Pearce | Temperley-Lieb Stochastic Processes[END_REF] for the Temperley-Lieb case k = 2, upon interpreting the action of Ři,i+1 (z i , z j ) (2.1) on a path as the identity with probability 1 -t i,j = (z j -q 2 z i )/(q 2 z j -z i ) and as e i /τ with probability t i,j , which (i) leaves the path unchanged (if π is convex at i, i + 1), (ii) adds a lozenge to it (if π is concave at i, i + 1) (iii) or shrinks it to a smaller path π such that C i,π,π = 1 (if π is flat or concave at i, i + 1), the last two operations weighted by 1/τ . The combination of entries P(π) = v π Ψ π /(v • Ψ) may then be interpreted as the invariant probability for the interface to be equal to the path π. These invariant probabilities read in the case k = 3, n = 2: { 5 13 , 4 13 , 3 26 , 3 26 , 1 13 } for the paths of (2.5). The above counterexample to integrality simply shows that these invariant probabilities will not be rational numbers in general.

It is still possible nevertheless that generalizations of the observables considered in [START_REF] Pearce | Temperley-Lieb Stochastic Processes[END_REF] could lead to simple rational expectation values. For instance, from our numerical data, we conjecture that the probability of convex transitions π i < π i+1 in the above interfaces reads C(2, k) = (k -1)(13k + 4) 2(2k -1)(4k + 1) (

for the A k-1 growth problem with n = 2. This is easily checked for k = 3, as the paths of We may also try to generalize the Pfaffian formula (4.6). The hope would be eventually to be able to define an A k-1 vertex model with a particular type of (domain-wall) boundary conditions on a finite domain, whose partition function exactly matches I As a side remark, the number of different objects enumerated by the ASM number A n is quite considerable: totally symmetric self-complementary plane partitions (TSSCPP), descending plane partitions (DPP), rhombus tilings of particular domains of the triangular DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ lattice, osculating walks on a square grid, ice configurations on a square with domain wall boundaries etc, and these give rise to many different formulas for A n , as determinants or Pfaffians. We may wonder which ones of these objects and/or formulas will be amenable to higher k generalizations, maybe all of them?

The other combinatorial point q = -1 (τ = 2) seems to be of a different nature. It corresponds to the rational limit of the above A k-1 IRF models, namely giving rise to a rational solution of the Yang-Baxter equation. Another rational solution of a similar nature was considered in [START_REF] Francesco | Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties[END_REF] and led to sum rules for the O(1) model of crossing loops. All these models seem to be related to the geometry of some specific matrix varieties, allowing in particular for interpreting in the homogeneous limit the various integer entries of Ψ at hand as degrees of the components of these varieties. The identification of these varieties for the open boundary cases is under way [15].

Fig. 1 :

 1 Fig.1:The regular-(a) and boundary-(b) qKZ equations. Each oriented line carries a spectral parameter. Each intersection between two such lines corresponds to a Ř operator. On the second line, we have represented the diagonal operators of respectively multiplication of the spectral parameter by s (shift), inversion of the spectral parameter with weight r (K (r) (z)) and weight rs (K (rs) (z)), repectively pictured as a crossed dot, empty dot, filled dot. The former is nothing but the composition of the two latter.

NN

  (z N ). The latter are DSM/SPhT-T05/135 http://www-spht.cea.fr/articles/T05/135/ obtained by showing that the operator Q in (2.9) is intertwined by K , as a direct consequence of the boundary Yang-Baxter equation. The boundary qKZ equation (2.11) has the advantage of being ridden of the cyclicity condition (second line of (2.10)), replaced by reflective boundary conditions, somewhat easier to deal with. We now restrict ourselves to vectors Ψ in the path representation above, namely look for Ψ = π Ψ π |π , π running over the set of A k-1 paths of length N = kn. When written in components, Eqs.(2.11) become

4. Sum rules 4 . 1 .

 41 Starting from a solution at N = kn, we may indeed send successively the first j spectral parameters to 0 (or equivalently to ∞), leaving us with only non-vanishing components of Ψ for paths π starting with the (unique) convex sequence 1, 2, 3, ..., j. This provides us with a solution to the boundary qKZ equation of z j+1 , ..., z N , hence in size N -j. Note that now in the corresponding representation the paths start at the point u 1 + u 2 + ... + u j in the Weyl chamber of SU (k) and end at its origin. This construction is valid for all j = 1, 2, ..., k -1 hence allows to exhaust all possible sizes.DSM/SPhT-T05/135http://www-spht.cea.fr/articles/T05/135/ Generalizations of the Razumov-Stroganov sum rule at q = -e iπ k+1

  nk (z 1 , ..., z nk |r) = sY (z 1 , ..., z nk |r)(4.8) where sY are "symplectic" Schur functions depending on the extra parameter r, and defined assY (z 1 , ..., z nk |r) = z 1 z 2 ...z kn det 1≤i,j≤kn z kn+1-j +j i -(r/z i ) kn+1-j +j det 1≤i,j≤kn z j i -(r/z i ) j (4.9)with Y as in (4.3), andi = n -1 -[(i -1)/k],is the number of boxes in the i-th row, i = 1, 2, ..., kn (we have added k bottom rows of 0 box for convenience). The sum rule (4.8

I

  1 , ..., z 2k |r) = k m=0 r m s Y m (z 1 , ..., z 2k ) (4.10) expressed in terms of ordinary Schur functions for the young diagrams Y m with k -m rows of 2 boxes and k rows of 1 box. Consequently, in the homogeneous limit where all z i → 1, we have

  higher rank generalizations of the Razumov-Stroganov sum rules for the open boundary O(1) spin chain. We have actually found a family of solutions depending on one parameter r playing the role of boundary magnetic field. The closed and open boundary cases of the Razumov-Stroganov sum rules are recovered for r = 0 and 1 respectively. Sum rules for the components of the solution à la Razumov-Stroganov are generically obtained at a unique point τ = 2 cos π k+1 for each k, where a covector v with the properties (4.1) and with positive entries can be constructed.

( 2 . 5 )

 25 have respectively {4, 3, 3, 3, 2} convex transitions among their 5 transitions, sum rules obtained in this paper produce nice integer sequences, which we expect to be counting generalizations of ASMs and of VSASMs. A first step in understanding what exactly is counted by these numbers could be to find some higher k > 2 analogues of the Izergin-Korepin determinant (or its open version (4.5)) for our sum rules I (k) kn (z 1 , ..., z kn |r).

  kn (z 1 , ..., z kn |r). Then the configurations of such a model would form the desired generalizations of ASMs or VSASMs. This might then open the route to generalizations of other symmetry classes of such objects, according to the symmetries of the above domain, in the spirit of[START_REF] Kuperberg | Symmetry classes of alternating sign matrices under one roof[END_REF].

  e j = e j e i , |i -j| > 1, e i e i+1 e i -e i = e i+1 e i e i+1 -e i+1 ,

	e 2 i = τ e i	(2.2)
	http://www-spht.cea.fr/articles/T05/135/
	DSM/SPhT-T05/135	1)

where the e i , i = 1, 2, ..., N -1 are the generators of the Hecke algebra H N (τ ), namely subject to the relations e i

  (P2) e i |π = π C i,π,π |π if π i ≥ π i+1 (π locally flat or concave), for some C i,π,π ∈ {0, 1} (P3) If C i,π,π = 1 then π is locally convex between steps i and i + 1, namely

	http://www-spht.cea.fr/articles/T05/135/
	DSM/SPhT-T05/135	locally convex)

Acknowledgments: We thank M. Bauer, P. Zinn-Justin and J.-B. Zuber for useful discussions. This work is partly supported by the Geocomp project (ACI Masse de Données) and the European network "ENIGMA", grant MRT-CT-2004-5652.

T05/135/

DSM/SPhT-T05/135

http://www-spht.cea.fr/articles/T05/135/

For arbitrary k, the homogeneous case of all z i = 1 yields again for

kn a reciprocal polynomial of r, with non-negative integer coefficients. The constant coefficient of this polynomial is the result of the homogeneous limit of the generalized Razumov-Stroganov sum rule computed in [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], and reads (k + 1) n(n-1)/2 times an integer

n , and which may be rewritten as

The numbers (4.14) are nothing but the properly normalized dimensions of the GL kn representations Y [START_REF] Fulton | Representation theory[END_REF]. At r = 1 the homogeneous value of I

kn reads (k + 1) n(n-1) times a number A (k) V (n) generalizing the number of (2n + 1) × (2n + 1) VSASMs A V (2n + 1). We find that

These numbers are the properly normalized dimensions of the SP 2kn representations Y [START_REF] Fulton | Representation theory[END_REF]. The numbers A (k)

V (n) are easily checked to reduce to A V (2n + 1) for k = 2 and all n, and to (4.13) for n = 2, and all k. The numbers

For illustration, for k = 3, n = 2 we have I

As mentioned in Sect.3.2, we have access to analogous sum rules for arbitrary sizes N not necessarily a multiple of k by sending spectral parameters to 0 and reducing the space of path states accordingly. In size nk -j, the sum rule simply reads (4.8), with say z 1 = z 2 = ... = z j = 0 and where the first j rows have been deleted from Y . This leads to more integer numbers in the homogeneous limit.

DSM/SPhT-T05/135

http://www-spht.cea.fr/articles/T05/135/ 4.2. More combinatorics at q = -1

In [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], a connection was established between the "rational limit" q → -1, z i → 1 of the U q (sl(k)) level 1 minimal polynomial solution of the qKZ equation without boundary and so-called extended Joseph polynomials, related to the geometry of the variety of upper triangular matrices with vanishing k-th power. As a particular corollary of this relation, the entries of Ψ were found to tend to positive integers in the rational homogeneous limit, in which one lets first all z i → 1, and then q → -1 after dividing out by the appropriate power of q + 1. These integers were interpreted as the degrees of the components of the abovementioned variety.

In the present case, we may also consider an analogous rational limit. Of course, at r = 0, we may reproduce the rational limit of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF]. If we want a different result from that of [START_REF] Francesco | Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials[END_REF], we need to let r → 1 as well. Substituting

into our minimal polynomial solution Ψ of the U q (sl(k)) level 1 boundary qKZ equation, and taking → 0, we obtain a two-parameter (a, ρ) family of polynomials of the w's.

At a = ρ = 0, these are some sorts of reflected versions of the Joseph polynomials [START_REF] Joseph | On the variety of a highest weight module[END_REF],

which await some algebro-geometric interpretation. A remarkable property is that for a = 1, ρ = 0, and in the homogeneous limit where w i → 0 for all i, the above polynomials reduce to non-negative integers, yet to be interpreted as degrees of components of some matrix variety.

Remarkably, for k = 2 and n arbitrary, we have found that for a suitable normalization of Ψ these would-be degrees sum to the total degree of the "Brauer scheme" considered in [START_REF] Knutson | A scheme related to the Brauer loop model[END_REF], suggesting a possible connection. More precisely, taking r = 1 and z i = 1 for all i in our solution Ψ, we have

where the numbers on the r.h.s. are the degree of the Brauer scheme {M ∈ M 2n (C I ), M • M = 0} where • stands for a suitable deformation of the ordinary matrix product [START_REF] Knutson | A scheme related to the Brauer loop model[END_REF], also obtained as sum rules in the O(1) crossing loop model on a semi-infinite cylinder [START_REF] Francesco | Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties[END_REF].

For illustration, in the cases n = 1, 2, 3 we find the following limits ψ

2n when q → -1 of Ψ(z i = 1|r = 1)/(q 2 -1) 2n(n-1) : ψ


Emmanuel Chautard
email: emmanuel.chautard@cjp.fr

Gaëlle Loubeau

Andrei Tchirkov

Jacques Chassagne

Claudine Vermot-Desroches

Laurent Morel

Pierre Verrelle

Andreï Tchirkov

Pierre Verrelle Centre

Jean Perrin

Jean Centre

Perrin

Akt signaling pathway: a target for radiosensitizing human malignant glioma

Keywords: Human malignant glioma, intrinsic radioresistance, Akt signaling pathway, STAT3 signaling pathway

come

Introduction

Gliomas are the most common primary tumors of the central nervous system. [START_REF] Legler | Cancer surveillance series [corrected]: brain and other central nervous system cancers: recent trends in incidence and mortality[END_REF][START_REF] Gurney | Brain and other central nervous system tumors: rates, trends, and epidemiology[END_REF][START_REF] Ohgaki | Epidemiology and etiology of gliomas[END_REF] Among these tumors, the most frequent and malignant type is glioblastoma (GBM).

Glioblastomas have been described as rapidly growing tumors associated with necrosis and endothelial proliferation. These neoplasms are extremely resistant to treatment, including radiotherapy and/or chemotherapy, and patient median survival does not exceed 1 year. [START_REF] Deangelis | Malignant glioma: who benefits from adjuvant chemotherapy?[END_REF][START_REF] Stupp | Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma[END_REF] Although radiotherapy increases patient survival, this treatment is not curative because of tumor regrowth inside the irradiated tumor volume. [START_REF] Walker | Randomized comparisons of radiotherapy and nitrosoureas for the treatment of malignant glioma after surgery[END_REF][START_REF] Walker | An analysis of dose-effect relationship in the radiotherapy of malignant gliomas[END_REF][START_REF] Bleehen | A Medical Research Council trial of two radiotherapy doses in the treatment of grades 3 and 4 astrocytoma. The Medical Research Council Brain Tumour Working Party[END_REF] This might be explained by the fact that interactions between tumor and microenvironnement are involved in tumoral radioresistance through angiogenesis, [START_REF] Shannon | Antiangiogenics and radiotherapy[END_REF] hypoxia 10 and immunosuppression. [START_REF] De Ridder | Macrophages enhance the radiosensitizing activity of lipid A: a novel role for immune cells in tumor cell radioresponse[END_REF][START_REF] Apetoh | Toll-like receptor 4-dependent contribution of the immune system to anticancer chemotherapy and radiotherapy[END_REF] Another part of tumor radioresistance is due to intrinsic radioresistance of tumor cells themselves. A molecular analysis in tumor samples of basal activation of different signaling pathway potentially involved in radioresistance could be of clinical interest. Phosphatidyl-inositol 3kinase (PI3K) / Protein Kinase B (Akt) and Janus Kinase (JAK) / Signal Transducer and Activator of Transcription (STAT) pathways serve to block apoptosis process, keeping cells alive in very toxic environments as chemotherapy or ionizing radiation (IR). [START_REF] Toker | Akt signaling and cancer: surviving but not moving on[END_REF][START_REF] Hodge | The role of IL-6 and STAT3 in inflammation and cancer[END_REF] In a bioclinical prospective study, Chakravarti et al. showed a significant correlation between the level of basal Akt phosphorylation and a poor prognosis in human glioma in a subset of patients treated by radiotherapy only. [START_REF] Chakravarti | The prognostic significance of phosphatidylinositol 3-kinase pathway activation in human gliomas[END_REF] Rahaman et al. reported experimental data demonstrating that inhibition of STAT3 signaling pathway was also associated with increased apoptosis and proliferation inhibition in malignant glioma. [START_REF] Rahaman | Inhibition of constitutively active Stat3 suppresses proliferation and induces apoptosis in glioblastoma multiforme cells[END_REF] The development of Akt and STAT3 inhibitors has been a goal of pharmaceutical companies since the discovery that these pathways are often activated in numerous human cancer such as melanoma, myeloma, brain cancer, breast cancer and ovarian cancer. [START_REF] Lopiccolo | Targeting Akt in cancer therapy[END_REF][START_REF] Yu | The STATs of cancer--new molecular targets come of age[END_REF] Combining drugs with radiation is common in cancer treatment, and aim at achieving better therapeutic effects than with single-modality therapy. Several chemical in vitro inhibitors have been developped against Akt [START_REF] Lopiccolo | Targeting Akt in cancer therapy[END_REF] or STAT3. [START_REF] Costantino | STAT 3 as a target for cancer drug discovery[END_REF] Akt Inhibitor IV (5-(2-Benzothiazolyl)-3ethyl-2-[2-(methylphenylamino)ethenyl]-1-phenyl-1H-benzimidazolium iodide) inhibits Akt phosphorylation by targeting the ATP binding site of a kinase upstream of Akt, but downstream of PI3K. [START_REF] Kau | A chemical genetic screen identifies inhibitors of regulated nuclear export of a Forkhead transcription factor in PTEN-deficient tumor cells[END_REF] Akt inhibitor IV sensitized human leukemic HL-60 cells to TRAIL (TNF-related apoptosis-inducing ligand). [START_REF] Han | 15-Deoxy-delta 12,14-prostaglandin J2 (15d-PGJ 2) sensitizes human leukemic HL-60 cells to tumor necrosis factor-related apoptosis-inducing ligand (TRAIL)-induced apoptosis through Akt downregulation[END_REF] JSI-124, Cucurbitacin I, is a triterpenoid compound that acts as a highly selective inhibitor of JAK/STAT3 signaling pathway. [START_REF] Blaskovich | Discovery of JSI-124 (cucurbitacin I), a selective Janus kinase/signal transducer and activator of transcription 3 signaling pathway inhibitor with potent antitumor activity against human and murine cancer cells in mice[END_REF] JSI-124 was recently shown to sensitize malignant glioma and medulloblastoma cells to temozolomide, 1,3-bis(2-chloroethyl)-1-nitrosourea and to cisplatin, with a synergy between JSI-124 and cisplatin. [START_REF] Lo | Constitutively activated STAT3 frequently coexpresses with epidermal growth factor receptor in high-grade gliomas and targeting STAT3 sensitizes them to Iressa and alkylators[END_REF] Here, we studied in human malignant glioma cell lines: (I) the relationship between intrinsic radioresistance and Akt or STAT3 basal activation; and (II) the impact of down modulation of Akt or STAT3 signaling on in vitro intrinsic radiosensitivity. Down modulation of Akt with a chemical inhibitor (Akt inhibitor IV) demonstrated a significant enhancement of radiation sensitivity on glioma cells in clonogenic survival assay. On the contrary, down modulation of STAT3 signaling with a chemical inhibitor (JSI-124) or a neutralizing gp130 antibody failed to radiosensitize glioma cells. The radioresistance was evaluated using clonogenic cell survival assay and the basal level of activation of signaling pathways using Western blot. These data indicate that Akt intercept node could be a more relevant therapeutic target than STAT3 for radiosensitizing human malignant glioma. Cell culture -Eight human malignant glioma cell lines were used in this study. SF763, SF767 and U251MG cell lines were kindly provided by Dr C. Delmas (Centre de Lutte Contre le Cancer Claudius Regaud, Toulouse, France). SW1783, SNB19, and U373MG were obtained from N. Auger (Institut Curie, Paris, France). T98G and CB193 cell lines were kindly provided by G. Pennarun (CEA, Grenoble, France). All cell lines were cultured in DMEM (with 4500mg/l glucose and L-glutamine) supplemented with Sodium Pyruvate 1%, Non Essential Amino Acids 1%, Gentamicin 10µg/ml and 10% Foetal Calf Serum in a humidified incubator containing 5% CO2 at 37°C. All cell lines were mycoplasm free after treatment with plasmocin (Invivogen, Toulouse, France).

Methods and Materials

Materials

Clonogenic cell survival assay -Cells in exponential phase of growth were trypsinised with a 0.25% Trypsin-PBS solution and were seeded at 1000 to 4000 cells per T25 flask. One day after, three flasks were irradiated per dose and irradiation was performed as single exposure doses delivered by a linear accelerator at room temperature. After 9 days of incubation, the content of the flasks were fixed with methanol and stained with a Giemsa stain solution (5%). The Plating Efficiency (PE) represents the percentage of cells seeded that grow into colonies of a given cell line. Colonies with more than 50 cells were counted by microscopic inspection, and plating efficiency as well as the radiation-surviving fraction (plating efficiency of experimental group/ plating efficiency of control group) was determined. Each experiment was repeated on three separate days, and each day triplicates of each dose were performed. The intrinsic radiosensitivity was evaluated by using two parameters: the Surviving Fraction at 2 Gy (SF2) and the Area Under survival Curve (AUC).

Survival curves were obtained by combinating data from three independent experiments (separate days) in accordance with linear-quadratic model (KaleidaGraph software 4.0).

Western blot -Cells in exponential phase of growth were harvested using Buffer C ( supplemented with NP-40, phosphatase inhibitors (NaF, Na2VO3) and protease inhibitor IP25X (Roche). Total proteins (30µg) were boiled in Laemmli sample buffer and subjected to SDS-PAGE. Proteins were then transferred to nitrocellulose membranes (Hybond ECL, Amersham Biosciences) followed by blocking in Tris-buffered saline 1X, 10% powdered milk, and incubated with indicated antibodies in the same buffer overnight at 4°C. Membranes were washed three times with 1X Tris-buffered saline, 0.05% Tween 20, and then incubated 1h with anti-rabbit (P.A.R.I.S) peroxidase-conjugated IgG. Detection was performed using ECL system (PerkinElmer Life Sciences). Three independent experiments were used for analysis with Quantity One (BioRad).

Chemical Akt and STAT3 inhibitors -Akt Inhibitor IV (B2311) was from Sigma and STAT3 inhibitor (JSI-124) was from Calbiochem (VWR, Fontenay Sous Bois, France). Stock solutions of Akt inhibitor IV (81.3µM) and of JSI-124 (19µM) were constituted using DMSO and stored at -20°C. These stock solutions were diluted to the desired concentration with culture medium. For Western blot analysis, cells were exposed 7 h to inhibitor prior protein extraction. For clonogenic survival assay in presence of chemical inhibitors, 1000 to 4000 cells were seeded per T25 flask. One day after, cells were pre-incubated with the different inhibitors (Akt inhibitor IV, JSI-124 or DMSO) for 7h prior to irradiation and then cells were then cultured with inhibitors for up to 24h. Data were normalized to control (treatment with inhibitor without irradiation) to exclude effect of inhibitor alone on surviving fraction.

Results

Radiosensitivity analysis of human glioma cell lines by clonogenic cell survival -

The intrinsic radiosensitivity of the glioma cell lines was investigated using standard clonogenic assay. Cell lines were irradiated with increasing doses of radiation (0, 2, 4, 6, 8 and 10 Gy). SF2 and AUC were used to measure intrinsic radiosensitivity. Survival data were fitted to the linear-quadratic model (fig. 1). As shown in Table1, PE values varied from 0.05 to 0.28. SF2 values ranged from 0.46 to 0.83 and AUC values ranged from 2.4 to 5.5. A significant correlation between SF2 and AUC (r = 0.951; p = 0.00043; linear regression) was observed.

Our data obtained are in agreement with previous results evaluating the radiosensitivity of human glioma. [START_REF] Delmas | Farnesyltransferase inhibitor, R115777, reverses the resistance of human glioma cell lines to ionizing radiation[END_REF][START_REF] Daido | Inhibition of the DNA-dependent protein kinase catalytic subunit radiosensitizes malignant glioma cells by inducing autophagy[END_REF][START_REF] Russell | Radiosensitization of human tumor cell lines induced by the adenovirus-mediated expression of an anti-Ras single-chain antibody fragment[END_REF][START_REF] Casper | Acetaminophen selectively reduces glioma cell growth and increases radiosensitivity in culture[END_REF] SF763 and SW1783 appeared to be respectively the most and the less radioresistant cell line with 0.83 and 0.46 for SF2; 5.5 and 2.4 Gy for AUC values.

Correlation between activation levels of Akt and STAT3 and radiosensitivity -Among the human glioma cell line panel, STAT3 phosphorylated Tyr705 and Akt phosphorylated Ser473 residues, which are known to be active phosphorylation sites, [START_REF] Alessi | Characterization of a 3-phosphoinositidedependent protein kinase which phosphorylates and activates protein kinase Balpha[END_REF][START_REF] De | Stats: transcriptional control and biological impact[END_REF] and STAT3 and Akt expression were analyzed by Western blot (fig. 2). Levels of activation were estimated by the phospho-protein/total protein expression ratio. A significant correlation was found between the pAkt/Akt ratio and SF2 (r = 0.764; p = 0.027; linear regression), but not between pSTAT3/STAT3 ratio and SF2. Note that the most radioresistant cell line, SF763, exhibited a high basal activation of both Akt and STAT3 signaling pathways. On the contrary, no activation of these pathways was observed in the SW1783 cell line, which is the most sensitive to ionizing radiation.

Impact of Akt or STAT3 signaling down modulation on glioma radioresistance -We

used chemical Akt and STAT3 inhibitors at lower doses that slightly affect plating efficiency in the absence of radiation in order to underline a radiosensitizing effect. In our study, the SF763 cell line exhibits an activation of both Akt and STAT3 signaling. The SF767 and SNB19 cell lines presents respecitvely only Akt and STAT3 activation pathway in basal conditions.

Firstly, SF763 cells were treated with a concentration range of Akt inhibitor IV, for 7h and Akt phosphorylation was investigated using Western blot analyses. As previously reported for other cell lines, [START_REF] Kau | A chemical genetic screen identifies inhibitors of regulated nuclear export of a Forkhead transcription factor in PTEN-deficient tumor cells[END_REF] we observed a specific decrease of Akt activation with a dose of 10µM in SF763 cells compared to cells treated with DMSO e . We also observed a lower decrease of Akt activation with 0.2 and 5µM (fig. 3A). Akt inhibitor IV, decreased PE in SF763 cells after 24h of exposure, in a dose dependent manner ranging from 0.86 for 0.2µM to 0.63 for 0.04µM (fig. 3B). When SF763 cells were exposed 24h with 0.2µM of Akt inhibitor IV and irradiated at 4 Gy after 7h of treatment, we observed a significant specific decrease in surviving fraction (p < 0.01, t-test) compared with the control (DMSO + 4 Gy, fig. 3C). Clonogenic survival assays ranging from 0 to 10 Gy demonstrated a highly significant enhancement of radiation sensitivity (p < 10 -7 , ANOVA) after treatment with 0.2µM of Akt inhibitor IV (fig. 3D).

To confirm Akt pathway involvement in glioma cell radioresistance we carried out experiments to know if down-modulation of Akt could increase radiation sensitivity of the SF767 cell line. SF767 cells were treated with a concentration range of Akt inhibitor IV, for 7h and Akt phosphorylation was investigated using Western blot. As for the SF763 cell line, we observed a specific decrease of Akt activation with a dose of 10µM in SF767 cells compared to cells treated with DMSO (control). We also observed a lower decrease of Akt activation with 0.1µM (fig. 4A). Akt inhibitor IV, decreased PE in SF767 cells after 24h of exposure, in a dose dependent manner ranging from 0.87 for 0.05µM to 0.20 for 0.3µM (fig. 4B). When SF767 cells were exposed 24h with 0.1µM of Akt inhibitor IV and irradiated at 2 Gy after 7h of treatment, we observed a significant specific decrease in surviving fraction (p < 0.05, t-test) compared with the control (DMSO + 2 Gy, fig. 4C). Clonogenic survival assays ranging from 0 to 10 Gy demonstrated a significant enhancement of radiation sensitivity (p < 10 -3 , ANOVA) after treatment with 0.1µM of Akt inhibitor IV (fig. 4D).

Secondly, SF763 cells were treated with a concentration range of JSI-124 for 7h and pSTAT3/STAT3 ratio was evaluated using Western blot analyses. As shown in fig. 5A, the exposure of cells to 0.2µM of JSI-124 for 7h induced a decrease of STAT3 pathway activation whereas lower doses seemed to have no impact. Although there was no decrease in pSTAT3 with JSI-124 0.01µM in these experimental conditions, clonogenic survival was altered. Thus, JSI-124 after 24h treatment proved to be an effective inhibitor of SF763 colony formation in vitro, in a dose dependent manner ranging from 0.87 for 0.01µM to 0.39 for 0.04µM (fig. 5B). To test a potential radiosensitizing effect of JSI-124, cells were treated with 0.01µM JSI-124 for 24h including irradiation at 4 Gy after 7h of treatment. As shown in fig. 5C, JSI-124 treatment failed to sensitize SF763 cells to ionizing radiation. STAT3 pathway activation involves gp130 recruitment usptream JAK2 activation [START_REF] De | Stats: transcriptional control and biological impact[END_REF] so another approach to inhibit STAT3 pathway was carried out using gp130 blocking antibody.

The exposure of SF763 cells to anti-gp130 blocking antibody (10µg/ml) did not affect plating efficiency (data not shown) but abrogated activation of STAT3 pathway after 24h of treatment as shown in figure 5D. In clonogenic survival assay, anti-gp130 or control antibody (IgG2a) was added when cells were attached in flasks (8h after seeding), before or after irradiation until the end of experiment. In all three conditions, a decrease of STAT3 pathway activation did not radiosensitize SF763 cells (representative results fig. 5E).

Because anti-gp130 antibody treatment does not affect colony forming with a striking inhibition of STAT3 in the SF763 cell line, we used this approach to test potential radiosensitizing effect of STAT3 inhibition in the SNB19 cell line. The exposure of SNB19 cells to anti-gp130 blocking antibody (10µg/ml) did not affect plating efficiency (data not shown) but abrogated activation of STAT3 pathway after 24h of exposure as shown in figure 6A. In clonogenic survival assay, anti-gp130 or control antibody (IgG2a) was added when cells were attached in flasks (8h after seeding). Despite the decrease of STAT3 pathway activation anti-gp130 antibody did not radiosensitize SNB19 cells (fig. 6B).

Discussion

Radiotherapy is a part of the gold standard treatment of glioblastomas, but these tumors are radioresistant. Targeting signaling pathways involved in GBM radioresistance may improve clinical results of radiotherapy. PI3K/Akt and JAK/STAT3 pathways are major cell survival pathways blocking apoptosis process and keeping cells alive in very toxic environments as chemotherapy or ionizing radiation. Here, we studied in human malignant glioma cell lines: (I) the relationship between intrinsic radioresistance and Akt or STAT3 basal activation; and (II) the impact of down modulation of Akt or STAT3 signaling on in vitro intrinsic radiosensitivity.

To date, little is known about direct relationship between Akt or STAT3 activation and tumor radioresistance. Our results showed significant correlation between basal Akt activation and Surviving Fraction at 2 Gy (SF2). Conversely, no such correlation was established between STAT3 activation and SF2 suggesting that Akt is a survival pathway associated with radioresistance. In our study, the most radioresistant glioma cell line was SF763, exhibiting an activation of both Akt and STAT3 signaling in basal conditions. SF763 was a good candidate to study effects of Akt or STAT3 down modulation on the radioresistance level. SF767 cell line presents only a high level of Akt and SNB19 cell line shows only an activation of STAT3 pathway so they are good models to study specific roles of respectively Akt and STAT3 inhibition in glioma radiosensitization.

Akt pathway down modulation and glioma radiosensitivity -Our results showed that

Akt pathway activation is closely related to human glioma radioresistance. Althought IR damages tumor cells through several mechanisms, IR is thought to kill cells primarily by causing DNA damage and, specifically, double strand breaks (DSBs). Prior to undergoing division, this leads to a DNA damage response to allow repair of the DNA damage. [START_REF] Willers | Repair of radiation damage to DNA[END_REF] The ability to repair is essential to cell survival because maintained DNA breaks induce apoptosis or senecence. [START_REF] Zhou | The DNA damage response: putting checkpoints in perspective[END_REF] A recent study by Kao et al. reported that PI3K/Akt signaling pathway down modulation led to persistence of unrepaired DSBs induced by radiation in a human glioblastoma cell line (U251), demonstrating that this pathway can modulate DNA damage repair in response to radiation and may be involved in radiation therapy efficiency. [START_REF] Kao | Inhibition of Phosphatidylinositol-2-OH Kinase/Akt Signaling Impairs DNA Repair in Glioblastoma Cells following Ionizing Radiation[END_REF] To explain this, one hypothesis could imply the DNA-dependent protein kinase (DNA-PK) which is one of the downstream targets of Akt signaling. [START_REF] Toulany | PI3K-Akt signaling regulates basal, but MAP-kinase signaling regulates radiation-induced XRCC1 expression in human tumor cells in vitro[END_REF] The greatest improvement in tumor control will also be through the understanding of stem cells radioresistance mechanisms. Down modulation of Akt signaling induced apoptosis, neurosphere formation suppression, and reduced motility and invasivness in brain tumor stem cells. [START_REF] Eyler | Brain cancer stem cells display preferential sensitivity to Akt inhibition[END_REF] Interestingly, Akt down modulation sensitizes medullobastoma stem cells located in the perivascular region to radiation-induced apoptosis, suggesting that Akt inhibitors may be an effective anti-cancer stem cell therapy. [START_REF] Hambardzumyan | PI3K pathway regulates survival of cancer stem cells residing in the perivascular niche following radiation in medulloblastoma in vivo[END_REF] Here, we showed that Akt inhibitor IV was able to inhibit in a dose dependent manner in vitro colony formation of malignant glioma cell line SF763. Clonogenic survival assays using AKT inhibitor demonstrated enhancement of radiation sensitivity when SF763 cells were exposed for 24h to 0.2µM of Akt inhibitor IV. Akt pathway activation involvement in glioma cell radioresistance was confirmed by significant enhancement of radiation sensitivity of SF767 cells after treatment with 0.1µM of Akt inhibitor IV. Many PI3K inhibitors have been developped in the last few decades to study PI3K signaling involvement in various biological process. A part of them were used to radiosensitize cancer cells, such as wortmannin [START_REF] Norman | Studies on the mechanism of phosphatidylinositol 3kinase inhibition by wortmannin and related analogs[END_REF] and LY294002 37 but they have limited clinical utility due to their severe toxicities. This could be explained in part by the fact that such drugs target all PI3K protein family. More recently Chen et al have shown that PI-103, a novel PI3K inhibitor, with less toxic properties, could radiosensitize PTEN-mutated cell lines whereas it could not radiosensitize glioma cell lines with PTEN wild type. [START_REF] Chen | Characterization of structurally distinct, isoformselective phosphoinositide 3'-kinase inhibitors in combination with radiation in the treatment of glioblastoma[END_REF] Re-establishment of PTEN was also a promising approach to radiosensitize glioma because PTEN is often mutated in these tumors. [START_REF] Kao | Inhibition of Phosphatidylinositol-2-OH Kinase/Akt Signaling Impairs DNA Repair in Glioblastoma Cells following Ionizing Radiation[END_REF][START_REF] Jiang | Phosphatase and tensin homologue deficiency in glioblastoma confers resistance to radiation and temozolomide that is reversed by the protease inhibitor nelfinavir[END_REF] In contrast to PI-103 [START_REF] Kao | Inhibition of Phosphatidylinositol-2-OH Kinase/Akt Signaling Impairs DNA Repair in Glioblastoma Cells following Ionizing Radiation[END_REF] or Nelfinavir [START_REF] Jiang | Phosphatase and tensin homologue deficiency in glioblastoma confers resistance to radiation and temozolomide that is reversed by the protease inhibitor nelfinavir[END_REF] , we reported in this work that Akt inhibitor IV radiosensitizes wild type PTEN glioma cells (SF763, SF767 [START_REF] Chen | Characterization of structurally distinct, isoformselective phosphoinositide 3'-kinase inhibitors in combination with radiation in the treatment of glioblastoma[END_REF] ), but these two cell lines were selected for experiments because of their high level of radioresistance regardless their PTEN status.

STAT3 pathway down modulation and glioma radiosensitivity -JSI-124, acts as a highly selective inhibitor of the JAK/STAT3 signaling pathway. [START_REF] Blaskovich | Discovery of JSI-124 (cucurbitacin I), a selective Janus kinase/signal transducer and activator of transcription 3 signaling pathway inhibitor with potent antitumor activity against human and murine cancer cells in mice[END_REF] Su et al have demonstrated that JSI-124 induced in glioma cells G (2) /M accumulation via downregulation of cyclin B1 and cdc2 expression. [START_REF] Su | JSI-124 inhibits glioblastoma multiforme cell proliferation through G(2)/M cell cycle arrest and apoptosis augment[END_REF] Down modulation of STAT3 signaling using JSI-124 was also associated with a decrease in local immunosuppression in a murine intracranial model of glioma. [START_REF] Fujita | Inhibition of STAT3 promotes the efficacy of adoptive transfer therapy using type-1 CTLs by modulation of the immunological microenvironment in a murine intracranial glioma[END_REF] JSI-124 was recently shown to sensitize malignant glioma and medulloblastoma cells to temozolomide, 1,3-bis(2-chloroethyl)-1-nitrosourea, and cisplatin with a synergy between JSI-124 and cisplatin, [START_REF] Lo | Constitutively activated STAT3 frequently coexpresses with epidermal growth factor receptor in high-grade gliomas and targeting STAT3 sensitizes them to Iressa and alkylators[END_REF] but to our knowledge, there is no study evaluating the ability of JSI-124 to alter cancer cell resistance to ionizing radiation. Here, we showed that JSI-124 alone was able to inhibit SF763 cells colony formation in a dose dependent manner, but that STAT3 down modulation, either using JSI-124 or anti-gp130 blocking antibody, did not modify SF763 cells sensitivity to ionizing radiation.

Despite the decrease of STAT3 pathway activation after anti-gp130 antibody exposure no radiosensitizing effect was observed in SNB19 cells. This is consistent with the absence of correlation between STAT3 activation and radioresistance level in the tested cell lines.

Moreover, a recent study reported that STAT3 can have a tumor-suppressive function that is regulated by the tumor suppressor PTEN. Nuclear constitutively activated form of Epidermal Growth Factor Receptor variant III (EGFRvIII) acts as a switch to convert this STAT3 from a tumor-suppressive to a pro-oncogenic protein. [START_REF] De La Iglesia | Identification of a PTEN-regulated STAT3 brain tumor suppressor pathway[END_REF] Altogether our results with these last data support that STAT3 involvement in glioma radioresistance remains unclear and would depend on PTEN and EGFRvIII status in glioma cells.

In summary, the results of the present study and other reports [START_REF] Kao | Inhibition of Phosphatidylinositol-2-OH Kinase/Akt Signaling Impairs DNA Repair in Glioblastoma Cells following Ionizing Radiation[END_REF][START_REF] Chen | Characterization of structurally distinct, isoformselective phosphoinositide 3'-kinase inhibitors in combination with radiation in the treatment of glioblastoma[END_REF][START_REF] Jiang | Phosphatase and tensin homologue deficiency in glioblastoma confers resistance to radiation and temozolomide that is reversed by the protease inhibitor nelfinavir[END_REF] strongly suggest that Akt is a valid target for glioma cell radiosensitization. Our conclusions are different from those of de la Pena et al, who obtained no radiosensitization effect on glioma cells by treatment with perifosine, a drug that down regulate Akt phosphorylation. [START_REF] De La Pena | Inhibition of Akt by the alkylphospholipid perifosine does not enhance the radiosensitivity of human glioma cells[END_REF] Further experiments are needed to understand the link between PTEN, EGFRvIII and STAT3 status in regard to glioma radioresistance. These data indicate that Akt intercept node could be a more relevant therapeutic target than STAT3 for radiosensitizing human malignant glioma.

The PI3K/Akt pathway could be activated by numerous cytokines or growth factors.

We reported that among these activators, interleukin-6 (IL-6) gene amplification and overexpression were associated with poor survival in patients with malignant gliomas. [START_REF] Tchirkov | Interleukin-6 gene amplification and shortened survival in glioblastoma patients[END_REF] We have previously reported that interleukin-6 gene is amplified in SF763 but not in SF767 cell line. [START_REF] Tchirkov | IL-6 gene amplification and expression in human glioblastomas[END_REF] PI3K/Akt could also be activated by EGFR which is often amplified in glioblastoma [START_REF] Chakravarti | Immunohistochemically determined total epidermal growth factor receptor levels not of prognostic value in newly diagnosed glioblastoma multiforme: report from the Radiation Therapy Oncology Group[END_REF][START_REF] Maher | Malignant glioma: genetics and biology of a grave matter[END_REF] but EGFR status in these cell lines is unknown. Furthermore, receptor tyrosine kinase independent activation of the PI3K/Akt pathway is commonly observed in many cancers, and can occur through multiple mechanisms, such as mutation or amplification of PI3K gene, amplification of Akt gene, activation of an upstream oncogene (e.g., RAS), or mutation or decreased expression of the tumor suppressor PTEN. [START_REF] Schuurbiers | The PI3-K/AKT-pathway and radiation resistance mechanisms in non-small cell lung cancer[END_REF] Because of these multiple possibilities, instead of inhibiting a single cell surface receptor, down modulation of signal transduction through such an intercept node will be a more effective approach to block radioresistance. [START_REF] Castellino | Mechanisms of disease: the PI3K-Akt-PTEN signaling node--an intercept point for the control of angiogenesis in brain tumors[END_REF] The PI3K/Akt node could be a relevant therapeutic target to radiosensitize tumor cells by inhibiting both anti-apoptotic mechanisms and DNA damage repair after radiation. A, Cells were treated with anti-gp130 blocking antibody (24h). Total proteins were electrophoresed by SDS-PAGE, followed by immunoblotting with anti-STAT3, anti-pSTAT3-Tyr705 and anti-β-actin antibody. B, Clonogenic survival of SNB19 cell line exposed to blocking anti-gp130 or to control (IgG2a) antibody when cells were attached in flasks (8h after seeding). Cells were irradiated to 2 Gy and the surviving fraction was compared with that of control (IgG2a). Representative experiment performed in triplicate is shown.

Captions

  -Akt (No 9272), phospho-Akt Ser473 (No 9271), STAT3 (No 4904), and phospho-STAT3 Tyr705 (No 9145) rabbit antibodies were from Ozyme (Saint Quentin Yvelines, France). Control rabbit IgG (No I5006) and β-actin (No A2066) were from Sigma (Saint Quentin Fallavier, France). Anti rabbit-FITC was from Dako (No F0054, Trappes, France) and anti rabbit-peroxidase was from P.A.R.I.S. (Compiègne, France). All culture reagents were purchased from GIBCO (Invitrogen, Cergy-Pontoise, France). Gp130 blocking antibody (No 852.060.000) and control (IgG2a, No 857.080.000) mouse antibody are from Diaclone (Besançon, France).

Fig. 1 .

 1 Fig. 1. (A-H) Clonogenic survival curves of human malignant glioma cell lines. Cells

Fig. 2 .

 2 Fig. 2. Akt and STAT3 basal signaling pathways activation. A, Cells in were harvested

Fig. 3 .

 3 Fig. 3. Impact of Akt pathway down modulation on SF763 cells.

Fig. 4 .

 4 Fig. 4. Impact of Akt pathway down modulation on SF767 cells.

Fig. 5 .

 5 Fig. 5. Impact of STAT3 pathway down modulation on SF763 cells.

Fig. 6 .

 6 Fig. 6. Impact of STAT3 pathway down modulation on SNB19 cells.

Acknowledgments

This work was supported by the Ligue Nationale Contre le Cancer (Comité du Puy De Dôme) and by the Region Auvergne. We are grateful to Nicolas Foray for assistance using Kaleidagraph software; Claude Beaudoin for critical reading of the manuscript ; Michele Manin for helpful advices for cell cultures; Angélique DeHaze, Julie Dufour, Antoine Nigon and Jean-paul Saru for technical assistance. E.C. was the recipient of a fellowship from the MENESR (Ministère de l'Education Nationale de l'Enseignement Supérieur et de la Recherche).