Word reading is a crystal morphism

We observe that word reading is a crystal morphism. This leads us to prove that, in the case of the complex special linear group, the map from all galleries to MV cycles is a morphism of crystals.


Introduction
In [GL05], Gaussent and Littelmann established a bijection between a certain set of LS galleries and the set of MV cycles associated to a simple representation of a complex connected reductive linear algebraic group. Establishing this correspondence initiated a project whose aim is to relate the path model and the basis of MV cycles achieved by the geometric Satake equivalence proven by Mirkovic and Vilonen in [MV07]. In [BG08], Baumann and Gaussent show that the latter map is in fact an isomorphism of crystals with respect to the crystal structure constructed on MV cycles by Braverman and Gaitsgory in [BG01].
In [GL12] one skeleton galleries are considered, which are piecewise linear paths in the associated real vector space spanned by the coweight lattice, establishing in this way an explicit connection with the path model. This further allowed the interpretation of galleries in terms of Young tableaux for types A, B, C. For the special linear group, the set of one skeleton LS galleries of a given type corresponds to the set of all semi-standard Young tableaux of a given shape.
In [GLN13], Gaussent, Littelmann and Nguyen show, in this context, that the map constructed in [GL05] and [GL12], which makes sense not only for LS galleries, assigns an MV cycle to any gallery.
In Theorem 3.1 of this paper we show, using their results and those of Baumann and Gaussent in [BG08], together with the observation about reading words mentioned in the title, that this map is a morphism of crystals. Moreover, the latter restricts to an isomorphism on each connected component.
for his helpful suggestions. The author has been supported by the Graduate School 1269: Global structures in geometry and analysis, financed by the Deutsche Forschungsgemeinschaft.
2. Galleries, words and crystals 2.1. Galleries and their words. A shape will be a finite sequence of positive integers i = (i 1 , ⋯, i r ), and by an arrangement of boxes of shape i we mean an arrangement of i j boxes positioned vertically, starting from right to left.
Fix a positive integer n. A gallery of shape i is a filling of an arrangement of boxes of the given shape with letters from the ordered alphabet A n ∶= {1, ⋯, n ∶ 1 < ⋯ < n} such that entries are strictly increasing along each column of boxes. We will denote the set of galleries of shape i by Γ(i).
Let W n denote the word monoid on A n . A word a 1 ⋯a k is considered as the gallery a k ⋯ a 1 . The word of a gallery δ of shape (m), m ∈ N is the word in W n which corresponds to reading the entries of δ from top to bottom and writing them down from left to right. It is denoted by w(δ). The word of a gallery is the concatenation of the words of each of its columns read from right to left. Concatenation of two galleries γ 2 * γ 1 is done from right to left, starting with γ 1 . is w(γ) = 25123, which in turn corresponds to the gallery 3 2 1 5 2 .
2.2. Crystals. For each i ∈ {1, ⋯, n − 1} we recall the action of Kashiwara's crystal operators on the set of all galleries, endowing the latter with a crystal structure. For basic facts about crystals that we need we refer to [BG01], Section 1. For a thorough treatment see [Kas95].
Let γ be a gallery of shape i = {i 1 , ⋯, i r }. To apply the crystal operator f i one does the following: a. Tag the columns of γ with a sign σ ∈ {+, −, ∅} (the resulting sequence of tags is sometimes called the i-signature of γ). If both i and i + 1 appear in the given column or if they do not appear in the column, then it is tagged with a (∅). If only i appears, it is tagged with a (+), and if only i + 1 appears, with a (−). b. Ignore the (∅)-tagged columns to produce a sub-gallery, and then ignore all pairs of consecutive columns tagged (− +), and get another sub-gallery. Continue this process, recursively obtaining subgalleries, until a final sub-gallery is produced with tags of the form To apply the operator f i (reap. e i ), modify the column corresponding to the right most + (respectively left most −) in the final subgallery tags, and replace the entry We also obtain that f 1 (γ) = 0.
The following fact is crucial for our purposes.
Proposition 2.5. The map is a crystal morphism.
Proof. Let γ be a gallery of shape (m), i.e., a single column. If two galleries γ 1 , γ 2 are labelled by (+) and (−) respectively, then the word associated to their concatenation γ 2 * γ 1 is in turn labelled by (− +). If the gallery γ is not labelled, then its word w(γ) (regarded as a gallery) is labelled wither by (− +) or by ∅.
Let γ = a 1 ⋯ a m with a 1 < ⋯ < a m . If γ is labelled by (∅) or by (−) then If γ is labelled by +, then, for some k ∈ {1, ⋯, r}, a k = i and since the column is labelled by only a (+), a k+1 > a k + 1. Hence, f i (γ) is obtained from γ by replacing i = a k by i + 1, with no need of reordering the entries, and therefore Example 2.6. A connected crystal of galleries and the crystal formed by its word-readings, regarded as galleries. is a semi-standard Young tableau. Note that the galleries considered in Example 2.6 are not.
We will denote by Γ(i) SSYT the set of all semi-standard Young tableaux of shape i.
Let γ, δ be two galleries, and let C(γ), C(δ) be the connected components they lie in, in the crystal of all galleries. We say that the galleries γ, δ are Remark 1. Two galleries δ, γ are equivalent if and only if their words w(δ), w(γ) are Knuth equivalent. This may be checked directly using the definition of the crystal operators on words, or via paths. For the latter, which includes the definition of the mentioned Knuth relations, see Section 13 in [Lit96].

Galleries and MV cycles
3.1. Notation. We assume throughout that G is the group SL n (C). Denote by X be the set of integral weights corresponding to the choice of a maximal torus T, and in it the set X + of dominant integral weights determined by fixing a Borel subgroup containing T. We denote the corresponding fundamental weights by ω i , for i ∈ {1, ⋯, n − 1}.
Consider the affine Grassmannian G associated to PSL n (C). To each dominant integral weight λ ∈ X + is associated a projective variety X λ ⊂ G which is defined as the closure of the PSL n (C[[t]])-orbit of λ in G.
Denote by Z(λ) the set of MV cycles in X λ , see Theorem 3.2 and Corollary 7.4 of [MV07]. For each i ∈ {1, ⋯, n − 1} we denote by e i , f i the set of crystal operators on Z(λ) defined by Braverman and Gaitsgory in [BG01].
To each shape i we may assign a dominant weight λ i = ω i 1 + ⋯ + ω ir as well as a Bott-Samelson like variety Σ i π i → X λ , see Section 8 of [GLN13]. It admits a C × -action with fixed points in bijection with the set Γ(i) of galleries of shape i, see Lemma 1 in [GLN13]. Given a gallery γ ∈ Γ(i), we denote by C γ ⊂ Σ(i) its corresponding Bialynicki-Birula cell. One of the main results in [GLN13], stated in Theorem 2 of the latter, establishes that the closure of the image π i (C γ ) is an MV cycle in Z(λ γ ), where λ γ is the dominant weight corresponding to the shape of the semi-standard Young tableau γ SS associated to γ as in Fact 2.8.

Galleries and MV cycles.
Theorem 3.1. Let λ ∈ X + be a dominant integral weight and let i be a type with associated dominant weight λ. Then, the map is a morphism of crystals.
Proof. Let λ ∈ X + , i and δ ∈ Γ(i) as in the statement of the Theorem. By Fact 2.8, we know that there exists a semi-standard Young tableau γ = δ SS of shape j such that δ ∼ δ SS . The weight λ δ mentioned in the paragraph above is precisely the dominant weight associated to j. Now, since by Proposition 2.5 word reading is a crystal morphism, we deduce that w(δ) ∼ w(γ). Theorem 2 b) in [GLN13] combined with Remark 1 then says π j (C w(γ) ) = π i (C w(δ) ). (1) We now make use of Proposition 2 in [GLN13], which states that the image closure π s (C η ) associated to a gallery η of shape s only depends on the word w(η). From this and (1) above we may deduce π j (C γ ) = π j ′ (C w(γ) ) = π i ′ (C w(δ) ) = π i (C δ ), where j ′ is the shape of w(γ) and i ′ is the shape of w(δ). Now, Theorem 5.8 in [BG08] says that the map is an isomorphism of crystals. Since, by definition, f k (δ) ∼ f k (γ) for all k ∈ {1, ⋯, n − 1}, this implies, together with equation (2): Corollary 3.2. Let λ ∈ X + be a dominant integral weight and let i be a type with associated dominant weight λ. Let Γ(i) δ be the connected component of the crystal of galleries Γ(i) where δ lies in. Then, the map induced from the restriction of the map in Theorem 3.1, is an isomorphism of crystals.