Following Schubert varieties under Feigin's degeneration of the flag variety

We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the degenerations in terms of symmetric group combinatorics. As a side result, we obtain an identification of some Schubert varieties with Richardson varieties in higher rank partial flag varieties.


Introduction
Let G be a complex simple Lie group and let P ⊂ G be a parabolic subgroup. In [Fei12], Feigin introduced a flat degeneration of the flag variety G/P , which is equipped with an action of the M-fold product of the additive group of the field (M being the dimension of a maximal unipotent subgroup of G). These degenerations of flag varieties (and some generalizations in type A) have been in the past years intensively studied from many different perspectives (see, for example, [Fei11], [CIFR12], [CIL15], [Fou16], [CIFF + 17], [LS19]).
In this paper, we deal with the effect of Feigin's degeneration on the Schubert varieties inside Fℓ n := SL n /B, for B the Borel subgroup of upper triangular matrices. In [Fei12] it is shown that in type A the degeneration Fℓ a n of Fℓ n can be embedded into the product of projective spaces, exactly as Fℓ n , and that the defining ideal is generated by degenerate Plücker relations. More precisely, the defining ideal I Fℓn of Fℓ n is generated by Plücker relations and the defining ideal I a Fℓn is obtained as the initial ideal in w (I Fℓn ) with respect to a weight vector w (whose components are indexed by Plücker coordinates), as described in Section 2.2.1. On the other hand, if v ∈ S n is a permutation, it is well-known that the ideal I v of the Schubert variety X v = BvB/B ⊆ Fℓ n is generated by the Plücker relations together with certain Plücker coordinates (see §2.3 for a more precise formulation). Thus it is natural to ask what happens to I v under Feigin's degeneration, that is to investigate in w (I v ).
From the first non-trivial example, it is already clear that not all Schubert varieties under Feigin's degeneration will stay irreducible: for n = 3, indeed, one of the six Schubert varieties degenerates to a reducible variety. Therefore, a consistent part of this paper is directed towards understanding this reducibility phenomenon.
We recall that the cohomology ring of Fℓ n can be identified (after doubling the degree) with its Chow ring, and the latter is generated by Schubert classes. Moreover, it is shown in [CIFR12] that Fℓ a n admits an affine paving and hence also its cohomology ring can be identified (up to doubling the degree) with its Chow ring. Therefore we expect the above mentioned reducibility phenomenon to be related to the surjectivity of the cohomology ring map ψ : H * (Fℓ a n , Z) → H * (Fℓ n , Z) proven in [LS19]. It would be interesting to investigate this relationship, and in particular deduce a description of the kernel of ψ in terms of Schubert classes. We should mention here that what we refer to as Feigin's degeneration is in fact a modified version of his original construction, which was coming from Lie theory. The version we deal with in this paper is the one which has been studied in [CIL15]. The variety one obtains in this way is isomorphic to Feigin's original degeneration, but in some sense it behaves better with respect to Schubert varieties. In fact, Caldero noticed in [Cal02] that there does not exist a (flat) toric degeneration of the flag variety under which all Schubert varieties degenerate to toric varieties. For n = 3 (which is the only case, apart from n = 2, in which Fℓ a n is toric) our version of the degeneration preserves irreducibility of all but one Schubert variety, while two Schubert varieties would become reducible under Feigin's original definition. This is why we feel that in this setting the definition we use is sort of optimal.
Before focusing on Schubert varieties which become reducible after degenerating, we first describe some cases in which they stay irreducible (see Section 3). In particular, we prove that there is a class of Schubert varieties (indexed by permutations which are less or equal than a distinguished Coxeter element) whose defining ideals are not affected by the degeneration (see Proposition 2).
Section 4 is devoted to sufficient conditions on the permutation v such that the initial ideal in w (I v ) is not prime. The strategy is to look for Plücker relations whose initial term is a (degree 2) monomial when considered modulo the Plücker coordinates which vanish on X a v := V (in w (I v )), which coincide with the ones vanishing on X v . The efficiency of some of the conditions we give is then tested by looking at the n = 4 and n = 5 examples, for which we can detect all initial ideals containing monomials (see Tables 1 and 2).
In previous joint work with Cerulli Irelli [CIL15], the second author proved that the degenerate flag variety Fℓ a n can be embedded in the flag variety SL 2n−2 /P of partial flags in C 2n−2 consisting of odd dimensional spaces (that is, P = P ω 1 +ω 3 +...ω 2n−3 ). Under this embedding, it was shown in [CIL15] that Fℓ a n is isomorphic to a Schubert variety. From this fact (together with classical results) one could obtain a new proof of projective normality, Frobenius splitting, and rationality of the singularities of Fℓ a n . In Section 5 we further exploit such an isomorphism and study the effect of Feigin's degeneration on Schubert varieties inside SL 2n−2 /P . The idea is to show irreducibility of the degeneration of some Schubert variety by proving that the abovementioned embedding sends it to a Richardson variety. Although our main focus is the analysis of Plücker relations (cf. Sections 4 and 3), for which there is no need to move to a higher rank (partial) flag variety, we decided to dedicate a section to the connection with Richardson varieties. By comparing Proposition 2 with Lemma 7 we obtain a realization of some Richardson varieties inside SL 2n−2 /P as Schubert varieties in a lower rank (complete) flag variety.
The last section of the paper deals with Schubert divisors, that is Schubert varieties of codimension one in Fℓ n . By applying our reducibility criteria from Section 4, we are able to prove that if n is even all Schubert divisors become reducible, while for n odd this happens for all but one. In this case, the remaining divisor is shown to be isomorphic to a Richardson variety inside SL 2n−2 /P , and hence irreducible.
We want to point out that our paper is very different in spirit from [Fou16], where irreducible flat degenerations of Schubert varieties corresponding to some special Weyl group elements (triangular elements) are produced by considering PBWdegenerations of Demazure modules V w (λ) and then realizing the desired degeneration as the closure of an appropriate G M a -orbit inside P(V w (λ)). So for any Schubert variety which is indexed by a triangular element (see [Fou16, Definition 1]) one can construct a flat irreducible degeneration via Fourier's procedure, while in this article we fix the degeneration (Feigin's) of the whole flag variety and study its effect on Schubert varieties (which are hence simultaneously degenerated).
Since a first draft of this paper appeared on the arxiv more research has been done regarding degenerations of Schubert varieties. Among them [CM20], where similar methods are employed to study Gröbner degenerations of Schubert varieties, and [CFF21,Kam22] which are very different in flavour and closely related to [Fou16].

Preliminaries and notation
2.1. Symmetric group combinatorics. The combinatorics of the symmetric group control many geometric properties of Fℓ n and its Schubert varieties, therefore we spend a little time here introducing the notation we will need later on.
For any two positive integers i, j ∈ Z ≥1 , with i ≤ j we denote by [i, j] := {a ∈ Z | i ≤ a ≤ j}. Moreover, we use the short hand notation [j] := [1, j]. We write [n] k for the set of subsets of cardinality k inside [n].
Let n ≥ 2 and denote by S n the symmetric group. Recall that the symmetric group S n admits a presentation as a Coxeter group, with set of simple reflections {s i | i = 1, . . . , n − 1}, where s i denotes the transposition (i, i + 1). We will use the standard terminology and say that a product s i 1 . . . s ir is a reduced expression for v ∈ S n if v = s i 1 . . . s ir and all other expressions of v as a product of simple reflections v = s j 1 . . . s jt are such that t ≥ r. In this case r = ℓ(w) is called the length of w. We denote by ≤ the Bruhat order on S n and recall the following equivalent characterization (see, for example, [BB05, Theorem 2.1.5]): For v ∈ S n and i, j ∈ [n] set In the following we will also need that if v ∈ S n and i ∈ [n − 1], then The symmetric group S n acts on [n] k for any k: This action is transitive, so that [n] k is identified with the S n -orbit of [k] and hence with the set of minimal length coset representatives in S n / s 1 , . . . , s k−1 , s k+1 , . . . , s n−1 . In this way the Bruhat order on S n induces a partial order on [n] k (see, for instance, [BB05, Proposition 2.5.1]) that we also denote by ≤. We will sometimes write elements v ∈ S n as [v(1), v(2), . . . , v(n)]. This is referred to as the one-line notation.
2.1.1. Sequences. In the following sections, we will often need to deal with sequences (i 1 , . . . , i k ) rather than sets {i 1 , . . . , i k }. We denote by S(n, k) the set of sequences of k pairwise distinct numbers between 1 and n.
If L ∈ S(n, d) and J = (j 1 , . . . , j k ) ∈ S(n, k), then the sequence L ′ = (L \ (l r 1 , . . . , l r k )) ∪ (j 1 , . . . j k ) ∈ S(n, d) is obtained from L by substituting the subsequence (l r 1 , . . . , l r k ) with (j 1 , . . . , j k ), that is l ′ a = l a if a ∈ {r 1 , . . . , r k } and l ′ a = j b if a = r b . There is a forgetful map By abuse of notation, if I ∈ S(n, k) and v ∈ S n , we will write ), etc., will have an analogous meaning).
2.2. Basics on the flag variety. Let n ≥ 2. We denote by Fℓ n the variety of complete flags in C n . Let (e i ) 1≤i≤n be the standard basis of C n . Let B ⊂ SL n be the Borel subgroup of upper triangular matrices. The group SL n acts (by base change) transitively on Fℓ n and we can identify the flag variety with the quotient SL n /B by looking at the SL n -orbit of the standard flag E • = ({0} ⊂ E 1 ⊂ · · · ⊂ E n−1 ⊂ C n ) ∈ Fℓ n with E i := span C {e 1 , . . . , e i } (i = 1, . . . n − 1).
Recall that under the left action of B, the flag variety decomposes as a union of Schubert cells indexed by the elements of the symmetric group S n : where, by abuse of notation, v in BvB/B denotes the corresponding permutation matrix in SL n . Finally, let X v be the Schubert variety, that is the closure BvB/B of a Schubert cell. Analogously, also B − , the Borel subgroup of lower triangular matrices, acts by left multiplication on SL n /B, providing the decomposition: We denote by X u the opposite Schubert variety B − uB/B. In §5, we will also consider Richardson varieties X u v := X v ∩ X u .
2.2.1. Plücker relations. Our main reference for Plücker coordinates and relations is [Ful97], while we refer to [Fei12] for the degenerate Plücker relations. We start by recalling the Plücker embedding of a Grassmannian. Recall that (e i ) 1≤i≤n is the standard basis of C n , so that is a basis of ∧ k C n . Let (∧ k C n ) * be the dual vector space, then the Plücker coordinate p i 1 ,...,i k ∈ (∧ k C n ) * for 1 ≤ i 1 < i 2 < . . . < i k ≤ n is defined to be the basis element dual to e i 1 ∧ . . . ∧ e i k . For i 1 , . . . , i k ∈ [n] pairwise distinct, but not necessarily increasing, the Plücker coordinate p i 1 ,...,i k has the following property Denote by p I the Plücker coordinate corresponding to a sequence I = (i 1 , . . . , i k ) ∈ S(n, k). In the following sections it will be sometimes convenient to simplify notation and index some Plücker coordinates by a set instead of a sequence. This has to be interpreted as being indexed by the sequence obtained by arranging the elements of the set in an increasing order.
We have obtained in this way the Plücker embedding The flag variety is embedded in the product of Grassmannians Fℓ n ֒→ Gr(1, C n ) × Gr(2, C n ) × · · · × Gr(n − 1, C n ).
By composing the latter embedding with the embedding (2.4) for each Grassmannian in the product, we get Fℓ n ֒→ PC n × P(∧ 2 C n ) × · · · × P(∧ n−1 C n ).
] with respect to this embedding. Then I Fℓn is generated by elements in {R k (j 1 ,...,je),(l 1 ,...,l d ) | e ≤ d, k ∈ [e]} given by where L = (l 1 , . . . , l d ) ∈ S(n, d), J = (j 1 , . . . , j e ) ∈ S(n, e), L ′ = (L \ (l r 1 , . . . , l r k )) ∪ (j 1 , . . . , j k ) and J ′ = (J \ (j 1 , . . . , j k )) ∪ (l r 1 , . . . , l r k ). The elements R k J,L will be referred to as Plücker relations. To simplify notation we set L k J,L = (J ′ , L ′ ) | ∃1≤r 1 <···<r k ≤#L, J ′ =(J\(j 1 ,...,j k ))∪(lr 1 ,...,lr k ), L ′ =(L\(lr 1 ,...,lr k ))∪(j 1 ,...,j k ) . (2.6) The weight vector w ∈ R ( n 1 ) +···+( n n−1 ) is defined componentwise by setting for k , the w-weight of the monomial r t=1 p It is r t=1 w It , while the initial form of a polynomial f consists of the sum of those monomials whose w-weight is minimal among the weights of all monomials in f . Given an ideal A (finite) set of elements in I whose initial forms generate in w (I) is called a Gröbner basis. In [Fei12, Theorem 3.13] computes a Gröbner basis for I Fℓn whose elements are the Plücker relations (2.5). Their initial forms are given by where the leading term is non-zero, only if We can choose J, L in such a way that (2.7) holds. Observe that for e = d, we always have in w (R k J,L ) = R k J,L since the condition (2.7) is empty. Definition 1 ( [Fei12]). The degenerate flag variety is the vanishing of the ideal in w (I Fℓn ), that is Fℓ a n := V (in w (I Fℓn )) ⊂ PC n × P(∧ 2 C n ) × · · · × P(∧ n−1 C n ). Remark 1. Feigin's original definition, valid for any simple Lie group, was different from the one we have just given, which is a characterization of the type A degenerate flag variety by [Fei12, Theorem 3.13]. As already mentioned in the introduction, we modify Feigin's definition to match the one considered in [CIL15]. Explicitly, to obtain our degeneration from Feigin's original one, a global shift by −1 (modulo n) to all indices is needed.

Ideals for Schubert varieties and their degeneration.
For v ∈ S n the defining ideal of the Schubert variety X v ⊂ Fℓ n is given by the vanishing of (p I ) I ≤v([#I]) . It is shown in [LLM98, §10.12] (see also [KR87,Theorem 3]) that by embedding Feigin's degeneration of the flag variety induces a degeneration X a v ⊂ Fℓ a n of any Schubert variety X v ⊂ Fℓ n : In what follows we study the initial ideals in w (I v ) in detail. Note that in w (p I ) = p I for all I ∈ S(n, d), for all d ∈ [n − 1]. Moreover, we have an inclusion: The following example shows that this inclusion may be strict. In the proof of Theorem 1 instead we will encounter examples of (2.10) being an equality.
Example 1. Consider the ideal I Fℓ 4 . Among its Plücker relations we have The first two terms have w-weight 2 while the last to have w-weight 3, so its initial form is p 4 p 123 − p 3 p 124 . Now consider v = s 1 s 2 s 3 ∈ S 4 , which in the one-line notation is [2, 3, 4, 1]. Hence, {p I } I ≤v([#I]) = {p 3 , p 4 , p 14 , p 24 , p 34 }. In particular, this implies that f := p 2 p 134 − p 1 p 234 ∈ I v and by definition its initial form lies in in w (I v ). As both monomials have the same w-weight, f agrees with its initial form. Notice however that f does not lie in (in w (R k J,L )) k,J,L + (p I ) I ≤v([#I]) . This demonstrates that the containment in (2.10) is strict in general.

Two classes of irreducible X a v
We investigate two classes of Schubert varieties which degenerate to irreducible varieties. In this section we use some basics on Gröbner bases which we summarize for completeness. For more details we refer to [HH11,Stu96].
A term order on C[x 1 , . . . , x n ] is a total order < on the set of monic monomials in Let in < (J) be a monomial initial ideal of the ideal J for some term order < on C[x 1 , . . . , x n ]. Then the set Let < be a term order on C[x 1 , . . . , x n ] and G = {g 1 , . . . , g s } a finite generating set of an ideal J. Then the S-polynomial of g i and g j is defined as Buchberger's criterion says that G is a Gröbner basis if and only if for all 1 ≤ i < j ≤ s the S-polynomial S(g i , g j ) reduces to zero with respect to {g 1 , . . . , g s }, see e.g [HH11, Theorem 2.3.2].
3.1. Small flag varieties. The main result of this section is the following.
Theorem 1. Let v ∈ S n be the minimal representative of the longest word in S n / s 1 , . . . , s i , s i+r , . . . , s n−1 for suitable i and r. Then In particular, X a v is irreducible. Before we prove the result, let us establish some useful lemmata. Note that written in one-line notation v is of form v = [1, 2, . . . , i, i + r, i + r − 1, . . . , i + 1, i + r + 1, . . . , n].
Lemma 1. For the Schubert variety we have X v ∼ = Fℓ r . In particular, the only non-vanishing Plücker coordinates besides p [s] for s ≤ n − 1 are associated with the index sets in Proof. There is a bijection Then ρ((R k J,L ) k;J,L∈Jv ) = I Fℓr . Next, we establish a connection between the defining ideal of the degenerate flag variety Fℓ a r and the initial ideal defining X a v . We keep the notation introduced in (3.2) and (3.1).
Lemma 2. Letw be the weight for Fℓ r , then ρ(in w ((R k J,L ) k,J,L∈Jv )) = inw(I Fℓr ). Proof. Let L = ((1, . . . , i), (l 1 , . . . , l d )) > J = ((j 1 , . . . , j e ), (1, . . . , i)). Consider the relation R k J,L . Without loss of generality we can assume that J and L are chosen in such a way that in w (R k J,L ) contains the monomial p J p L . All other monomials In what follows we use Feigin's standard monomial basis given by semistandard PBW-tableaux. As we work throughout the paper with conventions for the weight vector w as in [CIL15] a global shift in the indices of all Plücker variables is needed before we can use Feigin's basis in our setting. Whenever we use the combinatorics from [Fei12] in this section we assume we have applied the global shift to our index sets.
Recall that by [ Moreover, f = p v R k J,L for a fixed monomial p v that divides p a T and certain k,J,L. Givenw and the partial order ≤ we define a term order on C[pĨ :Ĩ ⊂ [r]] as follows: (3)w·u =w·v, p u and p v are not comparable with respect to ≤, and p u < lex p v . Here < lex denotes the lexicographic order on C[pĨ :Ĩ ⊂ [r]] with underlying lexicographic order on the variables corresponding to their index sets. Our term order ≺ is a refined version of a term order induced by a weight (see, for example the order ≺ w in [Stu96, page 4]). In particular, [Stu96, Proposition 1.8] holds also in our case and we have From [Fei12, Proof of Lemma 4.9] it follows that for p a T and f as above we have In particular, the cosets of the standard monomials, i.e. p u ∈ in ≺ (I Fℓ r ), form a (standard monomial) basis for C[pĨ :Ĩ ⊂ [r]]/I Fℓr . By (3.3) they also form a basis for C[pĨ :Ĩ ⊂ [r]]/ inw(I Fℓr )), denote it by B ≺ . In particular, we deduce from (3.4) that every standard monomial corresponds to a semistandard PBW-tableaux. Hence, B ≺ ⊂ B PBW . But as both are bases for the same algebra they have to be equal. This implies the first claim. The second follows as f = p v R k J,L , and so in particular p a T ∈ (in ≺ (R k J,L )) k,J,L . Proposition 1. The set {R k J,L } k,J,L∈Jv ∪ {p I } I ∈Jv is a Gröbner basis for I v and w, denoted by G v;w .
Proof. We use ≺ as defined in the proof of Lemma 3 and the map ρ from (3.2) to define a term order on C[p I : I ⊂ [n]]: By definition of < and Lemma 2 we have in . Moreover, as the R k J,L constitute a Gröbner basis for I Fℓr and ≺ by Lemma 3, it follows from Buchberger's criterion that the S-polynomials of pairs of these elements reduce to zero. Given the map ρ, the same must be true for S-polynomials of elements R k J,L with J, L ∈ J v with respect to the term order <. Hence, in order to verify the claim we only need to compute S-polynomials of the relevant Plücker relations and the vanishing Plücker variables. Consider R k J,L with J, L ∈ J v and p I with I ∈ J v . Then in < (R k J,L ) and in < (p I ) are relatively prime. So by [HH11, Lemma 2.3.1] their S-polynomials reduces to zero over R k J,L and p I . As the same is true for the Spolynomials of variables p I , p I ′ with I, I ′ ∈ J v , the claim follows by Buchberger's criterion.
Proof of Theorem 1. We need to show that the isomorphism of X v and Fℓ r induced by ρ survives the degeneration. This is true as by Lemma 2 and Proposition 1 ρ maps the initial ideal defining X a v to the ideal defining Fℓ r . Lastly, by [Fei12, §5.1] the degenerate flag variety is the closure of a homogeneous space and therefore irreducible. As X a v ∼ = Fℓ a r by the above, the claim follows.
. We set m := min{i}, M := max{i}, and r := M − m + 1. Let v ∈ s i 1 , · · · , s ir ⊂ S n and denote by v the element s i 1 −m+1 · · · s ir−m+1 ∈ S r . In this notation, from the proof of Theorem 1 we can deduce the following result, which in this case allows one to reduce to smaller rank flag varieties.
and v ∈ s i 1 , · · · , s ir ⊂ S n . Then for X a v ⊂ Fℓ a n we have 3.2. Isomorphic degenerate and original Schubert varieties. In the following we present another instance in which a Schubert variety stays irreducible under Feigin's degeneration of Fℓ n . In fact, for the class of varieties we deal with in this section a stronger property holds: the degeneration process does not deform them, that is X a v is isomorphic to the original Schubert variety X v . Recall that we denote by c ∈ S n the special Coxeter element c = s n−1 s n−2 · · · s 2 s 1 . We claim that in this case ) . Hence, Consider any term order < so that in < (I v ) = in < (in w (I v )). Such a term order exists as I v is homogeneous. Then the reduced Gröbner basis for I v with respect to < is also a reduced Gröbner basis for in w (I v ) with respect to <. As reduced Gröbner bases are unique the claim follows.

Criteria for reducibility
In this section we examine when Schubert varieties become reducible after degenerating. We give a number of sufficient conditions for certain monomials of degree two to be contained in the initial ideal in w (I w ) for w ∈ S n making repeated use of (2.10).
, is called an honest monomial if f has degree at least 2 and f ∈ (p I ) I ≤w([#I]) .
The following Lemma is straightforward: Lemma 4. Let w ∈ S n . If in w (I w ) contains an honest monomial then it is reducible.
4.1. Relations between Gr(1, C n ) and Gr(2, C n ). We start the discussion by focusing on very special Plücker relations, namely those between Plücker coordinates on Gr(1, C n ) and on Gr(2, C n ). In this case, we can classify the w ∈ S n for which in w (I w ) contains an honest monomial of this form.
For v ∈ S n denote by v the minimal length representative of the coset of v in S n / s 2 , s 3 . . . . s n−1 and v the minimal length representative of the coset of v in S n / s 1 , s 3 , s 4 , . . . , s n−1 .
The conditions on v and v in Theorem 2 are depicted for S 4 with j = 2, k = 4 in Figure 1. Proof. To simplify notation, for a ∈ [n] we denote p a := p (a) , and for a, b ∈ [n] we write p a,b instead of p (a,b) . We will only consider Plücker coordinates corresponding to increasing sequences in this proof and hence adapt the signs. Consider for 1 ≤ i < j < k ≤ n the Plücker relation R 1 (i),(j,k) = p i p j,k −p j p i,k +p k p i,j . Note that if in w (R 1 (i),(j,k) ) = R 1 (i),(j,k) the relation will not produce an honest monomial in in w (I w ) for any w ∈ S n as I w is prime. Note that R 1 (i),(j,k) = in w (R 1 (i),(j,k) ) only if i = 1. In this case in w (p 1 p j,k − p j p 1,k + p k p 1,j ) = −p j p 1,k + p k p 1,j .
As j < k, if p j vanishes on the Schubert variety X v , then so does p k . Hence, both monomials are zero on X v . Similarly, if p 1,j vanishes on X v , then so does p 1,k . Our aim is to determine v ∈ S n such that one of the two terms of in w (R 1 (i),(j,k) ) lies in (p I ) I ≤v([#I]) but the other does not as in this case, the ideal in w (I v ) contains an honest monomial. A priori, there are two cases for the restriction of p k and p 1,k to X v : (1) p 1,k = 0 and p k = 0, (2) p 1,k = 0 and p k = 0. We will show that in fact the second case can never happen. Both cases yield conditions on v and v (keeping also in mind that we do not want p j and p 1,j to vanish). In the first case we have the following conditions s j−1 s j−2 · · · s 2 s 1 ≤ v ≤ s k−2 s k−3 · · · s 2 s 1 and s k−1 s k−2 · · · s 3 s 2 ≤ v, (4.1) respectively, in the second case we have s k−1 s k−2 · · · s 2 s 1 ≤ v and s j−1 s j−2 · · · s 3 s 2 ≤ v ≤ s k−2 s k−3 · · · s 3 s 2 . Assume v ∈ S n is chosen such that the minimal length representatives of the cosets fulfill the inequalities in (4.2). Then for some x ∈ s 1 , s 3 , . . . , s n−1 . Observe that s k−1 · · · s 1 (1) = k and With the notation as in (2.1) this implies (s k−1 · · · s 1 ) 1,k = 1 > (s k−2 · · · s 2 x) 1,k = 0. But s k−1 · · · s 1 ≤ s k−2 · · · s 2 x, contradicting (2.2). Hence, case (4.2) never applies.
Remark 2. Theorem 2 is enough to detect all Schubert varieties in Fℓ 3 ֒→ Gr(1, C 3 )× Gr(2, C 3 ) which become reducible under Feigin's degeneration. In fact, the only Schubert variety having this property is the one indexed by s 1 s 2 . All the other permutations but the longest element (which indexes the Schubert variety corresponding to the irreducible variety Fℓ a n ) are ≤ c = s 2 s 1 and hence, by Proposition 2, are irreducible.

4.2.
Monomials from other relations. Theorem 3 (1) to (5) provide sufficient conditions on w ∈ S n for the initial ideal in w (I w ) to contain a degree two honest monomial originating from a Plücker relation between Plücker coordinates on adjacent Grassmannians, that is Gr(k, C n ) and on Gr(k + 1, C n ) for suitable k. Notice that here we are only producing sufficient conditions, so that for k = 1 we clearly obtain a weaker result than Theorem 2. Theorem 3 (6) and (7) deal with Plücker relations between not necessarily adjacent Grassmannians.
Table 1 (resp. Table 2 in the appendix) show to which permutations w ∈ S 4 (resp. S 5 ) each one of the points of Theorem 3 applies. The computations for these were performed in Sage [Dev16] and Macaulay2 [GS].
Let w ∈ S n . In the following, it will be convenient to set w([0]) := ∅. Moreover, since in w (I e ) = I e , we can exclude the case w = e right away in the following theorem.
(2) Assume such i, l, x exist. Let J be any sequence such that F (J) = (w([i − 1]) ∪ {w(i + 1)}) \ {i − 1} and j 1 = x, and let L be any sequence such that Taking the initial form with respect to w we obtain ) ∪ {w(i + 1)} and so restricting to X w we have p (L\(l))∪(x) = 0 as ws i > w. So in w (I w ) contains the monomial p J p L .
, but restricting to X w we have p (J\(i))∪(j) = 0 as j > i + 1. Hence, in w (I w ) contains the monomial p J p L . Consider the relation R 1 J,L : Taking the initial form with respect to w yields (6) First note that as w = e we have that w(i) = i in particular implies i < n.
Consider J any sequence such that As w(i) ∈ [j − 1] implies w(i) > j − 1 and so w(j) > w(i) > j − 1, then the set [j − 1] ∪ {w(j)} has cardinality j. So, Taking the initial form with respect to w yields in w (R 1 J,L ) = p J p L − p (w(j),[i−1]) p (L\(w(j)))∪(w(i)) . Since w(j) > w(i), the coordinate p (w(j),[i−1]) vanishes in the coordinate ring of X w , so that in w (R 1 J,L ) ∈ in w (I w ) is a monomial.
, and hence we get ) p (L\(w(j)))∪(i) . Further observe that w(j) > w(i + 1) ≥ i, which implies that p (w(j),w([i])) vanishes in the coordinate ring of X w . Then R 1 J,L produces a monomial.
Remark 4. In the points (6) and (7) of Theorem 3, such a j need not exists, in which case the criterion would simply not apply.
4.2.1. Efficiency of the various criteria from Theorem 3. We want to comment here on how efficient the various criteria of Theorem 3 are, based on the data we have collected for S 4 (see Table 1) and S 5 (see Table 2). The data can be found at the homepage: https://www.matem.unam.mx/~lara/schubert/. For n = 4, there are 11 permutations w such that at least one Plücker relation degenerates to a monomial. In the S 5 -case, this happens for 85 permutations.
Among the criteria collected in Theorem 3, point (6) seems to be the most powerful: it detects 9 out of 11 permutations for S 4 , and 65 out of 85 for S 5 . To cover the missing two permutations for S 4 it is enough to combine Theorem 3 (6) with one of the points (1),(4),(7) and one between (2) and (5). So that it is enough to apply three of our criteria to find all w ∈ S 4 such that in w (I w ) contains a Plücker relation which degenerates to a monomial.
Theorem 3 (1) picks 9 out of 11 permutations in S 4 , and 64 out of 85 for S 5 . Theorem 3 (3) covers 8 out of 11 permutations yielding monomial initial ideals for S 4 and 57 out of 85 for S 5 .
Finally, Theorem 3 (7) applies to only one permutation, resp. 8 permutations, in the n = 4, resp. n = 5, case, but it is necessary to cover all the permutations in S 5 containing monomial degenerate Plücker relations. For example, it is the only one among our criteria which can be applied to s 1 s 2 s 3 s 4 s 3 s 1 s 2 s 1 .

4.3.
Plücker relations not degenerating to monomials. In this section we study some cases in which none of the Plücker relations produces a monomial in the defining ideal in w (I w ). Clearly, this does not have to be equivalent to the irreducibility of the degeneration, but it turns out to be the case for n = 3 (by Remark 2) and n ∈ {4, 5} (by Macaulay2 computations). We do not know whether such an equivalence holds in general.
We have seen in §3.2, that if v ≤ c = s n−1 s n−2 · · · s 2 s 1 , then the initial ideal in w (I w ) coincides with I w . In the following proposition we will show that if we multiply c on the right by simple reflections s k 1 , . . . , s kr which commute pairwise and each appear at most once, then none of the Plücker relations degenerates to a monomial in in w (I cs k 1 ···s kr ).  Table 2 in the appendix) show which statements apply to which elements of S 4 (resp. S 5 ).
Proposition 3. For any h ∈ [n − 1], none of the Plücker relations degenerates to a monomial in in w (I cs h ).
Proof. First of all notice that if h = 1, then cs 1 < c and the claim follows from Proposition 2, which says that in w (I c ) = I c .
Let #L = p > k + 1, then by (4.3) we have F (J) = [k − 1] ∪ {k + 1, j} and F (L) = [p − 1] ∪ {l} for j 1 = j ∈ [k + 2, n] and l ∈ [p, n]. Note that j ∈ J is the only possible element to swap for elements in L non-trivially, so that we impose j ∈ L (otherwise R m J,L = 0 for any m). Remember that we may assume j ∈ [p, n]. Then in w (R 1 J,L ) = p J p L − p (J\(j))∪(l) p (L\(l))∪(j) − p (J\(j))∪(k) p (L\(k))∪(j) . for j = j 1 , l ∈ [k + 2, n] and j ∈ L in order for the relation to be non-trivial. We obtain Proof. First note that if h = 1, then by Proposition 2 in w (I cs 1 ) = I cs 1 . The Plücker relations involving non-vanishing Plücker coordinates on X cs 1 are for q < p ≤ j < l ≤ n the following pure differences Notice that the index sets of the Plücker coordinates in the above equation (as well as in the rest of this proof) are sets, and hence by convention, as sequences they are arranged in an increasing order, while in the proof of the previous result we always had j = j 1 . This only affect the relation by a global sign.
Remark 5. Note that while in w (I w ) and I w have the same generators for w ≤ c, this is not true for cs k+1 with k ≥ 1. Here taking the initial ideal with respect to w modifies the generators.
The following proposition generalizes Proposition 3 to a product of pairwise distinct commuting simple reflections.
Proposition 4. Take k 1 , . . . , k r ∈ [n − 1] with |k i − k j | > 1 for all i = j, then none of the Plücker relations degenerates to a monomial in in w (I cs k 1 ···s kr ).
Proof. We may assume k 1 < k 2 < . . . < k r without loss of generality. Moreover, since we are multiplying by pairwise distinct commuting reflections, and as Plücker relations only involve pairs of Grassmannians, it is enough to consider the cases r = 1, 2. The case r = 1 was dealt with in Proposition 3, so we are left with r = 2.
We consider two cases: firstly, we deal with the case k 1 = 1, and then we suppose k 1 = 1.
If k 1 = 1, cs 1 < c can be identified with the Coxeter elementc =s n−2 . . .s 1 in S n−1 (via s i →s i−1 for i ∈ [2, n − 1]). In this case, cs 1 s k 2 ∈ s 2 , . . . , s n−1 and, by Corollary 1, we have in w (I cs 1 s k 2 ) = in w (Ics k 2 ). We then apply Proposition 3 to obtain the claim. Now denote k 1 := k + 1 and k 2 := g + 1 and recall, that by assumption k < g + 1. As in the proof of Proposition 3, we only have to deal with Plücker relations R m J,L with #J = #L, where J cs k+1 s g+1 ([#J]) or L cs k+1 s g+1 ([#L]). We can further reduce to the case #J = k + 1, j 1 = j, and #L = g + 1, otherwise the Plücker relations are the same as the ones considered in Proposition 3, and the result has been proven above.
Lemma 5 below shows that the Coxeter word c = s n−1 · · · s 2 s 1 is in fact special among all Coxeter words regarding the degeneration.
Lemma 5. Let w ∈ S n have a reduced expresion w = s ir · · · s i 1 with i k = i l for all k = l. Then none of the Plücker relations degenerates to a monomial in in w (I w ) if and only if w ≤ c.
Proof. "⇐" by Proposition 2. "⇒" Assume w = s ir . . . s i 1 is a product of pairwise distinct simple reflections. First note that w ≤ c implies there exists an i k ∈ {i 1 , . . . , i r } such that i k + 1 = i l for l < k. We choose i = i k , such that k is minimal with this property. In particular, if there exists t with i t + 1 = i then t < k. Since s i commutes with all reflections s im with m > k, as in this case i m = i ± 1 by minimality of k, we observe w = s i s ir . . . s i k+1 s i k−1 . . . s i 1 ∈ s i s 1 , s 2 . . . , s i−1 , s i+1 , . . . s n−1 .
We deduce that w([i]) = [i − 1] ∪ {i + 1}. Moreover, notice w(i + 1) ≥ i + 2, since i + 1 is moved only by s i and s i+1 , but we apply s i+1 first and by hypothesis there are no other occurrences of s i+1 . We can now produce the degree two monomial in in w (I w ) by choosing as J any sequence such that F (J) = w([i]) and j 1 = i + 1, and as L any sequence with F (L) = [i] ∪ {i + 2}, so that ) the second term vanishes on X w . 4.4. More and more monomials. If we can write a permutation u ∈ S n as a product of two permutations v, w belonging to two distinct parabolic subgroups which centralize each other, then we can check how a Plücker relation degenerates on I u by looking at the ideals I v and I w . Lemma 6 concerns defining ideals for Schubert varieties and allows us to deduce Corollary 3, which suggests an inductive procedure on n to find Schubert varieties that become reducible under Feigin's degeneration.
Lemma 6. Let v, w ∈ S n assume there exist two sets of simple reflections S v = {s i 1 , . . . , s ir } and S w = {s j 1 , . . . , s js } such that |i h − j l | > 1 for all h ∈ [r], l ∈ [s] with v ∈ S v and w ∈ S w . Then for all sequences J, L with k ≤ #J we have Let v, w ∈ S n assume there exist two sets of simple reflections S v = {s i 1 , . . . , s ir } and S w = {s j 1 , . . . , s js } such that |i h − j l | > 1 for all h ∈ [r], l ∈ [s] with v ∈ S v and w ∈ S w . Then (1) None of the R k J,L degenerates to a monomial nor in in w (I w ) neither in in w (I v ), if and only if none of the R k J,L degenerates to a monomial in in w (I vw ).
(2) If in w (I w ) or in w (I v ) contains a monomial degenerate Plücker relation, then so does in w (I vw ).
Remark 6. From the previous corollary we see that the bigger n is, the more Schubert varieties become reducible after degenerating themà la Feigin, since there are several ways of embedding S m into S n for m < n as a parabolic subgroup. Indeed, the number of permutations v ∈ S n such that at least one Plücker relation degenerates to a monomial in in w (I v ) is 0,1,11,85 for n = 2, 3, 4, 5, respectively. As a curiosity, we mention here that there is exaclty one sequence in the On-Line Encyclopedia of Integer Sequences [Slo, Sequence A129180] whose first four terms are 0, 1, 11, 85, namely the Total area below all Schroeder paths of semilength n.

Degenerate Schubert and Richardson varieties
In this section we explore how degenerate Schubert varieties behave under the embedding of the degenerate flag variety Fℓ a n into a larger partial flag variety given by Cerulli Irelli and the second author in [CIL15]. 5.1. Degenerate flag varieties and flag varieties of higher rank. We start by introducing some notation and recalling the main result of [CIL15].
Let ω i denote the i-th fundamental weight for SL 2n−2 and consider the parabolic subgroup P := P ω 1 +ω 3 +···+ω 2n−3 of SL 2n−2 . Then, SL 2n−2 /P is the variety of (partial) flags in C 2n−2 whose points are flags of vector spaces of odd dimensions. Its Schubert varieties X w are indexed by minimal length coset representatives w ∈ S 2n−2 /W P , where W P is the Weyl group of the Levi of P . More precisely, if s i ∈ S 2n−2 denotes the simple transposition (i, i + 1), then W P = s 2 , s 4 , . . . s 2n−4 . Let w n ∈ S 2n−2 be defined by The following Theorem can be found in [CIL15].
Theorem 4 ( [CIL15]). The degenerate flag variety Fℓ a n is isomorphic to the Schubert variety X wn ⊂ SL 2n−2 /P . 5.1.1. Translation into Plücker coordinates. We describe here the isomorphism of Theorem 4 in terms of Plücker coordinates. Recall that whenever we index Plücker coordinates by a set, we really mean the associated sequence obtained by increasingly ordering the elements of the given set. Let In order to give the translation of the isomorphism in terms of coordinate rings, we need to set some notation. Let k ∈ [n − 1], we denote by {≤ w n } (2k−1) the set of where τ k : [n + k − 1] → [n] is given by For a sequence I = (i 1 , . . . , i k ) ∈ S(n, k) we set τ k (I) := (τ k (i 1 ), . . . τ k (i k )) ∈ S(n, k). If ρ k : [n] → [k, n + k − 1] is given by then the inverse map to (5.2) is given by On the level of sequences, this lifts to a map S(n, k) (1, 2, . . . , k − 1, ρ k (i 1 ), . . . , ρ k (i k )) Fix an ordered basis (ẽ j ) j∈[2n−2] of C 2n−2 , then the linear algebraic description of X wn is Denote by (e i ) i∈[n] an ordered basis for C n . For k ∈ [n − 1] define the projection operator (which we also denote by π k as in [CIL15]) Then there is an isomorphism, which we denote by the same symbol, of algebraic varieties and the desired isomorphism (cf. [CIL15]) is given by (5.3) ξ : X wn → Fℓ a n , (W 2k−1 ) k∈[n−1] → (π k (W 2k−1 )) k∈[n−1] . Remark 7. In [CIL15], an embedding of ζ : Fℓ n ֒→ SL 2n−2 /P is given, and hence the isomorphism from Theorem 4 is rather the inverse of the isomorphism ξ we consider here. We prefer to work with ξ instead of ζ since in this way we obtain an induced map from the coordinate ring of Fℓ a n to the coordinate ring of X wn , which we make explicit in the following.
As π * k is compatible with Plücker relations, we have an isomorphism ξ * : C[Fℓ a n ] → C[ X wn ], p I → π * #I (p I ). Notice that even if I is ordered increasingly, ρ k (I) needs not be ordered increasingly. To get an increasing sequence we have to multiply by some sign. While keeping track of the sign is fundamental to check that Plücker relations are satisfied, it is not relevant to us, as we only deal with vanishing of certain Plücker coordinates, which of course vanish independently of their sign. 5.2. Richardson varieties in SL 2n−2 /P . Let u, v ∈ S 2n−2 be minimal length coset representatives of S 2n−2 /W P and assume that u ≤ v. We denote by X u v := X v ∩ X u ⊆ SL 2n−2 /P the corresponding Richardson variety. Recall that its defining ideal in C[p I | #I ≡ 1(mod 2), I ⊂ [2n − 2]] is (5.4) I u v = (R k J,L ) + (p I ) I ≤v([#I]) + (p I ) I ≥u([#I]) . In the following we will show that for appropriate permutations x ∈ S n , u, v ∈ S 2n−2 with u ≤ v ≤ w n , the isomorphism ξ * induces an isomorphism between the coordinate rings C[X a x ] → C[ X u v ]. To stress out the fact that such an isomorphism really comes from the embedding ζ, we will express it as ζ(X a x ) = X u v . Since C[X a x ] = C[Fℓ a n ]/(p I | I x([#I])) and C[ X u v ] = C[SL 2n−2 /P ]/(p K | K v([#K]), K u([#K])), the claim will be proven by verifying that An important role will be played by the following permutation y n ∈ S 2n−2 :  if k > m.
Combining Lemma 7 with Proposition 2 we obtain the following corollary.

Schubert divisors
In this section we focus on Schubert divisors and apply the results from previous sections to them. In this case we can completely answer the question whether or not they stay irreducible under the degeneration.
Let w 0 ∈ S n be the longest element, then all Schubert divisors are indexed by permutations of the form w = w 0 s i for i ∈ [n − 1]. Note that The following Theorem 5 is an application of Theorem 3 (1) and (2).
Theorem 5. Let n > 2 and w ∈ S n be such that ws i = w 0 . If n is odd assume i = n+1 2 , for even n there is no additional assumption. Then X a w is reducible. Proof. We consider four cases separately: i < n 2 , i = n 2 , i ≥ n+3 2 , and i = n+2 2 . Notice that they cover all possiblities, since i > n 2 together with the assumption i = n+1 2 implies i > n+1 2 , hence i ≥ n+2 2 . We will deal with the first two cases by applying Theorem 3 (1), while we will use Theorem 3 (2) for the remaining two.