Queer supercrystal structure for increasing factorizations of fixed-point-free involution words

We show that the set of increasing factorizations of fixed-point-free (FPF) involution words has the structure of queer supercrystals. By exploiting the algorithm of symplectic shifted Hecke insertion recently introduced by Marberg (http://arxiv.org/abs/1901.06771v3), we establish the one-to-one correspondence between the set of increasing factorizations of fixed-point-free involution words and the set of primed tableau (semistandard marked shifted tableaux) and the latter admits the structure of queer supercrystals. In order to establish the correspondence, we prove that the Coxeter–Knuth related FPF-involution words have the same insertion tableau in the symplectic shifted Hecke insertion, where the insertion tableau is an increasing shifted tableau and the recording tableau is a primed tableau.


Introduction
Recently, Marberg [22] has introduced the symplectic shifted Hecke insertion, which is the symplectic shifted analogue of Hecke insertion introduced by Buch, Kresch, Shimozono, Tamvakis, and Yong [3] in their study of Grothendieck polynomials.By his insertion algorithm, Marberg established a bijection between increasing factorizations of symplectic Hecke words and the pair of shifted Young tableaux of the same shape such that the left is an increasing shifted tableau and the right is a set-valued shifted tableau, which are called insertion and recording tableaux, respectively.Restricted to reduced symplectic Hecke words, which are called fixed-point-free (FPF) involution words, set-valued shifted tableaux turn out to be primed tableaux (semistandard marked shifted tableaux).On the other hand, Assaf and Oguz [1,2] and the author [18] showed independently that the set of primed tableaux admits the structure of crystals for queer Lie superalgebra or simply queer supercrystals discovered by Grantcharov et al. [9,10,11,12].Combining these results, it is expected that the set of increasing factorizations of FPF-involution words is a queer supercrystal and the affirmative answer is given in this paper.To do so, we prove that Coxeter-Knuth related (Coxeter braid related in the terminology of [22]) FPF-involution words have the same insertion tableau, which ensures the one-to-one correspondence between the set of increasing factorizations of FPF-involution words and the set of primed tableaux.
The paper is organized as follows.In Section 2, we review crystals for the queer Lie superalgebra.Definitions of increasing shifted tableaux and primed tableaux are given in Section 3. In Section 4, we explain the algorithm of FPF-involution Coxeter-Knuth insertion (symplectic shifted Hecke insertion restricted to FPFinvolution words) introduced in [22] and show that Coxeter-Knuth related FPFinvolution words have the same insertion tableau (Theorem 4.4).The proof is so lengthy that it is relegated to Appendix A. In Section 5, we show that the set of increasing factorizations of FPF-involution words admits the structure of queer supercrystals (Theorem 5.1).The odd Kashiwara operators are given in Lemmas 5.3 and 5.4.We briefly mention the results of the "orthogonal" version without proofs in Appendix B.
Definition 2.2 (tensor product rule).Let B 1 and B 2 be gl(n)-crystals.The tensor product B 1 ⊗ B 2 is defined to be the set whose crystal structure is defined by Remark 2.2.Note that this definition is different from the one by the convention of Kashiwara.Since we use Kashiwara operators on primed tableaux which are constructed by the anti-Kashiwara convention for the tensor product rule [17], we adopt Definition 2.2.
Next, let us describe crystals for the queer Lie superalgebra q(n) or simply q(n)crystals introduced in [10,11].Definition 2.3.A q(n)-crystal is a set B together with the maps wt : B → P , ε i , ϕ i : B → Z ≥0 and ẽi , fi : B → B ⊔ {0} for i ∈ I := {1, . . ., n − 1, 1} satisfying the following conditions: (1) B is a gl(n)-crystal with respect to wt, ε i , ϕ i , ẽi and fi the operators ẽ1 and f1 commute with ẽi and fi , The crystal associated with an irreducible highest weight module V (λ) in the category O ≥0 int [9,12] of tensor representations over the q(n)-quantum group U q (q(n)) is denoted by B n (λ).We identify the partition λ with λ = n i=1 λ i ǫ i ∈ P + in a usual way [19].In B n (λ), the relevant partition λ is a strict partition λ = (λ 1 > λ 2 > . . .> λ l > λ l+1 = 0), where l ≤ n.A crystal B can be viewed as an oriented colored graph with colors i ∈ I when we define b This graph is called a crystal graph.The crystal graph of B n ( ), i.e., B n (λ) for λ = ǫ 1 ∈ P + , is given by Under these rules, B 1 ⊗ B 2 is also a q(n)-crystal.
Remark 2.3.These rules are different from the ones given in [6,10,11] because we adopt the anti-Kashiwara convention for the tensor product rule.
Let B be a q(n)-crystal and suppose that B is in a class of normal gl(n)crystal [4], i.e., every connected component in B is in one-to-one correspondence with a dominant integral weight.Here, the connected components in B are referred to as the maximal subcrystals of B where all the elements are connected by even Kashiwara operators.We define the automorphism S i on B by Let w be an element of Weyl group W of gl(n) which is generated by simple reflections s i for i = 1, . . ., n − 1.Then, there exists a unique action S w : where . These operators together with ẽ1 and f1 are called odd Kashiwara operators, while ẽi and fi (i = 1, . . ., n−1) are called even Kashiwara operators.
Theorem 2.1 ( [10,11]).Let B n (λ) be a q(n)-crystal.There is a unique element b ∈ B n (λ) such that ẽi b = ẽī b = 0 for all i = 1, . . ., n − 1, which is called a q(n)highest weight vector and there is a unique element b ∈ B n (λ) called a q(n)-lowest weight vector such that S w0 b is a q(n)-highest weight vector, where w 0 is the longest element of W .

Shifted tableaux
Let P + denote the set of strict partitions, λ = (λ 1 > λ 2 > • • • > λ l > λ l+1 = 0).For λ ∈ P + , the length l(λ) of λ is defined as the number of positive parts of λ.The shifted diagram of shape λ ∈ P + is defined to be the set A filling T of S(λ) with letters is called a shifted tableau where the entry at (i, j)position is denoted by T (i,j) .Definition 3.1.An increasing shifted tableau T of shape λ is a filling of S(λ) with letters from the alphabet {1, 2, . ..} such that entries are strictly increasing across rows and columns.
The row reading word of an increasing shifted tableau T of shape λ, denoted by row(T ), is the sequence of entries, T l(λ) T l(λ)−1 • • • T 1 , where T i is the sequence of entries of the ith row of T read from left to right (i = 1, 2, . . ., l(λ)).A primed tableau T of shape λ is a filling of S(λ) with letters from the alphabet, (1) the entries are weakly increasing across row and columns, (2) each row contains at most one i ′ for i = 1, . . ., n, (3) each column contains at most one i for i = 1, . . ., n, and (4) there are no primed letters on the main diagonal.
We denote by PT n (λ) the set of primed tableaux of shape λ.
A primed tableau is called standard if the set of entries consists of 1, 2, . . ., n each appearing either primed or unprimed exactly once for some n.
Example 3.2.See Example 4.5, where the right tableau is the standard primed tableau and the left is not.
Remark 3.1.A primed tableau is also called a semistandard marked shifted tableau [5] and is a set-valued shifted tableau [20,24] whose entries are all singleton sets.

The FPF-involution Coxeter-Knuth insertion
Let us start by recalling the definition of reduced words for an element in S ∞ , the symmetric group generated by simple transpositions s i (i = 1, 2, . ..).We identify the product of simple transpositions with their sequence of indices or the word.For a word w, we denote by |w| the number of letters in w and define l(w) = min{l | ∃i 1 , . . ., i l , w = s i1 • • • s i l }.A word w with |w| = l(w) is referred to as a reduced word.Two reduced words w and w ′ are called Coxeter-Knuth equivalent, denoted by w CK ∼ w ′ , if w ′ can be obtained from w by a finite sequence of Coxeter-Knuth relations on three consecutive letters (a + 1)a(a + 1) ∼ a(a + 1)a, bac ∼ bca, and acb ∼ cab, where a < b < c [4].Note that Coxeter-Knuth relations here are called Coxeter braid relation in [22] and are not the Coxeter-Knuth relations in [22].It is confusing but we follow the terminology of [4] here.
Two FPF-involution words, w and w ′ are called equivalent, denoted by w Sp ∼ w ′ , if w ′ can be obtained from w by a finite sequence of relations on consecutive letters ab ∼ ba (|a − b| > 1) and a(a + 1)a ∼ (a + 1)a(a + 1) and the relation with i Theorem 4.1 ([22]).A symplectic Hecke word is an FPF-involution word if and only if its equivalence class contains no words with equal adjacent letters.
From w, we recursively construct a sequence of pairs of tableaux, (∅, ∅) = (P (0) , Q (0) ), (P (1) , Q (1) ), . . ., (P (l) , Q (l) ) = (P Sp (w), Q Sp (w)), where P (k) is an increasing shifted tableau and Q (k) is a primed tableau.To obtain the tableau P (k) , insert the letter u k into P (k−1) as follows: First insert u k into the first row of P (k−1) .The rules for inserting a into a row or column, denoted by L, of the increasing shifted tableau T are as follows: To obtain 1) at the terminated position if the insertion terminated with row (resp.column) insertion.Tableaux P Sp (w) and Q Sp (w) are referred to as the insertion tableau and the recording tableau, respectively.This algorithm is called the FPF-involution Coxeter-Knuth insertion [22] and denoted by the map H Sp : w → (P Sp (w), Q Sp (w)).This is equivalent to the symplectic shifted Hecke insertion [22] restricted to FPF-involution words and is the shifted analogue of Edelman-Greene insertion [7].The insertion process of a letter x or a word u 1 u 2 • • • into an increasing shifted tableau T is denoted by .When we specify a row or column indicated by an arrow, we do not attach the symbol Sp in the arrow.This convention is also used in Appendix A. The following results are due to Marberg [22].

Theorem 4.2 ([22]
). Suppose that T is an increasing shifted tableau and a is a letter such that row(T )a is an FPF-involution word, then row(T This is one of our first main results.The proof of Theorem 4.4 is relegated to Appendix A. Note that if w ∈ RFPF (z), then a word w ′ such that w CK ∼ w ′ is also an FPF-involution word; w ′ ∈ RFPF (z).
The FPF-involution Coxeter-Knutk insertion is reversible.The letter ) by the following algorithm: Let (i k , j k ) be the position of the box containing the largest entry x k of Q (k) .Let y k be the entry at the position (i k , j k ) of P (k) .Remove the box of Q (k) at the position (i k , j k ).Let y k be the entry of the box of P (k) at the position (i k , j k ).Remove that box of P (k) and reverse insert y k into the row above if x k in Q (k) is unprimed and into the column to the left if x k is primed.
The rules for the reverse insertion of y into a row or column of P (k) are as follows: Let x be the largest entry of L with x < y.
(1) If L is a column and x is the last entry of L and x ≡ y (mod 2), then leave L unchanged and insert x − 2 to the row above.(2) If x + 1 = y, then leave L unchanged and insert x the row above if L is a row or to the next column to the right if L is a column.(3) In all other cases, replace x by y in L and insert x to the next column to the left if L is a column or to the row above if L is a row or x was on the main diagonal.
If we are in the first row of P (k) and the last step was a reverse row insertion or we are in the first column and the last step was a reverse column insertion, then u k is the letter bumped out.Theorem 4.5 ([22]).Let z ∈ F ∞ .Then, the map H Sp gives a bijection between RFPF (z) and the set of pairs (P, Q), where P is an increasing shifted tableau with row(P ) ∈ RFPF (z) and Q is a standard primed tableau with the same shape as P .
Given w ∈ RFPF (z), an increasing factorization of w is a factorization which is obtained by disregarding the grouping into blocks with |w| = w 1 + • • • + |w m | and each factor w i is strictly increasing.In the sequel, we consider w 1 w 2 • • • w m as the increasing factorization or the word w = w 1 w 2 • • • w m interchangeably.We denote by RF m FPF (z) the set of all increasing factorizations with m blocks of all FPF-involution words RFPF (z).We say that two factorizations w 1 w 2 • • • w m and w1 w2 w) by the following rule: Let x be the entry in Q Sp (w), which appears when a letter in w i is inserted.Then, x is replaced by i if x is unprimed and by i ′ if x is primed.We apply this procedure for all entries in Q Sp (w).
The algorithm of constructing the pair of tableaux (P Sp (w), are referred to as the insertion tableau and the recording tableau, respectively.
The semistandard FPF-involution Coxeter-Knuth insertion is also reversible.To reverse the insertion, we first standardize the recording tableau and apply the reverse FPF-involution Coxeter-Knuth insertion.The rest procedure is obvious.The standardization of a primed tableau T , denoted by st(T ), is given by the following procedure: We first replace all 1s appearing T , read from left to right, by 1, 2, . . ., i, where i is the number of 1s in T .Then, replace all 2 ′ s appearing T , read from top to bottom, by the primed numbers (i + 1) ′ , (i + 2) ′ , . . ., (i + j 1 ) ′ , where j 1 is the number of 2 ′ s in T .Then replace all 2s appearing T , read from left to right, by the unprimed numbers (i + j 1 + 1), (i + j 1 + 2), . . ., (i + j 1 + j 2 ), where j 2 is the number of 2s in T , and so on.
Then, the map H ′ Sp gives a bijection between RF m FPF (z) and the set of pairs (P, Q), where P is an increasing shifted tableau with row(P ) ∈ RFPF (z) and Q is a primed tableau with the same shape as P .

Queer supercrystal structure for increasing factorizations of FPF-involution words
It is established that PT m (λ) is a q(m)-crystal [1,2,18].The gl(m)-crystal structure of PT m (λ) together with (even) Kashiwara operators ẽP i and f P i (i = 1, . . ., m − 1) is given by Hawkes, Paramonov, and Schilling [17] and by Assaf and Oguz [1,2].The actual algorithm is quite involved.For details we refer the reader to [17] and [1,2].The odd Kashiwara operators f P 1 and ẽP 1 on PT m (λ) are given by the following two lemmas [1,2,18].Lemma 5.1.For T ∈ PT m (λ), the odd Kashiwara operator ẽP 1 on T is given by the following rule: If Lemma 5.2.For T ∈ PT m (λ), the odd Kashiwara operator f P 1 on T is given by the following rule: If the first row of T does not contain letters 1, then f P 1 T = 0. Let (1, i) be the position of the rightmost 1 in the first row. ( Sp is a bijection (Theorem 4.6), we can translate the gl(m)-crystal structure of PT m (λ) to that of RF m FPF (z).That is, Kashiwara operators ẽF i and f F i on RF m FPF (z) are given by First, we construct odd Kashiwara operators.Let Let us denote by cont(w i ) the set of letters appearing in w i .
FPF (z) be an increasing factorization of w ∈ RFPF (z) for z ∈ F ∞ .The action of the odd Kashiwara operator f F 1 on w 1 w 2 • • • w m is given by the following rule: If Let us write FPF (z) be an increasing factorization of w ∈ RFPF (z) for z ∈ F ∞ .The action of the odd Kashiwara operator ẽF 1 on w 1 w 2 • • • w m is given by the following rule: , where w1 = v 1 and cont( w2 ) = cont(w 2 )\{v 1 } with v 1 being the first entry of w 2 .
Let us write We give the proof of Lemma 5.3 only.The proof of Lemma 5.4 is similar.
Proof of Lemma 5.3.For cases (1) and ( 2), the statement is obvious.Let us assume that w 1 ≥ 2 and w 2 = 0 and write , then the configuration of the recording tableau up to the insertion of the letter v 1 has the configuration, This portion does not change under the subsequent insertions.By Lemma 5.2, which is also not an FPF-involution word because u 1 +1 is an odd letter.If u 1 < v 1 , then the first row of Q has one of the following two, Note that column insertions never happen in the insertion of w 2 to the increasing tableau, By Lemma 5.2, the configuration of the first row of f If u 2 > u 1 + 1, then the increasing factorization that is equivalent to w 1 w 2 and gives the above configuration is (u then the increasing factorization which is equivalent to w 1 w 2 and gives the above configuration is (u We omit the proof for the case when w 2 = 0; it is much simpler.This completes the proof.

By construction, two vertices connected by f F
1 or ẽF 1 in the same connected component of q(m)-crystal RF m FPF (z) have the same insertion tableau P Sp (w) and f F 1 or ẽF 1 are independent of the choice of P Sp (w).Note that the first letter of f The set RF m FPF (z) is a gl(m)-crystal with Kashiwara operators ẽF i and f F i given by Morse and Schilling [4,23].Before giving the verification of this claim, we restate Morse-Schilling's construction of their Kashiwara operators with appropriate alterations in our setting.
The crystal operators ẽF i and f F i only act on the block w i w i+1 (i = 1, . . ., m−1).We define the pairing of w i and w i+1 as follows: Pair the largest b ∈ cont(w i+1 ) with the smallest a > b in cont(w i ) and if no such a exists, then b is unpaired.We proceed in decreasing order on letters in w i+1 , ignoring previously paired letters in w i .Define ) is defined by replacing blocks w i w i+1 by wi wi+1 such that cont( wi ) = cont(w i )\{c} and cont( wi+1 ) = cont(w i+1 ) ∪ {c + s} for c = max L i and s = min ) is defined by replacing blocks w i w i+1 by wi wi+1 such that cont( wi ) = cont(w i ) ∪ {c − t} and cont( wi+1 ) = cont(w i+1 )\{c} for c = min R i and In this example, L 1 = {2, 3}, c = max L 1 = 3, and s = 1 so that cont( w2 ) = {3, 4}.

The weight of w
Thus, our claim has been verified.This property is compatible with Eq (5.1), i.e, the gl(m)-crystal structure of RF m FPF (z) is in one-to-one correspondence with that of PT m (λ), and ẽF i and f F i (i = 1, . . ., m − 1, 1) satisfy conditions in Definition 2.3.Thus, we can adopt Kashiwara operators above as the proper even Kashiwara operators on RF m FPF (z).Theorem 5.1.Let z ∈ F ∞ .Then, the set RF m FPF (z) admits a q(m)-crystal structure.The even Kashiwara operators are ẽF i and f F i (i = 1, 2, . . ., m − 1), whereas the odd Kashiwara operators are given in Lemma 5.3 and 5.4.This is our second main result.
Conjecture 5.1.The even Kashiwara operators given by Eq (5.1) agree with those given by Morse and Schilling.
It is not hard to check that the q(m)-crystal graph of RF m FPF (z) satisfies the local queer axioms introduced by Assaf and Oguz [1,2] and Gillespie, Hawkes, Poh, and Schilling [8].We omit the details Appendix A.
In this Appendix, T Sp ← w is taken to be the resulting tableau by the insertion unless stated otherwise, where T is an increasing tableau and w is a letter or word.
Proof of Theorem 4.4.The assertion is the direct consequence of Lemmas A.6, A.7, and A.8.
Lemma A.1.Let T be an increasing shifted tableau of shape λ such that row(T ) is an FPF-involution word.Then entries in the main diagonal of T are even letters.
Proof.Let T ′ i be the portion of T below its ith row and let T (i,i) = d i (i = 1, 2, . . ., l(λ) − 1).Since all entries of T ′ i are strictly greater than ) so that d i must be an even letter.It is clear that d l(λ) is an even letter.This is also obvious from the algorithm of FPF-involution Coxeter-Knuth insertion.
Lemma A.2. Let T be an increasing shifted tableau and a be a letter such that row(T )a is an FPF-involution word.Let T (1,1) = x and suppose that a < x.If Proof.Let T 1 be the first row of T and T ′ be the portion of T below the first row.
Let us write T 1 as x y A .The insertion T Sp ← a replace y by a + 2 so that x = a + 1.
First, let us consider the case when y = a + 2.Then, the entry a + 3 exists in T ′ .Otherwise, all the entries in T ′ are greater than or equal to a + 4 so that row(T )a = row(T ′ )T (1,1) T (1,2) row(A)a Sp ∼ (a + 1)(a + 2)row(T ′ )row(A)a Sp ∼ (a + 1)arow(T ′ )row(A)a Sp ∼ (a + 1)row(T ′ )row(A)aa, which implies that row(T )a is not an FPF-involution word and contradicts the assumption of Lemma A.2.The entry a + 3 exists only in the position (2, 2).If (a) p = a + 1.
b p .The word pacb is not an FPF-involution word in other cases.
Lemma A.5.Let T be a tableau p such that pa(a + 1)a is an FPF-involution word.Then, T Sp ← a(a + 1)a = T Sp ← (a + 1)a(a + 1).
. The word pa(a + 1)a is not an FPF-involution word in other cases.
Lemma A.6.Let T be an increasing shifted tableau of shape λ = (λ 1 , λ 2 , . ..) such that row(T )bac is an FPF-involution word, where a < b < c.Then, Proof.Let T 1 be the first row of T and T ′ be a portion of T below the first row.
(1) T (1,1) ≤ a.Let a ′ be the smallest entry in T 1 such that a < a ′ (if it exists).Then, , where ã is either a + 1 or a ′ and T Let us write T 1 as x y A .By Lemma A.2, x = a + 1 and , where Let us write T 1 as x A .Then, (1) T (1,1) = b and a ≡ b (mod 2).
, where the row insertion of b + 1 is followed by the column insertion of a + 2. From this configuration, we have that b = a + 1.Let us write the first row of T as , where the leftmost entry of A is strictly greater than a + l + 1.Then, It is easy to see that a ≡ b (mod 2) and It is easy to see that a ≡ b (mod 2) and Let b ′ be the smallest entry in T 1 such that b < b ′ .Let us denote T Sp ← b by T and let a ′ (resp.c ′ ) be the smallest entry in ( T ) 1 , the first row of T , such that where p = a ′ or a + 1, q = b ′ or b + 1, r = c ′ or c + 1 and p < q < r.Suppose that T ′ Sp ← qpr = T ′ Sp ← qrp with p < q < r.This assumption is satisfied when T ′ is a tableau of a single box (Lemma A.3).We claim that T Sp ← bac = T Sp ← bca by induction.
A column insertion is called the insertion to the top when the inserting letter x is greater then or equal to x ′ , the first entry of this column, or the column insertion to the empty column.If it is not the column insertion to the empty column and x < x ′ , then we call such an insertion the strong insertion to the top.
Suppose that an insertion to the top, which is not the column insertion to the empty column, begins at some column position in the course of insertion T Sp ← x.This implies the strong insertion to the top begins at the same column position in the course of insertion T ′ Sp ← x ′ , where x ′ is the inserting letter corresponding to x.Such a column insertion is called the initial insertion to the top and the inserting letter is called the initial column inserting letter.The initial insertion to the top triggers the subsequent insertions to the top.If an insertion to the empty column appears, then it is the last column insertion.If such a column insertion is not the subsequent insertion followed by some insertion to the top, then it is called the initial empty column insertion and the inserting letter is also called the initial column inserting letter but the initial empty column insertion does not trigger the subsequent insertions to the top.The initial empty column insertion must appear earlier than any other initial insertions to the top and once it appears no further initial empty column insertions appear.It is obvious that the sequence of initial column inserting letters in T Sp ← bac (resp.T Sp ← bca) and that of T ′ Sp ← qpr (resp.
Let us suppose that the sequence of initial column inserting letters is xy in T Sp ← bac with x < y and that the corresponding insertions are not the initial empty column insertions.The insertions to the top go as follows, where we assume that x < x ′ and y < y ′ .
The corresponding insertions to the top in T ′ Sp ← qpr go as follows.
x ↑ . . .Proof.Let T 1 be the first row of T and T ′ be a portion of T below the first row.
Case 1: This case must be excluded because row(T )a is not an FPF-involution word.
Let a ′ be the smallest entry such that a < a ′ .Then, , where ã is either a + 1 or a ′ .
Let a ′ (resp.b ′ ) be the smallest entry such that a < a ′ (resp.b < b ′ ).
Case 5: This case must be excluded because row(T )c is not an FPF-involution word.
(1) T (1,1) > a and T (1,1) ≡ a (mod 2).By Lemma A.2, the first row of T is either where the leftmost entry in A is greater than or equal to a + 4 and the leftmost entry in A ′ is greater than or equal to a + 3. (a) c = a + 2.
If T 1 = T 1 , then 1 , then the leftmost entry in A ′ is a + 3. Otherwise, all the entries in T ′ are greater than or equal to a + 5 and all the entries in A ′ are greater than or equal to a + 4 so that which is not an FPF-involution word.
The argument similar to Case 4 (1) in the proof of Lemma A. Any entry in T is greater than or equal to c + 2 so that row(T )cab Sp ∼ cabrow(T ).Since this is an FPF-involution word, c is an even letter so that x ≡ c (mod 2).(i) p = c + 2 and q = c + 3.
which is not an FPF-involution word.
Let w (resp.w ′ ) be the sequence of three letters bumped out from the first row of T , which is inserted to the second row of T , in the insertion T Lemma A.8. Let T be an increasing shifted tableau of shape λ = (λ 1 , λ 2 , . ..) such that row(T )a(a + 1)a is an FPF-involution word.Then, T Sp ← a(a + 1)a = T Sp ← (a + 1)a(a + 1).
Proof.Let T 1 be the first row of T and T ′ be a portion of T below the first row.If T (1,1) ≥ a, then a ≤ T (1,1) ≤ a + 2. Otherwise, row(T )a(a + 1)a is not an FPF-involution word.
The first row of T has the configuration, a a + 1 A .Otherwise, row(T )a is not an FPF-involution word.Let a ′ = T (1,i) be the smallest entry in T 1 such that a ≤ a ′ (i = 1).(a) a ′ = a.
The entry T (1,i+1) must be a + 1, since otherwise row(T )a is not an FPF-involution word.

Example 3 . 1 .
The row reading word of Find the smallest entry b of L with a ≤ b.If no such entry exists, then add a to the end of L. Stop.Otherwise, (1) If a = b, then leave L unchanged and insert a + 1 to the row below if L is a row or to the next column to the right if L is a column.(2) If L is a row and b is the first entry of L and a ≡ b (mod 2), then leave L unchanged and insert a + 2 to the next column to the right.(3) In all other cases, replace b by a in L and insert b to the row below if L is a row or to the next column to the right if L is a column or b was on the main diagonal.

Example 4 . 2 .
which also denotes the resulting tableau.The insertion steps in

a
≡ x (mod 2), then T (1,1) = a + 1, T (1,3) ≥ a + 4 and the insertion T Sp ← a changes the first row of T only.If a ≡ x (mod 2), then the insertion T Sp ← a changes the first row of T only.
a is not an FPF-involution word.Hence, T (1,3) ≥ a+4, which implies that the insertion T Sp ← a yields no bumped letter to be inserted to the second row; the first row ofT Sp ← a is a + 1 a + 2 a + 3 A .Next, let us consider the case when y > a + 2. It is obvious that the insertion T Sp ← a yields no bumped letter to be inserted to the second row;the first row of T Sp ← a is a + 1 a + 2 y A .(2) a ≡ x (mod 2).Let us write T 1 as x A .It is obvious that the insertion T Sp ← a yields no bumped letter to be inserted to the second row; the first row of T Sp ← a is a x A .Lemma A.3.Let T be a tableau p such that pbac is an FPF-involution word, where a < b < c.Then, T Sp ← bac = T Sp ← bca.Proof.By direct computation, we have (1) p < a. a < p < b.(a) p = a + 1. T Sp ← bac = T Sp ← bca = a + 1 a + 2 b c .(b) p ≡ a (mod 2).T Sp ← bac = T Sp ← bca = a p b c .(3) b < p < c.(a) b = a + 1 and p = a + 2. T Sp ← bac = T Sp ← bca = a + 1 a + 2 a + 3 c .(b) b = a + 1 and p > a + 2. T Sp ← bac = T Sp ← bca = a + 1 a + 2 p c .(c) a ≡ b (mod 2).T Sp ← bac = T Sp ← bca = a b p c .(4) c < p.(a) b = a + 1. T Sp ← bac = T Sp ← bca = a + 1 a + 2 p c .(b) a ≡ b (mod 2).T Sp ← bac = T Sp ← bca = a b p c .The word pbac is not an FPF-involution word in other cases.Lemma A.4.Let T be a tableau p such that pacb is an FPF-involution word, where a < b < c.Then, a < p < b.

↑←.Lemma A. 7 .
Sp ← qpr = T ′ Sp ← qrp by the assumption of induction, the insertions to the top in T ′ Sp ← qrp must be the same as above or y , the insertions to the top in T Sp ← bca go as follows.bca.The verifications of other cases are similar.We omit the details.Example A.1.We have that 327 CK cases, the insertions to the top go as follows.Let T be an increasing shifted tableau of shape λ = (λ 1 , λ 2 , . ..) such that row(T )acb is an FPF-involution word, where a < b < c.Then, T Sp ← acb = T Sp ← cab.

Sp←
acb (resp.T Sp ← cab).Then, we have that w CK ∼ w ′ by a case by case analysis.Contrary to Case 5 of the proof of Lemma A.6, we have all three types of Coxeter-Knuth related words w and w ′ .The same argument as in Case 5 of the proof of Lemma A.6 shows T Sp ← acb = T Sp ← cab.
• • • wm , both of which are elements of RF m FPF (z), are equivalent if two words w = w 1 w 2 • • • w m and w = w1 w2 • • • wm are equivalent; w