A multiplicative property of quantum flag minors

Ph. Caldero^{∗}

January 23, 2002

Abstract

We study the multiplicative properties of the quantum dual canonical basis B^{∗} associated to a semi-simple complex Lie group G. We provide a subset D of B^{∗} such that the following property holds : if two elements b, b′ in B^{∗} q-commute and if one of these elements is in D, then the product bb′ is in B^{∗} up to a power of q, where q the quantum parameter. If G is SL_{n}, then D is the set of so-called quantum flag minors and we obtain a generalization of a result of Leclerc-Nazarov-Thibon, [11].

0 Introduction

0.1 Let G be a semisimple complex Lie group and fix a maximal unipotent subgroup U^{−} of G. Let 𝔤 and 𝔫^{−} be respectively the Lie algebras of G and U^{−}. G. Lusztig and M. Kashiwara have constructed the so-called canonical basis B of the enveloping algebra U(𝔫^{−}) of 𝔫^{−}, which has properties of compatibility with standard filtrations.

Let ℂ[U^{−}] be the ℂ-algebra of regular functions on U^{−}. Then, the action of U^{−} on itself by left multiplication provides an action of U(𝔫^{−}) on ℂ[U^{−}] by differential operators. Now, consider the pairing U(𝔫^{−}) × ℂ[U^{−}] → ℂ, δ × f ↦ δ(f)(e), where e is the identity of U^{−}. Then, this pairing provides the so-called dual canonical basis B^{∗} of ℂ[U^{−}]. This article is concerned with some multiplicative properties of this basis.

0.2 Let q be an indeterminate and let U_{q}(𝔫^{−}), ℂ_{q}[U^{−}] be respectively the quantum analogue of the classical objects U(𝔫^{−}) and ℂ[U^{−}]. We still note B, resp. B^{∗}, the canonical basis, resp. the dual canonical basis, of U_{q}(𝔫^{−}), resp. ℂ_{q}[U^{−}]. We say that two elements b and b′ of B^{∗} q-commute if bb′ = q^{m}b′b, for an integer m. We say that they are multiplicative if q^{n}bb′ belongs to B^{∗} for an integer n. It is known, see [19], that if two elements are multiplicative, then they q-commute. The converse was believed to be true until Bernard Leclerc found counter examples.

Suppose that two elements b and b′ of the dual canonical basis q-commute. Let’s discuss now in which cases they are known to be multiplicative.

1) for all b, b′, if 𝔤 is of type A_{n}, n ≤ 3, B_{2}, [1], see also [4].

2) if 𝔤 is of type A_{n} and b is a small quantum minor, [19].

3) if 𝔤 is of type A_{n} and b, b′ are quantum flag minors, [11].

0.3 Let’s present the results of this article. Let W be the Weyl group of 𝔤 and let w_{0} be its longest element. For each reduced decomposition w̃_{0} of w_{0}, we have constructed in [4] a subalgebra Aw̃_{0} of ℂ_{q}[U^{−}] such that:

1) The space Aw̃_{0} is generated by a part of B^{∗},

2) pair of elements in B^{∗} ∩ Aw̃_{0} are multiplicative,

3) the algebras Aw̃_{0} and ℂ_{q}[U^{−}] are equal up to localization. They are called adapted algebras associated to a reduced decomposition of w_{0}. They are connected to the more general theory of cluster algebras, [8], see 5.2.

The main result of the article is the following:

∗Supported in part by the EC TMR network “Algebraic Lie Representations”, contract no. ERB FMTX-CT97-0100

Theorem. Let w̃_{0} be a reduced decomposition corresponding to an orientation of the Coxeter graph of 𝔤. Let b and b′ be two q-commuting elements of the dual canonical basis. If b is in Aw̃_{0} then b and b′ are multiplicative.

As a particular case, we obtain:

Corollary. Let 𝔤 be of type A_{n}. Let b be a quantum flag minor and let b′ be any element of B^{∗} which q-commutes with b, then b and b′ are multiplicative.

We refer to [11] for motivations of this result. Actually, it provides a criterion of irreducibility for module on the affine Hecke algebra of type A which are induced by the so-called evaluation modules. Let’s sketch the proof of the theorem. If two elements b and b′ of the dual canonical basis q-commute, then, the only property we have to obtain in order to prove that b and b′ are multiplicative is

(0.3.1)  q^{n}bb′ ∈ b′′ + qL^{∗},

where n is an integer, b′′ is in B^{∗} and where L^{∗} is the ℤ[q]-lattice generated by B^{∗}. The natural question is: how to control the powers of q in the multiplications of elements of the dual canonical basis ? The control of these powers are based on two main ideas:

1) Kashiwara proved that bases of integrable modules of the quantized enveloping algebra U_{q}(𝔤) cristallizes at q = 0, with compatibility with the tensor product. To be more precise, let P^{+} be the semigroup of integral dominant weights and let \overline{b}, \overline{b}′ be the corresponding elements in the crystal bases B(λ), B(λ′), of the integrable modules of highest weight, respectively, λ and λ′ in P^{+}. We can assert that if \overline{b}⊗\overline{b}′ belongs to the connected component of the crystal B(λ) ⊗ B(λ′) corresponding to B(λ + λ′), then (0.3.1) holds. The property \overline{b} ⊗ \overline{b}′ ∈ B(λ + λ′) can be checked easily via the Littelmann’s path model of the crystal basis, [13], [14], by comparing chains of elements of the Weyl group for the Bruhat ordering.

2) The quiver approach of the algebra ℂ_{q}[U^{−}] enables to interpret powers of q which appear in multiplications in terms of dim Hom(M, N) and dim Ext^{1}(M, N), where M and N are representations of a quiver. An important tool is that the map dim Hom(M,?) is increasing for the so-called degeneration ordering, see [3].

1 Notations.

1.1 Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n with Cartan matrix A = (a_{ij}). We fix a Cartan subalgebra 𝔥 of 𝔤. Let 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫 be the triangular decomposition and set N := dim𝔫. Let {α_{i}}_{i} be a basis of the root system R resulting from this decomposition and let R^{+} be the set of positive roots. Let P be the weight ℤ-lattice generated by the fundamental weights ϖ_{i}, i ∈ I := ℤ ∩ [1, n] and set P^{+} := ∑_{i} ℕϖ_{i}. Let W be the Weyl group, generated by the reflections corresponding to the simple roots s_{i} := s_{α_{i}}, with longest element w_{0}. We note < , > the W -invariant form on P.

1.2 In this section we define the quantized enveloping algebra of 𝔤 and the properties of its Poincaré-Birkhoff-Witt basis. We refer to [6] for precisions and proofs. Let q be an indeterminate. Let U_{q}(𝔤) be the quantized enveloping Hopf ℚ(q)-algebra as defined in [6]. Let U_{q}(𝔫), resp U_{q}(𝔫^{−}), be the upper, resp lower, “nilpotent” subalgebra of U_{q}(𝔤). The algebra U_{q}(𝔫), resp U_{q}(𝔫^{−}), is generated by E_{i}, resp F_{i}, 1 ≤ i ≤ n with quantum Serre relations. For all λ in Q := ⊕_{i}ℤα_{i}, let K_{λ} be the corresponding element in the algebra U_{q}^{0} = ℚ(q)[Q] of the torus of U_{q}(𝔤).

For all μ ∈ Q, let U_{q}(𝔫)_{μ} be the subspace of U_{q}(𝔫) generated by the products E_{i_{1}}^{n_{1}} … E_{i_{k}}^{n_{k}} such that ∑_{l} n_{l}α_{l} = μ. An element X of U_{q}(𝔫)_{μ} will be called (homogeneous) element of weight μ. We set wt(X) := μ.

Recall the triangular decomposition U_{q}(𝔤) = U_{q}(𝔫^{−})⊗U_{q}^{0}⊗U_{q}(𝔫). We define the following subalgebras of U_{q}(𝔤):

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

As in [22], [16], we introduce Lusztig’s automorphisms T_{i}, 1 ≤ i ≤ n, which define a braid action on U_{q}(𝔤) by (1.2.1)

T_{i}(E_{i}) = −F_{i}K_{α_{i}}, T_{i}(E_{j}) = ∑_{s=0}^{−a_{ij}}(−1)^{−a_{ij}−s}q_{α_{i}}^{a_{ij}+s}E_{i}^{(s)}E_{j}E_{i}^{(−a_{ij}−s)}, 1 ≤ i, j ≤ n, i ≠ j

(1.2.2)

T_{i}(F_{i}) = −K_{−α_{i}}E_{i}, T_{i}(F_{j}) = (−1)^{−a_{ij}−s} ∑_{s=0}^{−a_{ij}} q_{α_{i}}^{−a_{ij}−s}F_{i}^{(−a_{ij}−s)}F_{j}F_{i}^{(s)}, 1 ≤ i, j ≤ n, i ≠ j

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

where E_{α}^{(k)} = \frac{1}{[k]_{qα}!}E_{α}^{k}, [k]_{q_{α}}! = [k]_{q_{α}}[k − 1]_{q_{α}} … [1]_{q_{α}},

[k]_{q_{α}} = \frac{q_{α}^{n}−q_{α}^{−n}}{q_{α}−q_{α}^{−1}} , q_{α} = q^{\frac{(α,α)}{2}} . Fix a reduced decomposition w̃_{0} = s_{i_{1}}s_{i_{2}} … s_{i_{N}} of w_{0}. For each k, 1 ≤ k ≤ N, set β_{k} := s_{i_{1}} … s_{i_{k−1}}(α_{k}). It is well known, [17], that the {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 R^{+}. This ordering will identify the semigroup ℤ_{≥0}^{R^{+}} with the semigroup ℤ_{≥0}^{N}. In the sequel, we note {e_{k}, 1 ≤ k ≤ N} the natural basis of this semigroup.

For all k, define Ew̃_{0} ; β_{k} = E_{β_{k}} = T_{i_{1}} … T_{i_{k−1}}(E_{α_{i_{k}}}). For all m = (m_{i}) ∈ ℤ_{≥0}^{N}, set

Ew̃_{0}(m) = E(m) := E_{β_{1}}^{(m_{1})} … E_{β_{N}}^{(m_{N})}. It is known that {E(m), m ∈ ℤ_{≥0}^{N}} is a basis of U_{q}(𝔫) called the Poincaré-Birkhoff-Witt basis, in short PBW-basis, associated to the reduced decomposition w̃_{0}. In the same way, we can define the PBW-basis {F (m), m ∈ ℤ_{≥0}^{N}} of U_{q}(𝔫^{−}).

In the sequel, we call a (left) factor of w̃_{0} a reduced decomposition w̃ = s_{i_{1}} … s_{i_{k}}, 1 ≤ k ≤ N. Conversely, we say that w̃_{0} is a (right) completion of w̃.

Let w be in the Weyl group and let w̃ = s_{i_{1}} … s_{i_{k}} be a reduced decomposition of w which is completed to w̃_{0} = s_{i_{1}}s_{i_{2}} … s_{i_{N}}. Let U_{q}(𝔫w̃) be the ℚ(q)-space generated by the Ew̃_{0}(m) such that m_{i} = 0 for i > k. By [7, 2.3], U_{q}(𝔫w̃) is a subalgebra of U_{q}(𝔫) which depends only on w and not on the reduced decomposition w̃. In the sequel, we shall note it simply U_{q}(𝔫_{w}). Recall the following theorem, [6, 1.7]:

Theorem. Fix a reduced decomposition w̃_{0} of w_{0} and set

F w̃_{0} ; m(U_{q}(𝔫)) = ⊕_{n≺m}ℚ(q)Ew̃_{0}(n), m ∈ ℤ_{≥0}^{N},

where ≺ is the right lexicographical ordering of ℤ_{≥0}^{N}. Then, the spaces F w̃_{0} ; m(U_{q}(𝔫)), m ∈ ℤ_{≥0}^{N}, define a ℤ_{≥0}^{N}-filtration of U_{q}(𝔫). The associated graded algebra Grw̃_{0}(U_{q}(𝔫)) is generated by Grw̃_{0}(E_{α}), α ∈ R^{+} with relations (up to a sign):

Grw̃_{0}(E_{α})Grw̃_{0}(E_{β}) = q^{<α,β>}Grw̃_{0}(E_{β})Grw̃_{0}(E_{α}), α < β.

As in [11, 4.2], we define the bilinear forms dw̃_{0} = d and cw̃_{0} = c on ℤ_{≥0}^{N} × ℤ_{≥0}^{N} such that

Gr(E(m))Gr(E(n)) ∈ q^{−d(m,n)}(±1 + qℤ[q])Gr(E(m + n))
Gr(E(m))Gr(E(n)) = q^{c(n,m)}Gr(E(n))Gr(E(m)),

up to a sign. To be more precise, we have,

d(m, n) = ∑_{i>j} < β_{i}, β_{j} > m_{i}n_{j} + \frac{1}{2} ∑_{i} < β_{i}, β_{i} > m_{i}n_{i}.

We define the ℤ[q]-lattice L generated by the {Ew̃_{0}(m), m = (m_{i}) ∈ ℤ_{≥0}^{N}}. From [15, Proposition 2.3], this lattice does not depend on the choice of w̃_{0}.

1.3 There exists, see [23], a unique non degenerate Hopf pairing ( , ) on U_{q}(𝔟) × U_{q}(𝔟^{−}) such that

(u^{+}, u_{1}^{−}u_{2}^{−}) = (Δ(u^{+}), u_{1}^{−} ⊗ u_{2}^{−}) , u^{+} ∈ U_{q}(𝔟) ; u_{1}^{−}, u_{2}^{−} ∈ U_{q}(𝔟^{−})
(u_{1}^{+}u_{2}^{+}, u^{−}) = (u_{2}^{+} ⊗ u_{1}^{+}, Δ(u^{−})) , u^{−} ∈ U_{q}(𝔟^{−}) ; u_{1}^{+}, u_{2}^{+} ∈ U_{q}(𝔟)

(K_{λ}, K_{μ}) = q^{−(λ,μ)} , λ, μ ∈ Q

(K_{λ}, F_{i}) = 0 , λ ∈ Q , 1 ≤ i ≤ n

(E_{i}, K_{λ}) = 0 , λ ∈ Q , 1 ≤ i ≤ n

(E_{i}, F_{i}) = \frac{1}{1 − q_{α_{i}}^{−2}} δ_{ij} , 1 ≤ i, j ≤ n,

where Δ is the comultiplication of the Hopf algebra U_{q}(𝔤).

Let {E(m)^{∗}, m ∈ ℤ_{≥0}^{N}} be the dual basis of {F (m), m ∈ ℤ_{≥0}^{N}} for this pairing.

From [12], we claim that:

Claim. For all m in ℤ_{≥0}^{N} there exists a function f_{m}(q) in ℚ(q) such that E(m)^{∗} = f_{m}(q)E(m) and f_{m}(0) = 1. Moreover, this function is an eigenvector for the ℚ-automorphism of ℚ(q) defined by q ↦ q^{−1}.

We set

L^{∗} = ⊕_{m∈ℤ_{≥0}^{N}}ℤ[q]E(m)^{∗}.

1.4 In this section, we give some results about Lusztig’s canonical basis B and its dual B^{∗}. First of all, for λ in P^{+}, let V_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ. Choose a highest weight vector v_{λ}. It is known that V_{q}(λ) verifies the Weyl character formula. For all w in W, let v_{wλ} be a (non zero) extremal vector of weight wλ and let V_{q,w}(λ) := U_{q}(𝔫).v_{wλ} be the Demazure module.

Let B be the Lusztig’s canonical basis of U_{q}(𝔫^{−}), [15], which coincides with Kashiwara’s global basis, [9]. It verifies the following property:

Theorem. Fix λ in P^{+}, and let B(λ) := {b ∈ B, bv_{λ} ≠ 0}. Then, the set B(λ).v_{λ} is a basis of V_{q}(λ). Moreover, for all w in W there exists a unique subset B_{w} of B, which does not depend on λ and such that (B(λ) ∩ B_{w}).v_{λ} is a basis of V_{q,w}(λ).

Remark. In the sequel, if no confusion occurs, we shall identify B(λ) with B(λ).v_{λ}.

Let B^{∗} ⊂ U_{q}(𝔫) be the dual basis in U_{q}(𝔫), i.e. (b^{∗}, b′) = δ_{b,b′}. Remark that this basis is not really canonical since it depends on the choice of a Hopf pairing ( , ). Nethertheless, we shall call it the dual canonical basis.

Let η be the ℚ-automorphism of U_{q}(𝔤) such that η(E_{i}) = E_{i}, η(F_{i}) = F_{i}, and η(q) = q^{−1}. Let σ be the ℚ(q)-antiautomorphism of U_{q}(𝔤) such that \overline{E}_{i} = E_{i}, \overline{F}_{i} = F_{i}. We can now give a characterization of B^{∗}, see [11, Proposition 16]. This characterization will give rise to the Lusztig’s parametrization of the dual canonical basis B^{∗}. It depends on the choice of a reduced decomposition of w_{0}.

Proposition. Fix a reduced decomposition w̃_{0} of w_{0}. Then, for each m in ℤ_{≥0}^{N}, there exists a unique homogeneous element X := Bw̃_{0}(m)^{∗} = B(m)^{∗} in U_{q}(𝔫) such that

ση(X) = (−1)^{tr(X)}q^{<wt(X),wt(X)>/2}q_{X}X, X ∈ E(m)^{∗} + qL^{∗},

where wt(X) = ∑_{i} k_{i}α_{i}, is the weight of X, q_{X} = ∏_{i} q_{α_{i}}^{k_{i}}, and tr(X) = ∑_{i} k_{i}.

Remark. First remark that the proposition implies that L, resp. L^{∗}, is the ℤ[q]-lattice generated by B, resp. B^{∗}. Remark also that the eigenvalue of X = B(m)^{∗} for ση only depends on the weight of X. Now, it can be easily seen that the first condition in the proposition can be replaced by “X is an eigenvector for ση”.

We note Bw̃_{0}(m) = B(m) the corresponding element in the canonical basis B. For w in W and λ in P^{+}, we set B_{w}^{∗} := {b^{∗} ∈ B^{∗}, b ∈ B_{w}}, and B(λ)^{∗} := {b^{∗} ∈ B^{∗}, b ∈ B(λ)}.

2 Preliminary results on the dual canonical basis.

2.1 In this section, we obtain a property of triangularity for the decomposition matrix of the dual canonical basis in a general PBW basis.

Proposition. Let w̃_{0} be a reduced decomposition of w_{0}. Then, for all α in R^{+}, the element (Ew̃_{0} ; α)^{∗} of the dual PBW-basis belongs to B^{∗}.

Proof. : By Proposition 1.4 and Remark 1.4, it is enough to prove that (Ew̃_{0} ; α)^{∗} is an eigenvector for ση. By Claim 1.3, it is sufficient to prove that this is true for Ew̃_{0} ; α.

Set w̃_{0} = s_{i_{1}} … s_{i_{N}}. We have Ew̃_{0} ; α = T_{i_{1}} … T_{i_{p−1}}(E_{i_{p}}) for p such that α = β_{p}. By

(1.2.1)-(1.2.3), we obtain that for all homogeneous element X in U_{q}(𝔫) of weight μ, we have σηT_{i}(X) = (−q_{α_{i}})^{tr(s_{i}μ−μ)}T_{i}(ση(X))). Remark that E_{i_{p}} is an homogeneous eigenvector for ση. By induction, this is also true for Ew̃_{0} ; α.

The following corollary generalizes [15, Theorem 9.13 (a)].

Corollary. Let w̃_{0} be a reduced decomposition of w_{0}. Then, for all m in ℤ_{≥0}^{N}, we have the following homogeneous decompositions.

(i) ση(Ew̃_{0}(m)) = e_{m}^{m}Ew̃_{0}(m) + ∑_{n≺m} e_{m}^{n}Ew̃_{0}(n), e_{m}^{n} ∈ ℤ[q, q^{−1}],

(ii) Bw̃_{0}(m) = F w̃_{0}(m) + ∑_{m≺n} d_{m}^{n}F w̃_{0}(n), d_{m}^{n} ∈ qℤ[q],

(iii) Bw̃_{0}(m)^{∗} = Ew̃_{0}(m)^{∗} + ∑_{n≺m} c_{m}^{n}Ew̃_{0}(n)^{∗}, c_{m}^{n} ∈ qℤ[q].

Proof. Let m be in ℤ_{≥0}^{N} and set ση(E(m)) = ∑_{n} e_{m}^{n}E(n), e_{m}^{n} ∈ ℤ[q, q^{−1}]. Up to a multiplicative scalar, we have from the previous proposition: ση(E(m)) = ση(E_{β_{1}}^{(m_{1})} … E_{β_{N}}^{(m_{N})})

= ση(E_{β_{N}}^{(m_{N})}) … ση(E_{β_{1}}^{(m_{1})}) = E_{β_{N}}^{(m_{N})} … E_{β_{1}}^{(m_{1})}. Hence, by Theorem 1.2, Grw̃_{0}(ση(E(m))) =Grw̃_{0}(E(m)), up to a multiplicative scalar. This implies (i).

By duality, it is sufficient to prove (iii). The coefficients c_{m}^{n} are in qℤ[q] by Proposition 1.3. Now, the property of triangularity comes from Proposition 1.4 and (i).

Remark. In the previous corollary, the lexicographical ordering ≺ can be replaced by a coarser ordering: the one generated by m ≤ e_{k} + e_{k′}, 1 ≤ k < k′ ≤ N ⇔ E(m) is a term of the PBW decomposition of E_{β_{k}}E_{β_{k′}} − q^{<β_{k′} ,β_{k}>}E_{β_{k′}} E_{β_{k}}.

2.2 The previous corollary implies the compatibility of the dual canonical basis with the space U_{q}(𝔫_{w}), w ∈ W.

Proposition. Let w be in W. Then, U_{q}(𝔫_{w}) is generated as a space by a part of B^{∗}.

Proof. Let w̃ be a reduced decomposition of w and let w̃_{0} be a reduced decomposition of w_{0} which completes w̃. Fix an element of the PBW-basis which belongs to U_{q}(𝔫_{w}). Then, by 1.2, all smaller elements of the PBW-basis, with the same weight, belong to U_{q}(𝔫_{w}). By Corollary 2.1, this implies the proposition.

3 Quantum flag minors.

3.1 For all λ in P^{+}, the weights of B(λ)^{∗} are the λ − μ, where μ runs over the weights of V_{q}(λ) (with multiplicity). The quantum flag minors, see [10], [2, 4.2], are elements of B(ϖ_{i})^{∗}, 1 ≤ i ≤ n, which correspond to extremal vectors. To be more precise, let w be in W and let w̃ = s_{i_{1}} … s_{i_{k}} be a reduced decomposition of w. There exists a unique element in B(ϖ_{i_{k}})^{∗} with weight (Id − w)(ϖ_{i_{k}}). Note Δ∗ ; w̃ this element. Δ∗ ; w̃ is a quantum flag minor and each quantum flag minor can be written in this way.

The following proposition generalizes some properties of the q-center of U_{q}(𝔫) = U_{q}(𝔫_{w_{0}}) proved in [4, Proposition 3.2] to U_{q}(𝔫_{w}).

Proposition. Fix an element w in W and let w̃ = s_{i_{1}} … s_{i_{k}} be a reduced decomposition of w. We have

(i) Let X_{μ} be an element of weight μ in U_{q}(𝔫_{w}), then Δ∗ ; w̃X_{μ} = q^{<(Id+w)ϖ_{i_{k}} ,μ>}X_{μ}Δ∗ ; w̃,

(ii) let b_{μ}^{∗} be an element of weight μ in B_{w}^{∗}, then q^{<ϖ_{i_{k}} ,μ>}b_{μ}^{∗}Δ∗ ; w̃ ∈ B^{∗} mod qL^{∗}.

Proof. Let’s prove (i). Set w̃′ = s_{i_{1}} … s_{i_{k′}}, for k′ ≤ k. From [5, (3.3.2)], we obtain

Δ∗ ; w̃Δ_{w̃′}^{∗} = q^{−<(Id−w)ϖ_{i_{k}} −2ϖ_{i_{k}} ,μ′>}Δ_{w̃′}^{∗}Δ∗ ; w̃,

where μ′ := (Id − w′)ϖ_{i_{k′}} is the weight of Δ_{w̃′}^{∗} and k′ < k. Hence

Δ∗ ; w̃Δ_{w̃′}^{∗} = q^{<(Id+w)ϖ_{i_{k}} ,μ′>}Δ_{w̃′}^{∗}Δ∗ ; w̃.

This formula holds for k′ = k because < (Id + w)ϖ_{i_{k}}, (Id − w)ϖ_{i_{k}} >=< ϖ_{i_{k}}, ϖ_{i_{k}} > − < wϖ_{i_{k}}, wϖ_{i_{k}} >= 0, by W -invariance. Now, (i) comes from the fact that the division ring generated by the Δ_{w̃′}^{∗}, 1 ≤ k′ ≤ k, is the division ring of U_{q}(𝔫_{w}), [5, corollary 3.2].

The proof of (ii) is a straightforward generalization of [4, Proposition 3.2]. Let’s sketch the proof. Let b_{μ} and Δw̃ be the elements in B corresponding respectively to b_{μ}^{∗} and Δ∗ ; w̃ and suppose b_{μ} ∈ B(λ), λ ∈ P^{+}. Then, Δw̃ ⊗b_{μ} ∈ B(ϖ_{i_{k}})⊗B(λ) and this element corresponds to an element of the crystal basis at q = 0. We know that b_{μ} is in B_{w}. Hence, by Littelmann’s path model, we can associate to b_{μ} a chain of elements in W which are lower than w for the Bruhat ordering. Moreover, the chain associated to Δw̃ is reduced to w. So, both chains can be compared for the Bruhat ordering. This implies that Δw̃ ⊗ b_{μ} ∈ B(ϖ_{i_{k}} + λ) at the crystal level. This implies (ii) by [11, Proposition 33].

3.2 By [7, Theorem 3.2] and Proposition 2.2, we have:

Lemma. Let w̃_{0} = s_{i_{1}} … s_{i_{N}} be a reduced decomposition of w_{0} and let w = s_{i_{1}} … s_{i_{k}}, 1 ≤ k ≤ N. Let m be in ℤ_{≥0}^{N} such that m_{k′} = 0 for k′ > k. Then, Bw̃_{0}(m) is in B_{w}.

Remark that this lemma can be directly proved by applying the Berenstein-Zelevinsky formula, [2, Theorem 3.7], for the transition map between the Lusztig parametrization and the string parametrization of the dual canonical basis. Indeed, by [9, Theorem 12.4], the elements of B_{w}^{∗} are characterized by a string parametrization.

We can now prove the key proposition:

Proposition. Let w̃_{0} = s_{i_{1}} … s_{i_{N}} be a reduced decomposition of w_{0} and let w = s_{i_{1}} … s_{i_{k}}, 1 ≤ k ≤ N. Fix m in ℤ_{≥0}^{N}. Then,

q^{d(nw̃,m)}Δ∗ ; w̃Ew̃_{0}(m)^{∗} ∈ Ew̃_{0}(nw̃ + m)^{∗} + qL^{∗},

where nw̃ is such that Δ∗ ; w̃ = Bw̃_{0}(nw̃)^{∗}.

Proof. We first suppose that m is in ℤ_{≥0}^{k} × {0}^{N−k}. Then, by Proposition 3.1 and the previous lemma, we have

q^{<ν−ϖ_{i_{k}} ,μ>}Δ∗ ; w̃Bw̃_{0}(m)^{∗} ∈ B^{∗} + qL^{∗},

where ν and μ are respectively the weights of Δ∗ ; w̃ and Bw̃_{0}(m)^{∗}.

Recall the notations of 1.2. By [11, 4.2], we have

d(nw̃, m) + d(m, nw̃) =< ν, μ >, d(nw̃, m) − d(m, nw̃) = c(nw̃, m).

Using Proposition 3.1, we obtain d(nw̃, m) =< ν − ϖ_{i_{k}}, μ > . Hence, there exists m′ in ℤ_{≥0}^{N} such that

q^{d(nw̃,m)}Δ∗ ; w̃Bw̃_{0}(m)^{∗} ∈ Ew̃_{0}(m′)^{∗} + ∑_{n} qℤ[q]Ew̃_{0}(n)^{∗}.

Using Theorem 1.2 and Claim 1.3, we obtain m′ = nw̃ +m. By Proposition 2.2, this implies

q^{d(nw̃,m)}Δ∗ ; w̃Ew̃_{0}(m)^{∗} ∈ Ew̃_{0}(nw̃ + m)^{∗} + qL^{∗}.

We can now study the general case. Let m be in ℤ_{≥0}^{N} and decompose m = n + p, with n ∈

ℤ_{≥0}^{k}×{0}^{N−k} and p_{l} = 0 for l ≤ k. We have q^{d(nw̃,m)}Δ∗ ; w̃E(m)^{∗} = q^{d(nw̃,n)}Δ∗ ; w̃E(n)^{∗}E(p)^{∗} ∈

(E(nw̃+n)^{∗}+∑ qℤ[q]E(r)^{∗})E(p)^{∗}, where r runs over ℤ_{≥0}^{k}×{0}^{N−k}. Hence, q^{d(nw̃,m)}Δ∗ ; w̃E(m)^{∗} ∈ E(nw̃ + n + p)^{∗} + ∑ qℤ[q]E(r + p)^{∗} ⊂ E(nw̃ + m)^{∗} + qL^{∗}. This ends the proof.

4 Quiver orientations and quantum flag minors

According to [4], Proposition 3.2 is almost what we need if we want to test the Berenstein-Zelevinsky conjecture when one element is a quantum flag minor. In fact, we have to prove an analogue of this proposition where Ew̃_{0}(m)^{∗} is replaced by Bw̃_{0}(m)^{∗}. This can be obtained if we prove some increasing property of the linear form dw̃_{0}(nw̃,?). By results of Reineke, [19], and Bongartz, [3], it is possible to prove those properties by the quiver approach, [20], of quantum groups.

4.1 This section refers to [21] for notations and definitions. Define the graph Γ as follows: if 𝔤 is simply laced, i.e. of type A-D-E, Γ is the Coxeter graph of 𝔤, if 𝔤 is not simply laced then Γ is the A-D-E graph such that the graph of 𝔤 is a quotient of this graph, see [16, par. 14], [18, par. 7]. Fix an orientation \overrightarrow{Γ} of Γ and let Modk\overrightarrow{Γ} be the category of k-representations of \overrightarrow{Γ}, where k is an algebraically closed field.

For M, N in Modk\overrightarrow{Γ}, set

ε(M, N) = dim_{k}Hom(M, N), ζ(M, N) = dim_{k}Ext^{1}(M, N).

Let Indk\overrightarrow{Γ} be the set of indecomposable modules of Modk\overrightarrow{Γ}. Let τ: Indk\overrightarrow{Γ} → Indk\overrightarrow{Γ}∪\{0\} be the Auslander-Reiten translation, with the convention that τ(M) = 0 if M is a projective module. Recall, [21], that

(4.1.1)  ζ(M, N) = ε(N, τ(M)), M, N ∈ Indk\overrightarrow{Γ}.

By the work of Ringel, [20], the algebra U_{q}(𝔫) can be realized as a deformation of the Hall algebra associated to \overrightarrow{Γ}. In particular, for a special reduced decomposition w̃_{0}(\overrightarrow{Γ}) of w_{0}, the PBW-basis of U_{q}(𝔫) can be naturally parametrized by Modk\overrightarrow{Γ}. The following proposition is a recollection of results which can be found in [19], [15, par 4]. It gives informations on the link between the Modk\overrightarrow{Γ}-parametrization and the Lusztig parametrization associated to w̃_{0}(\overrightarrow{Γ}).

Proposition. There exists a reduced decomposition w̃_{0}(\overrightarrow{Γ}) = s_{i_{1}}s_{i_{2}} … s_{i_{N}}, defined up to commutations, and a ℤ_{≥0}-linear bijection ι: Modk\overrightarrow{Γ} → ℤ_{≥0}^{N} such that:

(i) if M is indecomposable non projective and if ι(M) = e_{k} , then ι(τ(M)) = e_{k′}, where k′ < k is maximal such that i_{k′} = i_{k}, (ii) d(ι(M), ι(N)) = ε(N, M) − ζ(M, N). Moreover, let i be a sink of \overrightarrow{Γ} then, up to commutations, i_{1} = i. Let \overrightarrow{Γ}_{i} be the oriented graph obtained from \overrightarrow{Γ} by reversing the arrows which go to i, we have w̃_{0}(\overrightarrow{Γ}_{i}) = s_{i_{2}} … s_{i_{N}} s_{i^{∗}}, where i^{∗} is such that w_{0}(α_{i}) = −α_{i^{∗}}.

4.2 In the following sections, we shall replace the lexicographical ordering ≺ by the coarser ordering discussed in Remark 2.1. We still note it ≺. If the reduced decomposition w̃_{0} is associated to a graph orientation, then this ordering is the Ext-ordering on Modk\overrightarrow{Γ} ≃ ℤ_{≥0}^{N}, [3], which is also the degeneration ordering, [loc. cit., Corollary 4.2, Proposition 3.2]. We are now ready to prove the expected “increasing property” of dw̃_{0}(nw̃,?).

Proposition. Let w̃_{0}(\overrightarrow{Γ}) = s_{i_{1}}s_{i_{2}} … s_{i_{N}} be the reduced decomposition of w_{0} associated to \overrightarrow{Γ} and let w̃ = s_{i_{1}}s_{i_{2}} … s_{i_{k}} be a subword of w̃_{0}(\overrightarrow{Γ}). Let nw̃ be as in 3.1. Then, the form d(nw̃,?) is increasing for ≺.

Proof. By [5, 3.2.2], we have nw̃ = ∑ e_{l} where l runs over the set of integers of [1, k] such that i_{l} = i_{k}. From Proposition 4.1, we obtain that

d(nw̃,?) = ∑_{i≥0} ε(?, τ^{i}(M_{k})) − ∑_{i≥0} ζ(τ^{i}(M_{k}),?),

where M_{k} is the indecomposable module corresponding to e_{k}. By (4.1.1), we obtain after elimination that d(nw̃,?) = ε(?, M_{k}), which is increasing by [3, Proposition 3.2].

4.3 Now, the natural question is to know if all quantum flag minor can be associated to a graph decomposition \overrightarrow{Γ}. In clear, fix u in W and a reduced decomposition ũ of u. Is there an orientation \overrightarrow{Γ} of Γ and a subword w̃ of w̃_{0} = w̃_{0}(\overrightarrow{Γ}) such that Δ∗ ; ũ = Δ∗ ; w̃. This is true for the A_{n} case, but not in general:

Claim. Let 𝔤 be of type A_{n}. Then, all quantum flag minors can be realized as Bw̃_{0}(nw̃)^{∗} where w̃_{0} is associated to an orientation of Γ.

Proof. It is known that for all k, 1 ≤ k ≤ n, the set of quantum minors in B(ϖ_{k})^{∗} is naturally indexed by a set of lines I = {i_{1} < i_{2} < … < i_{k}}. We generally note it Δ_{q}(I, J) where J = {1, …, k} is the set of its columns. This (flag) quantum minor corresponds to an extremal vector in V_{q}(ϖ_{k}) associated to

w = s_{(i_{1}−1)} … s_{1}s_{(i_{2}−1)} … s_{1} … s_{(i_{k}−1)} … s_{1}W^{k},

where W^{k} := {u ∈ W, uϖ_{k} = ϖ_{k}}. We claim that this reduced decomposition w̃ = s_{(i_{1}−1)} … s_{1}s_{(i_{2}−1)} … s_{1} … s_{(i_{k}−1)} … s_{1} can be completed to a reduced decomposition of w_{0} which is associated to an orientation of Γ. Indeed, this follows from the last assertion of Proposition 4.1 and the well known fact: w̃_{0} := s_{1}s_{2}s_{1}s_{3}s_{2}s_{1} … s_{n}s_{n−1} … s_{2}s_{1} is a reduced decomposition associated to the orientation \overrightarrow{Γ} of A_{n} such that all arrows are oriented to the left. Indeed, with the notations of Proposition 4.1, w̃ can be completed to the reduced decomposition of w_{0} associated to the quiver \overrightarrow{Γ}_{i}, where i is the following sequence of reversing arrows

i = 121 … (n − k) … 1(n − k + 1) … i_{1}(n − k + 2) … i_{2} … n … i_{k}.

Remark. The claim is not true if we take 𝔤 of type D_{4}. With the standard notations, the quantum flag minor associated to the reduced decomposition w̃ = s_{2}s_{1}s_{3}s_{2} can not be realized from a quiver orientation.

5 Adapted algebras associated to a quiver orientation.

5.1 We first recall some basic facts on adapted algebras associated to a reduced decomposition of w_{0}, see [4]. Fix a reduced decomposition w̃_{0} of w_{0} and let W̃ be the set of left subwords of w̃_{0}. Then, the quantum flag minors Δ∗ ; w̃, w̃ ∈ W̃, q-commute. Moreover, they generate a ℚ(q)-algebra Aw̃_{0} such that

1) Aw̃_{0} is a q-polynomial algebra of GK-dimension N = Card W̃,

2) As a space, Aw̃_{0} is generated by a part of the dual canonical basis, namely the monomials

in the Δ∗ ; w̃, w̃ ∈ W̃.

Now, we reach the main theorem:

Theorem. Fix an orientation \overrightarrow{Γ} of Γ and set w̃_{0} = w̃_{0}(\overrightarrow{Γ}). Suppose that b^{∗} in Aw̃_{0} ∪ B^{∗} and (b′)^{∗} in B^{∗} q-commute, then they are multiplicative, i.e. their product is an element of the dual canonical basis up to a power of q.

Proof. Set w̃_{0} = s_{i_{1}}s_{i_{2}} … s_{i_{N}}. The elements b^{∗} = Bw̃_{0}(m)^{∗} and (b′)^{∗} = Bw̃_{0}(m′)^{∗} qcommute. Hence, in order to prove that they are multiplicative, it is sufficient, see [11, 5.1], to prove that

(5.1.1)  q^{dw̃_{0} (m,m′)}Bw̃_{0}(m)^{∗}Bw̃_{0}(m′)^{∗} ∈ Bw̃_{0}(m + m′)^{∗} + qL^{∗}.

Recall that, by Corollary 2.2,

(5.1.2)  Bw̃_{0}(m′)^{∗} = Ew̃_{0}(m′) + ∑_{n≺m′} c_{m′}^{n}Ew̃_{0}(n)^{∗},

with c_{m′}^{n} ∈ qℤ[q]. Moreover, Bw̃_{0}(m)^{∗} is a product of Δ∗ ; w̃, where w̃ runs over a subset W̃_{0} of W̃. This implies that dw̃_{0}(m,?) = ∑ dw̃_{0}(nw̃,?), where w̃ runs over W̃_{0} with multiplicity. By Proposition 4.2, this asserts that the form dw̃_{0}(m,?) is increasing for ≺. Hence, (5.1.1) is a consequence of Proposition 3.2 and (5.1.2).

Remark. It is likely that the theorem works for all reduced decomposition of w_{0}. In fact, all we have to prove is a generalization of Proposition 4.2 for general w̃_{0}.

By Claim 4.3, we deduce the following corollary which generalizes a result of [11]:

Corollary. Let 𝔤 be of type A_{n}. Let b^{∗} and (b′)^{∗} be q-commuting elements in B^{∗}. If b^{∗} is a quantum flag minor, then b^{∗} and (b′)^{∗} are multiplicative.

5.2 In order to understand a “multiplicative” description of the dual canonical basis, S. Fomin and A. Zelevinsky have defined the so-called cluster algebras, [8]. Roughly speaking, cluster algebras are algebras equipped with a distinguished set of generators called cluster variables and this set is divided into a union of subsets called clusters. The cluster variables verify the so-called exchange relations. For each symmetrizable Cartan datum, Fomin and Zelevinsky associate a cluster algebra.

As a conclusion, we would like to enlight the link between the results of 5.1 and the theory of cluster algebras. This follows some discussions with A. Zelevinsky. One of the most amazing fact in the Fomin-Zelevinsky theory is that many algebras, as ℂ[U^{−}], ℂ[G], encountered in the representation theory of semi-simple Lie groups seem to be realized in the algebraic framework of cluster algebras. In particular the subalgebras Aw̃_{0} should specialize at q = 1 onto a subalgebra of ℂ[U^{−}] generated by a cluster. The connection with our problem is the following : it is reasonable to think that if b and b′ are q-commuting elements of the dual canonical basis, and if b is specialized at q = 1 onto a cluster variable, then b and b′ are multiplicative. What we proved in 5.1 is a particular case of this conjecture when the cluster is associated to some reduced word.

We would like to thank Bernard Leclerc, Eric Vasserot and Andrei Zelevinsky for usefull conversations.

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