REALISATION OF LUSZTIG CONES

PHILIPPE CALDERO, ROBERT MARSH, AND SOPHIE MORIER-GENOUD

Abstract. Let U_{q}(𝔤) be the quantised enveloping algebra associated to a simple Lie algebra 𝔤 over ℂ. The negative part U^{−} of U_{q}(𝔤) possesses a canonical basis B with favourable properties. Lusztig has associated a cone [20] to a reduced expression i for the longest element w_{0} in the Weyl group of 𝔤, with good properties with respect to monomial elements of B. The first author [8] has associated a subalgebra A_{i} of U^{−}, compatible with the dual basis B^{∗}, to each reduced expression i. We show that, after a certain twisting, the string parametrisation of the adapted basis of this subalgebra coincides with the corresponding Lusztig cone. As an application, we give explicit expressions for the generators of the Lusztig cones.

1. Introduction

Let U = U_{q}(𝔤) be the quantum group associated to a semisimple Lie algebra 𝔤. The negative part U^{−} of U has a canonical basis B with favourable properties (see Kashiwara [14] and Lusztig [18, §14.4.6]). For example, via action on highest weight vectors it gives rise to bases for all the finite-dimensional irreducible highest weight U-modules. The dual canonical basis B^{∗} of the positive part U^{+} has good multiplicative properties. Two elements of B^{∗} are said to be multiplicative if their product also lies in B^{∗} up to a power of q.

The first author has shown that for each reduced expression i = (i_{1}, i_{2}, …, i_{N}) for the longest element w_{0} (see § 2.2) in the Weyl group of 𝔤, there is a corresponding subalgebra A_{i} of U^{+}, known as a standard adapted subalgebra, with basis given by A_{i} ∩ B^{∗}, consisting entirely of elements which are pairwise multiplicative. The subalgebras A_{i} are q-polynomial algebras, i.e. algebras given by generators and qcommuting relations, with GK-dimension N = l(w_{0}). Note that adapted algebras were introduced for the Berenstein-Zelevinsky conjecture and are connected with the larger theory of cluster algebras [11].

By a Lusztig cone of U, we mean the cone L_{i} ⊆ ℕ^{N} associated by Lusztig (see [20, §16]) to each reduced expression i for w_{0}. In [20] these cones arise naturally from the linear term of a nonhomogeneous quadratic form associated to i which is used by Lusztig to give a positivity condition for a monomial

(1.1)  F_{i_{1}}^{(a_{1})}F_{i_{2}}^{(a_{2})} ⋯ F_{i_{N}}^{(a_{N})}

to lie in the canonical basis. Here the F_{i} are the standard generators of U^{−}. Monomials of this form with (a_{1}, a_{2}, …, a_{N}) lying in the Lusztig cone corresponding to i lie in the canonical basis in types A_{1}, A_{2} and A_{3} [20], in type A_{4} [22] and in type B_{2} [31]; see also [27]. Counter-examples of M. Reineke [27] and N. H. Xi [32] show that this fails in general in type A_{5}. Recently, R. Bedard [2] has analysed the quadratic forms associated to these monomials and, as an application, was able to compute some interesting examples in types D_{4}, A_{5} and affine A_{1}.

Note that Lusztig cones are used to describe regular functions on a reduced real double Bruhat cell of the corresponding algebraic group [33], they have links with primitive elements in the dual canonical basis (this can be seen using [4]) and therefore with the representation theory of affine Hecke algebras [17], and they are known to correspond to regions of linearity of the Lusztig reparametrisation functions (see [10]).

Given a reduced expression i for w_{0}, elements of the dual canonical basis can be parametrised via the string parametrisation in direction i (see [4, §2], [15] and [25, §2]), which we denote by c_{i} : B^{∗} → C_{i}, where C_{i} ⊆ ℕ^{N} is known as the string cone corresponding to i.

Date: December 10, 2003.

Our main result is that the set of string parameters (in direction i) of a certain twisting of the standard adapted subalgebra of the dual canonical basis corresponding to i coincides with the Lusztig cone corresponding to i. The twisting is done with the help of the Schützenberger involution. This gives a realisation of all Lusztig cones in terms of the dual canonical basis. It also implies that all Lusztig cones are simplicial (generalising results of Bedard [1] and the second author [23]), and enables us to give an explicit description of their spanning vectors; see Theorem 8.10.

The paper is organised as follows. Sections 1 and 2 give preliminary results on quantum groups and the canonical basis, including its parametrisations associated to a reduced word, and adapted algebras. In Section 4, we introduce the Schützenberger involution ϕ and its action on the dual canonical basis. In Section 5, we recall some facts on geometric lifting of the canonical basis in order to give a formula which describes ϕ in terms of the parametrisation of the dual canonical basis. A remarkable property is that, with a good choice of parametrisations, the action of the Schützenberger involution on the dual canonical basis is given by an affine map.

In Sections 6 and 7, we apply the results from previous sections to describe explicitly the twisted standard adapted subalgebra associated to a reduced word i, in terms of i-string parametrisation. By the multiplicative property of the adapted subalgebra and the ”affine map” property, this can be provided by an N × N matrix and a column vector. A combinatorial argument, together with the known PBWparametrisation of the adapted basis of a standard adapted subalgebra, allows us to prove the main theorem: in Section 8, we realise the Lusztig cones in terms of the string parametrisation of twisted standard adapted subalgebras. As an application, we give an explicit formula for the generators of the cones.

2. Notation and preliminaries.

2.1. Let A = (a_{ij})_{1≤i,j≤n} be the Cartan matrix of a finite dimensional semi-simple Lie algebra 𝔤 over ℂ. Let 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫 be a triangular decomposition, where 𝔥 is a Cartan subalgebra and where 𝔫^{−}, 𝔫 are opposite maximal nilpotent subalgebras of 𝔤. Let {α_{i}}_{i} be the set of simple roots of the root system Δ resulting from this decomposition. The set of positive roots is denoted by Δ^{+}.

Let P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n. Set P^{+} := ∑_{i} ℤ_{≥0}ϖ_{i}, endowed with the ordering ∑_{i} λ_{i}ϖ_{i} ≤ ∑_{i} μ_{i}ϖ_{i} ⇔ λ_{i} ≤ μ_{i}. The Weyl group W is generated by the reflections s_{i} corresponding to the simple roots. We denote by < , > the W -invariant form on P; we have a_{ij} =< α_{j}, α_{i}^{∨} > for all i, j, where the α_{i}^{∨}’s are the simple coroots.

For n a nonnegative integer and α a positive root, we set : q_{α} = q^{<α,α>/2}, [n]_{α} = \frac{q_{α}^{n}−q_{α}^{−n}}{q_{α}−q_{α}^{−1}} , [n]_{α}! = [n]_{α}[n − 1]_{α} … [1]_{α}.

2.2. Let W be the Weyl group of 𝔤, with Coxeter generators s_{1}, s_{2}, …, s_{n} and corresponding length function. An expression s_{i_{1}}s_{i_{2}} ⋯ s_{i_{m}} for an element of w is called reduced if it is of minimal length; we identify such an expression with the tuple i = (i_{1}, i_{2}, …, i_{m}). Set N := dim 𝔫. It is known that N is the length of the longest element w_{0} of the Weyl group. Let R be the set of reduced expressions for w_{0}.

Fix i in R. Let L_{i} be the set of points (c_{1}, …, c_{N}) ∈ ℤ_{≥0}^{N} with the following property : for any two indices p < p′ in {1, …, N} such that i_{p} = i_{p′} = i and i_{q} ≠ i whenever p < q < p′, we have

c_{p} + c_{p′} + ∑_{p<q<p′} a_{i_{p},i_{q}}c_{q} ≤ 0.

The cone L_{i} is the so-called Lusztig cone associated to the reduced expression i. This is defined in [20, §16] for the simply-laced case. We use here a natural generalisation to the general case which also appears implicitly (for type B_{2}) in [31].

2.3. Let d be an integer such that < P, P >⊂ (2/d)ℤ. Let q be a indeterminate and set 𝕂 = ℂ(q^{1/d}). We define the simply connected quantised enveloping 𝕂-algebra U_{q}(𝔤) as in [12]. Set d_{i} =< α_{i}, α_{i} > /2 and q_{i} = q^{d_{i}} for all i. Let U_{q}(𝔫), resp. U_{q}(𝔫^{−}), be the subalgebra generated by the canonical generators E_{i} := E_{α_{i}}, resp. F_{i} := F_{α_{i}}, of positive, resp. negative, weights, subject to the quantum Serre relations.

For all λ in P, let K_{λ} be the corresponding element in the algebra U_{q}^{0} = 𝕂[P] of the torus of U_{q}(𝔤) and K_{i} := K_{α_{i}}. We have the triangular decomposition U_{q}(𝔤) = U_{q}(𝔫^{−}) ⊗ U_{q}^{0} ⊗ U_{q}(𝔫). Set

U_{q}(𝔟) = U_{q}(𝔫) ⊗ U_{q}^{0}, U_{q}(𝔟^{−}) = U_{q}(𝔫^{−}) ⊗ U_{q}^{0}.

The algebra U_{q}(𝔤) is endowed with a structure of Hopf algebra with comultiplication Δ, antipode S and augmentation ε given by

ΔE_{i} = E_{i} ⊗ 1 + K_{i} ⊗ E_{i}, ΔF_{i} = F_{i} ⊗ K_{i}^{−1} + 1 ⊗ F_{i}, ΔK_{λ} = K_{λ} ⊗ K_{λ},
S(E_{i}) = −K_{i}^{−1}E_{i}, S(F_{i}) = −F_{i}K_{i}, S(K_{λ}) = K_{−λ},
ε(E_{i}) = ε(F_{i}) = 0, ε(K_{λ}) = 1.

Let ( , ) be the Hopf bilinear form, [28], on U_{q}(𝔟) × U_{q}(𝔟^{−}), uniquely defined by

(E_{i}, F_{j}) = δ_{ij}(1 − q_{i}^{2})^{−1}, 1 ≤ i, j ≤ n,
(XK_{λ}, Y K_{μ}) = q^{<λ,μ>}(X, Y), X ∈ U_{q}(𝔫), Y ∈ U_{q}(𝔫^{−}).

2.4. In this section, we define automorphisms of the quantised enveloping algebra and the Poincaré-Birkhoff-Witt basis. The automorphisms T_{i}, 1 ≤ i ≤ n, as in [18], are given by :

T_{i}(E_{i}) = −K_{i}^{−1}F_{i}, T_{i}(E_{j}) = ∑_{k+l=−a_{ij}}(−1)^{k} \frac{q_{α_{i}}^{−k}}{[k]_{α_{i}}![l]_{α_{i}}!}E_{i}^{k}E_{j}E_{i}^{l}, 1 ≤ i, j ≤ n, i ≠ j,

T_{i}(F_{i}) = −E_{i}K_{i}, T_{i}(F_{j}) = ∑_{k+l=−a_{ij}} \frac{q_{α_{i}}^{k}}{[l]_{α_{i}}![k]_{α_{i}}!}F_{i}^{l}F_{j}F_{i}^{k}, 1 ≤ i, j ≤ n, i ≠ j,

T_{i}(K_{α_{j}}) = K_{s_{i}(α_{j})}, 1 ≤ i, j ≤ n.

It is known that the T_{i}’s define a braid action on U_{q}(𝔤). Fix i = (i_{1}, …, i_{N}) in R. For each k, 1 ≤ k ≤ N, set β_{k} := s_{i_{1}} … s_{i_{k−1}}(α_{k}). It is well known that {β_{k}, 1 ≤ k ≤ N} is the set of positive roots and that

β_{1} < β_{2} < … < β_{N}

defines a so-called convex ordering on Δ^{+}. This ordering identifies the semigroup ℤ_{≥0}^{Δ^{+}} with the semigroup ℤ_{≥0}^{N}. In the sequel, we denote by {e_{k}, 1 ≤ k ≤ N} the natural basis of this semigroup.

For all k, define E_{β_{k}}^{i} = E_{β_{k}} = T_{i_{1}} … T_{i_{k−1}}(E_{α_{i_{k}}}). For all t = (t_{i}) ∈ ℤ_{≥0}^{N}, set E^{i}(t) = E(t) :=

E_{β_{1}}^{(t_{1})} … E_{β_{N}}^{(t_{N})}, where E_{β_{k}}^{(t_{k})} := \frac{1}{[t_{k}]_{β_{k}}!}E_{β_{k}}^{t_{k}}. It is known that \{E(t), t ∈ ℤ_{≥0}^{N}\} is a basis of U_{q}(𝔫) called the Poincaré-Birkhoff-Witt basis, in short PBW-basis, associated to the reduced expression i. In the same way, we can define the PBW-basis {F (t), t ∈ ℤ_{≥0}^{N}} of U_{q}(𝔫^{−}). We define now the automorphisms ¯ (over ℂ), ω

(over 𝕂), and the antiautomorphism σ (over 𝕂) of U_{q}(𝔤) by

Ē_{i} = E_{i}, K̄_{i} = K_{i}^{−1}, F̄_{i} = F_{i}, q̄ = q^{−1}, 1 ≤ i ≤ n,
ω(E_{i}) = F_{i}, ω(K_{i}) = K_{i}^{−1}, ω(F_{i}) = E_{i}, ω(q) = q, 1 ≤ i ≤ n,
σ(E_{i}) = E_{i}, σ(K_{i}) = K_{i}^{−1}, σ(F_{i}) = F_{i}, σ(q) = q, 1 ≤ i ≤ n.

Note that ω is a coalgebra antiautomorphism.

2.5. For any (left) U_{q}(𝔤)-module M, and any weight μ in P, let M_{μ} = {m ∈ M : K_{λ}.m = q^{<λ,μ>}m, ∀λ ∈ P } be the subspace of M of weight μ. For all λ in P^{+} let V_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ and highest weight vector v_{λ}. The module V_{q}(λ) satisfies the Weyl character formula. For λ in P^{+}, V_{q}(λ)^{∗} is naturally endowed with a structure of right U_{q}(𝔤)-module. Let η_{λ} be its weight element such that η_{λ}(v_{λ}) = 1. For λ, μ in P^{+}, the module V_{q}(λ) ⊗ V_{q}(μ), endowed with the diagonal action of U_{q}(𝔤), has a unique component of type V_{q}(λ + μ). Hence the restriction map provides a map r_{λ,μ} : V_{q}(λ)^{∗} ⊗ V_{q}(μ)^{∗} ≃ (V_{q}(λ) ⊗ V_{q}(μ))^{∗} → V_{q}(λ + μ)^{∗}. Since P^{+} is a free abelian semigroup, one may assume that the v_{λ} are normalised such that r_{λ,μ}(η_{λ} ⊗ η_{μ}) = η_{λ+μ}, λ, μ ∈ P^{+}, see [12, 9.1.10].

Set R^{+} := ⨁_{λ∈P^{+}} V_{q}(λ)^{∗} ⊗ v_{λ}. The space R^{+} can be equipped with a structure of algebra by the multiplication rule : (ξ ⊗ v_{λ}).(ξ′ ⊗ v_{λ′}) = r_{λ,λ′}(ξ ⊗ ξ′) ⊗ v_{λ+λ′}, ξ ∈ V_{q}(λ)^{∗}, ξ′ ∈ V_{q}(λ′)^{∗}.

Let j be the map R^{+} → U_{q}(𝔟^{−})^{∗}, ξ ⊗ v_{λ} ↦ ξ(?v_{λ}), ξ ∈ V_{q}(λ)^{∗}. The following lemma is standard, we give a proof for completion.

Lemma 2.1. The map j is an embedding of algebras.

Proof. The map j is a morphism of vector space. Now, for λ, λ′ in P^{+}, η in V_{q}(λ), η′ in V_{q}(λ′), and b in U_{q}(𝔟^{−}):

j((η ⊗ v_{λ})(η′ ⊗ v_{λ′}))(b) = r_{λ,λ′}(η ⊗ η′)(b.v_{λ+λ′}) = (η ⊗ η′)(b.(v_{λ} ⊗ v_{λ′}))
= η(b_{(1)}.v_{λ})η′(b_{(2)}.v_{λ′}) = (j(η ⊗ v_{λ}).j(η′ ⊗ v_{λ′}))(b),

using the Sweedler notation. Hence, j is an algebra morphism.

Now, let’s prove that j is an embedding. Suppose that j(∑_{k} η_{k} ⊗ v_{λ_{k}}) = 0 for a non zero element ∑_{k} η_{k} ⊗ v_{λ_{k}} in R^{+}. One may assume that the λ_{k} are distinct in P^{+} and that η_{1} is non zero. Let v in V_{q}(λ_{1}) such that η_{1}(v) ≠ 0. Since ⊕_{k}V_{q}(λ_{k}) is a semisimple U_{q}(𝔤)-module, the Jacobson density theorem asserts that there exists an a in U_{q}(𝔤) such that a.v_{λ_{1}} = v and a.v_{λ_{k}} = 0 for k ≠ 1. By the triangular decomposition, one may assume that a is in U_{q}(𝔟^{−}). We have j(∑_{k} η_{k} ⊗ v_{λ_{k}})(a) = η_{1}(v) ≠ 0. this yields a contradiction. □

Define the map

β : U_{q}(𝔟) → U_{q}(𝔟^{−})^{∗}, β(u)(v) = (u, v).

Then we have:

Theorem 2.2. [7] The map β is an injective antihomomorphism of algebras which maps K_{λ} to η_{λ} ⊗ v_{λ} ∈ R^{+} ⊆ U_{q}(𝔟^{−})^{∗}. There exists a unique subspace E_{λ} of U_{q}(𝔫) such that β(E_{λ}K_{λ}) = V_{q}(λ)^{∗} ⊗ v_{λ}.

3. Recollection on canonical bases and adapted algebras.

The recollection on canonical bases is from [5]. Results on adapted algebras can be found in [8] and [9].

3.1. The following theorem defines the canonical basis and its Lusztig parametrisation, see [21, Proposition 8.2].

Theorem 3.1. [21] Fix i in R. For all t in ℤ_{≥0}^{N}, there exists a unique element b = b_{i}(t) in U_{q}(𝔫^{−}) such that b̄ = b and b − F^{i}(t) ∈ q^{−1} ∑ ℤ[q^{−1}]F^{i}(t′). The map t ↦ b_{i}(t) defines a bijection from ℤ_{≥0}^{N} to a basis B of U_{q}(𝔫^{−}) which does not depend on the choice of i.

The basis B is called the canonical (or global) basis and the map b_{i} : t ↦ b_{i}(t) is the Lusztig parametrisation of B associated to the reduced expression i. We can define the action of the Kashiwara operators on the canonical basis as follows : for 1 ≤ i ≤ n, there exists a unique injective map f̃_{i} : B → B, such that for all i with i_{1} = i, we have

f̃_{i}(b_{i}(t_{1}, t_{2}, …, t_{N})) = b_{i}(t_{1} + 1, t_{2}, …, t_{N}).

For 1 ≤ i ≤ n, let ẽ_{i} : B → B ∪ {0} be such that ẽ_{i}(b) = b′ if there exists b′ such that f̃_{i}(b′) = b and ẽ_{i}(b) = 0 if not.

Let ε_{i}(b) =max{k, ẽ_{i}^{k}(b) ≠ 0} and let E : B → P^{+}, b ↦ ∑_{i} ε_{i}(b)ϖ_{i}. Now, the basis B is stable under σ. For all λ in P^{+}, set B(λ) = {b ∈ B, E(σ(b)) ≤ λ}.

A nice theorem of compatibility of the canonical basis with the Weyl modules V_{q}(λ) can be stated as follows :

Theorem 3.2. [13] Fix λ in P^{+}. Then, for b in B, we have b · v_{λ} ≠ 0 if and only if b ∈ B(λ). Moreover, B(λ) · v_{λ} is a basis of V_{q}(λ).

In the sequel, we will identify B(λ) with its image in V_{q}(λ).

3.2. We now introduce the string parametrisation of the canonical basis and the various transition maps.

Fix a reduced expression i in R and b in B. The string of b in the direction i is the sequence of integers c_{i}(b) := (t_{1}, …, t_{N}) defined recursively by :

t_{1} = ε_{i_{1}}(b), t_{2} = ε_{i_{2}}(ẽ_{i_{1}}^{t_{1}}(b)), …, t_{N} = ε_{i_{N}} (ẽ_{i_{N−1}}^{t_{N−1}} … ẽ_{i_{1}}^{t_{1}}(b)).

The map c_{i} defines a bijection from B onto the set of integral points of a rational convex polyhedral cone C_{i} in ℝ^{N}.

We can now define :

R_{i}^{i′} = (b_{i′})^{−1} ∘ b_{i} : ℤ_{≥0}^{N} → ℤ_{≥0}^{N},
R_{−i}^{−i′} = c_{i′} ∘ (c_{i})^{−1} : C_{i} → C_{i′},
R_{−i}^{i′} = (b_{i′})^{−1} ∘ (c_{i})^{−1} : C_{i} → ℤ_{≥0}^{N},
R_{i}^{−i′} = c_{i′} ∘ b_{i} : ℤ_{≥0}^{N} → C_{i′}.

3.3. Let B^{∗} ⊂ U_{q}(𝔫) be the basis dual to B with respect to the form ( , ) on U_{q}(𝔫) × U_{q}(𝔫^{−}). We call it the dual canonical basis. For b in B, we denote by b^{∗} the corresponding element in B^{∗}. Since we work with the dual canonical basis, in the sequel we shall regard b_{i} as a map from ℤ_{≥0}^{N} to B^{∗} (rather than B), using the identification b ↔ b^{∗}. Similarly we shall regard c_{i} as a map from B^{∗} to ℤ_{≥0}^{N}.

The set B^{∗} is stable under σ ”up to a power of q”. To be more precise, for b in B, there exists an integer m such that σ(b^{∗}) = q^{m}σ(b)^{∗}.

For λ in P^{+} and b in B(λ), let π_{λ}(b)^{∗} be the element of V_{q}(λ)^{∗} such that π_{λ}(b)^{∗}(b′.v_{λ}) = δ_{b,b′}, for all b′ ∈ B(λ), where δ is the Kronecker symbol. It is easily seen from the definitions that :

Lemma 3.3. For all λ in P^{+} and b in B(λ), we have β(b^{∗}K_{λ}) = π_{λ}(b)^{∗} ⊗ v_{λ}.

In the notation, we will sometimes omit π_{λ}.

The lemma implies that the spaces E_{λ} defined by Theorem 2.2 are compatible with the dual canonical basis. By the Weyl character formula, for all w in W and λ in P^{+}, there is a unique element of B^{∗} ∩ E_{λ} with weight λ − wλ. We denote this element by b_{w,λ}^{∗} and the corresponding element in the canonical basis by b_{w,λ}.

In the sequel, {π_{λ}(b)^{∗} ⊗ v_{λ}, b ∈ B(λ), λ ∈ P^{+}} will be called dual canonical basis of R^{+}. By a misuse of language, its Lusztig, resp. string, parametrisation will be the Lusztig, resp. string, parametrisation of the corresponding element b in B(λ).

3.4. Two elements of the dual canonical basis are called multiplicative if their product is an element of the dual canonical basis, up to a power of q. By [26], multiplicative elements q-commute. We start with the definition of adapted algebras.

Definition 3.4. A subalgebra A of U_{q}(𝔫), resp. R^{+}, is called adapted if

1) the intersection of A and the dual canonical basis is a basis of A, called adapted basis,

2) the elements of this basis are pairwise multiplicative.

We define standard adapted subalgebras of R^{+}, associated to a fixed reduced expression i for w_{0}; see [8]. Set y_{λ} := η_{λ} ⊗ v_{λ}. Let A_{i} be the subalgebra of R^{+} generated by y_{i} := y_{ϖ_{i}}, 1 ≤ i ≤ n, and c_{k}^{i} :=

b_{s_{i_{1}} ...s_{i_{k}} ,ϖ_{i_{k}}}^{∗} ⊗ v_{ϖ_{i_{k}}}, 1 ≤ k ≤ N. (This notation is standard, but shouldn’t be confused with c_{i} which is used for the string parametrisation). Using the antihomomorphism β, we obtain from [8] the following proposition :

Proposition 3.5. Fix i ∈ R. The algebra A_{i} is an adapted subalgebra of R^{+} and the adapted basis is given by monomials in the y_{i} and the c_{k}^{i}, up to a power of q.

As σ is an antiautomorphism which preserves the dual canonical basis up to a power of q, we can now define another family of adapted algebras by twisting the standard ones.

Note that σ(b_{s_{i_{1}} ...s_{i_{k}} ,ϖ_{i_{k}}}) ∈ B(μ_{k}), where μ_{k} := E(b_{s_{i_{1}} ...s_{i_{k}} ,ϖ_{i_{k}}}). Let A_{i}^{σ} be the subalgebra generated by the y_{i}, 1 ≤ i ≤ n, and the c_{k}^{iσ} := σ(b_{s_{i_{1}} ...s_{i_{k}} ,ϖ_{i_{k}}})^{∗} ⊗ v_{μ_{k}}, 1 ≤ k ≤ N. These elements q-commute by 2.2. Moreover, by 3.3 and [8, 2.2], we have

Proposition 3.6. Fix i ∈ R. Then,

(i) the algebra A_{i}^{σ} is an adapted subalgebra of R^{+}, (ii) the adapted basis is given by monomials in the y_{i} and the c_{k}^{iσ}, up to a power of q, (iii) the Lusztig parametrisation of y_{i} is zero, (iv) the Lusztig parametrisation of c_{k}^{iσ} is ∑ e_{l} where l runs over {l ≤ k, i_{l} = i_{k}}.

3.5. In this section, we are concerned with the lowest weight vectors in R^{+}. For all λ in P^{+}, set λ^{∗} := −w_{0}λ. Let v_{w_{0}λ} be the the unique element of weight w_{0}λ in the canonical basis of V_{q}(λ). Then, z_{λ} := v_{w_{0}λ}^{∗} ⊗ v_{λ} is an element of the dual canonical basis of R^{+}. It is known (see [8]) that these elements belong to all of the standard adapted algebras defined in the section above.

Lemma 3.7. The elements z_{λ}, λ ∈ P^{+} satisfy z_{λ}z_{μ} = z_{λ+μ}, λ, μ ∈ P^{+}.

Proof. Let b, b′ be two elements of the canonical basis of U_{q}(𝔫^{−}), with b ∈ B(λ). By [9, Proposition 3.1], we have b^{′∗}b^{∗} ∈ q^{−<λ,ν′>}B^{∗}, where ν′ is the weight of b^{′∗}.

Applying this formula when b^{∗}, resp. b^{′∗}, is the element of B(λ)^{∗}, resp. B(μ)^{∗}, of weight λ + λ^{∗}, resp. μ + μ^{∗}, and using Theorem 2.2, we obtain that

z_{μ}z_{λ} = q^{−<λ,μ+μ^{∗}>}q^{<λ,μ+μ^{∗}>}z_{λ+μ} = z_{λ+μ}.

This proves the lemma. □

In the sequel, we set z_{i} = z_{ϖ_{i}}.

4. The Schützenberger involution.

We can now define an involution ϕ of R^{+} by twisting the dual Weyl modules by the automorphism ω. This involution generalises the Schützenberger involution, [29], up to a diagram automorphism.

4.1. Let M be a U_{q}(𝔤)-module. With the help of the automorphism ω, we define a twisted U_{q}(𝔤)-module structure on M^{∗} via

x.ξ(m) = ξ(ω(x)m), ξ ∈ M^{∗}, x ∈ U_{q}(𝔤), m ∈ M.

Let M_{ω}^{∗} be the space M^{∗} endowed with this U_{q}(𝔤)-module structure.

Fix λ in P^{+}. Then, V_{q}(λ)_{ω}^{∗} is a simple right module. By dualising [18, Chap XXI], we obtain

Proposition 4.1. The module V_{q}(λ)_{ω}^{∗} is isomorphic to V_{q}(λ^{∗})^{∗}. There exists a unique right U_{q}(𝔤)-module isomorphism ϕ_{λ} which sends the dual canonical basis of V_{q}(λ)_{ω}^{∗} to the dual canonical basis of V_{q}(λ^{∗})^{∗}. It sends highest weight vectors to lowest weight vectors and conversely.

4.2. We define a map ϕ : R^{+} → R^{+}, such that

ϕ(ξ ⊗ v_{λ}) = ϕ_{λ}(ξ) ⊗ v_{λ^{∗}}, ξ ∈ V_{q}(λ)^{∗}, λ ∈ P^{+},

where ϕ_{λ} is as above. Then,

Proposition 4.2. The map ϕ is an involutive antiautomorphism of the algebra R^{+} which preserves the dual canonical basis of R^{+}.

Proof. Recall that ω is involutive. So, by Proposition 4.1, ϕ_{λ^{∗}}ϕ_{λ} is the identity and this implies that ϕ is involutive.

Now, let R_{λ}^{+} be the λ-component of R^{+}, which is isomorphic to V_{q}(λ)^{∗} as a right U_{q}(𝔤)-module. As noted in 2.4, ω is a coalgebra antiautomorphism. Hence, the map m :

m : R_{λ}^{+} ⊗ R_{μ}^{+} → R_{λ+μ}^{+}, m(a ⊗ b) = ϕ^{−1}(ϕ(b)ϕ(a))

is a morphism of right U_{q}(𝔤)-modules, where R_{λ}^{+} ⊗ R_{μ}^{+} is endowed with the diagonal action. Using Lemma 3.7, we obtain

m(y_{λ} ⊗ y_{μ}) = ϕ^{−1}(z_{λ+μ}) = y_{λ+μ} = y_{λ}y_{μ}.

As dimHom_{U_{q}(𝔤)}(R_{λ}^{+} ⊗ R_{μ}^{+}, R_{λ+μ}^{+}) = 1, this proves that m is the multiplication of R^{+} and thus that ϕ is an algebra antiautomorphism. The last assertion of the proposition is clear by Proposition 4.1. □

Corollary 4.3. Fix a reduced expression i in R. Then, ϕ(A_{i}^{σ}) is an adapted subalgebra of R^{+}.

The aim of the remaining sections is to prove that the Lusztig cone L_{i} is the i-string parametrisation of the adapted basis of ϕ(A_{i}^{σ}). We note that this is given by monomials in the z_{i} and the ϕ(c_{k}^{iσ}) up to a power of q.

5. Geometric Lifting and parametrisation.

Let’s fix a dominant weight λ. In this section, we study the geometric lifting of the morphism ϕ_{λ}. By using results of [5], we obtain an explicit formula for the morphism b_{i}^{−1}ϕ_{λ}c_{i}^{−1} which gives the Lusztig parametrisation t′ = (t′_{1}, ⋯, t′_{N}) of the element ϕ_{λ}(b) in terms of the string t = (t_{1}, ⋯, t_{N}) of b, where b is in the dual canonical basis of R^{+}.

5.1. We give here notation and recollection of [5]. Let G be the semisimple simply connected complex Lie group with Lie algebra 𝔤. For all i, 1 ≤ i ≤ n, we denote by φ_{i} : SL_{2} ↪ G the canonical embedding corresponding to the simple root α_{i}. Consider the one parameter subgroups of G defined by

x_{i}(t) = φ_{i} ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) , y_{i}(t) = φ_{i} ( \begin{matrix}1 & 0 \\ t & 1 \end{matrix} ) , t ∈ ℂ,

and

t^{α_{i}^{∨}} = φ_{i} ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) , t ∈ ℂ^{∗}.

The x_{i}(t), (resp. y_{i}(t), t^{α_{i}^{∨}}) generate subgroups N, (resp. N^{−}, H). We have the following commutation relations :

t^{α_{i}^{∨}}x_{j}(t′) = x_{j}(t^{a_{ij}}t′)t^{α_{i}^{∨}}, t^{α_{i}^{∨}}y_{j}(t′) = y_{j}(t^{−a_{ij}}t′)t^{α_{i}^{∨}}. (5.1)

We define two involutive antiautomorphisms of G, x → x^{T} and x → x^{ι}, by :

x_{i}(t)^{T} = y_{i}(t), y_{i}(t)^{T} = x_{i}(t), (t^{α_{i}^{∨}})^{T} = t^{α_{i}^{∨}},
x_{i}(t)^{ι} = x_{i}(t), y_{i}(t)^{ι} = y_{i}(t), (t^{α_{i}^{∨}})^{ι} = t^{−α_{i}^{∨}}.

The first one is called transposition and the second one is called inversion.

Let G_{0} := N^{−}HN be the set of elements in G which have a (unique) gaussian decomposition; we write x = [x]_{−}[x]_{0}[x]_{+} for the gaussian decomposition of x in G_{0}.

For all reduced expressions i = (i_{1}, ⋯, i_{N}) and all N-tuples t = (t_{1}, ⋯, t_{N}) in ℂ^{N}, we set:

x_{i}(t) := x_{i_{1}}(t_{1}) ⋯ x_{i_{N}} (t_{N}), and x_{−i}(t) := y_{i_{1}}(t_{1})t_{1}^{−α_{i_{1}}^{∨}} ⋯ y_{i_{N}} (t_{N})t_{N}^{−α_{i_{N}}^{∨}} .

The x_{i} and the x_{−i} parametrise subvarieties of G.

Theorem 5.1. [5] There exists a subvariety of G, L_{>0}^{e,w_{0}}, resp. L_{>0}^{w_{0},e}, such that for all i in R, the map x_{i}, resp. x_{−i}, is a bijection from ℝ_{>0}^{N} to L_{>0}^{e,w_{0}}, resp. L_{>0}^{w_{0},e}.

We denote by R̃_{i}^{i′} := x_{i′}^{−1} ∘ x_{i} and R̃_{−i}^{−i′} := x_{−i′}^{−1} ∘ x_{−i} the transition maps. A remarkable result of [5] asserts that R̃_{i}^{i′} (respectively, R̃_{−i}^{−i′}) is a geometric lifting of the map R_{i}^{i′} (respectively, R_{−i}^{−i′}), which was defined in the first section. Let’s be more precise.

By using results on semifields (see [3]), the authors define the so-called tropicalisation, denoted by [.]_{T rop}. The map [.]_{T rop} is a map from the semifield ℚ_{>0}(t_{1}, ⋯, t_{N}) to the set of maps ℤ^{N} → ℤ. The elements of ℚ_{>0}(t_{1}, ⋯, t_{N}) are called subtraction-free rational expressions in the t_{1}, ⋯, t_{N}. Tropicalising a subtraction-free expression means replacing the multiplication by the operation a ⊙ b := a + b and the sum by the operation a ⊕ b = min(a, b). Let’s give an example from [3].

Example 5.2. Let x, y be two indeterminates and set f = x^{2} − xy + y^{2}. Then, f is a subtraction-free expression because f = \frac{x^{3}+y^{3}}{x+y}. We have : [f]_{T rop} : ℤ^{2} → ℤ, with

[f]_{T rop}(m, n) = Min(3m, 3n) − Min(m, n) = Min(2m, 2n).

A geometric lifting is an element of the inverse image of this map. We can see in the example above that it is in general not unique.

The following theorem is a result of [5]. Recall that the Langlands dual of G is the semisimple Lie group G^{∨} corresponding to the transpose Cartan matrix A^{T}. We can identify the simple roots (resp. coroots) of G^{∨} with the simple coroots (resp. roots) of G. The Weyl groups are naturally identified. In the theorem, the notation (.)^{∨} means that we consider the applications defined in the same way but for G^{∨}, and the notation [.]_{T rop} is the componentwise tropicalisation.

Theorem 5.3. Fix two reduced expressions i, i′ in R. Then (R̃_{i}^{i′})^{∨}, resp. (R̃_{−i}^{−i′})^{∨}, is a geometric lifting of R_{i}^{i′}, resp. R_{−i}^{−i′} :

(i) [(R̃_{i}^{i′})^{∨}(t)]_{T rop} = R_{i}^{i′}(t), (ii) [(R̃_{−i}^{−i′})^{∨}(t)]_{T rop} = R_{−i}^{−i′}(t).

5.2. Let ζ : L_{>0}^{w_{0},e} → L_{>0}^{e,w_{0}} be the map defined by

ζ(x) := [x^{ιT}]_{+}.

By 5.1, we obtain that the map ζ is well defined and

Proposition 5.4. Let i = (i_{1}, ⋯, i_{N}) be a reduced expression for w_{0}, and suppose (t′_{1}, ⋯, t′_{N}) = (x_{i}^{−1} ∘ ζ ∘ x_{−i})(t_{1}, ⋯, t_{N}), t_{i} ∈ ℂ. Then, we have

t′_{k} = t_{k}^{−1} ∏_{j>k} t_{j}^{−a_{i_{j} i_{k}}} .

The following theorem and its corollary are a result of [24] but we give here a sketch of the proof. The description of the geometric lifting of ϕ_{λ} can be given in terms of parametrisations :

Theorem 5.5. Fix two reduced expressions i, i′ in R. Then, (x_{i}^{−1} ∘ ζ ∘ x_{−i′})^{∨}(t) is a subtraction-free expression and

b_{i}^{−1}ϕ_{λ}c_{i′}^{−1}(t) = [(x_{i}^{−1} ∘ ζ ∘ x_{−i′})^{∨}(t)]_{T rop} + b_{i}^{−1}ϕ_{λ}(η_{λ}).

Set (l_{1}, ⋯, l_{N}) := b_{i}^{−1}ϕ_{λ}(η_{λ}). We obtain the following tropicalised formula :

Corollary 5.6. For (t′_{1}, ⋯, t′_{N}) = b_{i}^{−1}ϕ_{λ}c_{i}^{−1}(t_{1}, ⋯, t_{N}),

t′_{k} = l_{k} − t_{k} − ∑_{j>k} a_{i_{k}i_{j}}t_{j}.

Remark 5.7. It is remarkable that this formula is affine. This is only true in the case i = i′. In general, the tropical term in the right hand side of 5.5 is piecewise linear.

Sketch of the proof. By Proposition 5.4, (x_{i}^{−1} ∘ ζ ∘ x_{−i}) is a subtraction-free expression. The first assertion of the theorem is obtained by composing with R̃_{i′}^{i}.

Let ϕ_{i,i′} : C_{i′} → ℤ^{N} be a family of maps labelled by two reduced expressions i and i′ such that the three following conditions are satisfied :

(1) ϕ_{i,i′}(0, ⋯, 0) = b_{i}^{−1}ϕ_{λ}(η_{λ}),
(2) ϕ_{i,i′} = R_{i′′}^{i} ∘ ϕ_{i′′,i′} = ϕ_{i,i′′} ∘ R_{−i′}^{−i′′},
(3) for ϕ_{i,i}(t_{1}, ⋯, t_{N}) = (t′_{1}, ⋯, t′_{N}), t′_{1} + t_{1} and t′_{k}, k ≠ 1, depend only on t_{2}, ⋯, t_{N}.

The theorem follows from the proposition:

Proposition 5.8. [24] We have, (i) (ϕ_{i,i′}) is a family satisfying (1), (2), (3) if and only if

ϕ_{i,i′} = b_{i}^{−1}ϕ_{λ}c_{i′}^{−1}.

(ii) The family (ϕ_{i,i′}) defined by

ϕ_{i,i′}(t) = [(x_{i}^{−1} ∘ ζ ∘ x_{−i′})^{∨}(t)]_{T rop} + b_{i}^{−1}ϕ_{λ}(η_{λ})

satisfies the conditions (1), (2), (3).

6. From the PBW-parametrisation to the string parametrisation

Let i = (i_{1}, ⋯, i_{N}) be a reduced expression for w_{0}. Our aim in this section is to describe the map u ↦ c_{i}ϕ_{λ}b_{i}(u), using Proposition 5.4 and Theorem 5.5.

Let S be the complex rational map extending

(t_{1}, t_{2}, …, t_{N}) ↦ (u_{1}, u_{2}, …, u_{N}) = (x_{i}^{−1} ∘ ζ ∘ x_{−i})^{∨}(t_{1}, t_{2}, …, t_{N}).

Then Theorem 5.5 states that S(t) is a subtraction-free expression and that

b_{i}^{−1}ϕ_{λ}c_{i}^{−1}(t) = [S(t)]_{T rop} + b_{i}^{−1}ϕ_{λ}(η_{λ}).

Proposition 5.4 gives an explicit expression for S(t) (although note that we need to take the expression for the dual root system by the definition of S).

Moreover, S is clearly birational. We first explain how to invert S.

Lemma 6.1. Let (u_{1}, u_{2}, … u_{N}) ∈ ℂ^{N}. Then S^{−1}(u_{1}, u_{2}, … u_{N}) = (t_{1}, t_{2}, … t_{N}) ∈ ℂ^{N}, where, for 1 ≤ k ≤ N, we have

(6.1)  t_{k} = u_{k}^{−1} ∏_{j>k} u_{j}^{<s_{i_{k+1}} ···s_{i_{j−1}} α_{i_{j}} ,α_{i_{k}}^{∨} >}.

Proof. We note that, by Proposition 5.4, if (t_{1}, t_{2}, …, t_{N}) ∈ ℂ^{N} and S(t_{1}, t_{2}, …, t_{N}) = (u_{1}, u_{2}, …, u_{N}) ∈ ℂ^{N}, then, for 1 ≤ k ≤ N, we have:

(6.2)  u_{k} = t_{k}^{−1} ∏_{j>k} t_{j}^{−a_{i_{k}i_{j}}} .

Since this map is a monomial transformation of ℂ^{N}, it is sufficient to show that substituting the expression (6.2) for u_{k} in terms of the t_{j} into right hand side of equation (6.1) reduces to the left hand side. Noting that

< s_{i_{k+1}} ⋯ s_{i_{j−1}}α_{i_{j}}, α_{i_{k}}^{∨} >=< α_{i_{j}}, s_{i_{j−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} >,

we obtain:

(t_{k}^{−1} ∏_{l>k} t_{l}^{−a_{i_{k},i_{l}}} )^{−1} ∏_{j>k} (t_{j}^{−1} ∏_{l>j} t_{l}^{−a_{i_{j} ,i_{l}}} )^{<α_{i_{j}} ,s_{i_{j−1}} ···s_{i_{k+1}} α_{i_{k}}^{∨} >} .

It is clear that the exponent of t_{k} in this expression is 1, and that, if l < k, then the exponent of t_{l} is zero. So we consider the case where l > k. The exponent of t_{l} is given by

a_{i_{k},i_{l}} + ( ∑_{l>j>k} −a_{i_{j},i_{l}} < α_{i_{j}}, s_{i_{j−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} >) − < α_{i_{l}}, s_{i_{l−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} > .

Since s_{i_{j}}(α_{i_{l}}) = α_{i_{l}} − a_{i_{j},i_{l}}α_{i_{j}}, this is equal to:

< α_{i_{l}}, α_{i_{k}}^{∨} > + ( ∑_{l>j>k} < s_{i_{j}}(α_{i_{l}}), s_{i_{j−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} > − < α_{i_{l}}, s_{i_{j−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} >)
− < α_{i_{l}}, s_{i_{l−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} > .

The sum telescopes to give:

< α_{i_{l}}, α_{i_{k}}^{∨} > + < α_{i_{l}}, s_{i_{l−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} > − < α_{i_{l}}, α_{i_{k}}^{∨} > − < α_{i_{l}}, s_{i_{l−1}} ⋯ s_{i_{k+1}}α_{i_{k}}^{∨} >= 0,

and we are done. □

Proposition 6.2. Fix a reduced expression i for w_{0}, and let u = (u_{1}, u_{2}, …, u_{N}) ∈ ℤ_{≥0}^{N}. Then

c_{i}ϕ_{λ}b_{i}(u) = [(S^{−1})(u)]_{T rop} + c_{i}ϕ_{λ}(η_{λ}),

where S^{−1}(u) is as in Lemma 6.1.

Proof. By Theorem 5.5, we have that

(6.3)  b_{i}^{−1}ϕ_{λ}c_{i}^{−1}(t) = [S(t)]_{trop} + b_{i}^{−1}ϕ_{λ}(η_{λ}).

It follows that

c_{i}ϕ_{λ^{∗}}b_{i}(u) = [S^{−1}(u − b_{i}^{−1}ϕ_{λ}(η_{λ}))]_{T rop},

noting that ϕ_{λ^{∗}}ϕ_{λ} is the identity map. We note that S is an invertible monomial map, so its tropicalisation is linear. Hence

c_{i}ϕ_{λ^{∗}}b_{i}(u) = [S^{−1}(u)]_{T rop} − [S^{−1}(b_{i}^{−1}ϕ_{λ}(η_{λ}))]_{T rop}.

Substituting t = c_{i}ϕ_{λ^{∗}}(η_{λ^{∗}}) into (6.3), we obtain

0 = b_{i}^{−1}η_{λ^{∗}} = [S(c_{i}ϕ_{λ^{∗}}(η_{λ^{∗}}))]_{T rop} + b_{i}^{−1}ϕ_{λ}(η_{λ}).

Hence

[S^{−1}(b_{i}^{−1}ϕ_{λ}(η_{λ}))]_{T rop} = −c_{i}ϕ_{λ^{∗}}(η_{λ^{∗}}).

Hence we have:

c_{i}ϕ_{λ^{∗}}b_{i}(u) = [S^{−1}(u)]_{T rop} + c_{i}ϕ_{λ^{∗}}(η_{λ^{∗}}),

giving the required result (since λ ↦ λ^{∗} is an involution). □

We can compute the constant term in this formula as follows:

Lemma 6.3. Let i be a reduced expression for w_{0}. Then c_{i}(ϕ_{λ}η_{λ}) = (v_{1}, v_{2}, …, v_{N}), where, for 1 ≤ k ≤ N, we have

v_{k} =< s_{i_{k−1}} ⋯ s_{i_{1}}λ, α_{i_{k}}^{∨} > .

Proof. This follows from [18, 28.1.4], since we are computing the string of the lowest weight vector in V_{q}(λ^{∗})^{∗}. □

We remark that ϕ_{λ}(η_{λ}) = z_{λ^{∗}}, so this lemma is computing the string c_{i}z_{λ^{∗}}.

7. The string parametrisation of a twisted standard adapted subalgebra

Recall (see Corollary 4.3) that the monomials in the elements z_{1}, z_{2}, …, z_{n} and the elements ϕ(c_{k}^{iσ}), k = 1, 2, …, N, form an adapted basis for the twisted standard adapted subalgebra ϕA_{i}^{σ} of R^{+}. Our aim is to compute the string parameters of these elements. We therefore use Proposition 6.2 to apply the map c_{i}ϕ_{μ_{k}}b_{i} to the element b_{i}^{−1}(c_{k}^{iσ}), for each 1 ≤ k ≤ N; see 3.4. These last vectors were described in Proposition 3.6(iv).

REALISATION OF LUSZTIG CONES 11

7.1. For convenience, we define a matrix V with columns given by these vectors. Definition 7.1. Let M_{N}(ℤ) denote the ring of N × N matrices with integer entries. Let V = (V_{jk}) ∈ M_{N}(ℤ) be defined by:

V_{jk} = \{ \begin{matrix}1 & j ≤ k, if i_{j} = i_{k}, \\ 0 & otherwise. \end{matrix}

Let v_{k} denote the kth column of V; this is b_{i}^{−1}(c_{k}^{iσ}) by Proposition 3.6(iv). 7.2. In order to apply Proposition 6.2 to compute c_{i}ϕ_{μ_{k}}b_{i}(v_{k}), for each 1 ≤ k ≤ N, we first need to apply (S^{−1})_{T rop} to v_{k}.

By Lemma 6.1, we have that (S^{−1})_{T rop}(u_{1}, u_{2}, …, u_{N}) = (t_{1}, t_{2}, …, t_{N}), where

t_{k} = −u_{k} + ∑_{j>k} < s_{i_{k+1}} ⋯ s_{i_{j−1}}α_{i_{j}}, α_{i_{k}}^{∨} > u_{j}.
= ∑_{j=1}^{N} T_{kj}u_{j},

where T = (T_{jk}) is the matrix defining the linear map (S^{−1})_{T rop}.

It follows that

T_{jk} = \{ \begin{matrix}−1 & j = k, \\ < s_{i_{j+1}} ⋯ s_{i_{k−1}}α_{i_{k}}, α_{i_{j}}^{∨} > & j < k, \\ 0 & otherwise. \end{matrix}

We note that T is the inverse of the matrix S defining the linear map S_{T rop}. Definition 7.2. Let C = (C_{jk}) ∈ M_{N}(ℤ) be the matrix given by:

C_{jk} = \{ \begin{matrix}< s_{i_{j+1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{i_{j}}^{∨} > & j ≤ k, \\ 0 & otherwise. \end{matrix} .

We have: Lemma 7.3. Let S, V and C be the matrices defined above. Then we have S^{−1}V = −C. Proof. We have that (S^{−1}V)_{jl} = ∑_{j≤k≤l} T_{jk}V_{kl}, and is zero if j > l. If j ≤ l then, using that < , > is W -invariant,

(7.1)  (S^{−1}V)_{jl} = T_{jj}V_{jl} + ∑_{j<k≤l,i_{k}=i_{l}} < s_{i_{k−1}} ⋯ s_{i_{j+1}}α_{i_{j}}^{∨}, α_{i_{l}} > .

Remark that T_{jj}V_{jl} = −δ_{i_{j},i_{l}}. Now, by calculating explicitly the coefficient of α_{i_{l}}^{∨} in s_{i_{l}} ⋯ s_{i_{j}}α_{i_{j}}^{∨}, we find that it is equal to the right hand side of 7.1. Hence,

(S^{−1}V)_{jl} =< ϖ_{i_{l}}, s_{i_{l}} ⋯ s_{i_{j}}α_{i_{j}}^{∨} >= − < s_{i_{j+1}} ⋯ s_{i_{l}}ϖ_{i_{l}}, α_{i_{j}}^{∨} >,

which is equal to −C_{jl} as required. □

We have the following Corollary: Corollary 7.4. The entries of C are nonnegative. Proof. In proof of Lemma 7.3, we noted that C_{jl} is equal to the negative of the coefficient of α_{i_{l}}^{∨} in the negative coroot s_{i_{l}} ⋯ s_{i_{j}}α_{i_{j}}^{∨}. □

12 PHILIPPE CALDERO, ROBERT MARSH, AND SOPHIE MORIER-GENOUD

7.3. We now would like to compute c_{i}ϕ_{μ_{k}}b_{i}(v_{k}) for each k (note that the v_{k} are the columns of V). In the following lemma we set ε_{l}(c_{k}^{i}) = ε_{l}(b_{s_{i_{1}} ...s_{i_{k}} ,ϖ_{i_{k}}}) by a misuse of notation. Lemma 7.5. For 1 ≤ l ≤ n and 1 ≤ k ≤ N, we have:

ε_{l}(c_{k}^{i}) = \{ \begin{matrix}− < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{l}^{∨} > & if < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{l}^{∨} >≤ 0, \\ 0 & otherwise. \end{matrix}

In particular,

μ_{k} = ∑_{1≤l≤n,<s_{i_{1}} ···s_{i_{k}} ϖ_{i_{k}} ,α_{l}^{∨}>≤0} − < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{l}^{∨} > ϖ_{l}.

Proof. This follows from the definition of the c_{k}^{i} and [18, 28.1.4] (and sl_{2}-representation theory). □

Let c_{k} be the kth column of C = −S^{−1}V, and let p_{k} = c_{i}(ϕ_{μ_{k}}η_{μ_{k}}). Let P be the matrix with columns p_{k}, k = 1, 2, …, N. Lemma 7.6. For 1 ≤ k ≤ N, we have

c_{i}ϕ_{μ_{k}}b_{i}(v_{k}) = −c_{k} + p_{k}.

In particular, the entries of −C + P are nonnegative. Proof. This follows immediately from Proposition 6.2 and Lemma 7.3. □

As a consequence, we have: Proposition 7.7. Let i be any reduced expression for w_{0}. Then c_{i}ϕ(A_{i}^{σ}) coincides with the nonnegative integer span of the columns of −C + P and the strings c_{i}(z_{i}), 1 ≤ i ≤ n. Proof. By Proposition 3.6, c_{i}ϕ(A_{i}^{σ}) is the nonnegative integer span of the c_{i}ϕ_{μ_{k}}b_{i}(v_{k}) together with the strings c_{i}(z_{i}), 1 ≤ i ≤ n. Since c_{i}(b^{∗}b^{′∗}) = c_{i}(b^{∗})+c_{i}(b^{′∗}) when the elements b^{∗} and b^{′∗} of the dual canonical basis are multiplicative [4, Cor. 3.3], the proposition follows from Lemma 7.6 and Corollary 4.3. □ 7.4. We also note the following formula for the entries of P: Lemma 7.8. For 1 ≤ j, k ≤ N, we have

P_{jk} = ∑_{1≤l≤n,<s_{i_{1}} ···s_{i_{k}} ϖ_{i_{k}} ,α_{l}^{∨}>≤0} − < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{l}^{∨} >< s_{i_{j−1}} ⋯ s_{i_{1}}ϖ_{l}, α_{i_{j}}^{∨} > .

Proof. By definition P_{jk} is the jth entry of c_{i}(ϕ_{μ_{k}}η_{μ_{k}}). Hence, the lemma results from Lemma 7.5 and Lemma 6.3. □ 7.5. We will next show that some of the columns of −C + P are entirely zero, and therefore can be neglected in Proposition 7.7. Definition 7.9. Given k ∈ {1, 2, …, N}, we set k(1) = min{j : j > k, i_{j} = i_{k}}, i.e. the first occurrence of i_{k} to the right of i_{k} in i. If there is no such occurrence, we set k(1) = N + 1. Lemma 7.10. Suppose that k(1) = N + 1. Then, for j = 1, 2, …, N, we have P_{jk} = C_{jk}, i.e. the kth column of P coincides with the kth column of C. Proof. We have

C_{jk} = \{ \begin{matrix}< s_{i_{j+1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{i_{j}}^{∨} > & j ≤ k, \\ 0 & j > k. \end{matrix}

Since k(1) = N + 1, we have

C_{jk} =< s_{i_{j+1}} ⋯ s_{i_{N}} ϖ_{i_{k}}, α_{i_{j}}^{∨} >,

(in either case). By Lemma 7.5,

μ_{k} = ∑_{1≤l≤n,<s_{i_{1}} ···s_{i_{k}} ϖ_{i_{k}} ,α_{l}^{∨}>≤0} − < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}α_{l}^{∨} > ϖ_{l}

= ∑_{1≤l≤n,<s_{i_{1}} ···s_{i_{N}} ϖ_{i_{k}} ,α_{l}^{∨}>≤0} − < s_{i_{1}} ⋯ s_{i_{N}} ϖ_{i_{k}}α_{l}^{∨} > ϖ_{l}

= ∑_{1≤l≤n,<w_{0}ϖ_{i_{k}} ,α_{l}^{∨}>≤0} − < w_{0}ϖ_{i_{k}}, α_{l}^{∨} > ϖ_{l}
= −w_{0}ϖ_{i_{k}}.

Hence, by Lemma 6.3

P_{jk} = − < s_{i_{j−1}} ⋯ s_{i_{1}}w_{0}ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= − < s_{i_{j}} ⋯ s_{i_{N}} ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= < s_{i_{j+1}} ⋯ s_{i_{N}} ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= C_{jk}.
□

7.6. We replace the zero columns in −C + P with vectors which we will see are the strings of the elements z_{1}, z_{2}, …, z_{n}, in order to obtain a matrix whose nonnegative integer span is the set of string parameters of the twisted standard adapted subalgebra of R^{+} corresponding to i. We call this matrix X:

Definition 7.11. Let X = (X_{jk}) ∈ M_{N}(ℤ) be the matrix defined as follows:

X_{jk} = \{ \begin{matrix}< s_{i_{j−1}} ⋯ s_{i_{1}}ϖ_{i_{k}}, α_{i_{j}}^{∨} >, & k(1) = N + 1, \\ −C_{jk} + P_{jk}, & otherwise. \end{matrix}

We note that the P_{jk} are given by Lemma 7.8, and that the C_{jk} are given in Definition 7.2. Also, it follows from Lemma 6.3 that if k(1) = N + 1 then the kth column of X is the string c_{i}z_{ϖ_{i_{k}}^{∗}}.

We have:

Proposition 7.12. Let i be any reduced expression for w_{0}. Then c_{i}ϕ(A_{i}^{σ}) coincides with the nonnegative integer span of the columns of X. In particular, the entries of X are nonnegative.

Proof. We note that the matrix X is the same as −C +P, except that if 1 ≤ k ≤ N and k(1) = N +1, then the kth column (which is zero by Lemma 7.10) is replaced by the string of z_{ϖ_{i_{k}}^{∗}}. The result now follows from Proposition 7.7. □

8. Lusztig cones and twisted standard adapted subalgebras

In this section, we will show that the cone of string parameters of c_{i}ϕ(A_{i}^{σ}), given by the nonnegative integer span of the columns of X, coincides with the Lusztig cone corresponding to i. At the same time we will show that the Lusztig cones are simplicial.

8.1. We first of all define a matrix L̃ whose rows include the defining inequalities of the Lusztig cone corresponding to i as a subset of ℕ^{N}. This matrix will later be modified to a matrix defining the Lusztig cone as a subset of ℤ^{N}.

Definition 8.1. Let L̃ ∈ M_{N}(ℤ) be the matrix defined by:

L̃_{jk} = \{ \begin{matrix}−1 & k = j or k = j(1) \\ −a_{i_{j},i_{k}} & j < k < j(1), \\ 0 & otherwise. \end{matrix}

Remark 8.2. Let r̃_{j} denote the jth row of L̃. The defining inequalities of the Lusztig cone L_{i} (as a subset of ℤ_{≥0}^{N}) are those inequalities of the form r̃_{j} · c ≥ 0 for those j such that j(1) ≤ N.

8.2. We will next show how this matrix is related to the matrices S and V already considered, i.e. that V^{−1}S = L̃. This will have the consequence that L̃S^{−1}V = I, in particular showing that the columns of S^{−1}V satisfy the defining inequalities of the Lusztig cone corresponding to i. We recall from Proposition 5.4 that S_{T rop} is defined by the matrix S = (S_{jk}) ∈ M_{N}(ℤ) where

S_{jk} = \{ \begin{matrix}−1 & j = k \\ −a_{i_{j},i_{k}} & j < k, \\ 0 & otherwise. \end{matrix}

We next need to compute the inverse of the matrix V.

Lemma 8.3. Let W = (W_{jk}) ∈ M_{N}(ℤ) be the matrix defined as follows:

W_{jk} = \{ \begin{matrix}1 & j = k \\ −1 & j < k, k = j(1), \\ 0 & otherwise. \end{matrix}

Then W = V^{−1}.

Proof. We show that W V = I, the identity matrix. The j, l-entry of W V is given by Z_{jl} = ∑_{k=1}^{N} W_{jk}V_{kl}. For this to be non-zero, we must have j ≤ k ≤ l and k = j or k = j(1). There are 5 cases: Case (a): If l < j, then clearly Z_{jl} = 0. Case (b): If l = j, then Z_{jl} = Z_{jj} = W_{jj}V_{jj} = 1 · 1 = 1. Case (c): If j < l < j(1), then Z_{jl} = W_{jj}V_{jl} = 1 · 0 = 0. Case (d): If l = j(1), then Z_{jl} = W_{jj}V_{j,j(1)} + W_{j,j(1)}V_{j(1),j(1)} = 1 · 1 + (−1) · 1 = 0. Case (e): If l > j(1), then Z_{jl} = W_{jj}V_{jl} + W_{j,j(1)}V_{j(1),l} = V_{jl} − V_{j(1),l} = 0 since V_{jl} = V_{j(1),l}. □

Lemma 8.4. Let V, S and L̃ be the matrices as defined above. Then V^{−1}S = L̃.

Proof. The j, l-entry of V^{−1}S is given by Y_{jl} = ∑_{k=1}^{N} W_{jk}S_{kl}. To be non-zero, we must have j ≤ k ≤ l and k = j or k = j(1). As before, we have the 5 cases: Case (a): If l < j then clearly Y_{jl} = 0. Case (b): If l = j, then Y_{jl} = Y_{jj} = W_{jj}S_{jj} = 1 · (−1) = −1. Case (c): If j < l < j(1), then Y_{jl} = W_{jj}S_{jl} = S_{jl} = −a_{i_{j},i_{l}}. Case (d): If l = j(1), then Y_{jl} = Y_{j,j(1)} = W_{jj}S_{j,j(1)} + W_{j,j(1)}S_{j(1),j(1)} = 1 · (−2) + (−1) · (−1) = −1. Case (e): If l > j(1), then Y_{jl} = W_{jj}S_{jl}+W_{j,j(1)}S_{j(1),l} = S_{jl}−S_{j(1),l} = −a_{i_{j},i_{l}}+a_{i_{j(1)},i_{l}} = −a_{i_{j},i_{l}}+a_{i_{j},i_{l}} = 0. We see that Y_{jl} = L̃_{jl} in every case. □

8.3. We now define a slightly altered version of the matrix L̃, whose rows will eventually be seen to be the defining inequalities of the Lusztig cone as a subset of ℤ^{N}. We will also see that this matrix is the inverse of the matrix X.

Definition 8.5. Let L = (L_{jk}) ∈ M_{N}(ℤ) be the matrix defined as follows:

L_{jk} = \{ \begin{matrix}−1 & k = j or k = j(1), \\ −a_{i_{j},i_{k}} & j < k < j(1), \\ 1 & j(1) = N + 1, s_{i_{1}}s_{i_{2}} ⋯ s_{i_{k−1}}(α_{i_{k}}) = α_{i_{j}}, \\ 0 & otherwise. \end{matrix}

Firstly, we show how some rows of L are related to the strings of lowest weight vectors:

Lemma 8.6. Let

v = c_{i}(ϕ_{λ}η_{λ}) = (v_{1}, v_{2}, …, v_{N}),

as in Lemma 6.3. Suppose that 1 ≤ j ≤ j(1) ≤ N. Let r_{j} be the jth row of L. Then we have r_{j} · v = 0.

REALISATION OF LUSZTIG CONES 15

Proof. Recall that for k = 1, 2, …, N, we have

v_{k} =< s_{i_{k−1}} ⋯ s_{i_{1}}λ, α_{i_{k}}^{∨} >,

by Lemma 6.3. We have

r_{j} · v = −v_{j} − v_{j_{(1)}} − ∑_{j<k<j_{(1)}} a_{i_{j},i_{k}}v_{k}
= − < s_{i_{j−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} > − < s_{i_{j(1)−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} > − ∑_{j<k<j(1)} a_{i_{j},i_{k}} < s_{i_{k−1}} ⋯ s_{i_{1}}λ, α_{i_{k}}^{∨} > .

We note that

s_{i_{k}}(α_{i_{j}}^{∨}) = α_{i_{j}}^{∨}− < α_{i_{k}}, α_{i_{j}}^{∨} > α_{i_{k}}^{∨} = α_{i_{j}}^{∨} − a_{i_{j},i_{k}}α_{i_{k}}^{∨},

so

r_{j} · v = − < s_{i_{j−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} > − < s_{i_{j(1)−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} >

+ ∑_{j<k<j(1)} < s_{i_{k−1}} ⋯ s_{i_{1}}λ, s_{i_{k}}(α_{i_{j}}^{∨}) − α_{i_{j}}^{∨} >

= − < s_{i_{j−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} > − < s_{i_{j(1)−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} >

+ < s_{i_{j(1)−1}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} > − < s_{i_{j}} ⋯ s_{i_{1}}λ, α_{i_{j}}^{∨} >= 0,

the sum telescoping. □

Next, we show that some entries of −C + P = S^{−1}V + P are zero: Lemma 8.7. Suppose that 1 ≤ j ≤ N and that s_{i_{1}}s_{i_{2}} ⋯ s_{i_{j−1}}(α_{i_{j}}^{∨}) = α_{r}^{∨} is a simple coroot. Then

P_{jk} = C_{jk}.

Proof. We note that, by Lemma 7.8,

P_{jk} = \{ \begin{matrix}− < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{r}^{∨} > & < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{r}^{∨} >≤ 0, \\ 0 & < s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{r}^{∨} >> 0, \end{matrix}

using the fact that < ϖ_{l}, α_{r}^{∨} >= δ_{lr}. Suppose first that j ≤ k. Then:

< s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{r}^{∨} > = < s_{i_{j+1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, s_{i_{j}} ⋯ s_{i_{1}}α_{r}^{∨} >
= < s_{i_{j+1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, s_{i_{j}}α_{i_{j}}^{∨} >
= − < s_{i_{j+1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= −C_{jk} ≤ 0,

by Corollary 7.4. So P_{jk} = C_{jk}. If j > k then

< s_{i_{1}} ⋯ s_{i_{k}}ϖ_{i_{k}}, α_{r}^{∨} > = < ϖ_{i_{k}}, s_{i_{k}} ⋯ s_{i_{1}}α_{r}^{∨} >
= < s_{i_{j−1}} ⋯ s_{i_{k+1}}ϖ_{i_{k}}, s_{i_{j−1}} ⋯ s_{i_{k+1}}s_{i_{k}} ⋯ s_{i_{1}}α_{r}^{∨} >
= < s_{i_{j−1}} ⋯ s_{i_{k+1}}ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= < ϖ_{i_{k}}, s_{i_{k+1}} ⋯ s_{i_{j−1}}α_{i_{j}}^{∨} > ≥ 0,

since s_{i_{k+1}} ⋯ s_{i_{j+1}}α_{i_{j}}^{∨} is a positive coroot. It follows that P_{jk} = 0 = C_{jk}. □ Remark 8.8. We note that the condition

s_{i_{1}} ⋯ s_{i_{k}}(α_{i_{j}}^{∨}) = α_{r}^{∨}

is equivalent to the condition

s_{i_{1}} ⋯ s_{i_{k}}(α_{i_{j}}) = α_{r}.

Secondly, this result shows that (S^{−1}V + P)_{jk} = 0 under this assumption, by Lemma 7.3.

8.4. We can now prove the following, as we have all the pieces we need:

Proposition 8.9. Let L and X be the matrices defined as above, so that the nonnegative integer span of the columns of X is the cone of the string parameters of the twisted standard adapted subalgebra corresponding to i. Then LX = I, the identity matrix.

Proof. Denote by x_{k} the kth column of X. Suppose first that 1 ≤ j ≤ j(1) ≤ N, and that 1 ≤ k ≤ k(1) ≤ N. Then r_{j} (the jth row of L) is the same as the jth row of L̃. By Lemma 8.4, L̃S^{−1}V = I. Hence r_{j} · (−c_{k}) = δ_{jk}, since −c_{k} is the kth column of S^{−1}V. By Lemma 8.6, r_{j} · p_{k} = 0 (where p_{k} is the kth column of P). It follows (from the definition of X) that r_{j} · x_{k} = δ_{jk}.

If 1 ≤ j ≤ j(1) ≤ N and k(1) = N + 1, then x_{k} = c_{i}z_{ϖ_{i_{k}}}^{∗}. By Lemma 8.6, we have that r_{j} · x_{k} = 0 (as required, noting that we must have j ≠ k).

If j(1) = N +1, then let 1 ≤ l ≤ N be defined by s_{i_{1}} ⋯ s_{i_{l−1}}α_{i_{l}} = α_{i_{j}}. Then L_{jl} = 1 is the only non-zero entry in r_{j}. It follows that r_{j}.x_{k} = X_{lk}. Case (a) Suppose that k(1) ≤ N. Then X_{lk} = −C_{lk} + P_{lk} = 0 by Lemma 8.7. Case (b) Suppose that k(1) = N + 1. Then

X_{lk} = < s_{i_{l−1}} ⋯ s_{i_{1}}ϖ_{i_{k}}, α_{i_{l}}^{∨} >
= < ϖ_{i_{k}}, s_{i_{1}} ⋯ s_{i_{l−1}}α_{i_{l}}^{∨} >
= < ϖ_{i_{k}}, α_{i_{j}}^{∨} >
= δ_{j,k},

as required (noting that j(1) = k(1) = N + 1). The proposition is proved. □

If c = (c_{1}, c_{2}, … c_{N}) ∈ ℤ^{N}, we write c ≥ 0 to denote c_{k} ≥ 0 for k = 1, 2, …, N. We have the following consequences.

Theorem 8.10. The Lusztig cone L_{i} is simplicial, defined by the matrix L:

L_{i} = {c ∈ ℤ^{N} : Lc ≥ 0}.

It coincides with the nonnegative integer span of the columns of the matrix X (see Definition 7.11).

Proof. By Proposition 7.12, the entries of X are nonnegative. By Proposition 8.9, XL = I, so nonnegative integer combinations of the rows of L are of the form (0, 0, …, 0, 1, 0, …, 0) (with a 1 in the kth position) and therefore correspond to inequalities of the form c_{k} ≥ 0. So

{c ∈ ℤ^{N} : Lc ≥ 0} ⊆ ℕ^{N}.

Since the inequalities corresponding to rows of L are either defining inequalities of L_{i} or inequalities of the form c_{k} ≥ 0, the claimed equality follows, and it is then immediate that L_{i} is spanned by the columns of L^{−1} = X. □

Remark 8.11. The fact that L_{i} is simplicial was already known for quiver-compatible reduced expressions for w_{0} for g simply laced [1] and for all reduced expressions for w_{0} in type A_{n} [23].

And we have:

Theorem 8.12. Let i be any reduced expression for w_{0}. Let L_{i} denote the Lusztig cone corresponding to i. Let ϕ(A_{i}^{σ}) denote the twisted standard adapted subalgebra corresponding to i. Then

c_{i}(ϕ(A_{i}^{σ})) = L_{i}.

Proof. This follows from Proposition 7.12 and Theorem 8.10. □

Example 8.13. Suppose that 𝔤 = sl_{4}(ℂ) (type A_{3}). Let i = (2, 3, 2, 1, 2, 3), a reduced expression for w_{0}. Then we have:

V = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , T = ( \begin{matrix}−1 & −1 & 1 & 0 & −1 & 1 \\ 0 & −1 & −1 & −1 & 0 & 1 \\ 0 & 0 & −1 & −1 & 1 & 0 \\ 0 & 0 & 0 & −1 & −1 & −1 \\ 0 & 0 & 0 & 0 & −1 & −1 \\ 0 & 0 & 0 & 0 & 0 & −1 \end{matrix} ) ,

C = −T V = ( \begin{matrix}1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) , P = ( \begin{matrix}1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) .

The matrix X of spanning vectors of the Lusztig cone corresponding to i and the defining matrix L are given by:

X = ( \begin{matrix}0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} ) , L = X^{−1} = ( \begin{matrix}−1 & 1 & −1 & 0 & 0 & 0 \\ 0 & −1 & 1 & 0 & 1 & −1 \\ 0 & 0 & −1 & 1 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) .

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Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France

   E-mail address: caldero@igd.univ-lyon1.fr

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England

   E-mail address: R.Marsh@mcs.le.ac.uk

Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France

   E-mail address: morier@igd.univ-lyon1.fr