Séminaire Lotharingien de Combinatoire 73 (2015), Article B73c

YOUNG TABLEAUX, MULTISEGMENTS, AND PBW BASES

JOHN CLAXTON AND PETER TINGLEY^{∗}

Abstract. The crystals for finite dimensional representations of 𝔰𝔩_{n+1} can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of each embedding B(λ) ↪ B(∞) in terms of these realizations. The infinity crystal is also parameterized by Lusztig’s PBW basis with respect to any reduced expression for w_{0}. We give an explicit description of the unique crystal isomorphism from PBW bases to multisegments in the case where w_{0} = s_{1}s_{2}s_{3} ⋯ s_{n}s_{1} ⋯ s_{1}s_{2}s_{1}, thus obtaining simple formulas for the actions of all crystal operators on this PBW basis. Our proofs use the fact that the twists of the crystal operators by Kashiwara’s involution also have simple descriptions in terms of multisegments, and a characterization of B(∞) due to Kashiwara and Saito. These results are to varying extents known to experts, but we do not think there is a self-contained exposition of this material in the literature, and our proof of the relationship between multisegments and PBW bases seems to be new.

Contents

1. Introduction  2
1.1. Acknowledgements  3
2. Background  3
2.1. The quantized universal enveloping algebra  3
2.2. Crystals  4
2.3. PBW bases and crystal bases  6
3. Multisegment and Young tableau realizations  8
3.1. Multisegment realization of B(∞)  8
3.2. Young tableau realization of B(λ)  13
3.3. Embeddings B(λ) ↪ B(∞)  14
4. Crystal isomorphism from the PBW basis to multisegments  14
4.1. The isomorphism  15
4.2. Some technical lemmas  15
4.3. Proof of Theorem 4.2  17
4.4. An example  18
References  20

2010 Mathematics Subject Classification. 17B37,05E15. Key words and phrases. crystal basis, PBW basis, Young tableaux, multisegment.

1. Introduction

Kashiwara’s crystals B(λ) are combinatorial objects corresponding to the highest weight representations of a symmetrizable Kac–Moody algebra. Here we will only consider the case when that algebra is 𝔰𝔩_{n+1}. Then the crystal B(λ) can be realized as the set of semistandard Young tableaux of a fixed shape along with some combinatorial operations.

We also consider the crystal B(∞) for U^{−}(𝔰𝔩_{n+1}), which is a direct limit of the B(λ) as λ → ∞. There is a combinatorial realization of B(∞) where the underlying set consists of multisegments (i.e., collections of “segments” [i, j] for various 1 ≤ i ≤ j ≤ n, allowing multiplicity). Importantly for us, the twists of the crystal operators by Kashiwara’s ∗-involution are also easy to describe in this realization. Furthermore, the weak crystal embeddings B(λ) ↪ B(∞) are easily understood in terms of Young tableaux and multisegments: Each box corresponds to a segment, and the tableau is sent to the collection of the segment corresponding to each box (see Theorem 3.11).

There is another realization of B(∞) which has as its underlying set Lusztig’s PBW monomials. Combinatorially, these are recorded by lists of exponents, called Lusztig data, which consist of an integer for each positive root. The construction depends on a choice of reduced expression for the longest word. The crystal operators are defined algebraically, and are somewhat difficult to work with in general.

The positive roots for 𝔰𝔩_{n+1} are naturally in bijection with segments: α_{i} +α_{i+1} +⋯+α_{j} corresponds to [i, j]. In this way Lusztig data and multisegments are in bijection. In most cases this bijection does not seem to have nice properties but, if we work with the reduced expression

(1.1)  w_{0} = s_{1}s_{2} ⋯ s_{n}s_{1}s_{2} ⋯ s_{n−1} ⋯ s_{1}s_{2}s_{1},

we show that it is a crystal isomorphism (see Theorem 4.2).

Much of the current work is to some extent understood by experts. The reduced expression (1.1) has been observed to have nice properties many times (see e.g. [Kam, §3.4.3], [BBF], [Lit, §5]), and the connection with Young tableaux has been made (see e.g. [BZ], [M, Prop. 2.3.13]). The map from tableaux to multisegments has been studied in e.g. [BZ, §8] and [Zel3, §7], although there it is not discussed in terms of crystals. Various relationships between the infinity crystal and multisegments have also been observed (see e.g. [LTV]). Kashiwara’s ∗ involution is well known in the context of multisegments: it is precisely the famous Zelevinsky multisegment duality first introduced in [Zel1, Zel2] (see also [Zel3]). Finally, in the affine 𝔰𝔩_{n+1} case, the embeddings B(Λ) ↪ B(∞) are described by Jacon and Lecouvey [JL]; our results from §3.3 can be derived from their results. In fact, much of the literature considers the affine case, partly because it is related to the such important topics as the p-adic representations theory of 𝔤𝔩_{n} and of certain Hecke algebras (see e.g. [BZ2, Vaz]), but this can obscure the simpler finite type case.

Ringel’s Hall algebra approach to quantum groups [Rin1] can also be used to see some of our results: By Gabriel’s theorem [Gab], in any finite type, U^{−}(𝔤) can be identified (as a vector space) with the split Grothendieck group of the category of representations of the quiver obtained by choosing an orientation of the Dynkin diagram. Ringel introduced a product in terms of the representation theory of the quiver that strengthens this relationship, and it was shown in [Lus1, Rin2] that the natural basis of the Grothendieck group consisting of isomorphism classes of representations coincides with Lusztig’s PBW basis for a reduced word adapted to the orientation. Reineke [Rei] gave an explicit description of the crystal operators acting on the PBW basis in terms of representations of quivers. In the 𝔰𝔩_{n+1} case, isomorphism classes of representation are naturally indexed by multisegments. Choosing the appropriate orientation of the quiver, Reineke’s work implies our results from §4.

In any case, we do not know a self-contained exposition of these results. Our methods are considerably more elementary and combinatorial than most of the references discussed above, and some of our proofs are new.

1.1. Acknowledgements. We thank Ben Salisbury, Monica Vazirani and Arun Ram for interesting discussions and for comments on an early draft. We also thank Tynan Greene who did some preliminary work with us in summer 2013. Finally, we thank the anonymous referee for providing some important references. Both authors received partial support from the NSF grant DMS-1265555.

2. Background

2.1. The quantized universal enveloping algebra. U_{q}(𝔰𝔩_{n+1}) is the quantized universal enveloping algebra for 𝔰𝔩_{n+1}. It is an algebra over ℂ(q) generated by {E_{i}, F_{i}, K_{i}^{±1}}, for 1 ≤ i ≤ n. Details can be found in e.g. [CP]. We will mainly work with U_{q}^{−}(𝔰𝔩_{n+1}), the subalgebra generated by the F_{i}. We first fix some notation

• W is the Weyl group, and w_{0} is the longest element in W.

• P is the weight lattice, and Q ⊂ P is the root lattice. P^{∨} and Q^{∨} are the co-weight and co-root lattices.

• {α_{i}}_{i=1,...,n}, {ω_{i}}_{i=1,...,n} are the simple roots and fundamental weights respectively; {α_{i}^{∨}}_{i=1,...,n} are the simple co-roots.

• ⟨·, ·⟩ is the pairing between the root lattice and the co-root lattice defined by

⟨α_{i}, α_{j}^{∨}⟩ = \begin{cases}2 if i − j \\ −1 if |i − j| = 1 \\ 0 otherwise. \end{cases}

• N = ( \begin{matrix}n \\ 2 \end{matrix} ) is the number of positive roots 𝔰𝔩_{n+1}.

• T_{i} is the algebra automorphism of U_{q}(𝔰𝔩_{n+1}) introduced by Lusztig (see [Lus2, 37.1.3]):

T_{i}(F_{j}) = \begin{cases}F_{j} i not adjacent to j \\ F_{j}F_{i} − qF_{i}F_{j} i adjacent to j \\ −K_{j}^{−1}E_{j} i = j. \end{cases}

T_{i}(E_{j}) = \begin{cases}E_{j} i not adjacent to j \\ E_{j}E_{i} − q^{−1}E_{i}E_{j} i adjacent to j \\ −F_{j}K_{j} i = j. \end{cases}

T_{i}(K_{j}) = \begin{cases}K_{j} i not adjacent to j \\ K_{i}K_{j} i adjacent to j \\ K_{j}^{−1} i = j. \end{cases}

These T_{i} define an action of the n-strand braid group on U_{q}(𝔰𝔩_{n+1}), which means

T_{i}T_{j}T_{i} = T_{j}T_{i}T_{j} if |i − j| = 1, and T_{i}T_{j} = T_{i}T_{i} otherwise.

There are embeddings of U_{q}(𝔰𝔩_{n}) ↪ U_{q}(𝔰𝔩_{n+1}) for all n, which just takes the generators of U_{q}(𝔰𝔩_{n}) to the generators with the same names in U_{q}(𝔰𝔩_{n+1}), and these are compatible with the braid group actions.

2.2. Crystals. Here we very briefly introduce Kashiwara’s crystals. The following definition is essentially from [Kas2, §7.2], although here we do not allow φ_{i}, ε_{i} to take the value −∞.

Definition 2.1. An abstract crystal is a set B along with functions wt: B → P (where P is the weight lattice), and, for each i ∈ I, ε_{i}, φ_{i}: B → ℤ and e_{i}, f_{i} : B → B ⊔ {0}, such that

(i) φ_{i}(b) = ε_{i}(b) + ⟨wt(b), α_{i}^{∨}⟩.

(ii) If e_{i}(b) ≠ 0, e_{i} increases φ_{i} by 1, decreases ε_{i} by 1 and increases wt by α_{i}.

(iii) f_{i}b = b′ if and only if e_{i}b′ = b. We often denote an abstract crystal simply by B, suppressing the other data.

Definition 2.2. A strict morphism of crystals is a map between two crystals that commutes with wt, e_{i}, f_{i}, ε_{i}, and φ_{i} for all i. A weak morphism is a map which commutes with all e_{i}, but not necessarily the other structure. An isomorphism of crystals is a strict morphism that has an inverse which is also a strict morphism.

Remark 2.3. In fact, a weak morphism ϕ : B_{1} → B_{2} must also have good properties with respect to the other structures. For instance, Definition 2.1 (iii) implies that, as long as b ∈ B_{1} satisfies f_{i}(b) ≠ 0, we have f_{i}(ϕ(b)) = ϕ(f_{i}(b)). It can however happen that f_{i}(b) = 0 but f_{i}(ϕ(b)) ≠ 0.

Definition 2.4. A highest weight abstract crystal is an abstract crystal which has a distinguished element b_{+} (the highest weight element) such that

(i) The highest weight element b_{+} can be reached from any b ∈ B by applying a sequence of e_{i} for various i ∈ I.

(ii) For all b ∈ B and all i ∈ I, ε_{i}(b) = max{n : e_{i}^{n}(b) ≠ 0}.

Notice that, since a highest weight abstract crystal is necessarily connected, it can have no non-trivial automorphisms.

The crystals we are interested in here are B(∞), which is related to U^{−}(𝔰𝔩_{𝔫+1}), and B(λ), which is related to a highest weight representation of 𝔰𝔩_{𝔫+1}. These are all highest weight abstract crystals. We don’t need details of how they are defined; instead we just use the characterization of B(∞) below, and the explicit description of B(λ) in terms of Young tableaux from §3.2.

The following notion is very convenient for us. It is a bit non-standard, but can be found in [TW].

Definition 2.5. A bicrystal is a set B with two crystal structures whose weight functions agree. We use the convention of placing a star superscript on all data for the second crystal structure, so e_{i}^{∗}, f_{i}^{∗}, φ_{i}^{∗}, etc. An element of a bicrystal is called highest weight if it is killed by both e_{i} and e_{i}^{∗} for all i.

The following is a rewording of [KS, Proposition 3.2.3] designed to make the roles of the usual crystal operators and the ∗-crystal operators more symmetric. See [TW] for this exact statement.

Proposition 2.6. Fix a bicrystal B. Assume (B, e_{i}, f_{i}) and (B, e_{i}^{∗}, f_{i}^{∗}) are both highest weight abstract crystals with the same highest weight element b_{+}, where the other data is determined by setting wt(b_{+}) = 0. Assume further that, for all i ≠ j ∈ I and all b ∈ B,

(i) f_{i}(b), f_{i}^{∗}(b) ≠ 0. (ii) f_{i}^{∗}f_{j}(b) = f_{j}f_{i}^{∗}(b). (iii) ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ ≥ 0 (iv) If ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ = 0 then f_{i}(b) = f_{i}^{∗}(b),

(v) If ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ ≥ 1 then ε_{i}^{∗}(f_{i}(b)) = ε_{i}^{∗}(b) and ε_{i}(f_{i}^{∗}(b)) = ε_{i}(b). (vi) If ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ ≥ 2 then f_{i}f_{i}^{∗}(b) = f_{i}^{∗}f_{i}(b).

Then (B, e_{i}, f_{i}) ≃ (B, e_{i}^{∗}, f_{i}^{∗}) ≃ B(∞), and e_{i}^{∗} = ∗e_{i}∗, f_{i}^{∗} = ∗f_{i}∗, where ∗ is Kashiwara’s involution from [Kas1, 2.1.1].

The following is immediate from conditions (iv), (v) and (vi) of Proposition 2.6.

Remark 2.7. The quantities in Proposition 2.6 have also been studied by Lauda and Vazirani [LV] for related reasons. For instance, there ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ is called jump_{i}(b).

Corollary 2.8. For any i ∈ I and any b ∈ B(∞) the subset of B(∞) that can be reached from b by applying sequences of the operators e_{i}, f_{i}, e_{i}^{∗}, f_{i}^{∗} is of the following form, where the solid and dashed arrows show the action of f_{i}, the dotted and dashed arrows show the action of f_{i}^{∗}, and the width of the diagram at the bottom is ⟨wt(b_{top}), α_{i}^{∨}⟩ for the top vertex b_{top} (in this example the width is 4).

\begin{matrix}• \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \\ • \end{matrix}

Furthermore, for any element b, the quantity ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ counts how many times one must apply f_{i} (or equivalently f_{i}^{∗}) to reach a dashed line. □

2.3. PBW bases and crystal bases. Fix a reduced expression w_{0} = s_{i_{1}} ⋯ s_{i_{N}}, and let i denote the sequence of indices i_{1}, …, i_{N}. It is well known that this gives an ordering of the positive roots of 𝔰𝔩_{n+1}: β_{1} = α_{i_{1}} and β_{j} = s_{i_{1}} ⋯ s_{i_{j−1}}α_{i_{j}} for 2 ≤ j ≤ N. Define

(2.1)  F_{β_{1}} = F_{i_{1}}, F_{β_{2}} = T_{i_{1}}F_{i_{2}}, …, F_{β_{j}} = T_{i_{1}} ⋯ T_{i_{j−1}}F_{i_{j}}, …

where the T_{i_{j}} are the braid group operators defined in §2.1. As shown by Lusztig [Lus2, Corollary 40.2.2], B_{i} = {F_{β_{N}}^{(a_{n})} ⋯ F_{β_{1}}^{(a_{1})}} is a basis for U_{q}^{−}(𝔤) (called the PBW basis). Here F_{β}^{(a)} means the quantum divided power

F_{β}^{(a)} = F_{β}^{a}/[a]!

where [a] = q^{−a+1} + q^{−a+3} + ⋯ + q^{a−3} + q^{a−1} and [a]! = [a][a − 1] ⋯ [2].

Let A = ℂ[q]_{0}, the algebra of rational function in q that do not have a pole at q = 0. By [Lus2, §42.1], L = span_{A}B_{i} does not depend on the reduced expression i. This A-module is called the crystal lattice. Furthermore, B_{i} + qL is a basis for L/qL which does not depend on i. We denote this basis of L/qL by B, and call it the crystal basis.

Definition 2.9. For each reduced expression i and each collection of non-negative integers a_{1}, …, a_{N}, let b_{a_{1},...,a_{N}}^{i} denote the element F_{β_{N}}^{(a_{n})} ⋯ F_{β_{1}}^{(a_{1})} + qL of the crystal basis B.

Definition 2.9 gives a parameterization of B by ℕ^{N} for each reduced expression i of w_{0}. We now define the crystal operators:

Definition 2.10. Fix i. Choose i so that i_{1} = i and i′ so that i′_{N} = n + 1 − i (so that β_{N} = α_{i}). For any a_{1}, …, a_{N}, define

f_{i}(b_{a_{1},...,a_{N}}^{i} ) = b_{a_{1}+1,...,a_{N}}^{i} ,  f_{i}^{∗}(b_{a_{1},...,a_{N}}^{i′}) = b_{a_{1},...,a_{N} +1}^{i},
e_{i}(b_{a_{1},...,a_{N}}^{i} ) = \begin{cases}b_{a_{1}−1,...,a_{N}}^{i} if a_{1} ≥ 1 \\ ∅ otherwise, \end{cases} e_{i}^{∗}(b_{a_{1},...,a_{N}}^{i} ) = \begin{cases}b_{a_{1},...,a_{N} −1}^{i′} if a_{N} ≥ 1 \\ ∅ otherwise, \end{cases}
ε_{i}(b_{a_{1},...,a_{N}}^{i} ) = a_{1},  ε_{i}^{∗}(b_{a_{1},...,a_{N}}^{i′}) = a_{N}.

Theorem 2.11 ([Sai, Theorem 4.1.2 and its proof]). B is the crystal basis for U_{q}^{−}(𝔰𝔩_{n+1}), as defined by Kashiwara [Kas2], and the e_{i}, e_{i}^{∗}, f_{i}, f_{i}^{∗} defined above are the crystal operators. In particular, B along with the operations from Definition 2.10 is a realization of the crystal B(∞) from §2.2.

Fix i and a reduced expression for w_{0} of the form i = (i, i_{2}, ⋯, i_{N}). Then i′ = (i_{2}, i_{3}, … i_{N}, n + 1 − i) is also a reduced expression. It is clear from the definitions that T_{i}^{−1} gives the bijection {b_{0,a_{2},...a_{N}} ⟶ b′_{a_{2},...a_{N} ,0}} between the subset of those b ∈ B(∞) where ε_{i}(b) = 0, and those b ∈ B(∞) where ε_{i}^{∗}(b) = 0. Define τ : B → {b ∈ B : ε_{i}(b) = 0} by τ : b_{a_{1},a_{2},...a_{N}}^{i} ↦ b_{0,a_{2},...a_{N}}^{i} and let σ_{i} = T_{i}^{−1} ∘ τ.

Proposition 2.12. For any b ∈ B(∞),

(2.2)  σ_{i}(b) = (e_{i}^{∗})^{max}f_{i}^{L}b for any L ≥ ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩.

Proof. It is immediate from the definitions that it suffices to consider the case ε_{i}(b) = 0. Then by [Sai, Corollary 3.4.8],

(2.3)  σ_{i}(b) = f_{i}^{φ_{i}^{∗}(b)}(e_{i}^{∗})^{ε_{i}^{∗}(b)}(b).

First assume L = ε_{i}(b)+ε_{i}^{∗}(b)+⟨wt(b), α_{i}^{∨}⟩, and refer to the diagram from Corollary 2.8 showing the part of the crystal reachable from b by applying e_{i}, f_{i}, e_{i}^{∗}, f_{i}^{∗}. By definition, φ_{i}^{∗}(b) = ⟨wt(b), α_{i}^{∨}⟩ + ε_{i}^{∗}(b) which is equal to ⟨wt(b), α_{i}^{∨}⟩ + ε_{i}(b) + ε_{i}^{∗}(b) since ε_{i}(b) = 0. By Corollary 2.8 we see that φ_{i}^{∗}(b) is the number of times one must apply f_{i}^{∗} to b to reach the dashed line. Thus (2.3) takes b, applies e_{i}^{∗} until it gets to the vertex, then applies f_{i} exactly φ_{i}^{∗} times. On the other hand, (2.2) takes b, applies f_{i} exactly φ_{i}^{∗} times to just reach the dashed lines, then applies e_{i}^{∗} the maximal number of times, thus reaching the top right boundary of the picture. Tracing this through, they agree.

If L > ε_{i}(b) + ε_{i}^{∗}(b) + ⟨wt(b), α_{i}^{∨}⟩ then (2.2) just applies some extra f_{i} which move along dashed lines, then some extra e_{i}^{∗}, which undoes this, and so the result does not change. □

The following key lemma will allow us to perform induction on rank.

Lemma 2.13. When w_{0} = s_{1}s_{2} ⋯ s_{n}s_{1} ⋯ s_{n−1} ⋯ s_{1}s_{2}s_{1}, the corresponding order on positive roots is

β_{1} = α_{1}, β_{2} = α_{1} + α_{2}, …, β_{n} = α_{1} + ⋯ + α_{n}, β_{n+1} = α_{2}, …, β_{N} = α_{n}.

Furthermore,

σ_{n} ⋯ σ_{2}σ_{1} (F_{α_{1}}^{(a_{1})} ⋯ F_{(α_{1}+···+α_{n})}^{(a_{n})}F_{α_{2}}^{(a_{n+1})} ⋯ F_{(α_{2}+···+α_{n})}^{(a_{2n−1})} ⋯ F_{α_{n}}^{(a_{N})})
= F_{α_{1}}^{(a_{n+1})} ⋯ F_{(α_{1}+···+α_{n−1})}^{(a_{2n−1})}F_{α_{2}}^{(a_{2n})} ⋯ F_{α_{n−1}}^{(a_{N})}.

Proof. This is a simple calculation using (2.1). □

3. Multisegment and Young tableau realizations

3.1. Multisegment realization of B(∞). We now define multisegments and their crystal structure, and prove that they realize B(∞). This is essentially the same as the realization discussed in [Sav, §4.1], although our proof is quite different, and there the term multisegment is not used. We have changed terminology to match [JL], where they consider the affine case. The term multisegment has also been used as we use it in e.g. [Zel3]. Our realization is also very similar to the one constructed in terms of marginally large tableaux in [HL] (see also [LS]), although we note that the ∗ operators are a bit easier to describe using the setup here.

Definition 3.1. (i) A segment is an interval [i, j] with i, j ∈ ℤ, 1 ≤ i ≤ j. (ii) A multisegment is a finite set of segments, allowing multiplicity. (iii) Given a multisegment M, M_{i,j} is the multiplicity of [i, j] in M. (iv) MS_{n} is the set of all multisegments where all segments [i, j] have j ≤ n.

(v) The height of a segment, [i, j] is j − i + 1.

(vi) The size |M| of a multisegment M is the sum over all segments of their heights.

We will represent a segment [i, j] with a columns of boxes containing the integers i to j. For example, [3, 5] will be drawn as

5 4 3 .

The size of a multisegment is the total number of boxes if you draw all the segments.

Definition 3.2. Given M ∈ MS_{n}, S_{i}(M) is the string formed as follows (see §4.4 for examples):

1. Order the segments of M from left to right, first in increasing order of height, then by largest to smallest bottom entry.

2. Place a “)” above each [h, i − 1] segment, and a “(” above each [h, i] segment.

S_{i}^{∗}(M) is the string formed as follows:

1. Order the segments of M from left to right, first by shortest to tallest, then by smallest to largest bottom entry.

2. Place a “)” below each [i + 1, j] segment, and a “(” below each [i, j] segment.

In these strings, we say brackets “(” and “)” are canceled if the “(” is directly to the left of “)”, or if the only brackets between them are canceled. uc_{i}(M) and uc_{i}^{∗}(M) are the strings formed from S_{i}(M) and S_{i}^{∗}(M) by deleting canceled brackets.

For M ∈ MS_{n} and 1 ≤ i ≤ n, define

f_{i}(M):= \begin{cases}(M\\{[h, i − 1]\}) ∪ \{[h, i]\} & if the right-most “)” in uc_{i}(M) is above an [h, i − 1] \\ M ∪ \{[i, i]\} & if there are no “)” in uc_{i}(M), \end{cases}

e_{i}(M):= \begin{cases}(M\\{[h, i]\}) ∪ \{[h, i − 1]\} & if the left-most “(” in uc_{i}(M) is above an [h, i], h ≠ i \\ M\\{[i, i]\} & if the left-most “(” in uc_{i}(M) is above a [i, i] \\ 0 & if there are no “(” in uc_{i}(M), \end{cases}

f_{i}^{∗}(M):= \begin{cases}(M\\{[i + 1, j]\}) ∪ \{[i, j]\} & if the right-most “)” in uc_{i}^{∗}(M) is below a [i + 1, j] \\ M ∪ \{[i, i]\} & if there are no “)” in uc_{i}^{∗}(M), \end{cases}

e_{i}^{∗}(M):= \begin{cases}(M\\{[i, j]\}) ∪ \{[i + 1, j]\} & if the left-most “(” in uc_{i}^{∗}(M) is below a [i, j], i ≠ j \\ M\\{[i, i]\} & if the left-most “(” in uc_{i}^{∗}(M) is above a [i, i] \\ 0 & if there are no “(” in uc_{i}^{∗}(M). \end{cases}

Remark 3.3. There is clearly symmetry between the unstarred and starred operators above. To make this precise, consider the map Flip : MS_{n} → MS_{n} which flips every segment over and re-indexes by 1 ↔ n, 2 ↔ n − 1, and so on. This sends S_{i}(M) to S_{n−i+1}^{∗}(M), and so interchanges the action of each f_{i} with the action of f_{n+1−i}^{∗}; in fact, this is the composition of Kashiwara’s ∗ involution with the Dynkin diagram automorphism. We will appeal to this symmetry often later on.

For any multisegment M, define

wt(M) := − ∑_{i=1}^{n}(# of i boxes)α_{i},

and notice that, and all i,

⟨wt(M), α_{i}^{∨}⟩ = (# of i − 1 boxes) + (# of i + 1 boxes) − 2(# of i boxes).

Proposition 3.4. (MS_{n}, e_{i}, f_{i}, e_{i}^{∗}, f_{i}^{∗}), with the weight function defined above and additional data determined as in Definition 2.4, is a highest weight bicrystal with the highest weight element the empty multisegment ∅.

Proof. We need to check that both structures satisfy the axioms in Definitions 2.1 and 2.4; we present the proof only for (MS_{n}, e_{i}, f_{i}), since the arguments for the other structure are the same. 2.1(i) and 2.4(ii) are true by definition, and 2.1(ii) easily follows from the definitions of ε_{i}, φ_{i}, and wt. For 2.1(iii) consider how applying f_{i} affects S_{i}: it either changes [h, i − 1] → [h, i] and a “)” to a “(” immediately to the right, or adds an [i, i] and a “(” as far left as possible. Removing canceled brackets, we have:

uc_{i}(M) : …)))) (((…

uc_{i}(f_{i}(M)) : …))) ((((…

where the red “(” has become the green “).” So e_{i} will act on the green bracket of uc_{i}(f_{i}(M)), reversing the effect of f_{i}.

Since multisegments are finite, to establish 2.4(i) it suffices to show that, for all nonempty M ∈ MS_{n}, there is some e_{k} such that e_{k}(M) ≠ 0. The segments are ordered the same way in all of the strings S_{i}(M), so they all have the same right-most segment, call it [j, k]. Clearly uc_{k} has an uncanceled “(”, and hence e_{k}(M) ≠ 0. □

The next few lemmas are needed to prove that this bicrystal satisfies the conditions in Proposition 2.6. The first one explains what ε_{i}(M) + ε_{i}^{∗}(M) + ⟨wt(M), α_{i}^{∨}⟩ means in terms of brackets.

Lemma 3.5. Given a multisegment M ∈ MS_{n}, for 1 ≤ i ≤ n let

ur_{i}(M) := the number of uncanceled “)” in S_{i}(M)
ur_{i}^{∗}(M) := the number of uncanceled “)” in S_{i}^{∗}(M)

Then

(3.1)  ε_{i}(M) + ε_{i}^{∗}(M) + ⟨wt(M), α_{i}^{∨}⟩ = ur_{i}(M) + ur_{i}^{∗}(M).

Proof. We proceed by induction on |M|, the base case M = ∅ being obvious. So, fix M ≠ ∅ and assume the result holds for all smaller multisegments. MS_{n} is a highest weight crystal by Proposition 3.4, so M = f_{j}(M′) for some j and some M′ with |M′| = |M| − 1, and by the inductive hypothesis the result holds for M′. We consider several cases.

Case 1: |i − j| > 1. Then ⟨wt(M), α_{i}^{∨}⟩ = ⟨wt(M′), α_{i}^{∨}⟩ and S_{i}(M) = S_{i}(M′), so ε_{i} and ur_{i} are unchanged. If f_{j} changes [i, j − 1] → [i, j] or [i + 1, j − 1] → [i + 1, j], these segments and their brackets will shift to the right in S_{i}^{∗}. This shift will either leave ur_{i}^{∗} unchanged, or change it by 1. We will use the notation ↑ 1 or ↓ 1 to record this change, so for example ur_{i} ↑ 1 means ur_{i}(M) = ur_{i}(M′) + 1.

• If ur_{i}^{∗} is unchanged, so is ε_{i}^{∗}.

• If ur_{i}^{∗} ↑ 1, then a “(” changed from canceled to uncanceled, so ε_{i}^{∗} ↑ 1.

• If ur_{i}^{∗} ↓ 1, then ε_{i}^{∗} ↓ 1.

Case 2: j = i − 1. Then ⟨wt(), α_{i}^{∨}⟩ ↑ 1. The string S_{i}^{∗} is not affected, so ε_{i}^{∗} and ur_{i}^{∗} are unchanged. One new “)” is created in S_{i}, which either increases ur_{i} by 1 or leaves it unchanged.

• If ur_{i} is unchanged, then a “(” that was previously uncanceled must be canceled. So ε_{i} ↓ 1.

• If ur_{i} ↑ 1, then no new “(” are canceled, so ε_{i} is unchanged.

Case 3: j = i + 1. Then ⟨wt(), α_{i}^{∨}⟩ ↑ 1.

• If f_{i+1} adds a [i + 1, i + 1], this has no affect on S_{i}, so ε_{i} and ur_{i} are unchanged. The fact that f_{i+1} acted this way implies that all “)” in S_{i+1}(M′) are canceled. In particular, M′_{i+1,i+1} ≥ M′_{i,i}, so adding an [i + 1, i + 1] must create an uncanceled “)” in S_{i}^{∗}. Hence ε_{i}^{∗} is unchanged and ur_{i}^{∗} ↑ 1.

• If f_{i+1} changes [i, i] → [i, i + 1], this shifts a “(” to the right in S_{i}^{∗}. As in the case |i − j| > 1, ur_{i}^{∗} and ε_{i}^{∗} are affected in the same way. In S_{i}, a “(” is removed. This can either increase ur_{i} or leave it fixed.

– If ur_{i} is fixed, there is one less uncanceled “(”, so ε_{i} ↑ 1. Also, ⟨wt(), α_{i}^{∨}⟩ ↑ 1.

– If ur_{i} ↑ 1, then ε_{i} is fixed. So ur_{i} ↑ 1 and ⟨wt(), α_{i}^{∨}⟩ ↑ 1.

• If f_{i+1} changes [h, i] → [h, i + 1] for h ≠ i this has no effect on S_{i}^{∗}, and the same effect on S_{i} as in the previous case.

Case 4: j = i. Then ⟨wt(), α_{i}^{∨}⟩ ↓ 2 and ε_{i} ↑ 1.

• If f_{i} adds an [i, i], the ur_{i}(M′) = ur_{i}(M) = 0 so ur_{i} is unchanged. In S_{i}^{∗}, a “(” is

added, this either decreases ur_{i}^{∗} or leaves it unchanged.

– If ur_{i}^{∗} is unchanged then there is a new uncanceled “(” so ε_{i}^{∗} ↑ 1.

– If ur_{i}^{∗} ↓ 1, then ε_{i}^{∗} is unchanged.

• If f_{i} changes [h, i − 1] → [h, i], this has no affect on S_{i}^{∗}, so ε_{i}^{∗} and ur_{i}^{∗} are

unchanged. In S_{i}, an uncanceled “)” is changed to a “(”, so ur_{i} ↓ 1. In all cases the two sides of (3.1) change by the same amount. □

Lemma 3.6. Let M ∈ MS_{n} and i ≠ j. For each h let ur_{i;h}(M) be the number of “)” in uc_{i}(M) corresponding to segments of height h.

(i) If f_{i}^{∗} applied to M acts on an [i + 1, j − 1] segment and ur_{j;j−i−1}(M) ≠ 0, then

(3.2)  ur_{j;j−i−1}(f_{i}^{∗}(M)) = ur_{j;j−i−1}(M) − 1
(3.3)  ur_{j;j−i}(f_{i}^{∗}(M)) = ur_{j;j−i}(M) + 1

(ii) If f_{i}^{∗} applied to M adds an [i, i], then ur_{i+1;1}(f_{i}^{∗}(M)) = ur_{i+1;1}(M) + 1.

(iii) In all other cases, ur_{j;h}(f_{i}^{∗}(M)) = ur_{j;h}(M).

Proof. (i): f_{i}^{∗} acts on a segment of height j − i − 1, and since ur_{j;j−i−1}(M) ≠ 0, this moves an uncanceled “)” in S_{j}(M) to the right, and (3.2) is clear. For (3.3), we need to show that the new [i, j − 1] is uncanceled in S_{j}(f_{i}^{∗}(M)). Since the bracket was originally uncanceled, it can only be canceled by a segment of height j − i. But ur_{j,j−i−1}(M) ≠ 0 implies that this cannot happen, so (3.3) holds.

(ii): Since f_{i}^{∗} adds an [i, i], all “)”in S_{i}^{∗}(M) are canceled. In particular, since [i+1, i+1] segments can only be canceled by [i, i] segments, M_{i+1,i+1} ≤ M_{i,i}. So in S_{i+1}(f_{i}^{∗}(M)), the new [i, i] corresponds to an uncanceled “).”

(iii): This breaks up into four cases

• If f_{i}^{∗} acts on a segment [i + 1, h] for h ≠ i − 1, i this clearly has no effect on S_{i} or uc_{i}.

• If f_{i}^{∗} acts on an [i + 1, j − 1] segment and ur_{j;j−i−1}(M) = 0, then a canceled “)” is moved to the right in S_{j}, and since it doesn’t move past any other “)”, it’s not hard to see that it remains canceled, so this has no effect on ur_{j}.

• If f_{i}^{∗} acts on an [i + 1, j] segment, this shifts a “(” to the right. The only way this can change ur_{j} is if an [i, j − 1] changes from canceled to uncanceled. But the fact that f_{i}^{∗} acts on an [i + 1, j] implies that M_{i,j−1} < M_{i+1,j}; so even after the shift, all [i, j − 1] segments in S_{j} are canceled.

• If f_{i}^{∗} adds an [i, i], this has no effect on S_{j} (so certainly it has no effect on ur_{j}), unless j = i + 1. Clearly ur_{i;h} cannot be affected for h > 1, and h = 1 is case (ii).

□

Lemma 3.7. f_{j}f_{i}^{∗}(M) = f_{i}^{∗}f_{j}(M) when i ≠ j.

Proof. For this equality to break, f_{i}^{∗} must change the length of the segment that f_{j} acts on, or vice-versa. By symmetry (see Remark 3.3), we can assume we are in the first case. By Lemma 3.6, there is only one way for this to happen: f_{i}^{∗} must act on M by changing a [i+1, j −1] to [i, j −1] (if j = i+1, this is interpreted as adding a new [i]), f_{j} must act on f_{i}^{∗}(M) by changing that [i, j −1] to an [i, j], and f_{j} must act on M by changing a different segment, which looking at the brackets in f_{i}^{∗}(M) must necessarily be a shorter segment. In fact, f_{j} must act on M by changing a [i + 1, j − 1], since if there are no uncanceled brackets corresponding to segments of that length in S_{i}(M), moving one of them to the right past some “(” cannot create an uncanceled bracket. From this one can quickly see that f_{j}f_{i}^{∗}(M) and f_{i}^{∗}f_{j}(M) both change an [i + 1, j − 1] to [i, j], so they agree. □

Proposition 3.8. MS_{n} along with the operators e_{i}, f_{i}, e_{i}^{∗}, f_{i}^{∗} is a realization of B(∞) as a bicrystal, where the second crystal structure is the twist of the first by Kashiwara’s involution.

Proof. We need to check that MS_{n} satisfies the conditions in Proposition 2.6. (i): This is clear from the definitions of f_{i} and f_{i}^{∗}. (ii): This is Lemma 3.7 (iii): This is clear from Lemma 3.5. (iv): By Lemma 3.5, if ε_{i}(M) + ε_{i}^{∗}(M) + ⟨wt(M), α_{i}^{∨}⟩ = 0, there are no uncanceled “)”

in either S_{i}(M) or S_{i}(M), so both f_{i} and f_{i}^{∗} add an [i, i].

(v): By Lemma 3.5, if ε_{i}(M) + ε_{i}^{∗}(M) + ⟨wt(M), α_{i}^{∨}⟩ ≥ 1, there is an uncanceled “)” in either S_{i}(M) or S_{i}^{∗}(M). Without loss of generality, assume it is in S_{i}(M). So f_{i} applied to M must change [h, i − 1] → [h, i] for some h ≤ i − 1. This has no affect on S_{i}^{∗}, so ε_{i}^{∗}(f_{i}(M)) = ε_{i}^{∗}(M). If there is also an uncanceled “)” in S_{i}^{∗}(M), the same argument shows that ε_{i}(f_{i}^{∗}(M)) = ε_{i}^{∗}(M). If there are no uncanceled “)” in S_{i}^{∗}, then f_{i}^{∗} adds an [i, i]. This creates a “(” all the way to the left in S_{i}(f_{i}^{∗}(M)). Since we assumed there was some uncanceled “)” in S_{i}(M), it must cancel the new “(”, and so ε_{i}(f_{i}^{∗}(M)) = ε_{i}(M).

(vi): By Lemma 3.5, if ε_{i}(M)+ε_{i}^{∗}(M)+⟨wt(M), α_{i}^{∨}⟩ ≥ 2, there are at least 2 uncanceled “)” between S_{i}(M) and S_{i}^{∗}(M). If there is at least one in each string, f_{i} and f_{i}^{∗} add to the tops and bottoms of segments respectively, so neither operator affects the other string, from which it is clear that f_{i}f_{i}^{∗}(M) = f_{i}^{∗}f_{i}(M).

Otherwise, one string has no uncanceled “)” and the other has at least 2. Assume without loss of generality that ur_{i}(M) = 0 and ur_{i}^{∗}(M) ≥ 2. As in (v), S_{i}(f_{i}^{∗}(M)) = 0 0 0 1 1 2 2 3 4 1 1 2 2 3 3 4 2 3 4 4 )

( )

(

(

Figure 1. A Young tableau b of shape (9, 7, 3, 1). The string of brackets S_{2}(b) is shown. f_{2}(b) is obtained by changing the right-most “1” on the second row to a “2”.

S_{i}(M), so f_{i} acts on segments of the same length in M and f_{i}^{∗}(M). Since ur_{i}(M) = 0 we see that f_{i}(M) = M ∪ {[i, i]}, so this creates a “(” all the way to the left in S_{i}^{∗}(f_{i}(M)), which cancels a “)”. But since there are at least 2 uncanceled “)” in S_{i}^{∗}(M), the rightmost one is still uncanceled in S_{i}^{∗}(f_{i}(M)). Therefore f_{i}^{∗} acts on segments of the same length in M and f_{i}(M). It follows that f_{i}f_{i}^{∗}(M) = f_{i}^{∗}f_{i}(M). □

3.2. Young tableau realization of B(λ). Recall that a partition λ = (λ_{1}, λ_{2}, ⋯, λ_{k}) is a non-increasing sequence of positive integers. The size of λ is |λ| = λ_{1} + ⋯ + λ_{k}. To each partition λ and n ≥ k − 1, we associate a weight for 𝔰𝔩_{n+1} by

λ → (λ_{1} − λ_{2})ω_{1} + ⋯ + (λ_{k−1} − λ_{k})ω_{k−1} + λ_{k}ω_{k},

where in the case n = k − 1 we drop the last term. As usual we will often simply write λ to mean this associated weight when the meaning is clear from context.

We associate to a partition its Young diagram, which consists of the |λ| boxes, arranged as a row of length λ_{1} above a row of length λ_{2}, etc. A semi-standard Young tableau of shape λ for 𝔰𝔩_{n+1} is a filling of the Young diagram of λ with the numbers {0, …, n}, which is weakly increasing along rows and strictly increasing down columns. See Figure 1. Denote the set of all such tableaux of a fixed shape by SSYT_{n}(λ).

Define operators f_{i} on SSYT_{n}(λ) for 1 ≤ i ≤ n as follows: f_{i} will change a i − 1 to a i, or else send the tableau to 0. To determine which i − 1 changes, place a ”)” above each column that contains a i − 1 but no i, and a “(” above each column that contains a i but no i − 1. Cancel the brackets as usual. f_{i} changes the i − 1 corresponding to the right-most uncanceled “)” to a i, or, if there is no uncanceled “),” then f_{i}(b) = 0. As with multisegments, e_{i} is calculated with the same string of brackets, and changes the left-most uncanceled i to an i − 1.

The following is well known. It can be found in a slightly different form in e.g. [Kas2, §5].

Theorem 3.9. SSYT_{n}(λ) with the operators defined above is a realization of the 𝔰𝔩_{n+1}- crystal B(λ). □

0 1 2 3 1 2 3 2 3 ( ) 3 (2 )13 2 (2 1 )3 2 1 ( 0 1 2 2 1 2 3 2 3^{3213 22 12 1}\begin{matrix}ψ \\ ψ \\ e_{3} \\ e_{3} \end{matrix}

Figure 2. Moving from Young tableaux to multisegments and applying e_{3}.

3.3. Embeddings B(λ) ↪ B(∞). The following map is essentially the map from tableaux to Kostant partitions from [LS], although presented a bit differently (and, as a warning, the term “segment” is used differently there). The same map is also studied in [Zel3, §7], although the fact that it is a crystal morphism is not explicitly discussed there.

Definition 3.10. For a Young tableau b ∈ SSYT_{n}(λ) define the corresponding multisegment M_{b} to be the collection containing the segment [i, j] for each j ≥ i in the i^{th} row of b. See Figure 2.

Theorem 3.11. The map ψ : b ↦ M_{b} from SSYT_{n}(λ) to MS_{n} is a weak embedding of crystals.

Proof. We must show that, for all b ∈ SSYT_{n}(λ) and all i, M_{e_{i}(b)} = e_{i}M_{b}, where on the left e_{i} is calculated as in §3.2, and on the right as in §3.1. Let S_{i}^{Y T} (b) denote the string of brackets from §3.2 and S_{i}^{MS}(M_{b}) denote the string of brackets from §3.1. It is immediate from the definitions that these differ only in the following ways:

• S_{i}^{Y T} (b) may have some “)” corresponding to “i − 1” in row i, which are all at the left end of the string. These are not present in S_{i}^{MS}(M_{b}).

• S_{i}^{MS}(M_{b}) may have canceling pairs of brackets corresponding to pairs of segments of the same length, say [h, i] canceling [h − 1, i − 1]. It can happen that these correspond to an i directly above an i − 1 in b, in which case this pair is not present in S_{i}^{Y T} (b).

Neither of these changes affect the uncanceled “).” □

4. Crystal isomorphism from the PBW basis to multisegments

For this section, all PBW bases are with respect to the reduced expression

(4.1)  i = (1, 2, ⋯, n, 1, 2 ⋯ (n − 1) ⋯ 1, 2, 3, 1, 2, 1).

Recall that the corresponding order on positive roots is

β_{1} = α_{1}, β_{2} = α_{1} + α_{2}, …, β_{n} = α_{1} + ⋯ + α_{n}, β_{n+1} = α_{2}, …, β_{N} = α_{n}.

To each positive root β = α_{i} + ⋯ + α_{j} associate the segment [β] = [i, j].

4.1. The isomorphism.

Definition 4.1. Let Φ be the map from PBW bases to multisegments that takes F_{β_{1}}^{(a_{1})} ⋯ F_{β_{N}}^{(a_{N})} to the multisegment with a_{j} copies of each [β_{j}].

Theorem 4.2. Φ is an isomorphism of 𝔰𝔩_{n+1} crystals.

The proof of Theorem 4.2 will occupy the rest of this section. The idea is to use Proposition 2.12, which says T_{i}^{−1} ∘ τ acts on B(∞) as realized using PBW bases in the same way as (e_{i}^{∗})^{max}f_{i}^{N} for large N (here τ means set the first exponent to 0). We consider T_{n}^{−1}τ ⋯ T_{1}^{−1}τ acting on PBW monomials and the corresponding crystal operators acting on multisegments, and show that these agree as they must if Φ is to be an isomorphism (see Proposition 4.6 below). The image of this map is PBW monomials/multisegments for 𝔰𝔩_{n} ⊂ 𝔰𝔩_{n+1}, and we can then use induction on rank.

The next section is fairly technical. We encourage the reader to look at Proposition 4.6 and the example in Section §4.4 before continuing.

4.2. Some technical lemmas.

Definition 4.3. Given M ∈ MS_{n}, define a multisegment M^{(k)} and integer a_{k} for each 1 ≤ k ≤ n inductively by

• a_{1} = ε_{1}(M) + ε_{1}^{∗}(M) + ⟨wt(M), α_{1}^{∨}⟩,
• M^{(1)} = f_{1}^{a_{1}}(M),
• a_{2} = ε_{2}(M^{(1)}) + ε_{2}^{∗}(M^{(1)}) + ⟨wt(M^{(1)}), α_{2}^{∨}⟩,
• M^{(2)} = f_{2}^{a_{2}}f_{1}^{a_{1}}(M),

and so on.

Lemma 4.4. Fix M ∈ MS_{n} and i ≤ k ≤ n.

(i) M_{i−1,k−1}^{(k)} = M_{i,k} (for i ≥ 2) and

(ii) M_{i,k}^{(k)} ≥ max_{1≤s≤n−k} {∑_{r=1}^{s} M_{i+1,k+r} − ∑_{r=1}^{s−1} M_{i,k+r}} . That is, M_{i,k}^{(k)} is at least the

number of uncanceled “)” in the substring of S_{i}^{∗}(M) consisting of those brackets that correspond to segments of length at least the length of [i, k].

Proof. We proceed by induction on k. By Lemma 3.5, a_{1} = ur_{1}(M) + ur_{1}^{∗}(M). So f_{1} is applied to M until there are no uncanceled “)” in S_{1}, and then applied ur_{1}^{∗}(M) more times, creating 1 s and “(”s all the way to the left. This proves (ii) for k = 1, and (i) is vacuous in this case.

Now assume the result holds for some k and all i ≤ k. Look at S_{k+1}(M^{(k)}):

k+1 (. . . . . . k+1 (k ). . . . . . k ). . . . . . ⋮ ; i+1 ; k+1 (. . . . . . ⋮ ; i+1 ; k+1 (⋮ ; i ; k ). . . . . . ⋮ ; i ; k ). . . . . . ⋮ ; 2 ; k+1 (. . . . . . ⋮ ; 2 ; k+1 (⋮ ; 1 ; k ). . . . . . ⋮ ; 1 ; k )⋮ ; 1 ; k+1 (. . . . . . ⋮ ; 1 ; k+1 (.

By the induction hypothesis, M_{i,k}^{(k)} ≥ M_{i+1,k+1} for all i. Therefore all the “(” over [i + 1, k + 1] segments cancel “)” over [i, k] segments, and no other “)” are canceled. Again using Lemma 3.5, a_{k+1} = ur_{k+1}(M^{(k)}) + ur_{k+1}^{∗}(M^{(k)}), so applying f_{k+1}^{a_{k+1}} changes all but M_{i+1,k+1}^{(k)} of the [i, k] segments to [i, k+1] segments, establishing (i) for k+1. Furthermore, assuming i ≤ k, this creates exactly M_{i,k}^{(k)} −M_{i+1,k+1}^{(k)} many [i, k +1]’s. Adding the original number of these segments,

M_{i,k+1}^{(k+1)} = M_{i,k+1} + M_{i,k}^{(k)} − M_{i+1,k+1}

≥ max_{1≤s≤n−k} {∑_{r=1}^{s} M_{i+1,k+r} − ∑_{r=1}^{s−1} M_{i,k+r}} − M_{i+1,k+1} + M_{i,k+1} (4.2)

= max_{1≤s≤n−k} {∑_{r=2}^{s} M_{i+1,k+r} − ∑_{r=2}^{s−1} M_{i,k+r}} ,

where (4.2) holds by the induction hypothesis. After shifting indices this gives (ii) for k + 1. Statement (ii) for M_{k+1,k+1}^{(k+1)} holds as in the case k = 1. □

Corollary 4.5. M^{(n)} is the multisegment obtained from M by

(i) removing all [1, i] segments for each i, (ii) replacing each segment of the form [i + 1, k + 1] for i ≥ 1 by [i, k], and (iii) adding some number of [i, n] segments for each i.

Proof. By Lemma 4.4(i) the number of segments [i, k] for each i ≤ k < n is given by (i) and (ii), since applying f_{ℓ} for ℓ > k does not change the number of [i, k] segments. Certainly [i, n] segments can only be created, not destroyed, so (iii) holds as well. □

Proposition 4.6. Fix a multisegment M. Then σ_{n} … σ_{2}σ_{1}(M) is the multisegment obtained by

(i) Removing all [1, i] segments

(ii) Shifting all remaining segments down by 1; i.e., for every [i, j] in M, there is a [i − 1, j − 1] in σ_{n} … σ_{2}σ_{1}(M).

Proof. By Proposition 2.6(ii) and Definition 2.1(iii), for i ≠ j we have (e_{i}^{∗})^{max}f_{j}(M) = f_{j}(e_{i}^{∗})^{max}(M). Thus

σ_{n} … σ_{2}σ_{1}(M) = (e_{n}^{∗})^{max} … (e_{2}^{∗})^{max}(e_{1}^{∗})^{max}f_{n}^{a_{n}} … f_{2}^{a_{2}}f_{1}^{a_{1}}(M),

where a_{i} is as in Definition 4.3. By Corollary 4.5, f_{n}^{a_{n}} … f_{2}^{a_{2}}f_{1}^{a_{1}}(M) is correct except that there may be extra segments [i, n] for each i. It remains to show that

(e_{n}^{∗})^{max} … (e_{2}^{∗})^{max}(e_{1}^{∗})^{max}

just deletes all the [i, n].

We proceed by induction on i, proving that each (e_{i}^{∗})^{max} changes all [i, n] segments into [i + 1, n] segments and does nothing else. For i = 1, we have

S_{1}^{∗}(M^{(n)}) : 1 (. . . . . . 1 (2 ). . . . . . 2 ). . . . . . ⋮ ; 1 ; j (. . . . . . ⋮ ; 1 ; j (⋮ ; 2 ; j+1 ). . . . . . ⋮ ; 2 ; j+1 ). . . . . . ⋮ ; 1 ; n (. . . . . . ⋮ ; 1 ; n (

By Corollary 4.5, the number of [1, j]’s in M^{(n)} is equal to the number of [2, j + 1]’s in M. Thus the “)” in S_{1}^{∗}(M) would be enough to cancel all the [1, j] in S_{1}^{∗}(M^{(n)}). Application of the various f_{k}^{a_{k}} can only have made these [2, j + 1] segments longer, and hence moved these “)” to the right, but not past the “(” coming from any [1, n] segment. It follows that the “(” corresponding to intervals [1, j] for j < n are in fact all canceled in S_{1}^{∗}(M^{(n)}), so these segments are not affected by (e_{1}^{∗})^{max}. Certainly the [1, n] intervals are all changed to [2, n] by (e_{1}^{∗})^{max}, so the i = 1 case has been established.

Now fix k ≥ 2 and assume that the claim holds for all i < k. By the induction hypothesis, the difference between S_{k}^{∗}((e_{k−1}^{∗})^{max} … (e_{1}^{∗})^{max}M^{(n)}) and S_{k}^{∗}(M^{(n)}) is only that the first string has some extra “(” corresponding to [k, n] segments. By the same argument as the base case, the uncanceled “(” in S_{k}^{∗}(M^{(n)}) are precisely those below [k, n] segments, so this remains true for S_{k}^{∗}((e_{k−1}^{∗})^{max} … (e_{1}^{∗})^{max}M^{(n)}). Hence (e_{k}^{∗})^{max} just changes all [k, n] segments to [k + 1, n]’s.

At the final step, e_{n}^{∗} clearly deletes all the [n, n] segments. □

4.3. Proof of Theorem 4.2. For each n let P BW_{n} denote the crystal of PBW monomials corresponding to the reduced expression (4.1). P BW_{n} ≃ B(∞) ≃ MS_{n} and B(∞) is connected, so there is a unique crystal isomorphism ϕ_{n} : P BW_{n} → MS_{n} for each n. We need to show that ϕ_{n} = Φ_{n}. We proceed by induction on rank, the 𝔰𝔩_{2} case being trivial. So, fix n ≥ 2 and assume ϕ_{n−1} = Φ_{n−1}.

Let ϕ_{n}(F_{1}^{(a_{1})}F_{α_{1}+α_{2}}^{(a_{2})} ⋯ F_{n}^{(a_{N})}) = { 1 _{c_{1}} ⋯ n _{c_{N}} } . We need to show that c_{i} = a_{i} for all

i. By Proposition 2.12,

(4.3)  ϕ_{n}(T_{n}^{−1}τ ⋯ T_{1}^{−1}τ(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})})) = (σ_{n} … σ_{1} ∘ ϕ_{n})(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})}),

where on the right side σ_{i} is calculated as (e_{i}^{∗})^{max}f_{i}^{N} for large N.

There is a natural copy of P BW_{n−1} ⊂ P BW_{n}; the monomials where the exponent of F_{α_{j}+···+α_{n}} is zero for all j. The image of ϕ_{n}|_{P BW_{n−1}} consists of exactly those multisegments with no segments of the form [i, n] for any i, which is naturally identified with MS_{n−1}. Certainly ϕ_{n}|_{P BW_{n−1}} is still a crystal isomorphism, so by induction is equal to ϕ_{n−1}. Thus the left side of (4.3) is:

ϕ_{n}(T_{n}^{−1}τ ⋯ T_{1}^{−1}τ(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})})) = ϕ_{n}(F_{1}^{(a_{n+1)}} ⋯ F_{n−1}^{(a_{N})}) by Lemma 2.13
= ϕ_{n−1}(F_{1}^{(a_{n+1})} ⋯ F_{n−1}^{(a_{N})})
= {1 _{a_{n+1}} ⋯ n−1 _{a_{N}} } by the induction hypothesis.

On the right side we have:

σ_{n} … σ_{1}(ϕ(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})})) = σ_{n} … σ_{1} ({ 1 _{c_{1}} ⋯ n _{c_{N}} })
= {1 _{c_{n+1}} ⋯ n−1 _{c_{N}} } by Proposition 4.6.

Hence c_{i} = a_{i} for i > n. Also

wt(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})}) = wt(ϕ(F_{1}^{(a_{1})} ⋯ F_{n}^{(a_{N})}))

implies

−(a_{1}β_{1} + ⋯ + a_{N}β_{N}) = −(c_{1}β_{1} + ⋯ + c_{n}β_{n} + a_{n+1}β_{n+1} + ⋯ + a_{N}β_{N}),

from which it follows that

a_{1}β_{1} + ⋯ + a_{n}β_{n} = c_{1}β_{1} + ⋯ + c_{n}β_{n},

since a_{i} = c_{i} for all other i. But β_{1} = α_{1}, β_{2} = α_{1} + α_{2}, …, β_{n} = α_{1} + ⋯ + α_{n} are linearly independent, so it follows that c_{i} = a_{i} for i ≤ n as well. □

4.4. An example. The main difficulty in proving Theorem 4.2 is establishing Proposition 4.6. As discussed in the proof of that proposition,

(4.4)  σ_{n} ⋯ σ_{1}M = (e_{n}^{∗})^{max} … (e_{2}^{∗})^{max}(e_{1}^{∗})^{max}f_{n}^{a_{n}} … f_{2}^{a_{2}}f_{1}^{a_{1}}(M).

Here we go through the reasons why the right hand side of (4.4) has the desired effect on

M = 1 2 2 3 2 1 2 1 3 2 4 3 4 3 4 3 2 5 4 3 2 .

The strings of brackets for i = 1 are:

S_{1}(M) : 3221 (4 34 33 22 12 14 3 25 4 3 2  S_{1}^{∗}(M) : 1 (2 )2 )32 1 (2 1 (3 2 )4 34 34 3 2 )5 4 3 2 ) .

By counting uncanceled “(” we see that ε_{1}(M) = 1 and ε_{1}^{∗}(M) = 0. We can also calculate ⟨wt(M), α_{1}^{∨}⟩ = 1, giving a_{1} = 2. As in Lemma 3.5, this is the number of uncanceled “)” in S_{1}(M) plus the number of uncanceled “)” in S_{1}^{∗}(M). Applying f_{1}^{2} creates two new 1 . So at the next step we get:

S_{2}(M^{(1)}): 32 (2 (1 )1 )1 )4 34 33 22 1 (2 1 (4 3 25 4 3 2  S_{2}^{∗}(M^{(1)}): 1112 (2 (3 )2 12 13 2 (4 3 )4 3 )4 3 2 (5 4 3 2 ( .

We see that a_{2} = 1, and f_{2}^{a_{2}} just changes the uncanceled 1 to a 2 1 . Continuing we find a_{3} = 3, a_{4} = 2, and a_{5} = 4. Application of the operators gives

M^{(5)} = 2113 23 22 15 4 35 4 35 4 33 2 14 3 2 15 4 3 25 4 3 2 .

Notice that the segments in M^{(5)} that do not contain 5 correspond exactly to the segments of M that do not contain 1, but shifted down by one. This is the content of Corollary 4.5.

Next we must apply the (e_{i}^{∗})^{max}. At the first step we have:

S_{1}^{∗}(M^{(5)}): 1 (1 (2 )2 1 (3 2 )3 2 )3 2 1 (5 4 35 4 35 4 34 3 2 1 (5 4 3 2 )5 4 3 2 ) .

There are no uncanceled “(” so ε_{1}^{∗} = 0 and (e_{1}^{∗})^{max} does nothing. Next,

S_{2}^{∗}((e_{1}^{∗})^{0}M^{(5)}): 112 (2 13 2 (3 2 (3 2 15 4 3 )5 4 3 )5 4 3 )4 3 2 15 4 3 2 (5 4 3 2 ( .

So e_{2}^{∗} is applied twice, deleting the 2s at the bottom of the right-most segments. Notice that all “(” corresponding to segments that do not contain a 5 are canceled; as discussed in the proof of Proposition 4.6 this will always be the case, so no segments are changed except those containing 5s. Certainly segments containing 5s always do change, and at the final step are deleted. We end up with

(e_{5}^{∗})^{max}(e_{4}^{∗})^{max}(e_{3}^{∗})^{max}(e_{2}^{∗})^{max}(e_{1}^{∗})^{max}f_{5}^{a_{5}}f_{4}^{a_{4}}f_{3}^{a_{3}}f_{2}^{a_{2}}f_{1}^{a_{1}}(M) = 1122 13 23 23 2 14 3 2 1 ,

as predicted by Proposition 4.6.

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Department of Mathematics and Statistics, Loyola University, Chicago, IL

  E-mail address: johnclaxton56@gmail.com

Department of Mathematics and Statistics, Loyola University, Chicago, IL

  E-mail address: ptingley@luc.edu