Combinatorics of semi-toric degenerations of Schubert varieties in type C

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand-Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko-Smirnov-Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.


Introduction
A goal of Schubert calculus is to compute the structure constants of the cohomology ring of a flag variety with respect to the basis consisting of Schubert classes; see [21,22,28] for the history of Schubert calculus.One approach to such computation is to realize Schubert classes as concrete combinatorial models such as Schubert polynomials.To study such combinatorial models, toric degenerations are useful.A toric degeneration is a flat degeneration of a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties.In the case of a flag variety, its toric degeneration with desirable properties induces degenerations of Schubert and opposite Schubert varieties to not necessarily irreducible torus-invariant closed subvarieties, called semi-toric degenerations.Kogan-Miller [24] constructed semi-toric degenerations of opposite Schubert varieties as some quotients of Knutson-Miller's semi-toric degenerations [22] of opposite matrix Schubert varieties which give a geometric proof of the pipe dream formula of Schubert polynomials.Since semi-toric degenerations can be constructed for Schubert and opposite Schubert varieties in general Lie type, they give us a hint to extend theories of Schubert calculus in type A to other Lie types.Using semi-toric degenerations of Schubert varieties for symplectic Gelfand-Tsetlin polytopes, the first named author [10] extended Kiritchenko-Smirnov-Timorin's combinatorial model [21] in type A to type C case.The purpose of the present paper is to develop combinatorics of this type C model of Schubert classes.
To be more precise, we first consider G = SL n+1 (C) (of type A).Then the Weyl group W of G is given as the symmetric group S n+1 .Let B ⊆ G be the subgroup of upper triangular matrices, and G/B the full flag variety.For w ∈ S n+1 , denote by ℓ(w) the length of w, and by X w ⊆ G/B (resp., X w ⊆ G/B) the Schubert variety (resp., the opposite Schubert variety) with dim C (X w ) = dim C (G/B) − dim C (X w ) = ℓ(w) (see [4,Section 1.2] for the precise definitions of X w and X w ).Then the cohomology class [X w ], called a Schubert class, coincides with [X w 0 w ] for each w ∈ S n+1 (see, for instance, [4,Section 1.3]), where w 0 ∈ S n+1 denotes the longest element.In addition, the set {[X w ] | w ∈ S n+1 } of Schubert classes forms a Z-basis of the cohomology ring H * (G/B; Z) of G/B.To study this basis, the Borel description of the (rational) cohomology ring H * (G/B; Q) is useful, which states that H * (G/B; Q) is isomorphic to the coinvariant algebra of S n+1 (see [28]).This description allows us to represent each Schubert class as a polynomial.Lascoux-Schützenberger [26] gave a specific choice {S w (x) | w ∈ S n+1 } of representatives, called Schubert polynomials, which have good combinatorial properties.Billey-Jockusch-Stanley [3] and Fomin-Stanley [9] gave an explicit combinatorial formula x D (1.1) of S w (x), called the pipe dream formula, where we used the notation in [22,Corollary 2.1.3].
A diagrammatic interpretation of the index set RP (w) was invented by Fomin-Kirillov [7] and developed by Bergeron-Billey [2] and by 29].Under this diagrammatic interpretation, each element of RP (w) is called an rc-graph or a reduced pipe dream.Let P + be the set of dominant integral weights, P ++ ⊆ P + the set of regular dominant integral weights, and GT (λ) the Gelfand-Tsetlin polytope for λ ∈ P + .By definition, each reduced pipe dream D ∈ RP (w) naturally corresponds to specific faces F D (GT (λ)) and F ∨ D (GT (λ)) of GT (λ), called a reduced Kogan face and a reduced dual Kogan face, respectively, such that w(F D (GT (λ))) −1 = w(F ∨ D (GT (λ))) = w in the notation of [21,Sections 3.3 and 4.3] (see [23,Section 2.2.1] and [24,Section 4]).For λ ∈ P ++ , Kogan-Miller [24] constructed a semi-toric degeneration of X w 0 ww 0 whose limit corresponds to the union of reduced dual Kogan faces F ∨ D (GT (λ)), D ∈ RP (w).Note that the union of reduced Kogan faces F D (GT (λ)), D ∈ RP (w), appears as a semi-toric degeneration of the Schubert variety X w 0 w −1 (see [10]).Relations between Schubert classes and reduced (dual) Kogan faces were studied in [23,17,21].Using an isomorphism between H 0 (SL n+1 (C)/B; Z) and the polytope ring of {GT (λ) | λ ∈ P + } (see [14]), Kiritchenko-Smirnov-Timorin [21] formulated the following equalities: We next consider G = Sp 2n (C) (of type C).Then the Weyl group W of G is given as the group of signed permutations (see Section 2 for more details).We use the notations similar to the case of type A such as G/B, X w , X w , P + , and P ++ .Denote by R(w) the set of reduced words for w ∈ W , and take i C ∈ R(w 0 ) as in (2.1).Then Littelmann [27] proved that the string polytope ∆ i C (λ) associated with i C and λ ∈ P + is unimodularly equivalent to the symplectic Gelfand-Tsetlin polytope SGT (λ).For λ ∈ P ++ , Caldero [6] constructed a toric degeneration of G/B to the normal projective toric variety Z(∆ i C (λ)) corresponding to ∆ i C (λ) (see also [15]).Using string parametrizations of Demazure and opposite Demazure crystals, Morier-Genoud [30] showed that Caldero's toric degeneration [6] induces semi-toric degenerations of X w and X w for w ∈ W .The first named author [10] determined the limits of X w and X w in the string polytope ∆ i C (λ), and generalized the realization (1.2) by Kiritchenko-Smirnov-Timorin [21] using such limits.When n = 2, 3, such generalization was previously studied in [18,20].Unlike the case of type A, the limit of X w and that of X w have different kinds of combinatorial properties.Indeed, the irreducible components of the limit of X w are parametrized by some reduced subwords of i C (see [10,Section 4]) while those of X w seem not have any good relation with reduced subwords even in the case G = Sp 4 (C) (see Example 3.8).In the present paper, we develop combinatorics of the irreducible components of the limit of X w , which inherit information on skew mitosis operators as we see below.These irreducible components can be parametrized by a certain set M (w) of skew pipe dreams (see Section 3 for the precise definitions of skew pipe dreams and M (w)).For each skew pipe dream D, let F D (∆ i C (λ)) denote the corresponding face of ∆ i C (λ).As we have seen above, the set M (w) and the corresponding faces F D (∆ i C (λ)), D ∈ M (w), have the following three geometric or representation-theoretic properties (see [10,Section 6] for more details): • for λ ∈ P ++ , the union D∈M (w) F D (∆ i C (λ)) corresponds to the limit of a semi-toric degeneration of X w ; • for λ ∈ P + , the set D∈M (w) (F D (∆ i C (λ)) ∩ Z N ) of lattice points gives a string parametrization of a Demazure crystal, where N := ℓ(w 0 ); • for λ ∈ P ++ , the Schubert class [X w ] can be realized as the sum through the theory of polytope rings.In the present paper, we introduce a combinatorial model of the set M (w) using a kind of pipe dream with self-crossings.More precisely, we associate to each skew pipe dream D a diagram G (D) of "n pipes with self-crossings" (see Definition 4.2 for the precise definition).The diagram naturally gives a signed permutation w D , where we have w D (j) < 0 if and only if the j-th pipe L j has an even number of self-crossings for 1 ≤ j ≤ n.For w ∈ W , a skew pipe dream D with w D = w is said to be reduced if the cardinality |D| is the minimum among skew pipe dreams D ′ with w D ′ = w.The following is the first main result of the present paper, which gives a path model to M (w).
Theorem 1 (see Theorems 4.5 and 5.1).For all w ∈ W , the set M (w) coincides with the set of reduced skew pipe dreams D such that w D = w.
The set RP (w) of reduced pipe dreams in type A have the following two kinds of combinatorial constructions: • the set RP (w) is stable under ladder moves and is obtained from a specific element D(w) ∈ RP (w) by applying sequences of ladder moves (see [2]); • the set RP (w) is obtained from the only one element of RP (w 0 ) by applying a sequence of (transposed) mitosis operators (see [22,29]).In the present paper, we show that these constructions can be generalized to the set M (w) in type C. The notion of transposed mitosis operators mitosis ⊤ j for skew pipe dreams, called transposed skew mitosis operators, was introduced by Kiritchenko [18], which implicitly leads to the definition of ladder moves for skew pipe dreams.In addition, the specific element D(w) in type A is generalized by the first named author [10] to an element D(w) of the set M (w) in type C. Let L (D(w)) be the set of skew pipe dreams obtained from D(w) by applying sequences of ladder moves.Then the following is the second main result of the present paper.
Theorem 2 (see Theorem 5.1).For w ∈ W , the set M (w) is stable under ladder moves.In addition, for (j 1 , . . ., j ℓ ) ∈ R(w), the following equalities hold: , where SY n is the only one element of M (e) for the identity element e ∈ W .
In our proofs of Theorems 1 and 2, we first prove the assertion of Theorem 1 for L (D(w)) in Theorem 4.5.Then we show Theorem 2 in Theorem 5.1, which implies Theorem 1 for M (w).Finally, we mention some previous works.Type C pipe dreams were already considered in the studies of Schubert polynomials [8,31] and of double Grothendieck polynomials [16].However, such type C pipe dreams are different from our reduced skew pipe dreams.The former inherits information on reduced subwords while the latter does not.Type C pipe dreams studied in [8,16,31] are closely related to semi-toric degenerations of opposite Schubert varieties (not of Schubert varieties); see [10,Section 4].A convex-geometric version of mitosis operators was introduced by Kiritchenko [18].Its relation with skew mitosis operators was studied in [19].
This paper is organized as follows.In Section 2, we review some basic definitions on symplectic groups and their Weyl groups.In Section 3, we recall some basic notions on skew pipe dreams and skew mitosis operators.Some combinatorial properties of D(w) are also proved in this section.In Section 4, we introduce a path model to reduced skew pipe dreams using a kind of pipe dream with self-crossings.In Section 5, we realize reduced skew pipe dreams using ladder moves and skew mitosis operators.
Acknowledgments.The authors are grateful to Tomoo Matsumura for useful comments and fruitful discussions.

Basic definitions on symplectic groups
In this section, we review some basic definitions on symplectic groups, following [25, Section 3] and [5,Section 8.1].Geometric and representation-theoretic parts of this and the next sections are not necessary to prove the main results of the present paper.Hence readers who are mainly interested in the proofs of the main results may skip them.Such parts are included because they give geometric and representation-theoretic applications of the main results.For n ∈ Z ≥2 , let G = SL 2n (C) be the complex special linear group of degree 2n, B ⊆ G the Borel subgroup consisting of the upper-triangular matrices, and H the maximal torus of B consisting of the diagonal matrices.Denoting by N G ( H) the normalizer of H in G, the Weyl group of type A 2n−1 is defined by W := N G ( H)/ H. Let {e 1 , . . ., e 2n } be the standard basis of C 2n .Under the standard representation of G on C 2n , the Weyl group W is regarded as the symmetric group on {Ce 1 , . . ., Ce 2n }.Let be an integer n × n anti-diagonal matrix whose anti-diagonal entries are all 1.We take an integer 2n × 2n anti-diagonal matrix w 0 as where O n denotes the n × n zero matrix.Define an algebraic group automorphism ω : for g ∈ G, where g T denotes the transpose of g.Then the fixed point subgroup of G coincides with the symplectic group with respect to the skew-symmetric matrix w 0 .This is the connected simply-connected simple algebraic group of type C n .We set C n 1 2 Then the Cartan matrix (c i,j ) i,j∈I of type C n is given by c As signed permutations, these are given as follows: where (i j) denotes the transposition of i and j.Note that W is generated by s 1 , s 2 , . . ., s n as a group.
and if r is the minimum number among such expressions of w.In this case, we call r the length of w, denoted by ℓ(w).Let R(w) be the set of reduced words for w.
which is called the longest element of W .Note that the skew-symmetric matrix w 0 is an element of N G (H), and we have ). (2.1) Since we have ℓ(w 0 ) = n 2 by Proposition 2.1, it follows that i C ∈ R(w 0 ).Define a rational convex polyhedral cone C i C to be the set of (a 2 , a 2 , a 2 , a n−1 , a (1)  n , . . ., a satisfying the following inequalities: [27,Theorem 6.1] proved that C i C coincides with the string cone associated with the reduced word i C (see also [1,Section 3.2]).Let P + be the set of dominant integral weights, and P ++ ⊆ P + the set of regular dominant integral weights.Denoting the set of fundamental weights by {ϖ i | i ∈ I}, we have for all 1 ≤ j ≤ N .Then we see by [27, Section 1] that ∆ i C (λ) coincides with the string polytope associated with i C ∈ R(w 0 ) and λ ∈ P + .We do not review the original definitions of string cones and string polytopes since we do not use them in the present paper, but note that these are defined from certain representation-theoretic objects, called Kashiwara crystal bases (see [13] for a survey on crystal bases).More precisely, the sets C i C ∩Z N and ∆ i C (λ)∩Z N of their lattice points give string parametrizations of crystal bases.See [27, Section 1] and [1, Section 3.2] for more details on string cones and string polytopes.The string polytope ∆ i C (λ) is an integral polytope for all λ ∈ P + , and it is N -dimensional if λ ∈ P ++ .Let us arrange the equations of the facets of C i C as Corollary 7] gave a unimodular affine transformation from the string polytope ∆ i C (λ) to the symplectic Gelfand-Tsetlin polytope SGT (λ) (also known as the type C Gelfand-Tsetlin polytope).Under this transformation, the face F k (∆ i C (λ)) of ∆ i C (λ) corresponds to a symplectic Kogan face of SGT (λ) which is an analog of a Kogan face of a type A Gelfand-Tsetlin polytope (see [10,Section 6]).For w ∈ W = N G (H)/H, take an element w ∈ N G (H) such that w = w mod H. Then we set X w := BwB/B ⊆ G/B, which is called a Schubert variety.The variety X w is an irreducible normal closed subvariety of G/B and does not depend on the choice of w (see, for instance, [11,Section II.14.15]).For λ ∈ P ++ , Caldero [6] constructed a flat degeneration of G/B to the normal projective toric variety Z(∆ i C (λ)) corresponding to ∆ i C (λ).Then Morier-Genoud [30] proved that Caldero's toric degeneration of G/B induces a degeneration of X w to a (not necessarily irreducible) closed subvariety of Z(∆ i C (λ)) that corresponds to a union of faces of ∆ i C (λ).Such degeneration and degenerated limit of X w are called a semi-toric degeneration and a semi-toric limit of X w , respectively.Let ∆ i C (λ, X w ) denote the union of faces of ∆ i C (λ) corresponding to the semi-toric limit of X w .Morier-Genoud [30] obtained the set ∆ i C (λ, X w ) using a Demazure crystal introduced in [12] that is a specific subset of a crystal basis.More precisely, the set ∆ i C (λ, X w ) ∩ Z N coincides with the string parametrizations of a Demazure crystal.

Skew pipe dreams and skew mitosis operators
In this section, we recall some basic notions on skew pipe dreams, following [18,10].We also prove some combinatorial properties of a specific skew pipe dream D(w) for w ∈ W .Let and regard it as a shifted Young diagram.For instance, we write SY 4 as Denote by SPD n the power set of SY n , and describe D ∈ SPD n by putting + in the boxes corresponding to the elements of D. For instance, D = {(1, 2), ( )} ∈ SPD 3 is described as We call an element of SPD n a skew pipe dream (see [18,Section 5.2]).Recall the reduced word i C = (i 1 , . . ., i N ) in (2.1).For w ∈ W , we write and denote by k w the minimum element of R(i C , w) with respect to the lexicographic order.If w is the identity element e, then we set k w = ∅.
Following [10, Section 6], we define the (symplectic) ladder move L i,j for (i, j) ∈ SY n , which is an analog of ladder moves for pipe dreams introduced by Bergeron-Billey [2].Let 1 ≤ k ≤ N be the integer given by (p k , q k ) = (i, j).Take D ∈ SPD n , and assume the following: and such that (p r , q r ), (p r , q r + 1) ∈ D for all k + 1 ≤ r ≤ ℓ − 1 with q r ∈ {j, 2n − j}.
Then the ladder move L i,j (D) ∈ SPD n of D at (i, j) is defined by Similarly, we define the inverse (symplectic) ladder move L −1 i,j for (i, j) = (p k , q k ) ∈ SY n as follows.Take D ∈ SPD n , and assume the following: ∈ D and such that (p r , q r ), (p r , q r +1) ∈ D for all ℓ+1 ≤ r ≤ k−1 with q r ∈ {j−1, 2n−j+1}.
Then the inverse ladder move L −1 i,j (D) ∈ SPD n of D at (i, j) is defined by ) is defined, and we have L −1 p,q (L i,j (D)) = D. Similarly, if the inverse ladder move L −1 i,j (D) = D ∪ {(p, q)} \ {(i, j)} is defined, then the ladder move L p,q (L −1 i,j (D)) is defined, and we have L p,q (L −1 i,j (D)) = D. Example 3.5.Let n = 3.Then we obtain the following: The notion of (transposed skew) mitosis operators for skew pipe dreams was introduced by Kiritchenko [18, Section 5.2], which is an analog of (transposed) mitosis operators for pipe dreams introduced by Knutson-Miller [22].The first named author [10, Section 6] used an analogous operator M i for i ∈ [n] to study string parametrizations of Demazure crystals.Let us recall their definitions.For D ∈ SPD n and i ∈ [n], we set and consider the following condition: If ( †) is not satisfied, then set M i (D) := ∅.If ( †) is satisfied, then the operator M i sends D to the set M i (D) of elements of SPD n obtained from D \ {(p r 0 , q r 0 )} by applying sequences of ladder moves L p,q such that q ∈ {n − i + 1, n + i − 1}.Note that by the definition of r 0 .Similarly, the transposed skew mitosis operator mitosis ⊤ i is defined as follows.If ( †) is not satisfied, then set mitosis ⊤ i (D) := ∅.If ( †) is satisfied, then we write start ).The operator mitosis ⊤ i sends D to the set mitosis ⊤ i (D) of elements of SPD n obtained from D \ {(p r 0 , q r 0 )} by applying sequences of ladder moves L p,q such that q ∈ {n − i + 1, n + i − 1} and such that p < start ⊤ i (D).By definition, we have mitosis ⊤ i (D) ⊆ M i (D).For a subset A ⊆ SPD n , we set M i (A) := D∈A M i (D) and mitosis ⊤ i (A) := D∈A mitosis ⊤ i (D).Remark 3.6.The transposed skew mitosis operator mitosis ⊤ i is obtained from Kiritchenko's mitosis operator in [18, Section 5.2] by rotating her skew pipe dreams counterclockwise at 90 degrees.
For w ∈ W , we define a set M (w) of skew pipe dreams by where we write k ′ D(w) = (k 1 , . . ., k ℓ ).By the definition of D(w), the set M (w) coincides with the set of skew pipe dreams of the form , where L r for 2 ≤ r ≤ ℓ is given by a sequence of ladder moves L pt,qt such that t < k r and such that q t ∈ {n − i kr + 1, n + i kr − 1}.
In particular, we have M (w) ⊆ L (D(w)), where L (D(w)) denotes the set of skew pipe dreams in SPD n obtained from D(w) by applying sequences of ladder moves.
Example 3.8 (see also [18,Example 2.9] and [10, Example 6.6]).Let n = 2. Then it follows that Comparing with Example 3.1, we see that M (w) seems not to have good relations with reduced subwords of i C .
Example 3.9.Let n = 3, and take w ∈ W as in Example 3.4.Then the set M (w) consists of the six skew pipe dreams given in Example 3.5.
For D ∈ SPD n and λ ∈ P + , we define a face ).The first named author [10, Theorem 6.8] proved that the set M (w) parametrizes irreducible components of the semi-toric limit of X w under the toric degeneration of G/B to Z(∆ i C (λ)) for λ ∈ P ++ .More precisely, the set ∆ i C (λ, X w ) corresponding to the semi-toric limit of X w is given by Proposition 3.10.For w ∈ W , the equalities L (D(w)) = M (w) = {D(w)} hold if and only if the following conditions are satisfied: Proof.Since D(w) ∈ M (w) ⊆ L (D(w)), the condition L (D(w)) = M (w) = {D(w)} is equivalent to the equality L (D(w)) = {D(w)}.We first prove the "only if" part, that is, let us show the conditions (i)-(iii) under the assumption that L (D(w)) = {D(w)}.If m(w, ℓ) ≤ m(w, ℓ + 1) ̸ = 0 for some 1 ≤ ℓ ≤ n − 1, then it is obvious that the ladder move L ℓ+1,ℓ+m(w,ℓ+1) (D(w)) of D(w) is defined, which contradicts to L (D(w)) = {D(w)}.Hence the condition (i) is satisfied.If m(w, i) < n − i + 1 for some 1 ≤ i ≤ n − 1 and m(w, i + 1) ̸ = 0, then we can define L i+1,i+m(w,i+1) (D(w)), which gives a contradiction.This proves the condition (ii).Similarly, we deduce the condition (iii).The "if" part of the proposition is an immediate consequence of the definition of ladder moves.This proves the proposition.□ Remark 3.11.Recall that the set M (w) parametrizes irreducible components of the semi-toric limit of X w under the toric degeneration of G/B to Z(∆ i C (λ)) for λ ∈ P ++ .Hence if w ∈ W satisfies the conditions in Proposition 3.10, then the semi-toric limit of X w is irreducible, that is, it is a toric degeneration of X w .
Example 3.12.Fix 1 ≤ k ≤ N , and set w : is defined since w does not satisfy the condition (iii).

A path model to skew pipe dreams
In this section, we introduce a path model to skew pipe dreams.Let SY n be the set of (i, j) ∈ Z 2 such that 1 ≤ j ≤ n and such that 1 ≤ i ≤ 2j − 1.We visualize SY n in a way similar to SY n .For instance, write SY 3 as Note that the Gelfand-Tsetlin pattern for G = Sp that is an extension of SY n .Let us define a path model G (D) of a skew pipe dream D ∈ SPD n by representing Ω(D) using a diagram of "pipes" as follows.
• We first replace all boxes + in Ω(D) with "crossings" .
• For (i, j) ∈ SY (ex) n \ Ω(D) with j ≤ n and i is odd or with j = n + 1 and i is even, the (i, j)-th empty box is changed to an "elbow joint" .
• For (i, j) ∈ SY (ex) n \ Ω(D) with j ≤ n and i is even or with j = n + 1 and i is odd, the (i, j)-th empty box is replaced with a "reversed elbow joint" .
For each 1 ≤ k ≤ n, let ℓ k denote the pipe in G (D) that has an edge at the left end of the (2k − 1)-st row of SY In particular, it follows that ℓ(w 0 ) − ℓ(w D ) = 9 − 3 = 6 = |D|.Hence D is reduced.
The following is the main result of this section.To prove Theorem 4.5, we prepare some lemmas.Lemma 4.6.For every w ∈ W , the skew pipe dream D(w) is reduced, and the equality w D(w) = w holds.
Proof.By the definition of D(w), it follows that |D(w)| = ℓ(w 0 ) − ℓ(w).Hence it suffices to show that w D(w) = w.We write w = w 1 w 2 • • • w n as in the proof of Proposition 3.3.For each 1 ≤ j ≤ n, focus on the (2j − 1)-st and 2j-th rows of Ω(D(w)).Then we define a subdiagram G (j) (D(w)) of G (D(w)) by forgetting the pieces corresponding to other rows.The subdiagram G (j) (D(w)) is given as in Figure 4.2.In particular, ℓ j does not have a self-crossing if and only if w −1 j (j) < 0. Define pipes In addition, since w −1 (j) < 0 if and only if w −1 j (j) < 0, it holds that w −1 (j) < 0 if and only if ℓ j does not have a self-crossing, which is also equivalent to the condition that w −1 D(w) (j) < 0. From these, we deduce that w D(w) = w.This proves the lemma.Lemma 4.7.Let D ∈ SPD n , and assume that the ladder move L i,j (D) is defined.Then the equality w L i,j (D) = w D holds.Similarly, if the inverse ladder move L −1 i,j (D) is defined, then it holds that w L −1 i,j (D) = w D .Proof.We prove the assertion only for the ladder move L i,j (D); a proof of the assertion for ) of the ladder move L i,j (D) is obtained from G (D) as in Figure 4.3.This implies the assertion when j = n.
Let j ̸ = n.We prove the assertion only when j < n; a proof of the assertion when j > n is similar.In this case, the diagram G (L i,j (D)) is obtained from G (D) as in Figure 4.4, where we write L i,j (D) = D ∪ {(p, q + 1)} \ {(i, j)}.Hence we know the assertion when j < n.This completes the proof of the lemma.□ Proof of Theorem 4.5.We see by Lemmas 4.6 and 4.8 that L (D(w)) ⊆ RSP (w).Hence it suffices to show that RSP (w) ⊆ L (D(w)).Take D ∈ RSP (w).If there is no (i, j) ∈ D such that (i, j − 1) ∈ SY n \ D, then the argument in the proof of Lemma 4.6 implies that (a) The case q < n.
(b) The case q ≥ n.Hence we may assume that there exists (i, j) ∈ D such that (i, j − 1) ∈ SY n \ D. Take 1 ≤ k ≤ N to be the minimum positive integer such that (p k , q k ) ∈ SY n \ D and such that (p k , q k + 1) ∈ D. Let us prove that the inverse ladder move L −1 (p k ,q k +1) (D) is defined.We take two cases: q k = n or q k ̸ = n.
If 4.5), which gives a contradiction since D is reduced.Hence we have (t, n) / ∈ D, which implies that the inverse ladder move L −1 (p k ,q k +1) (D) is defined.Let q k ̸ = n.We prove the assertion only when q k < n; a proof of the assertion when q k > n is similar.Set By the definitions of k and k ′ , it follows that (p ℓ , q ℓ ), (p ℓ , q ℓ + 1) ∈ D for all k ′ < ℓ < k such that q ℓ ∈ {q k , 2n − q k }.If (p k ′ , q k ′ ) ∈ D, then w D = w D ′ for D ′ := D \ {(p k , q k + 1), (p k ′ , q k ′ )} (see Figure 4.6), which gives a contradiction since D is reduced.Hence we have (p k ′ , q k ′ ) / ∈ D, which implies that the inverse ladder move L −1 (p k ,q k +1) (D) is defined.Continue this argument by replacing D with L −1 (p k ,q k +1) (D).Then, by a sequence of inverse ladder moves, we can change D to D such that there is no (i, j) ∈ D with (i, j − 1) ∈ SY n \ D. Since inverse ladder moves preserve RSP (w) by Lemma 4.8, we have D ∈ RSP (w), which implies that D = D(w).Hence D is obtained from D(w) by a sequence of ladder moves, that is, D ∈ L (D(w)).This proves the theorem.□

Mitosis recursion for reduced skew pipe dreams
In this section, we prove that the set M (w) can be constructed by transposed skew mitosis operators.The following is the main result of this section.Theorem 5.1.For w ∈ W and 1 ≤ j ≤ n such that ℓ(w) < ℓ(ws j ), it holds that L (D(ws j )) = M j (L (D(w))) = mitosis ⊤ j (L (D(w))).In particular, for (j 1 , . . ., j ℓ ) ∈ R(w), the following equalities hold:  To prove Theorem 5.1, we prepare some lemmas.
If j > 1, then by removing {(p r 0 , q r 0 )} ∈ D, the diagram G (D) is changed to G (D \ {(p r 0 , q r 0 )}) as in Figure 5.2.Hence we have D \ {(p r 0 , q r 0 )} ∈ RSP (ws j ) since D ∈ L (D(w)) = RSP (w).This completes the proof of the lemma.□ For 1 ≤ j ≤ n, set SY n , the set SPD n of skew pipe dreams can be regarded as a subset of SPD (j) n .For (p, q) ∈ SY n , define the ladder move L (j) p,q on SPD (j) n in a way similar to the ladder move L p,q on SPD n .When we consider SPD (j) n , we allow ladder moves that move The case q r0 ≥ n. for (p, q) ∈ SY n .The inverse ladder move (L (j) p,q ) −1 on SPD (j) n for (p, q) ∈ SY (j) n is similarly defined.Definition 5.3.Let x, y be indeterminates, and Z[x, y] the polynomial ring with Z-coefficients.= .
By the definition of mitosis ⊤ j , the argument in the proof of Lemma 5.8 can also be applied to the polynomial F (j) w (x, y), which implies the following.Lemma 5.9.The polynomial F (j) w (x, y) is invariant under the action of T x,y .Lemma 5.10.If ℓ(w) < ℓ(ws j ), then it holds that F w (x, y) + yF ws j (x, y) = T x,y (F w (x, y)) + xF ws j (x, y) = F (j)  w (x, y).x D = F (j) w (x, y). (5.3) In addition, it follows that T x,y (F w (x, y)) + xF ws j (x, y) = T x,y (F w (x, y)) + T x,y (yF ws j (x, y)) (by Lemmas 5.6 and 5.8) = T x,y (F (j) w (x, y)) (by (5.3)) = F (j)  w (x, y) (by Lemma 5.8).

Hence we deduce that
D∈mitosis ⊤ j (L (D(w))) x D = F ws j (x, y) = D∈L (D(ws j )) x D .
By comparing the numbers of terms, we conclude that |mitosis ⊤ j (L (D(w)))| = |L (D(ws j ))|.This proves the theorem.□ and H is a maximal torus of G. Hence the quotient variety G/B is the full flag variety of type C n .We identify the set I of vertices of the Dynkin diagram with [n] := {1, 2, . . ., n} as follows: and c i,j = 0 when |i − j| ≥ 2. Let N G (H) denote the normalizer of H in G, and W := N G (H)/H the Weyl group of type C n .Since ω( H) = H and ω(N G ( H)) = N G ( H), the automorphism ω induces an automorphism ω : W ∼ − → W . Then the Weyl group W of type C n is identified with the fixed point subgroup ( W ) ω .Write f −n := e 1 , . . ., f −1 := e n , f 1 := e n+1 , . . ., f n := e 2n .By the action of W on {Ce 1 , . . ., Ce 2n
ascending order.For instance, a skew pipe dream D = {(1, 2),(1,4),(1,5), (2, 4)} ∈ SPD 3 gives a sequence k ′ D =(1,3,4,7,9).For w ∈ W , let D(w) ∈ SPD n denote the skew pipe dream determined by the condition that k ′ D(w) = k w .If w = e, then we define D(e) to be SY n .It follows by[10, Proposition 6.3] that there exist unique elements m(w, 1), . . ., m(w, n) ∈ Z ≥0 such that where m ′ (w, n − k + 1) := 2k − 1 − m(w, n − k + 1).Then we see by the definition of D(w) that w = w ≤n .Let us prove that m(w, i) = m(w ≤k , i) for all n − k + 1 ≤ i ≤ n by induction on k.If k = 1, then the element w ≤1 = w 1 is the identity element e or s 1 depending on whether (n, n) ∈ D(w) or not.Hence the assertion is obvious in this case.If k ≥ 2, then we see that w ≤k−1 (j) = j for all j ∈ {±q | k ≤ q ≤ n} by the definition of w ≤k−1 .Hence it holds that m(w ≤k−1 , n − k + 1) = 2k − 1.By the definition of w k , it follows that Then it is obvious that w satisfies the conditions in Proposition 3.10.In addition, the face of ∆ i C (λ) corresponding to D(w) is precisely the string polytope associated with λ and the reduced word (i 1 , i 2 , . . ., i k ) for w (see[27, Section 1]).Hence the toric degeneration of X w in Remark 3.11 is precisely the one given in [6,Section 3].Example 3.13.Let n = 3, and w = s 1 s 2 s 1 s 3 s 2 ∈ W . Then we have D(w) = + + + + , which satisfies the conditions (i) and (ii).However, the ladder move L 2,2 (D(w)) = + + + + 2n (C) in[27, Section 6]  is arranged as the boxes in SY n .Denote by SPD n the power set of SY n , and describe D ∈ SPD n by putting + in the boxes corresponding to the elements of D. For each D ∈ SPD n , we define its Gelfand-Tsetlin type rearrangement Ω(D) ∈ SPD n as follows.For k ∈ Z >0 , the (2k − 1)-st row of Ω(D) is defined to be the first (n − k + 1) boxes in the k-th row of D. The 2k-th row of Ω(D) is given by reversing the last (n − k) boxes in the k-th row of D. Example 4.1.Let n = 3, and D = {(1, 1), (1, 2), (1, 5), (2, 2), (2, 3), (3, 3)} ∈ SPD 3 .Then we have

Definition 4 . 3 .
there exists a unique permutation v D : [n] → [n] such that the pipe ℓ k has an edge at the top of the (n + 1 − v D (k))-th column of SY (ex) n .We write ℓ k = L v D (k) .For D ∈ SPD n , we define w D ∈ W by w D (j) := (−1) sign(L j )+1 v −1 D (j) and w D (−j) = −w D (j) for j ∈ [n], where sign(L j ) denotes the number of self-crossings of the pipe L j in G (D).The skew pipe dream D is said to be reduced if |D| = ℓ(w 0 ) − ℓ(w D ).By definition, we have ℓ |w D (j)| = L j for all j ∈ [n].

Theorem 4 . 5 .
For all w ∈ W , the set L (D(w)) coincides with the set of reduced skew pipe dreams D ∈ SPD n such that w D = w.

Figure 4 . 2 .
Then it follows by Figure 4.2 that ℓ j = L (j)
SY n ⊔ {(0, n + j)}, and denote by SPD (j) n the power set of SY (j) n .Since SY n ⊆ SY (j)

D∈LLemma 5 . 8 .
(j) (D(w))x D .Define a Z-algebra automorphism T x,y : Z[x, y] → Z[x, y] by permuting x and y, that is,T x,y (f (x, y)) := f (y, x) for f (x, y) ∈ Z[x, y].The polynomial F (j)w (x, y) is invariant under the action of T x,y .