Antoine Duval

Vincent Pasquier

Pieri rules, vertex operators and Baxter Q-matrix *

We use the Pieri rules to recover the q-boson model and show it is equivalent to a discretized version of the relativistic Toda chain. We identify its semi infinite transfer matrix and the corresponding Baxter Q-matrix with half vertex operators related by an ω-duality transformation. We observe that the scalar product of two higher spin XXZ wave functions can be expressed with a Gaudin determinant.

Introduction

In this paper, we establish a correspondence between some lattice integrable models, the q-bosons and the Toda lattice, and multi-variable symmetric orthogonal polynomials. Our starting observation is that the Pieri rules [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] provide a family of commuting transfer matrices which are the main tool to study lattice integrable models. They are the multi-variable generalization of the three terms recursion relations of orthogonal polynomials [START_REF] Szego | Orthogonal Polynomials[END_REF]. Given a family of elementary symmetric functions organized into a generating function Ω(u), and a basis of symmetric orthogonal polynomials P λ , they tell us how the product Ω(u)P µ decomposes onto the P λ -basis. The branching rule of this decomposition Ω(u)P µ = λ P λ T λµ (u) define a commuting family of transfer matrices and can be interpreted as the eigenvalue equation for T (u).

In favorable cases, the transfer matrix is a product of local Lax operators (L matrices [START_REF] Faddeev | [END_REF]) and one can deduce from it the local Boltzmann weights of a lattice model. The lattice model we analyze in this respect is the q-boson model [START_REF] Kulish | [END_REF] where Bosonic particles occupy the sites of a one dimensional chain. The dynamics is controlled by a transfer matrix which at each transfer time lets the particles simultaneously hop in one direction in between their position and the next occupied one with an intensity controlled by the deformation parameter of the q-bosons. When the chain is closed, the number of particles is conserved, and it can be diagonalized by coordinate Bethe ansatz. The components of its eigenvectors are Hall-Littlewood polynomials P µ evaluated at roots of the Bethe equations.

We study this model on a semi infinite chain where there is no quantization condition (Bethe equations). We introduce two different dynamics. In one, the particles simultaneously hop to their right with possible creation at the origin. In the other, they hop to their left with possible annihilation at the origin. The corresponding transfer matrices are obtained as the limit of finite size matrices whose commutations relations are dictated by the Yang-Baxter equation. In the limit, these commutations simplify and can be represented with simple Vertex operators as was discovered by by Tsilevich [5]. When the deformation parameter is equal to zero, using the Frobenius correspondence between partitions and Fock space configurations, one recovers a free fermion model. The components of the eigenvectors are the Hall-Littlewood polynomials obeying a Pieri dynamics as described above, generated by the eigenvalue of the Bethe equations.

A remarkable property of the Hall-Littlewood polynomials is the existence of a second Pieri rule (related to the Hall algebra). We repeat the same analyses for its dynamics, relabeling the configurations in terms of a discretized version of the Toda lattice. Unlike the q-bosons case, the Hall-Pieri transfer matrix is not of a Lax type but is instead the Baxter Q-matrix [START_REF] Baxter | Exactly Solved Models in Statistical Mechanics dover publications inc[END_REF] generating the Bäcklund Toda lattice dynamics [START_REF] Sklyanin | [END_REF]. We construct it as an alternative to the coordinate Bethe ansatz, to derive the Toda chain Bethe equations.

We use its open variant to recover the open Toda chain wave functions in the same way as Gerasimov Lebedev and Oblezin [8]. We identify it with a vertex operator, the matrix elements of which are the branching coefficients of the Hall-Pieri rules and we recover this relation as a matrix element of the Yang-Baxter equation.

The open Toda chain wave functions can be deduced by an ω-duality transformation from the Hall-Littlewood polynomials. We have extensively used the λ-ring formalism [START_REF]Lascoux A Symmetric Functions and Combinatorial Operators on Polynomials[END_REF] [START_REF] Garsia | Séminaire Lotharingiens de Combinatoire B42m[END_REF] unifying the two type of Pieri rules and enabling to relate them simply to Cauchy identities.

The q-boson model appears as the infinite spin limit of an XXZ (or six vertex) higher spin chain which can be diagonalized by coordinate Bethe ansatz. In the last section, we observe that the scalar product of two semi infinite chain wave functions is independent of the spin and can be expressed with a Gaudin determinant [START_REF] Gaudin | [END_REF].

The q-bosons

The q-bosons [START_REF] Kulish | [END_REF][12] [13] are defined by their commutation relations:

tS S -SS = t -1, (1)
and are realized with a Weyl pair τ, S, τ S = tSτ as:

S = S -1 (1 -τ ). (2)
There are two possible Hermitian conjugations which preserve the relations (1) depending on the value of the parameter t, t * = t or t * = t -1 . We shall only consider the first one here S † = S, with t a real number between 0 and 1.

S and S act on q-bosons states which can be identified with S polynomials, S n = |n ( S equal to 0 when sitting to the right of S n ) with n a nonnegative integer as S|n = (1t n )|n -1 , τ |n = t n |n . The scalar product such that S is the adjoint of S, (τ = τ ) is:

n|m = (t) m δ n,m , (3) with:
(a

) m = (1 -a) • • • (1 -at m-1 ). (4)
The configurations space of the q-boson spin chain of length N consists in states where sites labeled by k are occupied by m k q-bosons,

S m 0 0 S m 1 1 • • • S m N-1 N -1
with k m k = n, the total number of Bosons. We can also characterize the state by the sequence of positions of the Bosons ordered decreasingly:

(N -1 ≥ µ 1 ≥ • • • ≥ µ n ≥ 0), |(N -1) m N-1 • • • 0 m 0 = |µ n • • • µ 1 where the particles labeled i (1 ≤ i ≤ n)
forgetting about their Bosonic nature occupies the sites µ i , (0 ≤ µ i ≤ N -1) of the chain. The scalar product between two Boson states is then given by:

µ|ν = δ µ,ν N -1 0 m k ! t . (5)
where (n)! t = (t) n is the t-factorial.

We shall also consider the trivial representation S 0 = S0 = 1, in which case, the occupation number starts at k = 1, µ i ≥ 1. Equivalently, the site 0 is occupied by an infinite number of bosons m 0 = ∞.

We consider the Lax-Matrix:

L k (z) = 1 z Sk S k z (6)
where the q-boson creation and annihilation operators S k , Sk act independently on spaces labeled k. It obeys the Yang-Baxter equation [START_REF] Faddeev | [END_REF]:

R 12 L 1 (u)L 2 (v) = L 2 (v)L 1 (u)R 12 , (7)
with

L 1 = L ⊗ 1, L 2 = 1 ⊗ L.
and the R matrix acting in the tensor product is:

R 12 (u, v) =     a . . . . b c . c b . . . . a     (8) a = ut -v, b = u -v, b = t(u -v), c = v(t -1), c = u(t -1). (9)
Everywhere, . = 0 in the following. It is straightforward to verify that the relations (8) with L given by ( 6) are equivalent to the q-boson relations (1) satisfied by S i , Si which commute for different values of i.

The monodromy matrix:

T N (z) = L 0 L 1 • • • L N -1 (10)
takes the form:

T N (z) = π 1 S 0 N -1 1 (1 + z Si-1 S i ) 1 z SN-1 , (11)
where the projector π requires we erase S i and Si in the terms of the z expansion each time they occur in pairs. In other words, π means we set S i Si = 1 in the formal expansion of the above product but do not identify Si with S -1 i . In particular, the transfer Matrix of the periodic chain is:

Λ N,x (z) = Tr T N (z)D ( 12
)
where D is the matrix:

D = 1 . . x (13)
define a commuting family of transfer matrices. It takes the form:

Λ N,x (z) = π(1 + z S0 S 1 ) • • • (1 + z SN-1 S N ), ( 14
)
where we require the periodicity condition:

S N +k = xS k . ( 15
)
The configuration space consists of decreasing sequences λ (1 ≤ λ i ≤ N), and the matrix elements Λ λµ can be nonzero only when each boson jumps to the right in between his position and the next occupied one:

λ n+1 ≤ µ n ≤ λ n ≤ µ n-1 • • • ≤ λ 1 = λ n+1 + N.
The one step translation T :

T S k = S k+1 T (16)
commutes with the transfer matrix: T Λ N,x = Λ N,x T , and in the n-Boson sector T N = x n . Everywhere, we use the symbol ǔ to represent the inverse of a variable: ǔ = 1/u. We require:

x = x, z = ž. ( 17
)
We can rewrite π

• • • (1 + z Si S i+1 ) as π • • • z Si (1 + žS i Si+1 )S i+1
, so that the Hermitian property ΛN,x (z) = xž N Λ N,x (z) holds. In other words, Λ H = ΛN,X (z)/ √ xz N is a Hermitian operator:

ΛH = Λ H . ( 18
)
This transfer matrix can be diagonalized by the coordinate Bethe ansatz. We obtain the eigenvectors and corresponding eigenvalues in the appendix A as a limit of the spin XXZ chain Bethe equations (146).

Finite and semi infinite open chains

Here, we consider the q-boson model on an open chain satisfying some specific boundary conditions. Our aim is to let the size of this chain go to infinity and relate it to some vertex operator constructions [14] [15]. The material of this section was essentially discovered in the precursor work of Tsilevich [5], and overlaps with similar constructions [16], [START_REF] Borodin | Commun[END_REF]. More specifically, the relation between the semi-infinite chain and vertex operators discussed here appears in the recent preprint of Wheeler and Zinn-Justin [18]. We follow the approach of Macdonald [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF], to emphasize the connections between the symmetric polynomials and the semi-infinite transfer matrices we introduce and to familiarize the reader with the λ-ring notations.

We take the trivial representation S 0 = S0 = 1 for L 0 and the q-boson representation for L i , i > 0, so T 11 = T 21 , T 12 = T 22 . There is no occupation on site 0 and the chain starts at site 1. The configuration space consists of partitions λ (1 ≤ λ i ≤ N). The matrix elements (T 11 ) λµ can be nonzero only when each boson jumps to the right in between his position and the next occupied one, and an additional boson labeled n + 1 can be injected before the first occupied site: 0

≤ λ n+1 ≤ µ n ≤ λ n ≤ µ n-1 • • • ≤ λ 1 ≤ N (No boson is injected if λ n+1 = 0,
and the number of Bosons n increases by one unit otherwise). We denote this condition µ ≺ λ, or λµ is a horizontal strip. Similarly, the matrix elements of T 12 can be nonzero only when each boson jumps to the left in between his position and the preceding occupied one, and the leftmost boson (at position µ n ) can disappear.

We define A N and its hermitian conjugated ĀN by:

(A N , z N +1 ĀN ) = ((T N +1 ) 11 , ((T N +1 ) 12 ) = (1, z)L 1 L 2 • • • L N (19)
Explicitly,

A N (z) = π(1 + zS 1 )(1 + zS 2 S1 ) • • • (1 + zS N SN-1 ) ĀN (z) = π(1 + ž S1 )(1 + ž S2 S 1 ) • • • (1 + ž SN S N -1 ), (20)
Some commutation relations which follow from the RLL relation [START_REF] Sklyanin | [END_REF] are:

[T 11 (u), T 11 (v)] = [T 12 (u), T 12 (v)] = 0 aT 11 (u)T 12 (v) = bT 12 (v)T 11 (u) + cT 11 (v)T 12 (u) (21)
independently of N.

We consider a semi infinite chain obtained by taking the N → ∞ limit of the monodromy matrix acting in a space with a finite number of q-bosons. In the limit N → ∞ of the finite N Hilbert space, we define the matrices (which coincide with the fermion limit of [15] for t = 0):

(Γ -, Γ + )(z) = lim N →∞ (A N , ĀN ). ( 22
)
Γ -is defined for |z| < 1 and coincides with the limit of A N (z); Γ + for |z| > 1 coincides with the limit of ĀN (z). They obey the same hermitian property as A N , ĀN :

Γ-(z) = Γ + (z). ( 23
)
Expressed in terms of A, Ā, the second line of (21) rewrites:

aA N (u) ĀN (v) = b ĀN (v)A N (u) + c(u/v) N +1 A N (v) ĀN (u). ( 24
)
In the N → ∞ limit, the third term behaves as |u/v| N << 1 compared to the two others. So, the commutations of Γ ± in the N → ∞ limit are:

[Γ -(v), Γ -(v ′ )] = [Γ + (u), Γ + (u ′ )] = 0 Γ + (u)Γ -(v) = 1 -tv/u 1 -v/u Γ -(v)Γ + (u) (25)
where |u|, |u ′ | > 1 and |v|, |v ′ | < 1. Γ + , Γ -respectively act as the identity on the left and right vacuum .| annihilated by S i and |. annihilated by Si .

The expression of the matrix elements of Γ -(z) are given by:

Γ -(z)|µ = µ≺λ z |λ-µ| ψ λ/µ |λ (26)
where:

ψ λ/µ = m j (λ)=m j (µ)-1 (1 -t m j (µ) ) (27)
agrees with the definition [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (5.8') p.229. Similarly, the matrix elements of Γ + (z) are given by:

Γ + (z)|λ = µ≺λ ž|λ-µ| φ λ/µ |µ (28)
where:

φ λ/µ = m j (λ)=m j (µ)+1 (1 -t m j (λ) ) ( 29
)
coincides with the definition [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (5.8) p.228. Define the state:

U| = .|Γ + (u 1 )Γ + (u 2 ) • • • Γ + (u n ) = .|Γ + (U) (30)
where

U = {u k , 1 ≤ k ≤ n} = u 1 + • • • + u n ,
meaning that the order of the u i is irrelevant (due to the commutations of Γ + (u i )). More generally, we denote

U + V | = U|Γ + (V ).
We can use the commutation relations [START_REF] Gaudin | Wave Function[END_REF] to check that U| is a left eigenstate of Γ -(z). Using λ-ring notations [START_REF]Lascoux A Symmetric Functions and Combinatorial Operators on Polynomials[END_REF] [10]:

U|Γ -(z) = Ω t ( Ǔz) U| (31)
where Ǔ = ǔ1 + • • •+ ǔn , and Ω(X) is the generating function of the complete symmetric functions [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (2.5) p 21:

Ω(X) = x k ∈X 1 1 -x k (32)
We also use λ-ring notations to define

Ω t (X) = Ω(X(1 -t)). Namely since Ω(X ± Y ) = Ω(X)Ω(Y ) ±1 , Ω t (X) = Ω(X)/Ω(tX).
Putting ( 26) and ( 31) together, we obtain the eigenvalue equation:

Ω t ( Ǔ z) U|µ = µ≺λ z |λ-µ| ψ λ/µ U|λ . (33)
Let Ω t (z Ǔ ) = ∞ r=0 q r ( Ǔ )z r where q r defines elementary symmetric polynomials of degree r. Define the polynomials Q µ ( Ǔ) symmetric in Ǔ (The symmetry property follows from the commutations of Γ + (U)), homogeneous of degree |µ| from the expansion:

U| = λ Q λ ( Ǔ ) λ| (34)
where λ| = λ|/ λ|λ is dual to |λ . Since the partitions λ's which occur in (34) all have their length l(λ) ≥ |U|, the polynomial Q λ ( Ǔ ) is null if the number of variables u i is less than l(λ). On the other hand, since Γ -(0) = 1, the number of variables in Q λ is immaterial as long as it is larger or equal to the length of λ and can therefore be sent to infinity. Expanding (33) in z, yields the decomposition of q r Q µ :

q r Q µ = µ≺λ,|λ-µ|=r ψ λ/µ Q λ ( 35
)
with λµ a horizontal r-strip. It coincides with [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (5.7') p.229, and allows to identify Q µ with the Hall-Littlewood polynomials.

Acting with Γ + (U ′ ) on a state Ū | (instead of .| as in (30) enables to define another symmetric function Q λ/µ ( Ǔ ) for µ ∈ λ:

Q λ ( Ǔ + Ǔ′ ) = µ⊂λ Q λ/µ ( Ǔ )Q µ ( Ǔ′ ). (36)
Equivalently:

μ|Γ

+ (U) = µ⊂λ Q λ/µ ( Ǔ) λ|. ( 37
)
By taking the adjoint of ( 30), we define the state:

|V = Γ -(v 1 )Γ -(v 2 ) • • • Γ -(v n )|. = Γ -(V )|. = λ P λ (V )|λ ( 38
)
where

P λ = Q λ / λ|λ and V = {v k , 1 ≤ k ≤ m}. It is a right eigenvector of Γ + (z): Γ + (z)|V = Ω t (žV )|V . (39)
More generally,

Γ -(v 1 )Γ -(v 2 ) • • • Γ -(v n )|µ = Γ -(V )|µ = λ P λ/µ (V )|λ (40)
If we introduce the sequences of partitions:

λ n = µ ≺ λ n-1 • • • ≺ λ 0 = λ which form the tableaux T of shape λ -µ [1] p.5
, and the quantities:

v T = v |λ 0 -λ 1 | 1 • • • v |λ n-1 -λn| n ψ T = ψ λ 0 /λ 1 • • • ψ λ n-1 /λn . (41)
Rewriting (40) in matrix form, we obtain the tableau expression of P λ/µ [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] 5.11' p.229:

P λ/µ (V ) = T ψ T v T (42)
where the summation is over all the tableaux T of shape λµ.

Inserting the identity 1 = λ |λ λ| into the scalar product U|V yields the Cauchy identity:

U|V = λ Q λ ( Ǔ )P λ (V ) = Ω t ( ǓV ). ( 43
)
The product P µ P ν can be decomposed on the P λ basis:

P µ P ν = λ f λ µν P λ . (44)
We have:

μ|Γ + (U)|V = Q λ/µ ( Ǔ )P λ (V ) = Ω t ( ǓV )P µ (V ). ( 45
)
Using the Cauchy identity to decompose Ω t ( Ǔ V ) and identifying the coefficient of P λ in the resulting expression, we recover the definition of Q λ/µ [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (5.2) p.227:

Q λ/µ = ν f λ µν Q ν . (46)
We can obtain a more explicit expression of Q µ by diagonalizing the infinite chain by Bethe Anzats [START_REF] Borodin | Commun[END_REF] in the sector of n q-bosons. The infinite chain transfer matrix is defined as:

A(z) = π -∞ -∞ (1 + zS k+1 Sk ), (47)
in which case the sites take integer values µ k ∈ Z defining a sequence of decreasing integers,

µ n ≥ • • • ≥ µ 1 .
The transfer matrix elements have the same form as [START_REF] Gaudin | La Fonction d'onde de Bethe[END_REF] with no positivity restriction on µ i :

A(z)|µ = µ≺λ z |λ-µ| ψ λ/µ |λ (48)
and λ, µ have length n. The Bethe-Lieb left eigenvectors, R|µ = R µ , corresponding to the eigenvalue (31):

λ R λ ( Ǔ)A λµ (z) = Ω t ( Ǔ z)R µ ( Ǔ ) (49)
has the expression (appendix (A):

R µ ( Ǔ ) = P (ǔ µ 1 1 • • • ǔµn n i<j ǔi -tǔ j ǔi -ǔj ) (50)
summed over the permutations of the ǔi . One recognizes the polynomial R µ ( Ǔ ) p. 204 of [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF], where we extend its definition to sequences of decreasing integers.

Consider the restriction of (48) to configurations of n q-bosons sitting on non-negative sites, |0 m 0 µ where µ is a partition, and m 0 = nl µ . Since (1 + zS 1 ) in Γ -(z) [START_REF] Gaudin | [END_REF] is replaced by (1 + zS 1 S0 ) in T (z), they are related by a similarity transform: T λµ = (t) n-lµ /(t) n-l λ (Γ -) λµ when restricted to these configurations. So, the eigenvectors of T (z) are deduced from those of Γ -(z) by:

(1 -t) n (t) m 0 R 0 m 0 µ ( Ǔ ) = Q µ ( Ǔ) (51)
where the normalization (1t) n is required for Q µ not to depend on m 0 . In this way, we recover the expression of Q µ ( Ǔ ) [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] p 211, and P µ (V ) p 208.

4 The half vertex-operators Γ a,±

In this section, we use more extensively the λ-ring formalism to introduce the vertex operators ω-dual to those of the preceding section. We introduce another Pieri rule and use its branching coefficients to construct a transfer matrix which we shall derive in a different way from its commutation relations in the section (6.2). The scalar product analogous to (5) on symmetric polynomials:

Q µ |Q ν = δ µ,ν k≥1 m k ! t (52)
Let us repeat the preceding construction with another Pieri rule defined in [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] p 215. Let Ω(X) be the generating function of the elementary symmetric functions [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (2.2) p 19:

Ω(X) = Ω(-ǫX) = x k ∈X (1 + x k ) ( 57
)
where the product ǫX means we make the change of variable x i → -x i in X and has a different meaning as -X after 32. Following the conventions of [20] 3.1, let us rename the previous operators Γ L,± , and similarly as in (54), define Γ R,± as:

Γ R,-(z)α(X) = Ω(zX)α(X), (58)
and the same argument as above for Γ R,+ (z) = ΓR,-(ž) yields:

Γ R,+ (z)α(X) = α(X -ǫž/(1 -t)). (59)
It follows from these relations that Γ a,+ (z) commutes with Γ a ′ ,+ (w) and acts as the identity on the right vacuum, Γ a,-(z) commutes with Γ a ′ ,-(w) and acts as the identity on the left vacuum. Moreover, the second line of [START_REF] Gaudin | Wave Function[END_REF] generalizes to:

Γ a 1 ,+ (u)Γ a 2 ,-(v) = Γ a 2 ,-(v)Γ a 1 ,+ (u)    Ω((1 -t)vǔ) Ω(vǔ) Ω((1 -t) -1 vǔ) a 1 = a 2 = L a 1 = a 2 a 1 = a 2 = R (60)
Let P , Q denote the Hall-Littlewood Polynomials, and P ω , Q ω their ω-duals [1] chapter 7, section 5.1:

P ω λ ′ (V ) = Q λ (-ǫV /(1 -t)) Q ω λ ′ (V ) = P λ (-ǫV /(1 -t)). (61)
It is shown in particular that P ω λ is the polynomial P (λ, t, 0) defined by (4.7) p 322, as the symmetric functions decomposing triangularity: P ω λ = m λ + µ<λ u λ,µ m µ on the symmetric monomial basis, and orthogonal for the scalar product dual to (55) defined by:

α(X)|Ω(XY /(1 -t)) ω = α(Y ). ( 62
)
The states |L, V , L, Ǔ | are defined in (53), and the states |R, V , R, Ǔ | by:

R, Ǔ| = Ω( Ǔ X) = λ P ω λ ′ ( Ǔ) λ| |R, V = Ω(V X) = λ Q ω λ ′ (V )|λ . ( 63
)
Notice that restricting Ǔ , V to N variables has the effect to truncate the summation imposing λ 1 ≤ N, in other words all the sites greater than N are empty.

Let

λ ′ 1 ≥ λ ′ 2 • • • ≥ λ ′ N ≥ 0 define the partition conjugated to the q-boson partition λ: m 1 = λ ′ 1 -λ ′ 2 , • • • , m N -1 = λ ′ N -1 -λ ′ N , m N = λ ′ N .
Let us introduce the following quantities attached to two partitions such that µ ′ ≺ λ ′ (in other words, λµ is a vertical strip) [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (3.2) p 215:

ψ ′ λ/µ = i≥1 (λ ′ i -λ ′ i+1 )! t (λ ′ i -µ ′ i )! t (µ ′ i -λ ′ i+1 )! t φ ′ λ/µ = i≥1 (µ ′ i -µ ′ i+1 )! t (λ ′ i -µ ′ i )! t (µ ′ i -λ ′ i+1 )! t . ( 64
)
Then, similarly as in ( 26), (28), the explicit expression of the transfer matrices Γ R(z),± acting on the semi infinite chains is:

Γ R,+ (z)|λ = µ ′ ≺λ ′ ž|λ-µ| ψ ′ λ/µ |µ Γ R,-(z)|µ = µ ′ ≺λ ′ z |λ-µ| φ ′ λ/µ |λ . (65)
We shall derive these expressions from (60) using the Yang-Baxter equation in section (6.2). Unlike Γ L,± , Γ R,± are not defined through a Lax matrix such as [START_REF] Baxter | Exactly Solved Models in Statistical Mechanics dover publications inc[END_REF], but correspond to spin chains. If we attach a decreasing sequence of nonnegative integers (spins)

µ ′ 1 ≥ µ ′ 2 • • • µ ′
N to the sites of a semi infinite chain, the transfer matrix elements (Γ R,-) λµ are nonzero only when each spin µ ′ k jumps in between his position and the position of the preceding spin:

µ ′ k ≤ λ ′ k ≤ µ ′ k-1 , (µ ′ 0 = ∞)
, so that the number of nonzero spins can increase by one at each step. The matrix elements, ψ ′ λ/µ , φ ′ λ/µ , are the products of local Boltzmann weights between neighboring spins.

In particular, as in (40) we have:

Γ R,-(V )|µ = λ Q ω λ ′ /µ ′ (V )|λ , (66)
from which, combining with (65), we obtain the tableau expression for Q ω λ ′ /µ ′ (V ) analogous to (42):

Q ω λ ′ /µ ′ (V ) = T φ ′ T v T (67)
where this time:

φ ′ T = φ ′ λ 0 /λ 1 • • • φ ′ λ n-1 /λn . ( 68
)
For example, the one and two spin states are equal to:

|R, {v 1 } = (S 1 v 1 ) -1 ∞ |R, {v 1 , v 2 } = (S 1 v 1 ) -1 ∞ (S 1 v 2 ) -1 ∞ (S 2 v 1 v 2 ) -1 ∞ . ( 69
)
Setting µ = 0 in (66), we obtain |R, V which is an eigenstate of Γ L,+ (z):

Γ L,+ (z)|R, V = Ω(V ž)|R, V . (70)
As it will become clear in the next section, Γ L,+ (z) is the transfer matrix of the open Toda chain, and the Tableau expression of |R, V coincides with the "Baxter-Gauss-Givenal" representation of the q-Toda wave functions due to Gerasimov Lebedev and Oblezin [8].

We can also recover the Product rule [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] p 215 by applying Γ R,+ (z) to

|L, V : Γ R,+ (z)|L, V = Ω(V ž)|R, V , (71)
and expanding in z:

λ ψ ′ λ/µ P λ = e r P µ , (72)
where λµ is a vertical r-strip.

5 Equivalence between the q-bosons and a Toda chain

In order to interpret the results of section (4), we find it convenient to rephrase them in the relativistic-Toda dynamics language [21], by demonstrating an equivalence between the two models. We will be mainly concerned with a discretized version discussed in [22][8] [23] and recover the expression of their eigenvectors.

For this, we need to to reinterpret the q-bosons configurations as Toda particle positions λ ′ i interacting with their nearest neighbors. Define

X k = S k /S k-1 (73) (S k = X 1 • • • X k ),
and for 1 ≤ k ≤ N -1:

τ k = x k /x k+1 (74)
obeying:

x k X k = tX k x k , (75)
and x k , X l commute if the labels k and l differ. Setting:

U k = 1 x k S k . , (76)
and:

L Toda k = 1 + zX k x k -zX k x -1 k . , (77)
we have:

U k-1 L Toda k = L k-1 U k ( 78
)
where L is defined in [START_REF] Baxter | Exactly Solved Models in Statistical Mechanics dover publications inc[END_REF]. Therefore, if we set:

T Toda N = L Toda 1 • • • L Toda N , (79)
we have:

U 0 T Toda N = T N U N ( 80
)
where T N is defined in [START_REF] Garsia | Séminaire Lotharingiens de Combinatoire B42m[END_REF].

The Periodic chain

The q-Toda chain transfer matrix [21] is defined as:

Λ Toda N,n (z) = tr T Toda N D Toda (81)
where:

D Toda = 1 . . t n (82)
is equal to the q-boson transfer Matrix Λ N,x (z) (12) restricted to the n-Boson sector if we require:

DU 0 = U N D Toda (83)
with D the q-boson twist D = (1, X) (13). Equivalently for:

x k-N = x k t n , S k+N = xS k . ( 84
)
The periodic-Toda particles are labeled by k, 1 ≤ k ≤ N and their positions take discrete values: x k = t λ ′ k with λ ′ k integer satisfying the periodicity conditions:

λ ′ k+N = λ ′ k -n. ( 85
)
The Boson occupation numbers are equal to the difference between successive λ's:

m k = λ ′ k -λ ′ k+1
, and X k adds one unit to λ ′ k :

X k |λ ′ l = δ k,l |λ ′ l + 1 . (86)
So, the periodic chain configuration space is characterized by

N integers |λ ′ with λ ′ 1 ≥ • • • ≥ λ ′ N ≥ λ ′ 1 -n. The momentum x = X 1 • • • X N translates the Toda positions λ ′
k globally by one unit:

|λ ′ 1 • • • λ ′ N = x|λ ′ 1 -1 • • • λ ′ N -1 , (87)
In the sector where it is equal to the q-boson twist parameter (13), the periodic q-Toda chain transfer matrix (81) coincides with the transfer matrix (14) restricted to the sector of n Bosons.

In particular, the z coefficient is the right-mover Hamiltonian:

H per 1 = S 1 S0 + S 2 S1 + • • • S N SN-1 = X 1 (1 -x 0 /x 1 ) + X 2 (1 -x 1 /x 2 ) + • • • + X N (1 -x N -1 /x N ). ( 88
)
As in (18), we can redefine a Hermitian transfer matrix:

Λ Toda,H N,n (z) = Λ Toda N,n (z)/ √ xz N . (89)
In particular, x H1 = H N -1 , so H 1 commutes with his adjoint and both can therefore be diagonalized simultaneously.

In the appendix (A), using the equivalence with the q-boson problem, we obtain the Bethe equations for the spectrum and the corresponding eigenvectors which are Hall-Littlewood polynomials specialized at the Bethe roots. Notice however that the q-boson scalar product is associated to a Hermitian conjugation different from the one considered by Ruijsenaars for t a root of unity, namely Xk = X k , xk = x k , and consequently, the diagonalization problem we solve is different from his. The Q-Matrix approach described in section (6.1) will be relevant in that case [START_REF] Babelon | Quantization of the Relativistic Toda Chain[END_REF].

The Open chain

In the open case, we can as well represent A N and its hermitian conjugated ĀN (19) with the Toda variables. We set the boundary variables to be:

S 0 = 1, x 0 = 0, S N +1 = ∞, x N +1 = 1, keeping the ratio S N +1 /X N +1 = X 1 • • • X N finite.
We consider the matrix element:

A N = (T B N U N +1 ) 12 = (T Toda N ) 11 , (90)
So, A N is the open q-Toda chain with N particles conserved quantities generating function [21]. In particular, the coefficient of z is the Hamiltonian:

H open 1 = S 1 + S 2 S1 + • • • S N SN-1 = X 1 + X 2 (1 -x 1 /x 2 ) + • • • + X N (1 -x N -1 /x N ). ( 91
)
The Toda configurations |λ ′ , with

λ ′ 1 ≥ λ ′ 2 • • • ≥ λ ′ N ≥ 0 define partitions conjugated to the q-boson partition, m 1 = λ ′ 1 -λ ′ 2 , • • • , m N -1 = λ ′ N -1 - λ ′ N , m N = λ ′ N . Similarly, ĀN (z) = z -(N +1) S -1 N +1 (T B N U N +1 ) 11 (19) is equal to: ĀN (z) = ( T Toda N (z)) 11 -( T Toda N (z)) 12 , (92)
where we factorize

S N +1 /X N +1 = X 1 • • • X N and define T Toda N (z) = z -N S -1 N T Toda N (z)
as in (79) with LToda

k = (zX k ) -1 L Toda k (z): LToda k (z) = 1 + ž Xk ž Xk x k -x k . ( 93
)
Notice that T Toda N (z) is not the hermitian conjugated of T Toda N (z). Restricting V to have N nonzero variables, V = V N , Γ L,+ (z) acts in the space with the first N sites occupied, and is equal to ĀN (z). From (70) we see that |R, V N are its eigenvectors with eigenvalue Ω(V N ž): The open Toda spin chain wave functions usually denote the (right) eigenvectors of A N (z) (20). To properly define them, it is necessary to relax the constraint λ ′ N ≥ 0 and allow the spins λ ′ k to take integer values in order to be able to diagonalize S N , its z N coefficient. So, the definition of λ ′ needs to be extended from partitions to sequences of decreasing integers. The homogeneity property [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] (4.17) p 325:

ĀN (z)|R, V N = Ω(V N ž)|R, V N , (94)
P λ+1 N (V N ) = v 1 • • • v N P λ (V N ) ( 95
)
enables to extend the definition of |R, V . From (63):

R, Ǔ|S N = λ P ω λ ′ +1 N ( Ǔ ) λ| = ǔ1 • • • ǔn R, Ǔ|
we deduce:

SN |R, V = v 1 • • • v N |R, V . (96)
We now can obtain the (right) eigenvectors of A N (z) (20) from those of ĀN as follows. Let us denote

v = v 1 • • • v N the eigenvalue of SN , and Ā′ N (z, v) the restriction of ĀN (z) to the space S N = v. So, in Ā′ N (z, v) we substitute v to SN . We can factor |R, V = (vS N ) -1 ∞ |R ′ , V to obtain the eigenstate |R ′ , V of Ā′ N (z, v).
Similarly, denoting |W h , V (W h for Whittaker) the eigenstate of A N (z), and A ′ N (z, v) the restriction of A N to the space A N = v where we substitute v to S N . We can factor out δ(S N /v) to obtain the eigenstate

|W ′ h , V of A ′ N (z, v). We recover A ′ N (z, v
) when we substitute S N -k /v to S k in the expression of Ā′ N (ž, v) . From this, the eigenvector of the Toda chain with the eigenvalue Ω( ǓN z) is given by:

|W h , U = δ(u 1 • • • u N S N )(u 1 • • • u N S N ) ∞ |R, U , (97)
which means the inverse Pochhammer symbol must be replaced by a delta function. For example, the two body wave function is obtained by replacing

(S 2 v 1 v 2 ) -1 ∞ by δ(S 2 v 1 v 2 ) in (69).
6 Baxter Q Matrix

Diagonalization of the closed Toda chain Transfer matrix

In this section, we recover the periodic q-boson, equivalently the closed discretized Toda-chain Bethe equations using Baxter Q-Matrix approach [START_REF] Baxter | Exactly Solved Models in Statistical Mechanics dover publications inc[END_REF] chapter 9, or [START_REF] Gaudin | Wave Function[END_REF][26] chapter 8.6. The results reported in this section have been obtained in collaboration with Olivier Babelon and Simon Ruisjenaars [START_REF] Babelon | Quantization of the Relativistic Toda Chain[END_REF]. The q-Toda Q-matrix (116) was obtained in a previous work with Michel Gaudin [START_REF] Gaudin | [END_REF] and related Q-matrices appear in recent works [16][28] [29].

The Bethe roots (178) and the corresponding eigenvalues (179) are recovered as the solutions of the following divisibility problem (here specialized to the q-boson limit s = 0):

Λ N (z)q n (z) = q n (tz) + xz N t n q n (z/t), ( 98
)
where Λ N (z) and q n (z) are respectively degree N and n polynomials. The strategy to diagonalize the closed chain transfer matrix Λ Toda N,n (z) (81) consists in obtaining (98) as a matrix equation where Λ N (z) = Λ Toda N,n (z), q n (w) form a commuting family of polynomial matrices of the degree N and n respectively. One then recovers (98) as the scalar equation for their joined eigenvalues.

In simplest nontrivial case, N = n = 2, the configuration space is three dimensional (λ ′ 0 = 2, λ ′ 2 = 0) with the basis |λ ′ 1 where λ ′ 1 takes the values 2, 1, 0, and the expressions of Λ 2 (z) and q 2 (z) are given by:

Λ 2 (z) =   1 + z 2 x (1 -t)z . (1 -t 2 )zx 1 + z 2 x (1 -t 2 )z . (1 -t)zx 1 + z 2 x   , (99)
q 2 (z) =   1 -z z 2 -(1 + t)zx 1 + z 2 x -(1 + t)z z 2 x 2 -zx 1   . ( 100
)
Our aim here is to prove (98) in the general case and to relate the q-matrix to the Hall Pieri rules. The first step is to obtain (98) as a vector equation. For this, we look for a null vector of a matrix conjugated to:

B N = (T D ) 12 , (101)
where T Toda D = T N D Toda and T N is defined as:

T N (z) = L Toda 1 • • • L Toda 2N +1 , (102)
and D Toda is the diagonal matrix: (1, t n ) (82). For future convenience, in the definition of T N , we have labeled the Toda Lax matrices (77) with odd numbers.

We perform a similarity transformation on the L Toda 2k+1 matrices defining:

L ′ 2k+1 (z) = M 2k L 2k+1 (z)M -1 2k+2 , (103)
where M 2k is a triangular two by two matrix:

M 2k = 1 x 2k . 1 , (104)
and

x k = t λ ′ k . So, if we denote T ′ D the matrix obtained by substituting L ′ 2k+1 to L 2k+1 in T D , we have T ′ D = M 0 T D M -1 0 .
We look for a null eigenvector ψ 2k+1 of (L ′ 2k+1 ) 12 , which amounts to the condition:

(zX 2k+1 (1 - x 2k x 2k+1 ) + (1 - x 2k+1 x 2k+2 ))ψ 2k+1 = 0, ( 105
)
with the solution:

ψ 2k+1 = λ ′ 2k+1 ψ λ ′ 2k ,λ ′ 2k+2 λ ′ 2k+1 (-z)|λ ′ 2k+1 , (106)
where the sum is for

λ ′ 2k ≥ λ ′ 2k+1 ≥ λ ′ 2k+2
, and:

ψ a,c b (z) = z b-c a -c a -b t (107)
with a ≥ b ≥ c. Then, evaluating the action of the diagonal elements yields:

(L ′ 2k+1 ) 11 ψ 2k+1 (z) = ψ 2k+1 (tz) (L ′ 2k+1 ) 22 ψ 2k+1 (z) = zX k ψ 2k+1 (z/t). (108)
So, given a set of decreasing integers

λ ′ 0 ≥ λ ′ 2 ≥ • • • ≥ λ ′ 2N -2 ≥ λ ′ 0 -n, imposing the periodicity condition M 0 = M 2N D Toda , (109)
equivalently λ ′ 2N = λ ′ 0 -n a solution of: (T ′ D ) 12 ψ {λ ′ 2k } = 0 (110)
is given by:

ψ {λ ′ 2k } = ψ 1 ⊗ ψ 3 • • • ⊗ ψ 2N -1 . (111)
Due to the triangularity of L ′ 2k+1 ψ 2k+1 :

L ′ 2k+1 ψ 2k+1 = ψ 2k+1 (tz) . ⋆ zX 2k+1 ψ 2k+1 (z/t) (112)
the action of T ′ D on ψ {λ ′ 2k } is a lower triangular two by two matrix the diagonal elements of of which are the respectively the direct product of the Lax matrix diagonal elements. Taking its trace, we recover (98) as a set of vector equations when we substitute ψ {λ ′ 2k } to q.

The second step is to organize the vectors ψ {λ ′ 2k } into a matrix, by requiring the commutation:

Λ(z)q(z) = q(z)Λ(z) (113)
For this, let us rewrite (98) in a manifestly conjugate invariant way:

Λ H q H (z) = (t n /xz N ) 1/2 q H (tz) + (t n xz N ) 1/2 q H (z/t) (114)
where q = z n/2 q H . So, due to the fact that two hermitian matrices commute if and only if their product is hermitian, (113) will hold if we can define q H hermitian obeying (114).

In the Toda notations, the scalar product ( 5) is given by:

µ ′ 1 • • • µ ′ N |λ ′ 1 • • • λ ′ N = δ λ ′ ,µ ′ N 1 (λ ′ k -λ ′ k+1 )! t . (115)
For example, the norm of the states λ ′ 1 = 2, 1, 0 in the N = n = 2 case are respectively (t) 2 , (t) 2 1 , (t) 2 . Consider the matrix q defined by:

q(z) = {λ ′ 2l } ψ {λ ′ 2l } λ ′ 2l | = {λ ′ l } |λ ′ 2l+1 N -1 k=0 (-z) λ ′ 2k -λ ′ 2k+1 (λ ′ 2k -λ ′ 2k+1 )! t (λ ′ 2k+1 -λ ′ 2k+2 )! t λ ′ 2l | (116)
where the sum is on decreasing sequences of integers

λ ′ 0 = n ≥ λ ′ 1 ≥ • • • ≥ λ ′ 2N = 0 ≥ λ ′ 2N +1 = λ ′ 1 -n.
We can bring λ ′ 1 to be equal to n in the bra using (87):

|λ ′ 1 • • • λ ′ 2N -1 = x -λ ′ 2N+1 |n, λ ′ 3 -λ ′ 2N +1 , • • • , λ ′ 2N -1 -λ ′ 2N +1 . (117)
It defines a degree n polynomial matrix in z, the zero degree matrix elements being for λ ′ 2k+1 = λ ′ 2k , and the maximal degree elements for λ ′ 2k = λ ′ 2k-1 . It commutes with the translation T (16) acting on Toda configurations as:

T |λ ′ k = |λ ′ k+2 (118) qT = T q (119)
and the matrix q H = z -N/2 q obeys the hermitian relation:

T qH = q H (120)
Since T commutes with Λ, (113) is satisfied. Moreover, since all matrices commute with the momentum x, the commutation holds in each finite dimensional momentum sector.

Returning to standard notations:

µ ′ k = λ ′ 2k+1 , ν ′ k+1 = λ ′ 2k , q (
z) is given by an expression similar to (65):

q(z)|ν = µ (-z) ν ′ k -µ ′ k N k=1 ν ′ k -ν ′ k+1 ν ′ k -µ ′ k+1 t |µ (121)
where the sequences ν, µ are interlaced as:

ν ′ 1 = n ≥ µ ′ 1 • • • ≥ µ ′ N ≥ 0.
Last, we must show that q(u) commutes with Λ(z) and q(v). Here, we only show the first commutation by adapting an argument of Sklyanin [START_REF] Sklyanin | [END_REF].

Denote N the diagonal matrix of the

N |µ = µ|µ |µ . (122)
Given a matrix Λ, we denote its conjugated by N as:

Λ = N ΛN -1 , (123)
and so, the commutation relation to be shown becomes:

Λ(z)q(u)N -1 = q(u)N -1 Λ(z) (124)
To show this relation, let us introduce a Lax operator L(z) and the corresponding monodromy matrix:

T N (z) = L 1 (z) • • • L N (z) (125)
to represent q N (z) as:

q N (z)N -1 = tr s (T N (-z)D s ), (126)
and use (81) to represent Λ. The condition:

R σs D σ D s = D σ D s R σs (127)
where D σ = diag(1, t n ) determines D s (126) to be:

D s = S -n (128)
From the definition of the hermitian conjugation, Λ = t Λ(ž), is obtained by substituting LToda (z) to L Toda (z) in (102), where LToda

(z) = 1 + zX k zx k X k -x -1 k . . (129)
The commutation relation will then follow from the intertwining Yang-Baxter relation [START_REF] Faddeev | [END_REF]:

R σs (z/u)L Toda σx (z)L sx (u) = L sx (u) LToda σx (z)R σs (z/u) (130)
The notation L σx , L sx , means the matrices act respectively in the direct product of the two dimensional σ space and the x spin space, the direct product of the s and x spin spaces. Multiplying (130) by the permutation P xs of x and s to the right, it rewrites:

R σs (z/u)L Toda σx (z) Ľ(u) = Ľ(u) LToda σs (z)R σx (z/u) (131)
where Ľ = L sx P sx . If we take:

R σs (z) = 1 + zS s -s -1 (1 -S) -1 , (132)
This R-matrix is the q-deformed R-matrix of [START_REF] Gaudin | Wave Function[END_REF] (14.54) used by Sklyanin [START_REF] Sklyanin | [END_REF] in the Toda case, it also appears in Korff in a similar context [16]. The above relation rewrites in matrix form as:

1 + z/uS s -s -1 (1 -S) -1 1 + zX x -zXx -1 . Ľ(u) = Ľ(u) 1 + zS zsS -s -1 . 1 + z/uX x -x -1 (1 -X) -1 (133)
which amounts to the following relations:

Ľx = x Ľ ĽSX = SX Ľ Ľx/s = x/s(1 -S) Ľ u Ľ(1 -s/x)S = S Ľ (134)
to be satisfied with: sS = tSs and xX = tXx. The solution is factorized as:

Ľ(u) = G(S)F u (s/x) (135)
where F, G satisfy Pochammer (q-Gamma) identity:

uF u (s)(1 -s)S = SF (s) (1 -S)G(S)s = sG(S) (136)
With these condition satisfied, the two equations (139) are conjugated. Dividing the first one by the norm matrix (N ) to its right and projecting with P N to its left, it becomes:

(1 + z ǔ)A L N (z) ĀR N (u)N -1 = ĀR N (u)N -1 ÃL N +1 (z) (141)
where:

A L N (z) = T Toda N (z) 11 ÃL N +1 (z) = T Toda N +1 (z) 11 -T Toda N +1 (z) 12 . (142)
We take ĀR N (u) equal to:

ĀR N (u) = (T N +1 (u)) ∞0 N . (143)
In matrix form:

ĀR N (u)|λ = µ N +1 k=1 ǔλ k -µ k (λ k -λ k+1 )! t (λ k -µ k )! t (µ k -λ k+1 )! t |µ , (144)
with

λ 1 ≥ µ 1 • • • ≥ µ N ≥ λ N +1 ≥ µ N +1 = 0. ĀR N (u)
satisfies the required commutation relations with P M and P N , and the corresponding N = ∞ matrix Γ R,+ agrees with the definition (65) of the Hall Pieri rules.

To prove (141), consider the Matrix element of the Yang-Baxter equation (130):

R σs (z ǔ)(T Toda N +1 ) σ (z)(T N +1 ) s (u) = (T N +1 ) s (u)( T Toda N +1 ) σ (z)R σs (z ǔ) (145)
taken between the states 1| ⊗ ∞| and |2 ⊗ |0 in the direct product of the two dimensional σ-space with the spin s-space. We eliminate R on the lefthand side with 1| ⊗ ∞|R = (1 + z ǔ) 1| ⊗ ∞|, and on the-right hand side with R|2 ⊗ |0 = (|1 -|2 ) ⊗ |0 . We also remove L Toda N +1 on the left-hand side using L Toda N +1 |2 = |1 , due to the zero value of the boundary spin: µ N +1 = 0. The resulting expression is equal to (141).

Notice that unlike the commutation relations between Γ L,-(z) and Γ L,+ (u), the commutation relations between Γ L,-(z) and Γ R,+ (u) are already satisfied in finite size. The commutations between Γ R,± and Γ L,± can be obtained along the same line and we shall study the commutations of Γ R,± between themselves elsewhere [30].

A remark on the Half infinite chain and the Gaudin determinant

In this section, we consider the scalar product of two XXZ Bethe states on a semi-infinite chain which Gaudin expressed using the determinant (155) [START_REF] Gaudin | [END_REF]. Recently, Kirillov and Noumi [31], and Warnaar [32] obtained the same determinant by applying the Sekuiguchi-Debiard operator to the Cauchy identity (43). Betea and Wheeler [33] reinterpreted it as the scalar product of two q-boson wave functions on the semi-infinite chain. We present here an interpolating scalar product (We do not give the proof here, but the result can already be inferred from the two particle case) by extending the argument of [START_REF] Gaudin | La Fonction d'onde de Bethe[END_REF][25] chapter 4. Appendix B to an arbitrary spin. We now consider the SL 2 lax matrix reducing to (6) for s = 0:

(1 + zs)L s (z) = 1 + zK zS - S + z + K (146)
where:

K = sτ, S -= S -1 (1 -τ ), S + = S(1 -s 2 τ ). (147)
The scalar product such that S -is the adjoint of S + :

n|m s = (t) m /(s 2 ) m δ n,m (148)
replaces [START_REF] Faddeev | [END_REF]. The Bethe-Lieb eigenvectors of the semi-infinite chain take the form (168):

R s µ ( Ǔn ) = 1 n k=1 (1 + sǔ k ) P B(ǔ i )ξ(ǔ 1 ) µ 1 • • • ξ(ǔ n ) µn (149)
where P permutes the ǔk , and ξ(ǔ k ), B(ǔ i ), are defined in (173), (175). The normalization prefactor is put for further convenience. Notice that R s µ is proportional to the symmetrizer of the Hecke algebra ⋒ acting on the monomial n 1 ξ(ǔ i ) µ i where ⋒ defined as:

f ⋒ (ǔ l ) = (1 -t) n n! t P f (ǔ 1 , • • • , ǔn ) i<j ǔi -tǔ j ǔi -ǔj (150)
and the Hecke algebra is generated by:

f ⋒ k (ǔ l ) = f (• • • ǔk , ǔk+1 • • • ) ǔk -tǔ k+1 ǔk -ǔk+1 + (ǔ k ↔ ǔk+1 ) (151)
We consider the scalar product of two Bethe wave functions on the semi infinite line µ k ≥ 0:

∆ s n ( Ǔn , V n ) = 0≤µ 1 ≤•••≤µn 1 µ|µ s R s µ ( Ǔn )R s µ (V n ). (152)
It is given by:

∆ s n ( Ǔn , V n ) = t n(n-1) 2 (1 -t) n D n δ n ( 153
)
where δ n is the Cauchy determinant:

δ n = | 1 1 -tǔ k v l |, (154)
∆ n the Gaudin determinant:

D n = | 1 (1 -ǔk v l )(1 -tǔ k v l ) |. (155)
Therefore, the scalar product which interpolates between the XXZ-chain for ts 2 = 1 and the Hall-Littlewood polynomials for s = 0 does not depend on the spin. Kirillov, Noumi and Warnaar obtained the q-deformation of the Hall-Littlewood formula as:

λ P qt λ (V n )Q qt λ ( Ǔn ) n k=1 (1 -q λ k t n-k+1 ) = Ω( 1 -t 1 -q qV n Ǔn )t n(n-1) 2 (1 -t) n D n δ n (156)
where P qt is the Macdonald polynomial [START_REF] Macdonald | Symmetric Functions and Hall Polynomials 2nd[END_REF] and λ 1 ≥ • • • ≥ λ n ≥ 0. Indeed, if one sets q = 0 in the preceding expression, using (51), it rewrites:

λ R λ (V n )R λ ( Ǔn ) λ|λ = t n(n-1) 2 (1 -t) n D n δ n (157)
which is (153) for s = 0. Lascoux [34] reduced the q = 0 case to q = 0 as follows. The only q dependence of the right hand side of (156) comes from the first factor, and we can also factor it out from the left hand side. Representing the product as the eigenvalue of a Dunkl operator [35]) enables to rewrite the left hand side as a modification of the Cauchy Kernel equal to:

Ω( 1 -t 1 -q Ǔn V n )(1 -tτ q ) • • • (1 -t n τ q )⋒ (158)
with ǔi τ q = ǔi+1 and ǔn+1 = qǔ 1 . Let X n = X k + Xk where

X k = {x 1 , • • • x k }, Xk = {x k+1 , • • • x n }. We have: Ω( 1 -t 1 -q X n )τ k q = Ω( 1 -t 1 -q (qX k + Xk ))
where τ q is defined as above with ǔ → x. Substituting qX k + Xk = qX n + (1q) Xk , we deduce it is also equal to:

Ω((1 -t)X n )τ k 0 Ω(q 1 -t 1 -q X n ). (159)
So, (158) is also equal to:

Ω((1 -t)V n Ǔn )(1 -tτ 0 ) • • • (1 -t n τ 0 ) ⋒ Ω( 1 -t 1 -q qV n Ǔn ) (160)
reducing q = 0 to:

Ω((1 -t)V n Ǔn )(1 -tτ 0 ) • • • (1 -t n τ 0 )⋒ = t n(n-1) 2 (1 -t) n D n δ n (161)
which is (157).

Conclusions and perspectives

In this paper, we have studied the q-boson model and a discretized version of the Toda lattice on finite and semi infinite chains in relation with the theory of the Hall-Littlewood multi-variable orthogonal polynomials. We have observed that the semi infinite transfer matrices are vertex operators with Cauchy type commutation relations. As a very interesting application, we observe that the q-boson model appears via the Hall-Pieri rules when studying the current fluctuations of the Asymmetric Simple Exclusion Process, as explained by Lazarescu in part III of his thesis [36].

Although most of the semi infinite chain results are straightforward to extend to the Macdonald case, the underlying statistical models seem not to be expressible in terms of local Boltzmann weights. However, some important simplifications occur with the Hall-Pieri rules [37] which suggest that these Macdonald lattice models are simpler than they appear. One can also run the process backwards to define them on finite lattices. It will be interesting to investigate how the finite size version is connected to the Yang-Baxter equation and the possibility to use the Bethe-ansatz to obtain its spectrum.

Another point not touched here is the fact that the components of the eigenvectors are also eigenstates of the Sekuiguchi-Debiard operators.

The very simple discretized Toda chain we have introduced here can be diagonalized by coordinate Bethe-Anzats. It does not satisfy the more physical Hermitian condition of the Ruijsenaars q-Toda chain. Although we cannot use the coordinate Bethe-ansatz in that case, we expect that the Q-matrix diagonalization of section (6) reveals useful as for the Toda chain [38] (see for example [39][24]). m i (µ) ≤ 1. Therefore, the only λ which contribute to the sum are those for which m i (λ) ≤ 2.

We define the Bolzman weights:

ω 1 = 1 + zs ω 3 = z + s ω 5 = z(1 -t) ω 2 = z + ts ω 4 = 1 + zts ω 6 = 1 -s 2 ω 0 = 1 -s 2 t. (165)
The matrix elements for λ's with all parts unequal (λ i = λ i+1 for all i) can be written as:

Λ s |µ = M 1 D µ i λ i Dλ i µ i-1 |λ (166)
where µ 0 = µ M + N, and:

D mn = ω n-m-1 3 ω 6 , m < n ω 4 /ω 5 , m = n Dnm = ω m-n-1 1 ω 5 , n < m ω 2 /ω 6 , n = m. (167)
To properly evaluate the matrix elements Λ s λµ when the partitions λ has equal parts, we must correct the boundary terms λ i = λ i-1 = µ i-1 by substituting ω 0 ω 5 to ω 2 ω 4 for each occurrence of Dµ i µ i D µ i µ i .

For the eigenvector, we take the ansatz:

R µ = P ∈S M B(P )ξ µ 1 P 1 • • • ξ µ M P M (168)
where B(P ) are amplitudes to be determined, and we assume it is valid for

µ 0 ≥ µ 1 • • • ≥ µ M .
Let us write the (incorrect) eigenvalue equation as if (166) was still valid for λ's with equal parts:

(RA s ) µ = P ∈S M B(P ) µ i ≤λ i ≤µ i-1 M 1 D µ i λ i Dλ i µ i-1 ξ λ i P i . ( 169
)
For a fixed permutation, we can effect the sum over each λ i independently in (169), and using:

m 2 ≤n≤m 1 D m 2 n Dnm 1 ξ n = ( ξω 5 ω 6 ω 1 -ξω 3 + ω 4 )ω m 2 -m 1 -1 1 ξ m 2 + (ω 2 - ω 5 ω 6 ω 1 -ξω 3 )ω m 2 -m 1 -1 3 ξ m 1 = X 2 + Y 1 (170)
where the subscript of X, Y coincides with that of the exponent of ξ, we get for P = 1:

µ i ≤λ i ≤µ i-1 M 1 D µ i λ i Dλ i µ i-1 ξ λ i i = (X M + Y M -1 )(X M -1 • • • Y 1 )(X 1 + Y 0 ) (171)
where it is understood that X i + Y i-1 is a function of ξ i . To recover the correct eigenvalue equation, the term ω 2 ω 4 in the products Y i X i needs to be replaced by ω 0 ω 5 .

We can choose the amplitudes B(P ) such that after the summation over the permutations, only the terms

X 1 • • • X M , Y 0 • • • Y M -1 ,
survive. this will be the case if we can cancel the terms Y i X i by requiring: B(P )Y i (ξ P i+1 )X i (ξ P i ) + B(P (i, i + 1))Y i (ξ P i )X i (ξ P i+1 ) = 0 (172)

where (i, j) denotes the permutation of i with j. The product (ξ P i ξ P i+1 ) λ i factors out, and using the Orbach parametrization:

ξ i = ξ(u i ) = u i + s 1 + u i s (173)
the condition (172) becomes:

B(P ) B(P (i, i + 1))

= -u P itu P i+1 u P i+1tu P i (174) with the solution B(P ) = B(u P 1 , • • • , u P M ):

B(u k ) = i<j u i -tu j u i -u j (175)
In the closed chain case, we must take into account the periodicity conditions S + k+N = XS + k , S - k+N = X -1 S - k , which impose:

R µ n+N ,µ 1 ,•••µ n-1 = XR µ 1 ,•••µn , (176)
or:

B(P )ξ N P 1 = XB(P C) (177)
where C k = k + 1. Then, using (174) to express B(P C)/B(P ) as a product over the two-body scatterings, this condition gives the Bethe equations determining the u i 's:

Xξ(u k ) -N = l =k u k -tu l tu k -u l , (178)

  and thus we have obtained the left eigenvectors of the open Toda chain transfer matrix.

[1] (4.9) p.225 enables to formally identify the dual states λ|, |λ with the dual polynomials P λ (X), Q λ (X) in an infinite number of variables |X| = ∞, so that using the Cauchy relation,(38) rewrites:|V = Ω t (V X)(53)and the transfer matrix Γ -(z) act multiplicatively as:Γ -(z)α(X) = Ω t (zX)α(X). (54)The scalar product can also be defined by its reproducing Kernel[START_REF] Garsia | Séminaire Lotharingiens de Combinatoire B42m[END_REF]:α(X)|Ω t (XY ) = α(Y ). (55)Γ + (z)is defined by the Hermitian condition (23) Γ + (z) = Γ-(z), and from Γ -(z)α|Ω t (XY ) = Ω t (zX)α|Ω t (XY ) = α|Ω t ((X + z)Y ) we deduce Γ + (z) acts by "adding ž to X"[START_REF] Garsia | Séminaire Lotharingiens de Combinatoire B42m[END_REF][19]:Γ + (z)α(X) = α(X + ž).(56)

Acknowledgement

V. Pasquier is grateful to Junichi Shiraishi for numerous friendly discussions, in particular for explaining him the Pieri rules.

We also thank Olivier Babelon, Jérémie Bouttier, Sylvie Corteel, Alexandre Lazarescu and Simon Ruisjenaars for illuminating discussions.

Part of this work was done during the workshop "Beyond Integrability" held at the CRM, Montreal.

and are represented as follow in the basis where s is diagonal s|n = t n |n :

In matrix form:

with µ s ≥ ν x ≥ ν s . Substituting (138), ( 128) in (126), we verify that q(z) agrees with our previous expression (121), concluding the proof that q(z) and Λ(w) commute.

The Hall Pieri rule

In this section, we obtain the expression of the vertex operators Γ R,± (65) by solving the two hermitian conjugated relations:

The first equation tells us that if we iterate the action of Γ L,-(v i ) on the vacuum, we create the state |L, V eigenstate of Γ R,+ (z) with the eigenvalue Ω(žV ). It is therefore equivalent to the Hall-Pieri rule (72). The second equation tells us that if we iterate the action of Γ R,-(v i ) on the vacuum, we create the ω-dual states |R, V , eigenstates of Γ L,+ (z) with the same eigenvalue Ω(žV ), closely connected to the open Toda chain eigenstates as we saw in section (5). Let P N denote the projector onto partitions |λ having the multiplicity of 1, λ 1 , less or equal to N: λ ′ N +1 = 0, and P M the projector onto partitions of length l(λ) = λ ′ 1 less or equal to M. By construction, Γ L,-P M = P M +1 Γ L,-P M , and the finite size q-boson transfer matrices defined in (19) are obtained by projection:

Define the finite size projections of Γ R,+ to be ĀR N where ĀR N = P N Γ R + = P N Γ R + P N +1 , and require Γ R + P M = P M Γ R + P M .

A Bethe-Lieb diagonalization

In this appendix, we partially diagonalize the transfer matrix (12) of the spin chain considered in section ( 7) by adapting the discussion of [START_REF] Gaudin | La Fonction d'onde de Bethe[END_REF] [25] section 7.3. or [START_REF] Baxter | Exactly Solved Models in Statistical Mechanics dover publications inc[END_REF] sections 8.2-4 to this case. The Bethe-Lieb technique has the advantage to provide an explicit expression of the eigenfunctions (which are studied thoroughly in [40] in the infinite chain limit). The monodromy matrix is:

where L s is defined by (146), we want to find the left eigenvectors of:

with D = (1, X) (13).

In the sector of M Bosons, the eigenvalue equation writes:

where l(µ) = l(λ) = M. We limit our treatment to the states µ such that the lattice sites are occupied by at most one Boson. All the parts µ k are distinct:

and the eigenvalue is the prefactor of

The preceding discussion only proves the validity of the ansatz (168) with B(P ) given by (175) for configurations λ with m k (λ) ≤ 2, and we will assume it is correct for m k (λ) arbitrary nonnegative integers.


David Pirtskhalava
email: david.pirtskhalava@sns.it

Luca Santoni
email: luca.santoni@sns.it

Enrico Trincherini
email: enrico.trincherini@sns.it

Filippo Vernizzi
email: filippo.vernizzi@cea.fr

Large Non-Gaussianity in Slow-Roll Inflation

Canonical models of single-field, slow-roll inflation do not lead to appreciable non-Gaussianity, unless derivative interactions of the inflaton become uncontrollably large. We propose a novel slow-roll scenario where scalar perturbations propagate at a subluminal speed, leading to sizeable equilateral non-Gaussianity, f equil NL ∝ 1/c 4 s , largely insensitive to the ultraviolet physics. The model is based on a low-energy effective theory characterized by weakly broken invariance under internal galileon transformations, φ → φ + bµx µ , which protects the properties of perturbations from large quantum corrections. This provides the unique alternative to models such as DBI inflation in generating strongly subluminal/non-Gaussian scalar perturbations.

Introduction:

The simplest, textbook version of inflation consists of a single canonical scalar field-the inflaton-slowly rolling down a sufficiently flat potential. It is a common feature of these models that the magnitude of non-Gaussianity is suppressed [1] well below the observable level for any foreseeable future; see, e.g., [2].

Generically, in the low-energy inflationary effective field theory (EFT), there are additional higher-derivative corrections to the canonical action, and one may wonder whether these can impact non-Gaussianity in any significant way. For example, consider the following Lagrangian

L = √ -g - 1 2
(∂φ) 2 -V (φ) + (∂φ) 4 Λ 4 + . . . , (1) where V (φ) denotes the inflaton potential and the last term provides a higher-order correction in the derivative expansion, governed by the EFT cutoff Λ. The virtue of such terms is that they do not renormalize the potential, while they do contribute to the non-Gaussianity [3],

f NL ∼ φ0 2 Λ 4 , (2)
where φ 0 denotes the background expectation value of φ and f NL is the nonlinearity parameter of scalar perturbations [26]. It is clear from eq. ( 2) that having f NL ∼ > 1 in this model implies going beyond the (at least apparent) regime of validity of the low-energy effective theory, which requires (∂φ) 2 ≪ Λ 4 . Therefore, unless the infinite number of derivative operators in the ellipses of eq. ( 1) can be resummed, one cannot trust values of non-Gaussianity greater than one. An example of such a resummation is provided by Dirac-Born-Infeld (DBI) inflation [4,5], where a higher-dimensional spacetime symmetry (nonlinearly realized on φ) protects the coefficients of the leading derivative operators from large quantum corrections. For a small speed of sound of scalar perturbations, c 2 s ≪ 1, DBI inflation predicts equilateral non-Gaussianity [6], with the amplitude f equil NL ∼ 1/c 2 s [5,7]. In this Letter, we propose a novel inflationary scenario, where the energy density of the early universe is dominated by the potential of a slowly-rolling scalar field, similarly to ordinary slow-roll theories. Yet, a definite set of higher-derivative interactions of the inflaton become relevant, leading to observably large non-Gaussianities. Nevertheless, the theory is predictive, since all the rest of the operators in the derivative expansion remain naturally small in the full quantum theory. These properties follow from the weakly broken [8] invariance under internal galileon transformations [9],

φ → φ + b µ x µ , (3)
that defines the Lagrangian our theory is based on. While the symmetry (3) has appeared in a variety of physical contexts, ranging from modified gravity [9][10][11][12] to scattering amplitudes [13], here we use it as a guideline for constructing largely UV-insensitive models of the early universe. To our knowledge, (3) is the only alternative to DBI-like symmetries for protecting the coefficients of strongly-coupled higher-derivative operators against large quantum corrections.

Just like in DBI inflation, enhanced scalar non-Gaussianity is generically associated with a reduced speed of sound of perturbations in our model; however, the enhancement is much stronger compared to the DBI case, the amplitude of equilateral non-Gaussianity growing as f equil NL ∝ 1/c 4 s for small c 2 s . The model: The theory we wish to study below is defined as a combination of an inflationary potential and the four Lagrangian terms [8], which, barring the factors of √ -g, can be written as follows

L 2 = Λ 4 2 G 2 (X) , (4)
L 3 = Λ 4 2 G 3 (X)[Φ] , (5)
L 4 = M 2 Pl G 4 (X)R + 2Λ 4 2 G ′ 4 (X) [Φ] 2 -[Φ 2 ] , (6)
L 5 = M 2 Pl G 5 (X)G µν Φ µν - 1 3 Λ 4 2 G ′ 5 (X) [Φ] 3 -3[Φ][Φ 2 ] + 2[Φ 3 ] . (7)
Here, Φ is a matrix, consisting of second derivatives of the inflaton, Φ µ ν ≡ ∇ µ ∇ ν φ/Λ 3 3 , and the brackets [. . .] denote the trace operator. Moreover, G a are arbitrary dimensionless functions of the dimensionless variable 1 X ≡ -g µν ∇ µ φ∇ ν φ/Λ 4 2 . For simplicity, we will assume that these functions can be Taylor-expanded around zero,

G a (X) = ∞ n=0 c (n)
a X n . Furthermore, the two scales in the theory are related to each other as Λ 4 2 = M Pl Λ 3 3 , so that the smaller of these, Λ 3 , can be regarded as the genuine cutoff of the underlying low-energy EFT (we assume M Pl is the parametrically highest scale in the problem).

In the limit M Pl → ∞, the Lagrangian terms in (4)-( 7) reduce to the galileons of Ref. [9], which are exactly invariant (up to a total derivative) under (3). In this limit, there is a non-renormalization theorem, according to which the galileon operators are not corrected, at least perturbatively, by quantum loops [14]. For a finite Planck mass, the operators (4)- (7) break the galileon symmetry, but only weakly [8]. The defining property of the theories with weakly broken invariance under (3) is a generalization of the non-renormalization theorem of Ref. [14], which renders the quantum corrections to the coefficients c

(n) a suppressed by positive (integer) powers of the tiny ratio Λ 3 /M Pl [8]. In addition, these theories belong to the so-called Horndeski class of scalar-tensor models [15], characterized by second-order equations of motion both for the scalar and the metric [16].

It has been shown in Ref. [8] that the properties of the theories with weakly broken galileon symmetry imply the possibility of a moderately coupled, yet predictive, regime characterized by

X = φ2 0 Λ 4 2 ∼ < 1 , Z ≡ H φ0 Λ 3 3 ∼ < 1 , (8)
for a homogeneous φ-profile on a FRW background with the Hubble rate H. From now on, X will be under-

1
We use the 'mostly plus' signature for the metric.

stood as evaluated on the background solution. Despite the moderate coupling, quantum corrections are under control when the scalar background profile satisfies (8)even in the case that these inequalities are saturatedand the predictions of the classical theory can be trusted.

The slow-roll backgrounds: As noted above, we will be interested in the potentially-dominated models of inflation, characterized by weakly broken invariance under the galileon transformations. These are governed by the following Lagrangian

L = √ -g M 2 Pl 2 R - 1 2 (∂φ) 2 -V (φ) + 5 i=2 L i + . . . , (9)
where, since we have extracted the canonical scalar and graviton kinetic terms, G 2 is assumed to start at least quadratic in X, while G 3 can have a linear piece. From now on we will set G 4 = G 5 = 0 for the sake of simplicity; generalization to the case of nonzero G 4 and G 5 is straightforward and will be commented on where appropriate. The ellipses in ( 9) denote an infinite number of other operators, present in the low-energy effective theory. We will assume that the potential V (φ) satisfies the ordinary slow-roll conditions, ǫ V ≪ 1 and

|η V | ≪ 1,
where the (potential-based) slow-roll parameters ǫ V and η V are defined as

ǫ V ≡ M 2 Pl 2 V ′ V 2 , η V ≡ M 2 Pl V ′′ V . (10)
The previous analysis of quantum loops, leading to the non-renormalization theorem summarized above, has concentrated on the case with a vanishing potential [8].

It is straightforward to show that the same results remain intact also in the presence of a nonzero, but sufficiently flat V (φ), satisfying (10).

For the flat FRW ansatz, ds 2 = -dt 2 + a 2 (t)d x 2 , the two Friedmann equations that follow from (9) read

3M 2 Pl H 2 = V -Λ 4 2 X 1 2 + G 2 X -F (X, Z) , (11)
2M 2 Pl Ḣ = -Λ 4 2 XF (X, Z) + 2M Pl XG ′ 3 φ0 , (12)
where H ≡ ȧ/a and the function F (X,

Z) ≡ 1 + 2G ′ 2 - 6ZG ′
3 has been introduced for later convenience. Moreover, in the slow-roll regime the homogeneous equation of motion of φ reduces to

3H φ0 F (X, Z) ≃ -V ′ (φ 0 ) . ( 13
)
We are interested in a regime where higher-derivative operators in (9) become important, while the quantum corrections are still under control. To this end, we assume Z ∼ 1, which also fixes the magnitude of the parameter X. Indeed, from the definition of X and Z, eq. ( 8), it follows that √ X = Λ 2 2 Z/(M Pl H); making use of eq. ( 12), one immediately obtains X ∼ √ ǫ, where ǫ ≡ -Ḣ/H 2 . Note the order-unity slowly varying function of time F (X, Z) in eqs. ( 11)- (13), which is strictly one in canonical slow-roll inflation. Apart from this minor modification, all the equations that describe the background solution are similar to those of ordinary slow-roll models (up to corrections of higher order in ǫ V and η V ). In particular, unlike e.g. the DBI case, the usual flatness conditions ǫ V ≪ 1 and |η V | ≪ 1 need to hold for sustaining the quasi-de Sitter phase in our model. It is precisely for this reason that we refer to it as "slow-roll". 2 At the level of perturbations, our scenario is of course very different from the canonical slow-roll inflation; for example, unlike the latter, the scalar perturbations become strongly coupled at an energy scale parametrically smaller than M Pl , something we discuss in greater detail below.

It follows from the Friedmann equations that the contributions from the derivative operators in (9) to the inflationary energy density and pressure are proportional to X ∼ √ ǫ. One may wonder therefore, whether loop corrections can outweigh these contributions for small values of ǫ. For Z ∼ 1, the leading quantum corrections to the background stress tensor scale as ∼ Λ 4 3 [8]. This should be much smaller than Λ 4 2

√

ǫ, which implies a lower bound on the slow-roll parameter, ǫ ≫ (H/M Pl ) 2 . This is the same bound on ǫ as the one that arises from requiring quantum fluctuations of the inflaton to be small [22,23].

Non-Gaussianity: Inflationary theories can be conveniently studied in a model-independent way using the EFT framework [19,20]. To this end, we decompose the metric in the ADM variables, (14) and work in the unitary gauge, where the constanttime hypersurfaces are chosen to coincide with those of uniform φ. The perturbed quantities are defined as δN ≡ N -1, and δK ≡ K -3H, where K denotes the trace of the extrinsic curvature of equal-time hypersurfaces. The action in eq. ( 9) (with G 4 = G 5 = 0) can be expanded to the cubic order in perturbations in the following way [21]

ds 2 = -N 2 dt 2 + γ ij N i dt + dx i N j dt + dx j ,
S = d 4 x √ -g M 2 Pl 2 R -M 2 Pl Ḣ 1 N 2 -M 2 Pl (3H 2 + Ḣ) + M 4 2 2 δN 2 + M 4 3 δN 3 -M 3 1 δKδN + M 3 2 δKδN 2 , ( 15
)
2 Inflationary models based on particular subsets of the Lagrangian terms in ( 4)-( 7) have been studied in Refs. [17,18]. However, these references have focused on kinetically-driven inflation, corresponding to Λ 4 2 ∼ M 2 Pl H 2 and X ∼ Z ∼ 1 in our notation.

where M 2 , M 3 , M1 and M2 are functions of time of canonical dimension one. In terms of the functions G 2 and G 3 introduced in eqs. ( 4) and ( 5), these are

M 4 2 = -2Λ 4 2 X 3ZG ′ 3 + 6ZXG ′′ 3 -2XG ′′ 2 -Y G ′ 3 , M 4 3 = -2Λ 4 2 X 3XG ′′ 2 + 2 3 X 2 G ′′′ 2 -Z 4G ′ 3 + 11XG ′′ 3 + 2X 2 G ′′′ 3 + Y G ′ 3 , M 3 1 = -2 Λ 4 2 X H ZG ′ 3 , M 3 2 = -2 Λ 4 2 X H Z (2G ′ 3 + XG ′′ 3 ) , (16)
where Y ≡ φ0 /Λ 3 3 . One can see from (16), that approximate invariance under galileon transformations imposes the following (radiatively stable) hierarchy among the various EFT coefficients:

M 4 2 ∼ M 4 3 ∼ M 2 Pl Ḣ, M 3 1 ∼ M 3 2 ∼ M 2 Pl Ḣ H . (17)
This is in stark contrast to what happens e.g. in solely shift-symmetric theories, where the coefficients that stem from higher-derivative operators such as M 3 1 and M 3 2 , are much stronger suppressed. The latter hierarchy motivates to define the dimensionless, order-one coefficients

α 1,3 ≡ - M 4 2,3 2M 2 Pl Ḣ , α 2,4 ≡ - M 3 1,2 H 2M 2 Pl Ḣ , ( 18
)
convenient for describing the parameter space of the theories at hand.

At sufficiently high energies (encompassing the Hubble scale), the dynamics of scalar perturbations is fully dominated by the dynamics of the adiabatic mode π, defined through φ( x, t) = φ 0 (t + π( x, t)) [19,20]. In the decoupling limit corresponding to this regime, the scalar part of the action (15) reads

S π = d 4 x a 3 (-M 2 Pl Ḣ) (1 + α 1 ) π2 -c 2 s (∂π) 2 a 2 + (α 2 -α 1 ) π (∂π) 2 a 2 -2(α 1 + α 3 ) π3 + 2 α 2 -α 4 H π2 ∂ 2 π a 2 + α 2 H (∂π) 2 ∂ 2 π a 4 , (19)
where the speed of sound is

c 2 s ≡ 1 + α 2 1 + α 1 . ( 20
)
It follows from eq. ( 20) that if, for whatever reason, the parameter α 2 happens to be close to -1, one can have strongly subluminal scalar perturbations. Most impor-tantly, the approximate galileon invariance guarantees that such an 'accidental' arrangement of the parameters is respected by loop corrections. This is qualitatively different from how the small c 2 s arises in models such as DBI inflation, as we discuss in detail below 3 .

It is worth stressing at this point that the operators in the last line of ( 19) can be rewritten in terms of those in the second line via a perturbative field redefinition [24]. This simply amounts to using the linear equation of motion in (19). After straightforward manipulations, one finds 4

S (3) π = d 4 x a 3 (-M 2 Pl Ḣ) γ 1 π (∂π) 2 a 2 + γ 2 π3 . (21)
The two operators in (21) are precisely those appearing in the decoupling limit of DBI theories and the bispectrum they produce is close to the equilateral shape [6].

The genuine difference arises once the magnitude of non-Gaussianities is concerned: instead of the f equil NL ∼ 1/c 2 s behaviour characteristic of DBI inflation [7], in theories with weakly broken galileon symmetry non-Gaussianity scales as f equil NL ∼ 1/c 4 s in the small-c 2 s limit. The latter scaling is due to the last operator in (19), whose precise contribution to the three-point function of the curvature perturbation ζ reads [18,25]

5 B ζ (k 1 , k 2 , k 3 ) = - 1 16
H 8 A 2 α 2 1 + α 1 1 c 10 s k 2 1 (k 2 1 -k 2 2 -k 2 3 ) k t (k 1 k 2 k 3 ) 3 × 1 + i>j k i k j k 2 t + 3 k 1 k 2 k 3 k 3 t + 2 perms . (22)
Here, A denotes the normalization of the π-kinetic term in the decoupling limit, A ≡ (-M 2 Pl Ḣ)(1 + α 1 ). The amplitude of non-Gaussianity can be directly read off from eq. ( 22)

f equil NL = 5 18 k 6 * B ζ (k * , k * , k * ) ∆ 2 ζ * = 65 162 α 2 1 + α 1 1 c 4 s , (23)
3 In the limit X → 0, using eq. ( 16) with G 4 = G 5 = 0 in eq. ( 18), one finds α 1 = 3α 2 . This gives a negative kinetic term to π for α 2 ≃ -1. The parameter X need not be very small, however; it is of order X ∼ √ ǫ ≃ a few × 0.1 in e.g. slow-roll models with monomial potentials. Moreover, the relation α 1 = 3α 2 no longer holds for G 4 = 0 or G 5 = 0. 4 Explicitly, the coefficients γ 1 and γ 2 read

γ 1 ≡ (c 2 s -1) 1 + 2 c 2 s + (1 + c 2 s )α 1 , γ 2 ≡ 2 1 - 1 c 2 s 2 + 1 c 2 s + 2 c 2 s (α 1 -2α 4 ) + 2α 1 -2α 3 .
5 We follow the standard definition of the comoving curvature perturbation, g ij = a 2 e 2ζ δ ij . The three-point function is de-

fined as ζ( k 1 )ζ( k 2 )ζ( k 3 ) = (2π) 3 δ (3) i k i B ζ (k 1 , k 2 , k 3 ), and kt ≡ k 1 + k 2 + k 3 .
where ∆

ζ ≡ k 3 P ζ (k) = H 4 /(4Ac 3
s ) is the dimensionless power spectrum, evaluated at a fiducial momentum scale k * . A significantly reduced speed of sound, α 2 ≃ -1 (see eq. ( 20)), implies a negative f NL . However, due to the strong dependence of f NL on c 2 s , even slightly subluminal perturbations can produce a sizeable amount of non-Gaussianity; for example, a 10 % tuning of the α 2 parameter can give rise to f equil NL ≃ -70 for α 1 = 1 and α 2 = -0.9. We stress that such a tuning is not 'unnatural' as a result of the non-renormalization theorem outlined above.

If the theory (19) is to be predictive, it is crucial that π is weakly coupled at energies of the order of the inflationary Hubble rate, Λ ⋆ ≫ H, where Λ ⋆ is the energy scale at which perturbative unitarity is violated in the 2 → 2 scattering of π. In the c 2 s ≪ 1 limit, Λ ⋆ is set by the last interaction term in the action (19), and is estimated as Λ ⋆ ∼ Λ 3 c 11/6 s . For α 1 ≃ 1 and α 2 ≃ -1, using the experimental value ∆ ζ ≃ 5π 2 × 10 -9 , one finds

Λ 3 ⋆ ∼ O(50) f equil NL (8H) 3 . (24)
Even for the largest f equil NL compatible with the current observational bounds [26], the strong coupling scale is fairly above H, but well below the symmetry breaking scale [27]

, Λ 3 b ∼ O(5) f equil NL Λ 3
⋆ . We close this section with a remark concerning the regime of validity of the decoupling limit. For the small values of the speed of sound we are interested in, one should be careful with mixing terms that involve spatial derivatives. For example, consider the δN δK operator in eq. ( 15). The most important mixing of scalar modes with gravity that arises from this operator is suppressed by a factor of ǫ/c 2 s compared to the π-kinetic term at horizon crossing. Therefore, the validity of the decoupling limit analysis requires that c 2 s ∼ > ǫ hold, which puts an upper bound (f NL ∼ < 1/ǫ 2 ) on the amplitude of non-Gaussianity attainable within the decoupling limit.

Discussion: In 'ordinary' inflationary theories, the statistical properties of perturbations are mostly determined by operators with the least number of derivatives, i.e. (∂φ) 2n or, in the EFT language, by operators of the form (δN ) n . The prototype example of this is provided by DBI inflation, where the background can be consistently strongly coupled, with an infinite number of operators of the above type becoming relevant for its dynamics. In the EFT of eq. ( 15), this corresponds to the large coefficient M 4 2 or, equivalently, to α 1 ≫ 1 in the notation of eq. ( 18), and implies a parametrically suppressed speed of sound, c 2 s ≃ 1/α 1 , see eq. ( 20) (higher-derivative operators are effectively negligible in DBI inflation, so one can set α 2 ≃ 0). Despite the strong coupling, the symmetries of DBI theories protect the structure of the Lagrangian from large quantum corrections. This mech-anism of obtaining strongly sub-luminal scalar perturbations has been extensively studied in the literature (see, e.g., [5,7,[27][28][29]) and provides an attractive way of generating large equilateral non-Gaussianity in single-field inflation.

In this work we have proposed an alternative scenario that allows for strongly subluminal scalar perturbations within a well-defined low-energy EFT. Our model relies on moderate coupling, i.e. α i ∼ 1, as a result of which more than one operator in eq. ( 15) become large enough to affect the scalar perturbations significantly. In the simplest realization considered above, the scalar twoand three-point functions are determined by the operators δN 2 and δN δK. Instead of the α 1 ≫ 1, α 2 ≪ 1 case characteristic of DBI inflation, our mechanism relies on an adjustment of the order-unity α 1 and α 2 , which results in a somewhat reduced speed of sound in (20). The theories we have studied in this work not only allow for such an adjustment, but also provide a way to protect it against loop corrections. The central reason behind the robustness of the results obtained in the classical theory is the weakly broken galileon symmetry [8], which inherits the remarkable quantum properties of the galileon operators [9], exactly invariant under (3).

Acknowledgements: We thank Daniel Baumann, Paolo Creminelli and Greg Gabadadze for valuable discussions and comments on the manuscript. The work of D.P. is supported in part by MIUR-FIRB grant RBFR12H1MW and by funds provided by Scuola Normale Superiore through the program "Progetti di Ricerca per Giovani Ricercatori". The work of E.T. is supported in part by MIUR-FIRB grant RBFR12H1MW. F.V. acknowledges partial support from the grant ANR-12-BS05-0002 of the French Agence Nationale de la Recherche.