Adapted algebras and standard monomials

P. Caldero^{∗} and P. Littelmann^{∗}

January 22, 2002

Abstract Let G be a complex semisimple Lie group. The aim of this article is to compare two basis for G-modules, namely the standard monomial basis and the dual canonical basis. In particular, we give a sufficient condition for a standard monomial to be an element of the dual canonical basis and vice versa.

Introduction

Let G denote a semisimple, simply connected, algebraic group defined over an algebraically closed field k of arbitrary characteristic. We fix a Borel subgroup B and a maximal torus T ⊂ B, denote W the Weyl group of G with respect to T. For a dominant weight λ let V (λ) the corresponding Weyl module and let Q be the parabolic subgroup which normalizes a highest weight vector. Let L_{λ} be the corresponding ample line bundle on G/Q. Consider the embedding G/Q ↪ ℙ(V (λ)) given by the global sections H^{0}(G/Q, L_{λ}) ≃ V (λ)^{∗}.

The aim of standard monomial theory is to give a presentation of the ring R = ⨁_{n≥0} H^{0}(G/Q, L_{nλ}), which is compatible with the natural subvarieties of G/Q as for example the Schubert varieties X(τ), the opposite Schubert varieties X^{κ} and the Richardson varieties X_{τ}^{κ} = X_{τ} ∩ X^{κ}, τ, κ ∈ W/W_{Q}. More precisely, the aim is to construct a basis B ⊂ H^{0}(G/Q, L_{λ}) such that certain (= standard) monomials of degree n in these basis elements form a basis B(nλ) for H^{0}(G/Q, L_{nλ}), the relations provide an algorithm to write a non-standard monomial as linear combination of standard monomials, and the restrictions {m|_{X_{τ}^{κ}} | m ∈ B(nλ), m ≢ 0 on X_{τ}^{κ}} of the standard monomials to a Richardson variety X_{τ}^{κ}, form a basis for H^{0}(X_{τ}^{κ}, L_{nλ}). Such a basis has turned out to be a powerful tool in the investigation of the geometry of Schubert (and related) varieties, see for example [13] and [16].

∗This research has been partially supported by the EC TMR network “Algebraic Lie Representations”, contract no. ERB FMRX-CT97-0100.

The concept of an adapted algebra has been introduced by the first author in [3, 4] in connection with the Berenstein-Zelevinsky conjecture on q-commuting elements of the dual canonical basis ℂ_{q}[U^{−}]. Let U ⊂ B be a maximal unipotent subgroup, let U^{−} be its opposite unipotent subgroup and let 𝔫^{−} be its Lie algebra. Denote U_{q}(𝔫^{−}) the quantized enveloping algebra and let B ⊂ U_{q}(𝔫^{−}) be the canonical basis. As in the classical case, one has a non-degenerate pairing between the quantized algebra of regular functions ℂ_{q}[U^{−}] on U^{−} and U_{q}(𝔫^{−}), so ℂ_{q}[U^{−}] is naturally equipped with a canonical basis, the dual B^{∗} ⊂ ℂ_{q}[U^{−}] of the canonical basis B. This basis has many nice properties, for example the natural injection V_{q}(λ)^{∗} ↪ ℂ_{q}[U^{−}] is compatible with the basis, i.e., B^{∗}(λ) = B^{∗} ∩ V_{q}(λ)^{∗} is a basis for V_{q}(λ)^{∗}. So the specialization at q = 1 provides a basis B_{1}^{∗}(λ) ⊂ H^{0}(G/Q, L_{λ}), of which is known that it is compatible with Schubert (and related) varieties.

So on the one hand, B_{1}^{∗}(λ) looks like a perfect candidate as a starting basis for a standard monomial theory. The problem with this approach is that the multiplicative structure of the dual canonical basis is hardly understood, see [7]. The adapted algebras occur in this context, they are maximal subalgebras of ℂ_{q}[U^{−}] which are spanned as a vector space by a subset S ⊂ B^{∗}, and all elements of S are multiplicative, i.e., with b, b′ ∈ S also bb′ ∈ S. In particular, their product is again an element of the dual canonical basis! The first author has constructed in [3] for every reduced decomposition w̌_{0} of the longest word w_{0} in the Weyl group an adapted algebra Aw̌_{0} ⊂ ℂ_{q}[U^{−}].

On the other hand, the construction in [18] of a standard monomial theory for the ring R provides in particular a basis for H^{0}(G/Q, L_{λ}), the socalled path vectors p_{π}, which also has nice representation theoretic features (see for example [13, 16, 20]), and it is a natural question to ask for the relationship between these two bases. Recall that the path model theory [17, 19] provides combinatorial model [8, 10] for the crystal base theory, so we can index the the elements b_{π} ∈ B_{1}^{∗}(λ) and the path vectors p_{π} by L–S paths of shape λ.

The main result of this article can be formulated as follows (for a more precise and more general formulation see section 5): In the q-commutative parts of B^{∗} given by the adapted algebras Aw̌_{0}, the two bases coincide up to normalization. I.e., if b_{π} ∈ B_{1}^{∗}(λ) is the specialization of an element of an adapted algebra Aw̌_{0} ⊂ ℂ_{q}[U^{−}] for some reduced decomposition w̌_{0} of w_{0}, then b_{π} = p_{π}, up to normalization (multiplication by a root of unity). Vice versa, if the chain (\underline{τ}) of Weyl group elements in the L–S path π = (\underline{τ}, \underline{a}) is compatible with some reduced decomposition of w_{0}, then b_{π} = p_{π}, up to normalization. In the last section, we provide some examples to illustrate the connection between the path basis and the dual canonical basis.

1 Notation

Let G be a complex semisimple, simply connected algebraic group with Lie algebra 𝔤. We fix a Cartan decomposition 𝔤 = 𝔫⊕𝔥⊕𝔫^{−}, where 𝔟 = 𝔥⊕𝔫 is a Borel subalgebra of 𝔤 with fixed Cartan subalgebra 𝔥. Let U^{−} be a maximal unipotent subgroup of G such that Lie U^{−} = 𝔫 and let ℂ[U^{−}] be the algebra of regular functions on U^{−}. The left multiplication of U^{−} on itself induces a natural morphism ϕ from the enveloping algebra U(𝔫^{−}) on the algebra of differential operators on ℂ[U^{−}]. This induces a natural non-degenerate pairing on U(𝔫^{−}) × ℂ[U^{−}], mapping (u, f) to the value of ϕ(u)(f) at the identity of U^{−}.

This pairing has an analogue in the quantum setting, i.e., one has a natural non-degenerate pairing on U_{q}(𝔫^{−}) × ℂ_{q}[U^{−}]. So the algebras can be viewed as one being the restricted dual of the other.

Kashiwara [9] and Lusztig [22] have constructed the so-called global crystal or canonical basis B of the algebra U_{q}(𝔫^{−}). By the dual canonical basis we mean the basis B^{∗} of ℂ_{q}[U^{−}] dual to B with respect to the pairing above.

Some elements of B^{∗} can be easily described, they correspond to socalled extremal weight vectors in representations. Let λ ∈ P^{+} be a dominant weight and let V_{q}(λ) be the corresponding irreducible highest weight representation. Fix a highest weight vector v_{λ}, and for τ ∈ W/W_{λ} let v_{τ} be the unique weight vector in V_{q}(λ) of weight τ(λ) obtained in the following way: let τ = s_{i_{1}} ⋯ s_{i_{r}} be a reduced decomposition of τ and set

v_{τ} = F_{α_{i_{1}}}^{(n_{1})} ⋯ F_{α_{ir}}^{(n_{r})}v_{λ},

where n_{1} = ⟨s_{i_{2}} ⋯ s_{i_{r}}(λ), α_{i_{1}}^{∨}⟩ , …, n_{r} = ⟨λ, α_{i_{r}}^{∨}⟩. It follows from the Verma relations for the generators of U_{q}(𝔫^{−}) that the vector is independent of the chosen decomposition. This weight space is one dimensional, let b_{τ}^{λ} ∈ V_{q}(λ)^{∗} be the unique element of weight −τ(λ) such that

b_{τ}^{λ}(v_{τ}) = 1.  (1)

This linear form is considered as a form on U_{q}(𝔫^{−}) by setting b_{τ}^{λ}(u) = b_{τ}^{λ}(uv_{λ}). The compatibility property of the canonical basis with highest weight representations shows immediately that these functions b_{τ}^{λ} are elements of B^{∗}. In the following we refer to the b_{τ}^{λ} as (λ, q)–minors. For 𝔤 = 𝔰l_{n} and λ a fundamental weight, these are exactly the quantum minors.

It is often more convenient to forget in the notation the stabilizer W_{λ} and, by abuse of notation, just write b_{τ}^{λ} for τ ∈ W and λ ∈ P^{+}. In this way we can formulate the following simple product rule (see [6]):

∀λ, μ ∈ P^{+} : b_{τ}^{λ}b_{τ}^{μ} = b_{τ}^{λ+μ} in ℂ_{q}[U^{−}].  (2)

More generally, for ξ ∈ V_{q}(λ)^{∗} let c_{ξ}^{λ} be the linear form on U_{q}(𝔫^{−}) obtained by setting c_{ξ}^{λ}(u) := c_{ξ}^{λ}(uv_{λ}). The following commutation relation can be found for example in [6], it is a direct consequence of [15], where the expression of the R-matrix is made explicit: suppose ξ, η are of weight ν_{1}, ν_{2}. Then

c_{ξ}^{λ}c_{η}^{λ} = q^{−(λ,λ)+(ν_{1},ν_{2})}c_{η}^{λ}c_{ξ}^{λ} + ∑ c_{η_{i}}^{λ}c_{ξ_{i}}^{λ}  (3)

where there exist p_{i} ∈ ℂ(q) and some non-constant monomials M_{i} such that

η_{i} ⊗ ξ_{i} = p_{i}(q)(M_{i}(E) ⊗ M_{i}(F))η ⊗ ξ.

2 (λ, q)–minors and adapted algebras

Recall that two elements a, b ∈ ℂ_{q}[U^{−}] are said to q-commute if ab = q^{m}ba for some m ∈ ℤ. Two elements b, b′ ∈ B^{∗} are called multiplicative if the product is a multiple of an element of the canonical basis, i.e., bb′ = rb′′ for some b′′ ∈ B^{∗}, r ∈ ℂ(q). The elements of the dual canonical basis are in general neither multiplicative nor q-commuting. Let ϖ_{1}, …, ϖ_{n} be the set of fundamental weights and fix a reduced decomposition w_{0} = s_{i_{1}} ⋯ s_{i_{N}} of the longest word in W, we write w̌_{0} to refer to this decomposition. Set

y_{j} = s_{i_{1}} ⋯ s_{i_{j}} and set b_{j} := b_{y_{j}}^{ϖ_{i_{j}}} ∈ B^{∗}.  (4)

Remark 1 Since b_{τ}^{λ} = b_{τ′}^{λ} for τ ≡ τ′ mod W_{λ}, it is easy to verify the following description of the elements above:

{b_{1}, …, b_{N}} = {b_{y_{j}}^{ϖ_{i}} | 1 ≤ i ≤ n, 1 ≤ j ≤ N}

These elements have been studied in great detail by the first author in [3], where he shows that they have the following remarkable properties:

i) they are multiplicative and q-commute, and, more precisely, ii) the monomials b_{1}^{m_{1}} ⋯ b_{N}^{m_{N}} are, up to a power of q, elements of B^{∗}.

Let Sw̌_{0} be the set of the monomials above and let Aw̌_{0} be the subalgebra of ℂ_{q}[U^{−}] spanned by Sw̌_{0}. Actually, Aw̃_{0} is generated as a space by Sw̃_{0}.

These properties motivate the following definition: a subalgebra A of ℂ_{q}[U^{−}] is called adapted if it is spanned by a subset P^{∗} ⊂ B^{∗} such that the elements in P^{∗} are multiplicative and A is maximal with this property, i.e., for all b ∈ B^{∗} − P^{∗} there exist a p ∈ P^{∗} such that b and p are not multiplicative.

Theorem 2 ([3]) The subalgebra Aw̌_{0} is adapted with the elements of Sw̌_{0} (up to a rescaling by a power of q) as spanning set. The set Sw̌_{0} is an Ore set in ℂ_{q}[U^{−}], and for the localizations one has: S−1 ; w̌_{0}ℂ_{q}[U^{−}] = S−1 ; w̌_{0}Aw̌_{0}.

3 Lusztig’s Frobenius maps

Let R be the ring of Laurent polynomials ℤ[q, q^{−1}]. We denote U_{R}(𝔫^{−}) the R-form of U_{q}(𝔫^{−}) generated by the divided powers F_{i}^{(n)}. Fix a primitive 2ℓ-th root of unity ξ, where ℓ itself is even, and denote ℤ_{ξ} the ring ℤ[ξ].

Let U(𝔫^{−}) be the ℤ_{ξ}-form of the enveloping algebra of 𝔫^{−}, i.e., it is obtained from the Kostant ℤ-form by the ring extension ℤ ↪ ℤ_{ξ}.

Let U_{ξ}(𝔫^{−}) be the ℤ_{ξ}-algebra obtained from U_{R}(𝔫^{−}) by specialization at q = ξ, i.e., it is obtained from the Lusztig R-form U_{R}(𝔫^{−}) by base change with respect to R → ℤ_{ξ}, q ↦ ξ.

Let A = (a_{i,j})_{1≤i,j≤n} be the Cartan matrix of G and denote by ^{t}A = (\overline{a}_{i,j}), \overline{a}_{i,j} := a_{j,i}, the transposed matrix. Let \underline{d} = (d_{1}, …, d_{n}), d_{i} ∈ ℕ, be minimal such that the matrix (d_{i}a_{i,j}) is symmetric. We denote by d the smallest common multiple of the d_{j}. To distinguish between the objects associated to A and ^{t}A, we add always a ^{t} in the notation. For example, we write ^{t}𝔤 = ^{t}𝔫^{−} ⊕^{t}𝔥 ⊕^{t}𝔫 for the Cartan decomposition of the semisimple Lie algebra associated to ^{t}A, and we write ^{t}F_{i} for the generators of U_{q}(^{t}𝔫^{−}).

In the following we assume that ℓ is divisible by 2d and we set \overline{ℓ} = ℓ/d and let ℓ_{i} ∈ ℕ be minimal such that ℓ_{i} \frac{d}{d_{i}} ∈ ℓℤ. In [22], Chapter 35, Lusztig defines two algebra homomorphisms:

\begin{aligned}Fr & : & ^{t}U_{ξ}(𝔫^{−}) & → & U(𝔫^{−}) & ^{t}F_{i}^{(m)} & ↦ & F_{i}^{(m/ℓ_{i})} \\ Fr′ & : & U(𝔫^{−}) & → & ^{t}U_{ξ}(𝔫^{−}) & F_{i}^{(m)} & ↦ & ^{t}F_{i}^{(ℓ_{i}m)} \end{aligned}

where we set F_{i}^{(m/ℓ_{i})} := 0 if ℓ_{i} does not divide m. The first map is in fact a Hopf algebra homomorphism, but the second is not.

The corresponding R-form of ℂ_{q}[U^{−}] is denoted R_{q}[U^{−}], and its specializations at ξ respectively 1 are denoted R_{ξ}[U^{−}] respectively R_{1}[U^{−}]. The two algebra homomorphisms Fr and Fr′ induce dual maps:

Fr^{∗} : R_{1}[U^{−}] → R_{ξ}[^{t}U^{−}]  Fr′^{∗} : R_{ξ}[^{t}U^{−}] ↦ R_{1}[U^{−}].

Note that Fr^{∗} is an algebra homomorphism since Fr is a Hopf algebra homomorphism. Further, since Fr ∘ Fr′ is the identity map it follows that Fr′^{∗} ∘ Fr^{∗} is the identity. In particular, Fr^{∗} is injective and maps R_{1}[U^{−}] isomorphically onto a commutative subalgebra of R_{ξ}[^{t}U^{−}]

The maps Fr respectively its dual Fr^{∗} are called the Frobenius maps, and the maps Fr′^{∗} and Fr′^{∗} are called the Frobenius splittings (for the connection with the algebraic geometric notion of Frobenius splitting in positive characteristic see [11] and [12]).

Of special interest are for us the (λ, q)–minors:

Proposition 3 The Frobenius map Fr^{∗} is an algebra homomorphism which, restricted to the (λ, q)–minors, is the \overline{ℓ}-th power map, i.e.,

Fr^{∗}(b_{τ}^{λ}) = (b_{τ}^{λ})\overline{ℓ} = b_{τ}^{\overline{ℓ}λ}.

The splitting Fr′^{∗} is a ℤ_{ξ}-module homomorphism which commutes with the image of Fr^{∗}, i.e., for f ∈ R_{1}[U^{−}] and g ∈ R_{ξ}[^{t}U^{−}] we have

Fr′^{∗}(Fr^{∗}(f)g) = fFr′^{∗}(g).

In particular,

Fr′^{∗}(b_{τ_{1}}^{m_{1}\overline{ℓ}} ⋯ b_{τ_{r}}^{m_{r}\overline{ℓ}}) = b_{τ_{1}}^{m_{1}} ⋯ b_{τ_{r}}^{m_{r}}.

Proof. The first part follows immediately from [18], we sketch for convenience the arguments: let V_{ξ}(\overline{ℓ}λ) be the U_{ξ}(^{t}𝔤)–Weyl module for the highest weight \overline{ℓ}λ, and let L(\overline{ℓ}λ) be the simple module of highest weight \overline{ℓ}λ. Then L(\overline{ℓ}λ) is nothing but the Weyl module V (λ) for U(𝔤), viewed as quantum module via the (extension of the) Frobenius map Fr : U_{ξ}(^{t}𝔤) → U(𝔤). One sees easily that this induces a dual map (also) denoted Fr^{∗} : V (λ)^{∗} → V_{ξ}(\overline{ℓ}λ)^{∗} which sends extremal weight vectors of V (λ)^{∗} to the corresponding extremal weight vectors of V_{ξ}(\overline{ℓ}λ)^{∗}. In particular, the b_{τ}^{λ} is sent to b_{τ}^{\overline{ℓ}λ}, which is equal to (b_{τ}^{λ})\overline{ℓ} by (2). The commutation part follows from [11], see also [12]. More precisely, the assumptions made on ℓ in these articles are, as mentioned in their introductions, just made for convenience. Suppose f ∈ R_{1}[U^{−}] and g ∈ R_{ξ}[^{t}U^{−}]. By choosing λ ∈ P^{+} generic enough, we can assume that we can identify f and g as elements of representations, i.e.,

f ∈ V (λ)^{∗} ↪ R_{1}[U^{−}], g ∈ V_{ξ}(\overline{ℓ}λ)^{∗} ↪ R_{ξ}[^{t}U^{−}],

and Fr′^{∗}(Fr^{∗}(f)g) ∈ V (2λ)^{∗} ↪ R_{1}[U^{−}]. Now the same arguments as in [11], section 4, or [12], section 6, go through to prove the compatibility Fr′^{∗}(Fr^{∗}(f)g) = fFr′^{∗}(g). The rest is obvious since Fr′^{∗} is a splitting. •

The canonical basis B ⊂ U_{q}(𝔫^{−}) spans U_{R}(𝔫^{−}) as an R-module, correspondingly its dual basis B^{∗} spans R_{q}[U^{−}] as an R-module. It follows that the images of this basis in R_{ξ}[^{t}U^{−}] respectively R_{1}[U^{−}] span the corresponding specializations.

Fix a reduced decomposition w_{0} = s_{i_{1}} ⋯ s_{i_{N}} of the longest word in W. As in section 2, set b_{j} := b_{y_{j}}^{ϖ_{i_{j}}} ∈ B^{∗} ⊂ U_{q}(𝔤), let Sw̌_{0} be the set of the monomials in the b_{j}. Still denote Aw̃_{0} the R-module generated by Sw̃_{0}. The R–algebra Aw̌_{0} specializes to a subalgebra Aξ ; w̌_{0} ⊂ R_{ξ}[^{t}U^{−}] respectively A1 ; w̌_{0} ⊂ R_{1}[U^{−}]. In both cases the algebra has the (images of the) monomials in Sw̌_{0} as a basis as ℤ_{ξ}-module.

As an immediate consequence of the proposition above one sees:

Corollary 4 Fr^{∗} induces an inclusion of algebras Fr^{∗} : A1 ; w̌_{0} → ^{t}Aξ ; w̌_{0} such

that Fr^{∗}(b_{1}^{m_{1}} ⋯ b_{N}^{m_{N}}) = b_{1}^{m_{1}\overline{ℓ}} ⋯ b_{N}^{m_{N} \overline{ℓ}}.

4 The path model, (λ, q)–minors and SMT

We come now first back to representations of G respectively U(𝔤). A combinatorial tool for the analysis of these representations is the path model, of which we recall quickly the most important features. Let λ ∈ P^{+} be a dominant weight.

Let \underline{τ} = (τ_{0}, …, τ_{r}) be a strictly decreasing sequence (with respect to the Bruhat order) of elements of W/W_{λ}, and let \underline{a} = (a_{1}, …, a_{r}) be a strictly increasing sequence of rational numbers such that 0 < a_{1} < … < a_{r} < 1. The pair π = (\underline{τ}, \underline{a}) is called a L-S path of shape λ if the pair satisfies the following integrality condition. For all i = 1, …, r:

• set r_{i} = l(τ_{i−1}) − l(τ_{i}). There exists a sequence β_{1}, …, β_{r_{i}} of positive roots joining τ_{i−1} and τ_{i} by the corresponding reflections, i.e.,

τ_{i−1} > s_{β_{1}}τ_{i−1} > s_{β_{2}}s_{β_{1}}τ_{i−1} > … > s_{β_{r_{i}}} ⋯ s_{β_{1}}τ_{i−1} = τ_{i},

and a_{i} ⟨τ_{i−1}(λ), s_{β_{1}} ⋯ s_{β_{j−1}}(β_{j}^{∨})⟩ ∈ ℤ for all j = 1, …, r_{i}.

In the following formulas we set always a_{0} = 0 and a_{r+1} = 1. The weight of an L-S path π = (\underline{τ}, \underline{a}) of shape λ is the convex linear combination

π(1) = ∑_{i=0}^{r} x_{i+1}τ_{i}(λ), where x_{i} = a_{i} − a_{i−1} for 1 ≤ i ≤ r + 1.

For more details on the combinatorics of L-S paths we refer to [17]. Let B(λ) be the set of L-S paths of shape λ, and let B(λ)_{τ} be the set of L-S paths π = (\underline{τ}, \underline{a}) of shape λ such that τ ≥ τ_{0}. The character of the Weyl module V (λ) of highest weight λ and the Demazure module V (λ)_{τ} can be calculated using the L-S paths:

Theorem 5 ([17])

Char V (λ) = ∑_{π∈B(λ)} e^{π(1)} and Char V (λ)_{τ} = ∑_{π∈B(λ)_{τ}} e^{π(1)}.

The path model of a representation is closely connected to the combinatorics of the crystal base. For the irreducible representation V_{q}(λ) fix a highest weight vector v_{λ} and let B(λ) = {bv_{λ} | b ∈ B, bv_{λ} ≠ 0} be the canonical basis of V_{q}(λ). Denote L_{q}(λ) the ℂ[q]–submodule of V_{q}(λ) spanned by the elements of the canonical basis B(λ) and set

B^{q}(λ) = {b mod L_{q}(λ) | b ∈ B(λ)}.

Recall ([10, 8]) that the Kashiwara crystal B^{q}(λ) (with the Kashiwara operators f_{α}, e_{α}) is isomorphic to the crystal given by the path model B(λ) (with the root operators). In the following we will often identify these two crystals and just write B(λ). For an element b ∈ B(λ) we write [b]^{λ} or just [b] for its class in B(λ), and for an element π ∈ B(λ) we write b_{π} for the element in B(λ) such that [b_{π}] = π.

The tensor products of crystals can be defined in the path model in terms of concatenations of paths or in terms of standard tuples:

Definition 6 Let π_{1} = (\underline{τ}, \underline{a}) ∈ B(λ_{1}), π_{2} = (\underline{κ}, \underline{b}) ∈ B(λ_{2}), …, respectively π_{m} = (\underline{σ}, \underline{c}) ∈ B(λ_{m}) be L-S paths of shape λ_{1}, λ_{2}, …, respectively λ_{m}, where

\underline{τ} = (τ_{0}, …, τ_{r}), \underline{κ} = (κ_{0}, …, κ_{s}), …, \underline{σ} = (σ_{0}, …, σ_{t}).

The tuple \underline{π} = (π_{1}, π_{2}, …, π_{m}) is called a standard tuple of shape \underline{λ} = (λ_{1}, …, λ_{m}) if there exist representatives τ̃_{0}, … τ̃_{r} ∈ W of the τ_{0}, …, τ_{r} ∈ W/W_{λ_{1}}, and representatives κ̃_{0}, …, κ̃_{s} ∈ W of the κ_{0}, …, κ_{s} ∈ W/W_{λ_{2}} etc., such that

τ̃_{0} ≥ … ≥ τ̃_{r} ≥ κ̃_{0} ≥ … ≥ κ̃_{s} ≥ … ≥ σ̃_{0} ≥ … ≥ σ̃_{t},

Such a sequence (τ̃_{0}, …, τ̃_{r}, κ̃_{0}, …, κ̃_{s}, …, σ̃_{0}, …, σ̃_{t}) of representatives is called a defining chain. The weight \underline{π}(1) is the sum π_{1}(1) + … + π_{m}(1) of the weights of the L–S paths.

Remark 7 If λ_{1} = … = λ_{m}, the condition on standardness reduces to the condition τ_{r} ≥ κ_{0} ≥ κ_{s} ≥ … ≥ σ_{0} in the Bruhat ordering on W/W_{λ}.

By the independence of the crystal structure of path models [19], we have a canonical bijection between the set B(\underline{λ}) of standard tuples of shape \underline{λ} and the set of L–S paths B(λ) of shape λ = λ_{1} + … + λ_{m}. In fact, the standard tuples correspond exactly to the elements in the Cartan component of the tensor product of the crystals. The notation b\underline{π} ∈ B(λ) for an element of the canonical basis corresponding to an element of a path model generalizes in the obvious way.

The following is a translation of [14], Proposition 33, into the language of standard tuples:

Proposition 8 Let b_{i} ∈ B(λ_{i}) be elements of the canonical basis and let π_{i} ∈ B(λ_{i}) be the L–S path such that [b_{i}] = π_{i}, i = 1, …, m. Suppose the tuple \underline{π} = (π_{1}, …, π_{m}) is standard, let π ∈ B(λ), λ = ∑_{i=1}^{m} λ_{i}, be the L–S path of shape λ that identifies with the standard tuple, and let b_{π} = b\underline{π} ∈ B(λ) be such that [b_{π}] = π. Then [b_{1}] ⊗ … ⊗ [b_{m}] identifies with [b_{π}], and for the dual basis elements there exists a s ∈ ℕ such that

q^{s}b_{1}^{∗}b_{2}^{∗} … b_{m}^{∗} = b_{π}^{∗} mod qL_{q}^{∗}(λ).

The dual representation V (λ)^{∗} can be geometrically realized as the space of global sections H^{0}(G/B, L_{λ}) of the line bundle L_{λ} on G/B. We recall now the construction of a basis of H^{0}(G/B, L_{λ}) using (λ, q)–minors. The restriction map induces a map between (tensor products of) dual Weyl modules of the quantum group U_{ξ}(^{t}𝔤):

V_{ξ}(λ)^{∗} ⊗ … ⊗ V_{ξ}(λ)^{∗} ⟶ V_{ξ}(\overline{ℓ}λ)^{∗}.

We write shortly b_{τ}^{λ} ⋯ b_{κ}^{λ} for the image of b_{τ}^{λ}⊗…⊗b_{κ}^{λ}. The dual of the Frobenius splitting induces a map between dual Weyl modules of the quantum group U_{ξ}(^{t}𝔤) and U(𝔤) (see [18]):

Fr′^{∗} : V_{ξ}(\overline{ℓ}λ)^{∗} ⟶ V (λ)^{∗}.

Note that these maps are actually defined over ℤ_{ξ}.

Let now π = (\underline{τ}, \underline{a}) ∈ B(λ) be an L-S path of shape λ, fix ℓ ∈ ℕ, ℓ > 0 minimal such that 2d divides ℓ and \overline{ℓ}a_{i} ∈ ℕ for all i = 1, …, r, and let ξ be a corresponding primitive 2ℓ-th root of unity.

Definition 9 The path vector p_{π} ∈ H^{0}(G/B, L_{λ}), π = (\underline{τ}, \underline{a}), is defined as:

p_{π} := Fr′^{∗} ((b_{τ_{0}}^{λ})^{\overline{ℓ}x_{1}}(b_{τ_{1}}^{λ})^{\overline{ℓ}x_{2}} ⋯ (b_{τ_{r−1}}^{λ})^{\overline{ℓ}x_{r}}(b_{τ_{r}}^{λ})^{\overline{ℓ}x_{r+1}}) ,

where x_{i} = a_{i} − a_{i−1} for 1 ≤ i ≤ r + 1, and the b_{τ_{j}}^{λ} are (λ, q)-minors in V_{ξ}(λ)^{∗} for j = 0, …, r.

To distinguish between extremal weight vectors for U(𝔤) and U_{ξ}(^{t}𝔤) we keep the notation b_{τ}^{λ} in the quantum case and write p_{τ}^{λ} for the classical case. The reader should think of the p_{π} as a kind of algebraic approximation of

p_{π} ∼ ^{ℓ}\sqrt{p_{τ_{0}}^{ℓx_{1}}p_{τ_{1}}^{ℓx_{2}}p_{τ_{3}}^{ℓx_{3}} … p_{τ_{r}}^{ℓx_{r+1}}}.

Though the expression above does not make sense in general, we will see soon that at least in some special cases the expression can be given a useful interpretation.

We will also consider standard monomials in the path vectors:

Definition 10 Let \underline{π} = (π_{1}, …, π_{m}) be a tuple of shape \underline{λ} = (λ_{1}, …, λ_{m}). We associate to \underline{π} the monomial p\underline{π} = p_{π_{1}}p_{π_{2}} ⋯ p_{π_{m}} in H^{0}(G/Q, L_{λ}), where λ = λ_{1} + … + λ_{m}. The monomial p\underline{π} is called a standard monomial of shape \underline{λ} if \underline{π} is a standard tuple.

The path vectors are defined over ℤ_{ξ} for some appropriate root of unity, so the collection 𝔹(\underline{λ}) of all standard monomials of shape \underline{λ} is defined over the ring S ⊂ ℂ generated by ℤ and all roots of unity.

Theorem 11 ([18]) The set of standard monomials 𝔹(\underline{λ}) of shape \underline{λ} forms a basis for the space of sections H^{0}(G/B, L_{λ}), λ = λ_{1} + … + λ_{m}.

For more information about applications of standard monomial theory (singularities of Schubert varieties, generators and relations for homogeneous coordinate rings of Schubert varieties, Koszul property, Pieri-Chevalley type formula, ...) see for example [13], [18], [16].

5 The path vectors and the dual canonical basis

Given an L-S path of shape λ, we use the same notation p_{π} for the path vector in H^{0}(G/B, L_{λ}) as well as the element c_{p_{π}}^{λ} ∈ ℂ[U^{−}]. The dual canonical basis B^{∗} is compatible with the injection H^{0}(G/B, L_{λ}) ↪ ℂ[U^{−}], i.e., a subset of B^{∗} forms a basis of the image of H^{0}(G/B, L_{λ}).

So it is natural to ask for a description of the transformation matrix between the basis given by the path vectors and the basis given by elements of the dual canonical basis.

Let π = (\underline{τ}, \underline{a}) be an L-S path of shape λ. We identify an element τ ∈ W/W_{λ} with its minimal representative in W. Fix a reduced decomposition w_{0} = s_{i_{1}} ⋯ s_{i_{N}} of the longest word of the Weyl group, we write w̌_{0} to refer to this decomposition. For j = 1, …, N, set y_{j} = s_{i_{1}} ⋯ s_{i_{j}} as in (4).

Definition 12 We say that π is compatible with w̌_{0} if \underline{τ} ⊂ \{y_{1}, …, y_{N}\}, i.e., {τ_{0}, …, τ_{r}} ⊂ {y_{1}, …, y_{N}}.

Let now \underline{π} = (π_{1}, …, π_{m}) be a standard tuple of shape \underline{λ} = (λ_{1}, …, λ_{m}) with a defining chain (see Definition 6).

τ̃_{0} ≥ … ≥ τ̃_{r} ≥ κ̃_{0} ≥ … ≥ κ̃_{s} ≥ … ≥ σ̃_{0} ≥ … ≥ σ̃_{t},

Definition 13 We say that \underline{π} is compatible with w̌_{0} if the defining chain can be chosen such that {τ̃_{0}, …, τ̃_{r}, κ̃_{0}, …, κ̃_{s}, …, σ̃_{0}, …, σ̃_{t}} ⊂ {y_{1}, …, y_{N}}.

The following theorem provides the connection between certain standard monomials and the adapted algebras:

Theorem 14 Let p\underline{π} = p_{π_{1}} ⋯ p_{π_{m}} ∈ H^{0}(G/B, L_{λ}) be a standard monomial monomial of shape \underline{λ} = (λ_{1}, …, λ_{m}). If there exists a defining chain (τ̃_{0}, …, τ̃_{r}, …, σ̃_{0}, …, σ̃_{t}) which is compatible with some reduced decomposition of w_{0}, then the inclusion H^{0}(G/B, L_{λ}) ↪ ℂ[U^{−}], λ = λ_{1} + … + λ_{m}, maps the standard monomial p\underline{π} onto the element b∗ ; \underline{π} of the dual canonical basis, up to multiplication by a root of unity.

Proof. We fix a reduced decomposition w̌_{0} of w_{0} and assume that the standard tuple \underline{π} = (π_{1}, …, π_{m}) has a defining chain compatible with w̌_{0}. Let A1 ; w̌_{0} ⊂ R_{1}[U^{−}] ⊂ ℂ[U^{−}] be the corresponding (specialization of the) adapted algebra defined in section 2. We will show that the p_{π_{j}} are elements of the set of monomials Sw̌_{0} spanning A1 ; w̌_{0}. This proves the theorem by Theorem 2.

So in the following it suffices to consider only one path vector p_{π}. Fix an appropriate ℓ ∈ ℕ as in section 4, Definition 9, for π = (τ_{0}, …, τ_{r}; a_{1}, …, a_{r}), and let ξ be a primitive 2ℓ-th root of unity. Set

m_{π} := (b_{τ_{0}}^{λ})^{\overline{ℓ}a_{1}}(b_{τ_{1}}^{λ})^{\overline{ℓ}(a_{2}−a_{1})} ⋯ (b_{τ_{r−1}}^{λ})^{\overline{ℓ}(a_{r}−a_{r−1})}(b_{τ_{r}}^{λ})^{\overline{ℓ}(1−a_{r})},

where the b_{τ_{j}}^{λ} are (λ, q)–minors in V_{ξ}(λ)^{∗} for j = 0, …, r. By definition, we have p_{π} = Fr′^{∗}(m_{π}). Let λ = ∑_{j=1}^{n} λ_{j}ϖ_{j} be the expression of λ as a linear combination of fundamental weights. Then, by (2), we know for arbitrary τ ∈ W:

b_{τ}^{λ} = ∏_{i=1}^{n}(b_{τ}^{ϖ_{j}})^{λ_{j}}

So we can rewrite m_{π} as:

m_{π} = (∏_{j=1}^{n}(b_{τ_{0}}^{ϖ_{j}})^{\overline{ℓ}a_{1}λ_{j}} ) (∏_{i=j}^{n}(b_{τ_{1}}^{ϖ_{j}})^{\overline{ℓ}(a_{2}−a_{1})λ_{j}} ) ⋯ (∏_{i=j}^{n}(b_{τ_{r}}^{ϖ_{j}})^{\overline{ℓ}(1−a_{r})λ_{j}} )

The defining chain of \underline{π} is compatible with w̌_{0}, it follows (see Remark 1) for the lifts τ̃_{j} of the τ_{j} that the bϖ_{i} ; τ̃_{j} are generators of ^{t}Aξ ; w̌_{0} ⊂ R_{ξ}[^{t}U^{−}]. Note if

τ̃ ∈ W is a lift for τ ∈ W/W_{λ}, then (b_{τ}^{ϖ_{j}})^{λ_{j}} = (bϖ_{j} ; τ̃)^{λ_{j}}. So b_{π} ∈ ^{t}Aξ ; w̌_{0}, and

by reordering the elements we can write with the notation as in (4):

m_{π} = cb_{1}^{r_{1}} ⋯ b_{N}^{r_{N}} ,

where c is some root of unity. If all the exponents r_{j} are divisible by \overline{ℓ}, then we know by Proposition 3 that

p_{π} = Fr′^{∗}(m_{π}) = cFr′^{∗}(b_{1}^{r_{1}} ⋯ b_{N}^{r_{N}}) = cb_{1}^{r_{1}/\overline{ℓ}} ⋯ b_{N}^{r_{N} /\overline{ℓ}}

is an element of Sw̌_{0}, the spanning set for A1 ; w̌_{0}.

To prove the divisibility property, let π = (τ_{0}, …, τ_{r}; a_{1}, …, a_{r}) be an L-S path of shape λ compatible with w̌_{0} and consider the following projection: write λ as a linear combination ∑_{α} λ_{α}ϖ_{α} of fundamental weights, fix a simple root α, let \overline{τ} be the class of τ in W/W_{ϖ_{α}}, and set

π_{α} = (\overline{τ}_{0}, …, \overline{τ}_{r}; a_{1}, …, a_{r}) = (σ_{0}, …, σ_{t}; c_{1}, …, c_{t}).

where the second pair of sequences is obtained from the first by omitting repetitions among the \overline{τ}_{j} as well as the corresponding b_{j}, i.e., if

σ_{0} = \overline{τ}_{0} … = \overline{τ}_{j−1} > σ_{1} = \overline{τ}_{j} = … = \overline{τ}_{k−1} > σ_{2} = \overline{τ}_{k} = … = \overline{τ}_{m−1} > …

then c_{1} = a_{j}, c_{2} = a_{k}, c_{3} = a_{m}, . . .. Set

m_{π_{α}} = (b_{σ_{0}}^{ϖ_{α}})^{\overline{ℓ}λ_{α}c_{1}} (b_{σ_{1}}^{ϖ_{α}})^{\overline{ℓ}λ_{α}(c_{2}−c_{1})} ⋯ (b_{σ_{t}}^{ϖ_{α}})^{\overline{ℓ}λ_{α}(1−c_{t})} ∈ R_{ξ}[^{t}U^{−}]

Then m_{π} = ξ^{p}m_{π_{α_{1}}} m_{π_{α_{2}}} ⋯ m_{π_{αn}} ∈ R_{ξ}[^{t}U^{−}] for some power p of ξ. So the divisibility property, and hence the first part of the theorem, is a consequence of the following lemma, and the fact that p_{π} and the canonical basis element b_{π} coincide (up to multiplication by an appropriate root of unity) follows from Proposition 8.

Lemma 15 Set c_{0} = 0 and c_{r+1} = 1. The exponents have the property λ_{α}(c_{j+1} − c_{j}) ∈ ℤ for all j = 0, …, r.

Proof. Let π_{α} = (\overline{τ}_{0}, …, \overline{τ}_{r}; a_{1}, …, a_{r}) = (σ_{0}, …, σ_{t}; c_{1}, …, c_{t}) be as above and suppose

σ_{i−1} = \overline{τ}_{j−1} > σ_{i} = \overline{τ}_{j} = \overline{τ}_{j+1} = … = \overline{τ}_{k−1} > σ_{i+1} = \overline{τ}_{k}

Corresponding to the fixed reduced decomposition w̌_{0} let y_{j} = s_{i_{1}} ⋯ s_{i_{j}} be as in (4). By assumption, π is compatible with w̌_{0}, so {τ_{0}, …, τ_{r}} ⊂ {y_{1}, …, y_{N}}. It follows: there exist indices such that

τ_{k} ≤ y_{t} < y_{t}s_{α} ≤ τ_{k−1} < … < τ_{j} ≤ y_{v} < y_{v}s_{α} ≤ τ_{j−1}

and y_{t}s_{α} and y_{v}s_{α} are elements of {y_{1}, …, y_{N}}. Set β = y_{t}(α), so s_{β}(y_{t}s_{α}) = y_{t}. The definition of an L-S path implies:

c_{i+1}λ_{α} = a_{k}λ_{α} = a_{k}⟨λ, α^{∨}⟩ = a_{k}⟨y_{t}(λ), β^{∨}⟩ ∈ ℤ.

Similarly, set δ = y_{v}(α), so s_{δ}(y_{v}s_{α}) = y_{v}. The definition of an L-S path implies:

c_{i}λ_{α} = a_{j}λ_{α} = a_{j}⟨λ, α^{∨}⟩ = a_{j}⟨y_{v}(λ), δ^{∨}⟩ ∈ ℤ.

It follows: λ_{α}(c_{i+1} − c_{i}) ∈ ℤ. •

6 Examples

In this section, we study some examples which illustrate our main theorem. Let 𝔤 be semisimple Lie algebra, let B ⊂ U_{q}(𝔫^{−}) be the canonical basis and denote L_{q}(𝔫^{−}) the ℂ[q]–submodule of U_{q}(𝔫^{−}) spanned by the elements of the canonical basis B. Let (B(∞), L_{q}(𝔫^{−})) be the crystal of U_{q}(𝔫^{−}) (see for example [9], or [8]), consisting of the ℂ[q]–module L_{q}(𝔫^{−}), the basis

B(∞) = {b mod L_{q}(𝔫^{−}) | b ∈ B},

and the Kashiwara operators {f_{α}, e_{α} | α a simple root} on B(∞) ∪ {0}.

In the same way as the dual canonical basis B^{∗} is compatible with the representations V_{q}(λ)^{∗}, the crystal of U_{q}(𝔫^{−}) can be seen as the limit of the crystals of the representations V_{q}(λ). Recall (section 4) that the path model can also be understood as a combinatorial model for the crystal of the representations.

Example 16 Let 𝔤 be of type A_{2}, denote α_{1}, α_{2} the simple roots, then every element b ∈ B^{∗} ⊂ ℂ_{1}[U^{−}] of the dual canonical basis is the image of some appropriately chosen standard monomial p\underline{π}. Proof. Let u_{∞} ∈ B(∞) be the highest weight element of B(∞) and let u_{b} be the class of b in B(∞). The element u_{b} ∈ B(∞) can be written as

u_{b} = f_{α_{1}}^{ℓ}f_{α_{2}}^{m}f_{α_{1}}^{n}u_{∞}, where m ≥ n, or u_{b} = f_{α_{2}}^{r}f_{α_{1}}^{s}f_{α_{2}}^{t}u_{∞} where s ≥ t,

and s = ℓ + n, r = max{n, m − ℓ} and t = min{m − n, ℓ} (see [21] or [2]).

Note if u_{b} = f_{α_{1}}^{ℓ}f_{α_{2}}^{m}f_{α_{1}}^{n}u_{∞} is such that ℓ < m−n, then in the corresponding expression u_{b} = f_{α_{2}}^{r}f_{α_{1}}^{s}f_{α_{2}}^{t}u_{∞} we have r = m − ℓ > n = ℓ + n − ℓ = s − t. So in the following we just write u_{b} = f_{α_{i}}^{ℓ}f_{α_{j}}^{m}f_{α_{i}}^{n}u_{∞} and suppose that i and j are chosen such that ℓ ≥ m − n ≥ 0.

Set λ = (ℓ + 2n − m)ϖ_{i} + (m − n)ϖ_{j}, and let \underline{π}\underline{λ} = (id, id, id) be the unique standard tuple of shape \underline{λ} = (nϖ_{i} , (m − n)ϖ_{j} , (ℓ + n − m)ϖ_{i}) of weight \underline{π}\underline{λ}(1) = λ. By applying the root operators (see [19]), we get for \underline{π} = f_{α_{i}}^{ℓ}f_{α_{j}}^{m}f_{α_{i}}^{n}π\underline{λ} the following standard sequence of shape \underline{λ}:

\underline{π} = (π_{1} = (s_{j}s_{i}) , π_{2} = (s_{i}s_{j}) , π_{3} = (s_{i})).

Note that the sequence (s_{i}s_{j}s_{i} , s_{i}s_{j} , s_{i}) is a defining chain for \underline{π} which is obviously compatible with the reduced decomposition w_{0} = s_{i}s_{j}s_{i}. It follows that the image of the standard monomial p\underline{π} = p_{s_{j}s_{i}}p_{s_{j}s_{i}}p_{s_{i}} of shape \underline{λ} in ℂ[U^{−}] is, up to multiplication by a root of unity, the element b = b\underline{π} of the dual canonical basis B^{∗}.

Example 17 The case 𝔤 = 𝔰l_{3} is very special, in general it is not possible to find for every element in B(∞) an “appropriate” standard tuple of L–S paths with a defining chain compatible with a reduced decomposition of w_{0}. An example is for 𝔤 = 𝔰l_{4} the element b ∈ B^{∗} such that for its class in B(∞) holds u_{b} = f_{α_{2}}f_{α_{1}}f_{α_{3}}u_{∞}. Nevertheless, it is easy to see by Theorem 14, that the other elements of B(∞) of the same weight

u_{b_{1}} = f_{α_{1}}f_{α_{2}}f_{α_{3}}u_{∞} , u_{b_{2}} = f_{α_{3}}f_{α_{2}}f_{α_{1}}u_{∞} , u_{b_{3}} = f_{α_{3}}f_{α_{1}}f_{α_{1}}u_{∞}

correspond all to elements of the dual canonical basis which are “standard monomials”: b_{1} = p_{(s_{1}s_{2}s_{3})} ∈ H^{0}(G/B, L_{ϖ_{3}}), b_{2} = p_{(s_{3}s_{2}s_{1})} ∈ H^{0}(G/B, L_{ϖ_{1}}) and b_{3} = p_{(s_{1}s_{3}s_{2})} ∈ H^{0}(G/B, L_{ϖ_{2}}).

But let now λ = ϖ_{1} + ϖ_{3}, then there exists an L–S path of shape λ that corresponds under the identification between crystal graphs and path models to u_{b}, it is the path π = (s_{2}s_{3}s_{1}, s_{3}s_{1}; \frac{1}{2}). The 𝔤-module V (λ) is isomorphic to 𝔤 endowed with the adjoint action. Via this isomorphism, the canonical basis of V (λ) is given (up to scalar multiples) by a Chevalley basis. As a function, p_{π} is the dual of the canonical basis element h_{α_{2}}, thus b = p_{π} can again be constructed as a “standard monomial” for an appropriate choice of λ, but this time not in the context of Theorem 14.

Example 18 We suppose again 𝔤 = 𝔰l_{3}. Set λ = 2ϖ_{1} + 2ϖ_{2}. We shall explicit the p_{π} corresponding to the zero weight space V (λ)_{0} of V (λ).

Let’s introduce some notation. Let

a = p_{s_{1}}^{ϖ_{1}}, b = p_{s_{2}}^{ϖ_{2}}, c = p_{s_{2}s_{1}}^{ϖ_{1}}, d = p_{s_{1}s_{2}}^{ϖ_{2}},

be the elements of ℂ[U^{−}] as defined in section 4. It is well known (see [3]), that the dual canonical basis is given by {a^{k}b^{l}c^{s}d^{t}, k, l, s, t ∈ ℕ, kl = 0}. Moreover, it is easy to see that ab = c + d. Indeed, ab decomposes in the base {c, d} of the subspace of ℂ[U^{−}] of weight α_{1} + α_{2}. Both coefficients in this base are equal by symmetry and equal to one by [5], Theorem 2.3.

The LS paths corresponding to V (λ)_{0} are

π_{1} = (s_{2}s_{1}, s_{1}; \frac{1}{2} ), π_{2} = (w_{0}, Id; \frac{1}{2} ), π_{3} = (s_{1}s_{2}, s_{2}; \frac{1}{2} ).

Remark that π_{2} is compatible with a reduced decomposition while π_{1} and π_{3} are not. As in the proof of Theorem 9, with l = 2, we obtain, up to a root of one,

p_{π_{1}} = cba, p_{π_{2}} = cd, p_{π_{3}} = dab.

Hence,

p_{π_{1}} = c^{2} + cd, p_{π_{2}} = cd, p_{π_{3}} = cd + d^{2}.

Remark that p_{π_{2}} is an element of the dual canonical basis while p_{π_{1}} and p_{π_{3}} are not. Under the identification of the path model with crystal basis elements we have in the dual canonical basis: b_{π_{1}} = c^{2}, b_{π_{2}} = cd and b_{π_{3}} = d^{2}, so the transformation matrix reads as: p_{π_{1}} = b_{π_{1}} + b_{π_{2}}, p_{π_{2}} = b_{π_{2}} and p_{π_{3}} = b_{π_{3}} + b_{π_{2}}.

Example 19 We suppose again 𝔤 = 𝔰l_{3}, set now λ = mϖ_{1} + mϖ_{2}. To simplify the notation of the L–S paths, we still write π = (\underline{τ}, \underline{a}) for \underline{a} = (a_{1}, a_{2}, a_{3}), even if we have equality in 0 ≤ a_{1} ≤ a_{2} ≤ a_{3} ≤ 1. In case of equality, the reader has to omit the corresponding parameters and Weyl group elements to avoid double counting.

We claim that the p_{π} have a simple expression in terms of the canonical basis: if π = (\underline{τ}, \underline{a}) is compatible with some reduced decomposition, then p_{π} = b_{π} is an element of the dual canonical basis (after a possible renormalization with a root of unity).

If π = (\underline{τ}, \underline{a}) is not compatible with a reduced decomposition, then let ℓ be even and minimal with the property that ℓa_{i} ∈ ℕ for all i. Again, after a possible renormalization (multiplication with a root of unity):

Proposition 20 If π = (\underline{τ}, \underline{a}) is not compatible with a reduced decomposition, then set t = min{m(a_{2} − a_{1}), m(a_{3} − a_{2})}. The following transition relation holds between the canonical basis and the path vectors:

p_{(\underline{τ};\underline{a})} = ∑_{j=0}^{⌊t⌋} \binom{tℓ}{jℓ}b_{(\underline{τ};a_{1}+ \frac{j}{m} ,a_{2},a_{3}− \frac{j}{m})}

Proof. To prove the statement, we divide the set of L–S paths in groups. The first group is the set B(λ)^{1} = B(λ)^{1,1} ∪ B(λ)^{1,2}, where:

\begin{matrix}B(λ)^{1,1} = \{π = (w_{0}, s_{1}s_{2}, s_{1}, id; a_{1}, a_{2}, a_{3}) | ma_{1}, ma_{2}, ma_{3} ∈ ℕ\} \\ B(λ)^{1,2} = \{π = (w_{0}, s_{2}s_{1}, s_{2}, id; a_{1}, a_{2}, a_{3}) | ma_{1}, ma_{2}, ma_{3} ∈ ℕ\} \end{matrix}

These paths are all compatible with a reduced decomposition, so the elements p_{π} are canonical basis elements: Set x_{1} = a_{1}, x_{2} = a_{2} − a_{1}, x_{3} = a_{3} − a_{2}, then, after renormalizing the p_{π} (multiplication by a root of unity), we have (with same notation as above):

\begin{matrix}b_{π} = p_{π} = a^{m(x_{2}+x_{3})}c^{mx_{1}}d^{m(x_{1}+x_{2})} for π ∈ B(λ)^{1,1} \\ b_{π} = p_{π} = b^{m(x_{2}+x_{3})}c^{m(x_{1}+x_{2})}d^{mx_{1}} for π ∈ B(λ)^{1,2} \end{matrix}

In other words, let a^{u}c^{v}d^{w} (respectively b^{u}c^{v}d^{w}) be an element of the dual canonical basis such that

v ≤ w ≤ u + v ≤ m (respectively: w ≤ v ≤ u + w ≤ m)

In both cases, this canonical basis element is equal to p_{π}, where

π = \{ \begin{matrix}(w_{0}, s_{1}s_{2}, s_{1}, id; \frac{v}{m}, \frac{w}{m}, \frac{u+v}{m}) & for a^{u}c^{v}d^{w} \\ (w_{0}, s_{2}s_{1}, s_{2}, id; \frac{w}{m}, \frac{v}{m}, \frac{u+w}{m}) & for b^{u}c^{v}d^{w} respectively \end{matrix}

The second set of L–S paths of shape λ is in this example the union B(λ)^{2} = B(λ)^{2,1} ∪ B(λ)^{2,2}, where:

\begin{matrix}B(λ)^{2,1} = \{π = (w_{0}, s_{2}s_{1}, s_{1}, id; a_{1}, a_{2}, a_{3}) | ma_{1}, 2ma_{2}, ma_{3} ∈ ℕ\} \\ B(λ)^{2,2} = \{π = (w_{0}, s_{1}s_{2}, s_{2}, id; a_{1}, a_{2}, a_{3}) | ma_{1}, 2ma_{2}, ma_{3} ∈ ℕ\} \end{matrix}

By construction, we get (up to multiplication by a root of unity): Let ℓ be minimal and even such that ℓa_{i} ∈ ℕ for all i, then, for π ∈ B(λ)^{2,1}, we have up to multiplication by a root of unity r̃:

p_{π} = \{ \begin{matrix}r̃Fr′^{∗}(c^{mx_{1}ℓ}d^{mx_{1}ℓ}c^{mx_{2}ℓ}(ab)^{mx_{3}ℓ}b^{m(x_{2}−x_{3})ℓ}) & if x_{2} ≥ x_{3} \\ r̃Fr′^{∗}(c^{mx_{1}ℓ}d^{mx_{1}ℓ}c^{mx_{2}ℓ}(ab)^{mx_{2}ℓ}a^{m(x_{3}−x_{2})ℓ}) & if x_{3} ≥ x_{2} \end{matrix}

and hence by Proposition 3 for some roots of unity r′:

p_{π} = \{ \begin{matrix}r′b^{m(x_{2}−x_{3})}c^{mx_{1}}d^{mx_{1}}Fr′^{∗}(c^{mx_{2}ℓ}(ab)^{mx_{3}ℓ}) & if x_{2} ≥ x_{3} \\ r′a^{m(x_{3}−x_{2})}c^{mx_{1}}d^{mx_{1}}Fr′^{∗}(c^{mx_{2}ℓ}(ab)^{mx_{2}ℓ}) & if x_{3} ≥ x_{2}. \end{matrix}

Replacing ab by (c + d) (note that c and d commute) we get after renormalizing (by a root of unity):

p_{π} = \{ \begin{matrix}∑_{j=0}^{mx_{3}ℓ} \binom{mℓx_{3}}{j}b^{m(x_{2}−x_{3})}c^{mx_{1}}d^{mx_{1}}Fr′^{∗}(c^{m(x_{2}+x_{3})ℓ−j}d^{j}) & if x_{2} ≥ x_{3} \\ ∑_{j=0}^{mx_{2}ℓ} \binom{mℓx_{2}}{j}a^{m(x_{3}−x_{2})}c^{mx_{1}}d^{mx_{1}}Fr′^{∗}(c^{m2x_{2}ℓ−j}d^{j}) & if x_{3} ≥ x_{2} \end{matrix}

Since 2mx_{2} respectively m(x_{2} + x_{3}) are integers, it follows that Fr′^{∗} maps a summand to zero for j not divisible by ℓ, and hence:

p_{π} = \{ \begin{matrix}∑_{j=0}^{⌊mx_{3}⌋} \binom{mℓx_{3}}{jℓ}b^{m(x_{2}−x_{3})}c^{mx_{1}}d^{mx_{1}}c^{m(x_{2}+x_{3})−j}d^{j} & if x_{2} ≥ x_{3} \\ ∑_{j=0}^{⌊mx_{2}⌋} \binom{mℓx_{2}}{jℓ}a^{m(x_{3}−x_{2})}c^{mx_{1}}d^{mx_{1}}c^{m2x_{2}−j}d^{j} & if x_{3} ≥ x_{2} \end{matrix}

and thus, after reordering, we get for π ∈ B(λ)^{2,1}:

p_{π} = \{ \begin{matrix}∑_{j=0}^{⌊mx_{3}⌋} \binom{mℓx_{3}}{jℓ}b^{m(x_{2}−x_{3})}c^{m(x_{1}+x_{2}+x_{3})−j}d^{mx_{1}+j} & if x_{2} ≥ x_{3} \\ ∑_{j=0}^{⌊mx_{2}⌋} \binom{mℓx_{2}}{jℓ}a^{m(x_{3}−x_{2})}c^{m(x_{1}+2x_{2})−j}d^{mx_{1}+j} & if x_{3} ≥ x_{2} \end{matrix}  (5)

One sees easily that in terms of the root operators one has

π = f_{1}^{m(x_{1}+2x_{2})}f_{2}^{m(2x_{1}+x_{2}+x_{3})}f_{1}^{mx_{1}}(id),

so, by Example 16, the corresponding element b_{π} in the dual canonical basis is

b_{π} = \{ \begin{matrix}b^{m(x_{2}−x_{3})}c^{m(x_{1}+x_{2}+x_{3})}d^{mx_{1}} & if x_{2} ≥ x_{3} \\ a^{m(x_{3}−x_{2})}c^{m(x_{1}+2x_{2})}d^{mx_{1}} & if x_{3} ≥ x_{2}. \end{matrix}

i.e., b_{π} is the summand for j = 0 in the expression above. Note that the exponents in the summands for p_{π} in (5) satisfy all the following inequalities:

\begin{aligned}2w & ≤ & u + v + w & ≤ & 2v & ≤ & 2m & for b^{u}c^{v}d^{w} \\ 2w & ≤ & v + w & ≤ & 2(u + v) & ≤ & 2m & for a^{u}c^{v}d^{w}. \end{aligned}

On the other hand, any triple (u, v, w) of integers satisfying these inequalities gives an element of the dual canonical basis of V (λ)^{∗} and is hence equal to b_{η} for some L–S path η of shape λ. We obtain the corresponding path by the following rules:

η = \{ \begin{matrix}(w_{0}, s_{2}s_{1}, s_{1}, id; \frac{w}{m}, \frac{u+v+w}{2m}, \frac{v}{m}) & for b_{η} = b^{u}c^{v}d^{w} \\ (w_{0}, s_{2}s_{1}, s_{1}, id; \frac{w}{m}, \frac{v+w}{2m}, \frac{u+v}{m}) & for b_{η} = a^{u}c^{v}d^{w} \end{matrix}

In the same way one gets for π ∈ B(λ)^{2,2}:

p_{π} = \{ \begin{matrix}∑_{j=0}^{⌊mx_{3}⌋} \binom{mℓx_{3}}{jℓ}a^{m(x_{2}−x_{3})}c^{mx_{1}+j}d^{m(x_{1}+x_{2}+x_{3})−j} & if x_{2} ≥ x_{3} \\ ∑_{j=0}^{⌊mx_{2}⌋} \binom{mℓx_{2}}{jℓ}b^{m(x_{3}−x_{2})}c^{mx_{1}+j}d^{m(x_{1}+2x_{2})−j} & if x_{3} ≥ x_{2} \end{matrix} (6)

As above, in terms of the root operators one has

π = f_{2}^{m(x_{1}+2x_{2})}f_{1}^{m(2x_{1}+x_{2}+x_{3})}f_{2}^{mx_{1}}(id),

so the corresponding element b_{π} in the dual canonical basis is the summand for j = 0 in the expression above:

b_{π} = \{ \begin{matrix}a^{m(x_{2}−x_{3})}c^{mx_{1}}d^{m(x_{1}+x_{2}+x_{3})} & if x_{2} ≥ x_{3} \\ b^{m(x_{3}−x_{2})}c^{mx_{1}}d^{m(x_{1}+2x_{2})} & if x_{3} ≥ x_{2} \end{matrix}

The exponents in the expression for p_{π} in (6) satisfy the following inequalities:

\begin{aligned}2v & ≤ & u + v + w & ≤ & 2w & ≤ & 2m & for b^{u}c^{v}d^{w} \\ 2v & ≤ & v + w & ≤ & 2(u + w) & ≤ & 2m & for a^{u}c^{v}d^{w} \end{aligned} ,

and for any triple (u, v, w) of integers satisfying these inequalities, we can obtain the corresponding path by the following rules:

η = \{ \begin{matrix}(w_{0}, s_{1}s_{2}, s_{2}, id; \frac{v}{m}, \frac{u+v+w}{2m}, \frac{w}{m}) & for b_{η} = a^{u}c^{v}d^{w} \\ (w_{0}, s_{1}s_{2}, s_{2}, id; \frac{v}{m}, \frac{v+w}{2m}, \frac{u+v}{m}) & for b_{η} = b^{u}c^{v}d^{w} \end{matrix}

As a consequence we can now describe in terms of the path model the sum expressions in (5) and (6): For all π = (\underline{τ}, \underline{a}) ∈ B(λ)^{2}, \underline{a} = (a_{1}, a_{2}, a_{3}), we get for t = min{mx_{2}, mx_{3}}

p_{(\underline{τ};\underline{a})} = ∑_{j=0}^{⌊t⌋} \binom{tℓ}{jℓ}b_{(\underline{τ};a_{1}+ \frac{j}{m} ,a_{2},a_{3}− \frac{j}{m})}  (7)
•

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P.C.: Institut Girard Desargues, UPRES-A-5028 Université Claude Bernard, Lyon I, Bat 101 69622 Villeurbane Cedex, France email: caldero@desargues.univ-lyon1.fr P.L: Fachbereich Mathematik Universität Wuppertal, Gauß-Straße 20 42097 Wuppertal, Germany email: littelmann@math.uni-wuppertal.de Keywords: Quantum groups, Canonical basis, Standard Monomial Theory