Noncommutative LR coefficients and crystal reflection operators

We relate noncommutative Littlewood-Richardson coefficients of Bessenrodt-Luoto-van Willigenburg to classical Littlewood-Richardson coefficients via crystal reflection operators. A key role is played by the combinatorics of frank words.


Introduction
Quasisymmetric Schur functions, introduced by Haglund-Luoto-Mason-van Willigenburg [14], form a prominent basis for the Hopf algebra of quasisymmetric functions QSym. The quasisymmetric Schur function indexed by the composition α, denoted by S α , is obtained by summing monomials attached to semistandard composition tableaux of shape α. This is reminiscent of the definition of Schur functions as sums of monomials corresponding to semistandard Young tableaux. As the name suggests, quasisymmetric Schur functions share many properties with classical Schur functions, and Mason's map ρ [24] connects the combinatorics of composition tableaux to that of Young tableaux. Understanding analogues of Schur functions and their generalizations has long been a theme in algebraic combinatorics; see [1,2,3,4,25,32] for recent work in this context.
The Hopf algebra of noncommutative symmetric functions NSym, introduced in the seminal paper [12], is Hopf-dual to QSym as shown by Malvenuto-Reutenauer [23]. Bessenrodt-Luoto-van Willigenburg [5] studied the dual basis elements s α corresponding to quasisymmetric Schur functions. The resulting functions, also indexed by compositions, are called noncommutative Schur functions. The inclusion of the Hopf algebra of symmetric functions Sym into QSym induces a projection χ : NSym Sym. This projection maps noncommutative Schur functions to classical Schur functions, and justifies the name of the former. The structure constants C γ αβ that arise in s α · s β = γ C γ αβ s γ (1) are called noncommutative Littlewood-Richardson (LR) coefficients. These coefficients turn out to be nonnegative integers that refine the classical LR coefficients c λ νµ that arise in the product of Schur functions More precisely, suppose ν and µ are partitions that rearrange to compositions α and β respectively. Applying χ to both sides in (1) and comparing the result with (2) implies (3) where the sum on the right runs over all compositions γ that rearrange to λ. Among the numerous combinatorial interpretations of c λ νµ , the one we focus on states that c λ νµ counts LR tableaux of shape λ/µ and content ν. Our primary goal in this article is to describe the summands on the right hand side in (3) in terms of LR tableaux. To this end, crystal reflection operators are key.
To state our result, we introduce the necessary notation briefly. The reader is referred to Section 2 for details. Given a composition α, we denote the partition underlying α by sort(α). Let LRT(λ, µ, ν) be the set of LR tableaux of shape λ/µ and content ν. Given a permutation σ, let LRT σ (λ, µ, ν) be the set of tableaux obtained by applying the crystal reflection operators corresponding to a reduced word of σ. Let α := σ · ν, and let β be any composition that satisfies sort(β) = µ. On applying the map ρ −1 β (which sends Young tableaux to composition tableaux, [22,Chapter 4]) to elements in LRT σ (λ, µ, ν), we obtain the disjoint decomposition where X γ αβ consists of all tableaux T ∈ LRT σ (λ, µ, ν) whose outer shape under ρ −1 β is given by the composition γ . Under this setup, our main theorem states the following.
Theorem 1.1. The noncommutative LR coefficient C γ αβ equals the cardinality of X γ αβ . The upshot of Theorem 1.1 is that starting from LRT(λ, µ, ν), we can compute all noncommutative LR coefficients C γ αβ , where α dictates the choice of crystal reflections to be performed and β determines the generalized ρ map to be applied. We also interpret Theorem 1.1 in terms of chains in Young's lattice indexed by certain frank words and obtain a rule for C γ αβ involving box-adding operators on compositions. In the case where α is either a partition or reverse partition, our interpretations yield the two LR rules in [6]. For an example demonstrating Theorem 1.1, see Subsection 3.2.
Note that LR coefficients can be interpreted as the number of lattice points in an appropriate convex polytope/cone [26]. Since Theorem 1.1 describes noncommutative LR coefficients by refining the set of LR tableaux, it is natural to ask if noncommutative LR coefficients count lattice points in some refinement of the corresponding polytope/cone. Data suggest that the analogue of the saturation conjecture, famously resolved by Knutson and Tao [15], holds for noncommutative LR coefficients as well. In fact (3) tells us that saturation of noncommutative LR coefficients implies that of the classical counterparts. This motivates studying these constants from a polytopal perspective, which is why the authors were interested in a statement like Theorem 1.1 in the first place.
Outline of the article. Section 2 sets up all the necessary combinatorial background. Section 3 describes our central result along with examples. Section 4.1 introduces growth words and frank words. Section 4.2 presents an outline of our proof strategy given its technical nature. Section 4.3 identifies LR tableaux with certain Algebraic Combinatorics, Vol. 4 #1 (2021) distinguished frank words called compatible words, drawing upon work by Remmel-Shimozono [29]. In Section 4.4, we describe the Lascoux-Schützenberger symmetric group action on frank words. Section 4.5 relates crystal reflections acting on LR tableaux to the aforementioned action restricted to compatible words. Section 5 relates the results in Section 4 back to noncommutative LR coefficients. We conclude our article with Corollary 5.4, which reinterprets our combinatorial interpretation for noncommutative LR coefficients in terms of box-adding operators on compositions.

Background
To keep this section brief, we assume knowledge of combinatorial structures and algorithms arising in symmetric functions theory such as partitions, skew shapes, Young tableaux, Robinson-Schensted-Knuth insertion and jeu-de-taquin. The reader is referred to standard texts [11,30,33] for further information.
Regarding notions relevant to the theory of quasisymmetric Schur functions, we adhere to the notation and conventions in [22]. We emphasize that what we call quasisymmetric Schur functions in this article are the Young quasisymmetric Schur functions in [22]. This choice, although cosmetic as both the functions are related via a simple involution, has its benefits. The tableau objects that we consider in the quasisymmetric/noncommutative setting align with the more prevalent notion of Young tableaux rather than reverse tableaux. Furthermore, it is the Young quasisymmetric Schurs that decompose the dual immaculate quasisymmetric functions [4] positively as shown recently in [2], thereby connecting two well-studied quasisymmetric analogues of Schurs. Consider w = w 1 · · · w n ∈ Z * + . We call n the length of w and denote it by |w|. The word w n · · · w 1 is denoted by w r . We say that w is reverse-lattice if every suffix of w contains at least as many i's as i + 1's for all i ∈ Z + . Let S n be the permutation group on [n] and denote the longest element in S n by w (n) 0 , or simply by w 0 if n is clear from context. The permutation σ ∈ S n acts on a sequence λ = (λ 1 , . . . , λ n ) by σ · λ = (λ σ −1 (1) , . . . , λ σ −1 (n) ). Note that this is a right action.

Compositions and partitions.
A finite list of nonnegative integers α = (α 1 , . . . , α ) is called a weak composition. If α i > 0 for all 1 i , then α is called a composition. If, in addition, we have α 1 · · · α > 0, then α is called a partition. Given α = (α 1 , . . . , α ) we call the α i the parts of α and the sum of the α i , denoted by |α|, is called the size of α. We denote the number of parts of α by (α) and call it the length of α. The unique composition of length and size zero is denoted by ∅. We denote the composition (α , . . . , α 1 ) by α r . Also, denote the transpose of a partition λ by λ t . The reader is warned that on occasion we will suppress commas and parentheses when writing our compositions.
The composition diagram of α = (α 1 , . . . , α ) is the left-justified array of boxes with α i boxes in row i from the bottom. If α is a partition, then the composition diagram of α coincides with the Young diagram of α in French notation. See Figure 1 for the composition diagram of (2, 1, 3).

Young tableaux.
A Young tableau T (henceforth YT) of skew shape λ/µ is a filling of the boxes of λ/µ with positive integers so that entries along the rows increase weakly read from left to right and entries along the columns increase strictly read from bottom to top. If the entries in T are all distinct and belong to [|λ/µ|], then we call T a standard Young tableau (henceforth SYT.) We denote the set of YTs of shape λ/µ by YT(λ/µ).
If T is a YT with maximal entry m, then the content of T , denoted by cont(T ), is the weak composition (α 1 , . . . , α m ) where α i for 1 i m counts the instances of i in T . The standardization of T , denoted by stan(T ), is obtained by replacing the α i entries in T equal to i by the integers 1 +   • α = (β 1 , . . . , β m , 1).
(1) The entries in each row increase weakly from left to right.

2.5.
Mason's ρ map. Next we discuss a crucial map that bridges the combinatorics of CTs and YTs. Let CT(−/ /β) denote the set of all CTs with inner shape β and YT(−/ β) denote the set of all YTs with inner shape sort(β). Then the map ρ β : CT(−/ /β) → YT(−/sort(β)), which generalizes the map for semistandard skyline fillings [24] and is introduced in [22, Chapter 4], is defined as follows. Given τ ∈ CT(−/ /β), obtain ρ β (τ ) by writing the entries in each column in increasing order from bottom to top and bottom-justifying these new columns on the inner shape sort(β), which might be empty.
The inverse map ρ −1 (1) take the set of i entries in the leftmost column of T and write them in increasing order in rows (β) + 1, 2, . . . , (β) + i above the inner shape β in the first column to form the leftmost column of τ , (2) take the set of entries in column 2 in increasing order and place them in the row with the largest index so that either • the box to the immediate left of the number being placed is filled and the row entries weakly increase when read from left to right, or • the box to the immediate left of the number being placed belongs to the inner shape, (3) repeat the previous step with the set of entries in column k for k = 3, . . . , m where m is the largest part of sort(β). In the case β = ∅, the map ρ β is Mason's shift map [24] (or ρ map), and we set ρ := ρ ∅ . One may verify that the YT on the left in Figure 2 maps to the CT in Figure 4 under ρ −1 (2,4) . 2.6. Reading words, jdt-equivalence and rectification. Given a word w = w 1 · · · w n , the Robinson-Schensted correspondence (via row insertion or column insertion) associates an ordered pair (P(w), Q(w)) of YTs of the same shape. We call P(w) and Q(w) the insertion tableau and recording tableau respectively. We call two words w 1 and w 2 Knuth-equivalent if P(w 1 ) = P(w 2 ). Intimately related to the notion of Knuth-equivalence is the notion of jdt-equivalence. For this we need to introduce reading words.
Definition 2.2 (Reading word). The reading word of T ∈ YT(λ/µ) (respectively τ ∈ CT(α/ /β)), denoted by rw(T ) (respectively rw(τ )), is obtained by reading the entries of T (respectively τ ) in every column in decreasing order, starting from the leftmost column and going to the right.
We declare YTs T 1 and T 2 to be jdt-equivalent if their reading words are Knuthequivalent, that is, P(rw(T 1 )) = P(rw(T 2 )). In view of Mason's map ρ β , all combinatorial notions discussed in the context of Young tableaux are inherited by composition tableaux. We focus on the notion of rectification given its importance in our context.
is a partition and rw(T ) is reverse lattice. The set of LR tableaux of shape λ/µ and content ν is denoted by LRT(λ, µ, ν).
To describe the noncommutative LR rule, we need noncommutative analogues of Schur functions or, equivalently, quasisymmetric analogues of skew Schur functions. Following [22, Proposition 5.2.6], we define the skew quasisymmetric Schur function indexed by α/ /β to be Here The noncommutative Schur functions are defined indirectly [22, Definition 5.6.1] as elements of the basis in NSym dual to the basis of quasisymmetric Schur functions in QSym. We now proceed to describe the LR rule for noncommutative Schur functions, equivalent to [22,Theorem 5.6.2].
Given a composition α = (α 1 , . . . , α k ), the canonical composition tableau τ α is constructed by filling the boxes in the i-th row of the composition diagram of α with consecutive positive integers from 1 + i−1 j=1 α j to i j=1 α j from left to right, for 1 i k. Figure 6 shows the canonical CT of shape (4, 2, 3, 1) on the right. where C γ αβ is the number of SCTs of shape γ/ /β that rectify to τ α . For the proof of the theorem in its original context, see [5,Theorem 3.5]. As mentioned in the introduction, one motivation for this article is to develop a polytopal perspective on noncommutative LR coefficients. Toward this end, relating them explicitly with LR tableaux is a worthwhile endeavour as the latter's description allows for a straightforward translation to linear inequalities [7,15,26].
2.9. Crystal reflection operators and LR tableaux. For in-depth exposition on crystal bases and their relevance in algebraic combinatorics and representation theory, we refer the reader to [8]. We proceed to describe crystal reflection operators.
Given a positive integer i, we define the crystal reflection operator s i acting on the set of Young tableaux as follows.
(1) Let T ∈ YT(λ/µ), and let w = w 1 . . . w n := rw(T ). Scan w from left to right and pair each i + 1 with the closest unpaired i that follows. (2) If no further pairing is possible, then change all unpaired i's to i + 1's or vice versa depending on whether the number of i's is greater than the number of i + 1's or not. Say the new word obtained via this procedure is w . (3) Define s i (T ) to be the unique YT of shape λ/µ such that rw(s i (T )) = w . Lascoux-Schützenberger [17] (see also [18,Section 3] and [28,Proposition 9]) proved that the operators s i give a well-defined action of the (infinite) symmetric group on YT(λ/µ) by verifying the Coxeter-Moore relations. In particular, we obtain σ(T ) by computing s i1 · · · s i k (T ) for any reduced word s i1 · · · s i k for σ.
The crucial insight behind our work is that even though LRT σ (λ, µ, ν) sheds no further light on classical LR coefficients, it "knows" about noncommutative LR coefficients. To motivate our upcoming results and to establish a connection with Theorem 2.5, we invite the reader to check that for any tableau T in Figure 7, we have that ρ −1 (rect(T )) is the distinguished CT in Figure 8 with the defining property that it is the unique CT with shape and content both equaling (1,4,3). Equivalently, we could say that ρ −1 (rect(stan(T ))) is the canonical CT of shape (1,4,3).
In Section 5, we will establish this property in general by exploiting the interplay between frank words and jeu-de-taquin. This will allow us to establish the veracity of our main theorem.

Main result and related discussion
Our main result provides a combinatorial description for C γ αβ using crystal reflection operators. We state it next and give a proof assuming results from Sections 4 and 5.
In contrast, Theorem 3.1 emphasizes the curious aspect that computing noncommutative LR coefficients is best done by passing through the commutative realm. Indeed, although computing classical LR coefficients is known to be #P-hard, we can appeal to one of many descriptions that they possess to compute them in a deterministic manner; see [7,15,20,37]. Once the tableaux in LRT(λ, µ, ν) are computed, it remains to apply an appropriate sequence of crystal reflections (determined by α) followed by an appropriate ρ β to determine all noncommutative LR coefficients C γ αβ . On a related noted, Bessenrodt-Tewari-van Willigenburg [6], in classifying symmetric skew quasisymmetric Schur functions, established Yamanouchi-type rulescalled left and right LR rules therein -for the C γ αβ in the special cases where α is either a partition or a reverse partition, although the proof therein was not uniform.

Algebraic Combinatorics, Vol. 4 #1 (2021)
It transpires that these rules are in fact two extremes of our Theorem 3.1. Indeed by picking σ to either be the identity or the longest word in S (ν) , we obtain the rules of [6]; see discussion at the end of Section 5 for details.

3.2.
Illustrating our main result. We discuss an example next to illustrate Theorem 3.1. Let λ = 8642, µ = 4321 and ν = 532. Figure 9 shows tableaux in LRT(λ, µ, ν), implying in particular that c 8642 532,4321 = 4. Example 3.2. Suppose we pick σ ∈ S 3 to be the identity permutation and β = µ. Thus we do not need to apply crystal reflections. By applying ρ −1 4321 to the LR tableaux in Figure 9 we obtain the CTs in Figure 10. By grouping these CTs according to outer shape, we infer that  Example 3.3. Sticking with the same σ as before, we now pick β = 3412. The CTs resulting from applying ρ −1 3412 are shown in Figure 11. Grouping them according to outer shape reveals a different decomposition of c 8642 532,4321 . More precisely, we obtain C 6824 532,3412 = 3, and C 4826 532,3412 = 1.

Frank words and LR tableaux
In order to prove Theorem 3.1, we need to understand the rectification of tableaux in LRT σ (λ, µ, ν) followed by an application of the generalized ρ map. To this end, it helps to explore the relation between growth words of these tableaux and frank words.

Growth words and frank words. Definition 4.1 (Growth word). Given an SYT T , its growth word gw(T ) is obtained by reading entries from largest to smallest and recording the columns they belong to.
We extend the definition of growth word to all YTs by setting gw(T ) := gw(stan(T )).
The reader may verify that for both tableaux in Figure 2, the growth word is 76564321531. Frank words were introduced by Lascoux-Schützenberger [18] in their investigation of key polynomials. Subsequently, Reiner and Shimozono [27] studied the combinatorics of frank words in depth in the context of a flagged LR rule, and we follow their exposition as far as notions in this section are concerned.
Definition 4.2 (Frank word). We say that w is frank if P(w) is of shape λ t where λ = sort(colform(w)). Given a composition α, we define Frank(α) to be the set of frank words w satisfying colform(w) = α.
A remarkable feature of frank words is that one knows the shape of the P-tableau without performing jeu-de-taquin. As an example of a frank word, consider w = 432 32 6531 . It is a 3-column word with colform(w) = (3,2,4). For clarity, we put frames around maximal column words. Figure 14 depicts P(w). Note that the shape underlying it is (4, 3, 2) t . Therefore, w ∈ Frank ((3, 2, 4)).

4.2.
Outline of proof. Given the technical nature of our argument, we give a brief overview of our strategy toward establishing Theorem 3.1.
(1) Given a skew shape λ/µ and a composition α |λ| − |µ|, we consider the distinguished subset of Frank(α) comprising λ/µ-compatible frank words. (2) The main result of this section, Proposition 4.6, identifies these words as precisely the growth words of tableaux in LRT σ (λ, µ, ν) where ν satisfies sort(α) = ν. To arrive at this result, we study a symmetric group action on frank words and establish that it really is the crystal reflection action in disguise. This is the content of Subsections 4.4 and 4.5. (3) Lemma 5.1 then states that the rectification of a Young tableau may be described in terms of its growth word using Schützenberger's evacuation. (4) Finally, Proposition 5.3 establishes that applying ρ −1 to rect(stan(T )) for T an element in LRT σ (λ, µ, ν) does indeed give the canonical tableau of shape σ · ν. From this, Theorem 3.1 follows.

λ/µ-compatible frank words and LR tableaux.
Definition 4.3. Given partitions λ, µ such that µ ⊆ λ and a composition α, we say that w ∈ Frank(α) is λ/µ-compatible if for every suffix w of w, we have that cont(w ) + µ t is a partition and that cont(w)

Definition 4.3 may be interpreted as saying that w ∈ Frank(α) is λ/µ-compatible if
it is the growth word of a Young tableau of shape λ/µ. Observe that we must have α |λ| − |µ|.
As an example, note that w = 621 76432 ∈ Frank(α) is λ/µ-compatible for λ = 76422, µ = 5521, and α = 35. The reader can easily verify that w is the growth word of the tableau in Figure 15. Note further that this tableau is an LR tableau.
There is an intimate link between LR tableaux and compatible words. To understand this link we go via an intermediate tableau object.
Given w ∈ Frank(λ, µ, α), let w (1) · · · w (m) be its maximal column word factorization where m := (α). Construct a Young tableau φ(w) of shape λ/µ and content α r as follows: Let λ (0) := λ and inductively define λ (i) for 1 i m to be such that λ (i−1) /λ (i) is a horizontal strip with boxes in columns given by letters appearing in w (i) . Subsequently, fill the boxes of the horizontal strips λ (i−1) /λ (i) with m + 1 − i to obtain φ(w). Note that λ (m) is µ and that φ(w) does indeed belong to YT(λ/µ). Figure 15 gives an example.  Proof. Let w ∈ Frank(λ, µ, ν r ). As colform(w) = ν r and w is frank, we know that w is the reading word of a tableau of skew shape rotate(ν t ), which is obtained from the shape ν by a 180 • rotation. It follows that φ(w) ∈ LRT(λ, µ, ν). This establishes FrankTab(λ, µ, ν r ) ⊆ LRT (λ, µ, ν). The opposite inclusion follows since the reading word of an LR tableau is reverselattice. In particular, the growth word of any T ∈ LRT(λ, µ, ν) is the reading word of a YT of skew shape rotate(ν t ). Figure 16 depicts growth words of LR tableaux in Figure 5 as reading words of YTs of skew shape rotate(ν t ). Here λ = 76432, µ = 644 and ν = 431. Now that we have established elements of FrankTab(λ, µ, ν r ) as LR tableaux in the usual sense, it remains to understand the case where ν r is replaced by an arbitrary composition.

4.4.
A symmetric group action on frank words. We proceed to describe a symmetric group action on frank words that we subsequently connect to the symmetric group action on Young tableaux via crystal reflections described earlier. This new action is best understood by focusing on 2-column frank words, very much in the manner that crystal reflections are understood by acting on consecutive letters.
If the former holds, define ι(w) to be the reading word of the unique tableau T of shape (β 1 , β 1 ) t /(β 1 − β 2 ) t that is jdt-equivalent to T (obtained by performing jeu-detaquin slides within the rectangle (β 1 , β 1 ) t ). If the latter holds, define ι(w) to be the reading word of the unique tableau T of shape (β 2 , β 1 ) t that is jdt-equivalent to T .
Clearly, ι is an involution on A. Equally importantly, w and ι(w) are Knuthequivalent. For instance, jdt-equivalence of tableaux in Figure 17 implies that ι( 76421 632 ) = 621 76432 . We employ the involution ι to construct the desired symmetric group action. Let λ be a partition and let m := (λ). Define Following [18], define an action of S m on F λ by describing the action of the generator s i for 1 i m − 1 as follows: Let w (1) · · · w (m) be the maximal col- (m) . Observe that sort(colform(v)) = λ. As v and w are Knuth-equivalent, we infer that v ∈ F λ . We define v to be s i (w). This given, we may define σ(w) for any σ ∈ S m by picking a reduced word for σ. Figure 18 depicts the S 3 -orbit of the frank word 4321 632 53 ∈ F 432 . We have chosen to represent the action on words via their tableau representatives. In particular, each column when read from top to bottom gives a maximal column word in the maximal column word factorization. Each s i performs a "local" jeu-de-taquin on two columns.

4.5.
Relating the two symmetric group actions. Our next lemma connects the action of crystal reflection operators on FrankTab(λ, µ, α) to the symmetric group action on λ/µ-compatible frank words in Frank(λ, µ, α).
Assume p < q. Let w (1) = a 1 . . . a p and w (2) = b 1 . . . b q . We have a 1 > · · · > a p and b 1 > · · · > b q . As w is frank, it is the reading word of a Young tableau of skew shape rotate((q, p) t ). Thus, a i b i for 1 i p.
Instead of computing ι(w) by rectifying the appropriate two-columned tableau, one may perform successive Schensted column insertions of the numbers a p down to a 1 starting from the single-columned tableau with column word w (2) . See Figure 19 for an example. Compare with Figure 17 which established the same fact using jeu-de-taquin.
In each intermediate step of this column insertion procedure, the number a i being inserted into the current tableau bumps a distinct element from {b 1 , . . . , b q }. Furthermore this bumped entry is guaranteed to be strictly greater than the entries in the second column in the current tableau. Therefore, the insertion tableau is completely determined by the entries that get bumped. More precisely, for i from p down to 1, define  the integer m(i) recursively as follows. We define m(p) to be the largest integer j such that a p b j . Subsequently, for i = p − 1, . . . , 1, define m(i) to be the largest integer j such that j < m(i + 1) and a i b j . Observe that in our Schensted column-insertion procedure, the entry a i bumps b m(i) . Therefore, the set of entries that get bumped is {b m(i) | 1 i p}. For the example in Figure 19, we have m(3) = 5, m(2) = 4 and m(1) = 2. Therefore the set of entries that get bumped is Consider rw(φ(w)) = u 1 · · · u n . The word v := v 1 · · · v n obtained by recording the column to which each u i belongs gives us the weakly increasing arrangement of letters in w. Furthermore, for 1 i p (respectively 1 i q) the letter in v corresponding to the ith 2 (respectively 1) from the left in rw(φ(w)) is equal to a i (respectively b i ). Recall that the crystal reflection operator s 1 acting on φ(w) begins by pairing each 2 in rw(φ(w)) to the closest unpaired 1 to its right. Equivalently, in our current context, a 2 corresponding to a i for some 1 i p gets paired with the 1 in rw(φ(w)) corresponding to b m(i) . We infer that the unpaired 1s in rw(φ(w)) correspond to those b j that are not bumped. These are precisely the b j that determine which 1s in φ(w) turn into 2s in computing s 1 (φ(w)). Thus we infer that s 1 (φ(w)) = φ(ι(w)). This establishes the claim in the case p < q. The case p q is similar and left to the reader.
To illustrate the ideas in the preceding proof, Figure 20 depicts the action of s 1 on the tableau φ(w) from Figure 15, where w = 621 76432 . From Figure 19, we see that the entries that do not get bumped are {4, 7}. Also, note that rw(φ(w)) = 22111211 where the unpaired 1s are highlighted. In terms of the tableau φ(w), we see that the unpaired 1s belong to columns 4 and 7. The tableau on the right in Figure 20 is easily verified to be φ(ι(w)) as ι(w) = 76421 632 . We are ready to give a precise relation between LRT σ (λ, µ, ν) and λ/µ-compatible frank words with a certain column form that generalizes Lemma 4.4.

Noncommutative LR coefficients and frank words
Now that we understand tableaux in LRT σ (λ, µ, ν) as certain λ/µ-compatible words with a prescribed column form, we are ready to establish the connection to noncommutative LR coefficients. Recall that our goal is to show that rect(stan(T )) for T ∈ LRT σ (λ, µ, ν) is equal to the canonical CT of shape σ · ν. We recast rect(stan(T )) in terms of the growth word of T .
Proof. We sketch the proof and follow the exposition in [21]. Let T = stan(T ) and suppose that T has n boxes. Consider the biword u v where u := u 1 · · · u n is the longest word in S n and v := v 1 · · · v n is obtained by recording the column in T to which u i belongs. In other words, v = gw(T ) = gw(T ). Consider biwords u v and u v , where u (respectively v ) in the weakly increasing rearrangement of u (respectively v) and v (respectively u ) is the rearrangement induced by the aforementioned sorting. Then we have that v = gw(T ) r and u = rw(T ).
Note that rect(stan(T )) = rect(T ) = P(rw(T )) = P(u For the leftmost tableau T in Figure 22, its standardization rectifies to the tableau in the middle. We have gw(T ) = 321 7621 5 , and Q(gw(T )) is shown on the right. We invite the reader to verify that evac(Q(gw(T ))) is indeed the tableau in the middle upon transposing.
We now appeal to a simple defining characterization of canonical composition tableaux that will be useful. Recall that the descent set of an SYT T with n boxes  Figure 22. A demonstration of the claim rect(stan(T )) = evac(Q(gw(T ))) t .
Proof. Observe that the first column of T when read from bottom to top is forced to be 1, 1 + α 1 , 1 + α 1 + α 2 , . . . , 1 + (α)−1 j=1 α j . The first claim follows from this observation, and our second claim follows from applying the map ρ to τ α . We omit the details.
We finally arrive at the key proposition that is utilized in the proof of our central result Theorem 3.1.
Note that the descent set of gw(T ) is {n − i | i ∈ [n − 1] set(α)}, which therefore is also the descent set of Q(gw(T )). It follows that the descent set of evac(Q(gw(T ))) is [n − 1] set(α), which in turn implies that the descent set of rect(stan(T )) = evac(Q(gw(T ))) t is set(α). Thus we have established that rect(stan(T )) has shape sort(α) and descent composition α. Lemma 5.2 proves the proposition.
This also completes the proof of our main theorem. A remarkable aspect of Proposition 5.3 is that the symmetric group action on LR tableaux via crystal reflection operators translates to the usual permutation action on the parts of the shape underlying the rectification, after applying the ρ map.

5.1.
Revisiting the LR rules in [6]. To conclude this article, we briefly describe an equivalent interpretation of our central result in the language of box-adding operators on compositions. By Proposition 4.6, we know that elements of LRT σ (λ, µ, ν) may be constructed by computing λ/µ-compatible words w satisfying colform(w) = (w 0 σw 0 ) · ν r , where w 0 is the longest word in S (ν) . Equivalently, these words are exactly the growth words of standardizations of tableaux in LRT σ (λ, µ, ν).
Thus, they may be identified with certain saturated chains in Young's lattice from µ to λ. By applying the map ρ β where sort(β) = µ, these chains may be interpreted as chains in L c from β to certain compositions γ that satisfy sort(γ) = λ. Fixing a γ and counting these chains allows us to compute C γ αβ where α = σ · ν.
Algebraic Combinatorics, Vol. 4 #1 (2021) We make the preceding discussion precise by phrasing the result in the language of box-adding operators on compositions introduced in [35] following the seminal work of Fomin [9]. Given i 1 and a composition α that has at least one part equaling i − 1, we define t i (α) to be the unique composition covering α in L c where the new box occurs in the i-th column. Given a word w = w 1 · · · w n , define t w := t w1 · · · t wn . We have the following corollary.