Cohomology classes of interval positroid varieties and a conjecture of Liu

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.


Introduction
where M k,n−k is the set of k × (n − k) complex matrices, and I k is the k × k identity matrix. This variety X D is called a diagram variety. For example, if D = {(1, 1), (1,2), (2, 2)}, k = 2, n = 4, then X D is the closure of the set of 2-planes in C 4 which are the rowspans of matrices of the form 0 0 1 0 0 * 0 1 .
One can associate a complex representation S D of the symmetric group S |D| to a diagram D, called the Specht module of D. These generalize the usual irreducible Specht modules, which occur when D is the Young diagram of a partition; the definition for general diagrams is due to James and Peel [8].
Each of these objects, diagram variety and Specht module, naturally leads to a class in the cohomology ring H * (Gr(k, n), Z). For the diagram variety, we take the Chow ring class of X D and use the natural isomorphism between H * (Gr(k, n), Z) and the Chow ring of Gr(k, n) to obtain a cohomology class [X D ] ∈ H 2|D| (Gr(k, n), Z).
As for the Specht module, let s D be the Frobenius characteristic of S D , meaning s D = λ a λ s λ if S D ≃ λ a λ S λ , where s λ is a Schur function. Here λ runs over partitions, and S λ is an irreducible Specht module. There is a surjective ring homomorphism φ from the ring of Date: October 29, 2014. The author was partially supported by grant DMS-1101017 from the NSF. symmetric functions to H * (Gr(k, n), Z), sending the Schur function s λ to the Schubert class σ λ = [X λ ], or to 0 if λ ⊆ (k n−k ). Hence we can consider the cohomology class φ(s D ).
Liu proved Conjecture 2.5, or the weaker variant claiming equality of degrees, for various classes of diagrams [13]. However, it turns out that this conjecture fails in general, as we show in Section 2.
Work of Kraśkiewicz and Pragacz [10], and of Reiner and Shimozono [16], shows that for the diagram D(w) of a permutation w ∈ S n , s D(w) is the Stanley symmetric function F w (or F w −1 , depending on conventions). Thus, if Conjecture 2.5 were to hold for D(w), then [X D(w) ] = φ(F w ). Here D(w) is the diagram with one cell (i, w(j)) for each inversion i < j, w(i) > w(j) of w.
Extending work of Postnikov [15], Knutson, Lam, and Speyer [9] have defined a collection of subvarieties Π f of Grassmannians called positroid varieties, indexed by certain affine permutations f . These are exactly the varieties obtained by projecting a Richardson variety in the flag variety Fl(n) to Gr(k, n). They show that the positroid variety Π f has cohomology class φ(F f ), whereF f is the affine Stanley symmetric function of f . Given an ordinary permutation w ∈ S n , say f w is the bijection Z → Z with and f (i + 2n) = f (i) + 2n. It is easy to show thatF fw = F w . Bearing Conjecture 2.5 in mind, this suggests a relationship between Π fw and X D(w) .
In fact, Conjecture 2.5 can fail even when D is a permutation diagram D(w), which makes this reasoning dubious. Nevertheless, we will give a degeneration of Π fw to a (possibly reducible) variety containing X D(w) as a component, which implies an upper bound on [X D(w) ].
Theorem (Theorem 5.6). The cohomology class φ(F w ) − [X D(w) ] is a nonnegative integer combination of Schubert classes.
For w ∈ S n , the varieties Π fw are a special case of the rank varieties considered by Billey and Coskun [1]. Rank varieties are themselves a special class of positroid varieties, namely the varieties obtained by projecting a Richardson variety in a partial flag variety Fl(k 1 , . . . , k ℓ ) to Gr(k ℓ , n). We work out necessary and sufficient conditions on f for Π f to be a rank variety.
Coskun [3] gave a recursive rule for computing the cohomology class of a rank variety. We give a different formula for this class, in terms of ordinary Stanley symmetric functions.
Theorem (Theorem 4.1). If X ⊆ Gr(k, n) is a rank variety, then [X] = φ(F w ) for some ordinary permutation w.
Example 4.2 below gives an example of a rank variety X = Π f whereF f is not even Schurpositive, much less equal to any ordinary Stanley symmetric function F w . Thus, Theorem 4.1 is not simply a trivial corollary of the result that [X] = φ(F f ). (1,2), (2, 2), (2, 3)}, k = 2, and n = 5, then

A counterexample to Liu's conjecture
the closure being in the Zariski topology.
Notice that X • D is an open dense subset of X D isomorphic to C k(n−k)−|D| . In particular, it is irreducible, so X D is also irreducible and has codimension |D|.
We now describe a representation of S |D| associated to each diagram D. Say |D| = m for convenience. A bijective filling of D is a bijection T : D → [m]. The symmetric group S m acts on bijective fillings of D by permuting entries. Fix a bijective filling T of D. Let R(T ) denote the group of permutations σ ∈ S m for which i, σ(i) are always in the same row of T . Let C(T ) be the analogous subgroup with "row" replaced by "column".
The Specht modules associated to general diagrams were studied by James and Peel [8]. As D runs over (Young diagrams of) partitions of m, the Specht modules provide a complete, irredundant set of complex irreducibles for S m ; more about these classical Specht modules can be found in [19] or [6]. It is easy to show that the isomorphism type of S D does not depend on the choice of T , and that it is unaltered by permuting the rows or the columns of D. If the rows and columns of D cannot be permuted to obtain a partition (equivalently, the rows of D are not totally ordered under inclusion), then S D will not be irreducible. For example, if λ/µ is a skew shape, then where c λ µν is a Littlewood-Richardson coefficient. In general it is an open problem to give a combinatorial rule for decomposing S D into irreducibles. The widest class of diagrams for which such a rule is known are the percentavoiding diagrams, studied by Reiner and Shimozono [18]; see also [12] and [17].  In particular, S (λ/µ) ∨ ≃ ν c λ µν S ν ∨ . Comparing this decomposition of S (λ/µ) ∨ to the expansion [X (λ/µ) ∨ ] = ν c λ µν σ ν ∨ discussed above suggests the next conjecture (and proves it when D = (λ/µ) ∨ ). [13]). For any diagram D, the cohomology classes [X D ] and φ(s D ) are equal.

Conjecture 2.5 (Liu
Liu proved Conjecture 2.5 in the case above where D ∨ is a skew shape, or when it corresponds to a forest [13] in the sense that one can represent a diagram D ⊂ [k] × [n − k] as the bipartite graph with white vertices [k], black vertices [n − k], and an edge between a white i and black j whenever (i, j) ∈ D. In [2], we proved Conjecture 2.5 when D ∨ is a permutation diagram and S D is multiplicity-free.
One gets a weaker version of Conjecture 2.5 by comparing degrees. The degree of a codimension d subvariety X of Gr(k, n) is the integer deg(X) defined by [X]σ . Under the Plücker embedding, this gives the usual notion of the degree of a subvariety of projective space, namely the number of points in the intersection of X with a generic d-dimensional linear subspace. One can check using Pieri's rule that deg(σ λ ) = f λ ∨ , the number of standard Young tableaux of shape λ ∨ . This is also dim S λ ∨ . Since degree is additive on cohomology classes, Conjecture 2.5 predicts the following.
Liu proved Conjecture 2.6 when D ∨ is a permutation diagram, and when D ∨ has the property that if (i, j 1 ), (i, j 2 ) ∈ D and j 1 < j < j 2 , then (i, j) ∈ D. In light of the assertion of Theorem 2.4 that taking complements in the decomposition of S D gives the decomposition of S D ∨ , one may be tempted to gloss over the issue of D versus D ∨ . In fact, the analogue of

Theorem 2.4 then says
On the other hand, an explicit calculation in Macaulay2 shows deg X D = 21384. Therefore Conjectures 2.6 and 2.5 both fail for D.
The discrepancy in degrees is 24024 − 21384 = 2640 = f 4422 , which hints at how to see this discrepancy more explicitly. Given a k-subset I of [n], write p I for the corresponding Plücker coordinate on Gr(k, n), so p I (A) is the minor of A in columns I. Let Y be the scheme determined by the vanishing of the Plücker coordinates p 1678 , p 2578 , p 3568 , p 4567 . These are exactly the Plücker coordinates which vanish on X D . One can check by computer that codim Y = 4, and so Y is a complete intersection. This implies that Since the four Plücker coordinates cutting out Y vanish on X • D , the diagram variety X D is contained in Y . However, Y has another component, namely the Schubert variety which is This Schubert variety has degree dim S (22 (3,3), (4, 4)} is not itself a permutation diagram, this counterexample leads directly to one of the form X D(w) . Take w = 21436587. Then D(w) = {(1, 1), (3, 3), (5, 5), (7, 7)} can be obtained from D by permuting rows and columns, and viewing D in a larger rectangle. Neither of these operations on diagrams affects s D or [X D ], identifying the latter with its pullback along an embedding of Gr(k, n) into Gr(k, n + 1) or Gr(k + 1, n + 1). Thus Conjecture 2.5 can fail for permutation diagrams.
More counterexamples to Conjecture 2.5 can be easily manufactured from this one. For two diagrams D 1 and ] and similarly that s D1·D2 = s D1 s D2 . Therefore if Conjecture 2.5 holds for D 1 but not D 2 , then it will fail for D 1 · D 2 . (3,3), (4, 4), (5, 5)}, and whether Conjecture 2.5 fails for D ′ . However, trying to repeat the analysis above runs into an immediate problem: the analogue of Y , which is the scheme Z cut out by p 1789(10) , p 2689(10) , p 3679(10) , p 4678(10) , p 56789 no longer even has the same codimension as X D (thanks to Ricky Liu for pointing this out). Indeed, X D has codimension 5 but Z contains the codimension 4 Schubert cell

Positroid varieties and rank varieties
for all i and some fixed n. WriteS n for the set of affine permutations with a particular n.
Note that the image of any set {a, a + 1, . . . , a + n − 1} completely determines an affine permutation. Call such an image a window. We will write affine permutations in one-line notation as the image of [n]: 14825 fixes 1, sends 3 to 8, 7 to 9, etc. Members of any window are all distinct modulo n, so n i=1 f (i) ≡ n(n + 1)/2 (mod n). Let av(f ) be the integer Warning. Affine permutations are usually required to satisfy av(f ) = 0, which ours need not. However, for a fixed k, affine permutations inS n satisfying av(f ) = k are in bijection with those satisfying av(f ) = 0 by subtracting k from each entry in a window. When we refer to constructions on affine permutations that require a Coxeter group structure (e.g. length, reduced words, Stanley symmetric functions), we are implicitly using this isomorphism to transport that structure from the "usual" affine permutation group {f ∈S n : av(f ) = 0}.
The length ℓ(f ) of an affine permutation f is the number of inversions i < j, f (i) > f (j), provided that we regard any two inversions i < j and i + pn < j + pn as equivalent.
Any affine permutation f has a permutation matrix, the Z × Z matrix A with A i,f (i) = 1 and all other entries 0. For any i, j ∈ Z, define Thus, #[i, j](f ) is the number of 1's strictly northeast of (i, j) in the permutation matrix of f , in matrix coordinates. Fix a basis e 1 , . . . , e n of C n . We will abuse notation by writing X both for the span of the vectors in X, if X ⊆ C n , and for the span of vectors let Proj X : C n → X be the projection which fixes those basis vectors e i with i ∈ X and sends the rest to 0. For integers i ≤ j, write [i, j] for {i, i + 1, . . . , j}. We interpret indices of basis vectors modulo n, so that [i, j] ⊆ C n even if i, j fail to lie in [1, n].  Knutson-Lam-Speyer also computed the cohomology class of Π f in terms of affine Stanley symmetric functions. These are a class of symmetric functions indexed by affine permutations introduced by Lam in [11], and we now give a definition.
LetS k n be the set of affine permutations with av(f ) = k. ThenS 0 n is a Coxeter group with simple generators s 0 , . . . , s n−1 , where s i interchanges i + np and i + 1 + np for every p. A reduced word for f ∈S 0 n is a word a 1 · · · a ℓ in the alphabet [0, n − 1] with s a1 · · · s a ℓ = f and such that ℓ is minimal with this property. Let Red(f ) denote the set of reduced words for f . A reduced word a = a 1 · · · a ℓ is cyclically decreasing if all the a i are distinct, and if whenever j and j + 1 appear in a (modulo n), j + 1 precedes j. An affine permutation is cyclically decreasing if it has a cyclically decreasing reduced word. For a partition λ, let m λ be the monomial symmetric function indexed by λ.
As mentioned above, subtracting k from each entry of a window for f ∈S k n gives an isomorphismS k n →S 0 n , which we use to extend the definition ofF f to all ofS n . Theorem 3.6 ([9], Theorem 7.1). For f ∈ Bound(k, n), the cohomology class The ordinary Stanley symetric functions indexed by members of S n , introduced by Stanley in [20], are a special case of affine Stanley symmetric functions. To be precise, we can view w ∈ S n as the affine permutation inS 0 n sending i + pn to w(i) + pn for 1 ≤ i ≤ n. Then the Stanley symmetric function F w of w isF w . This is Proposition 5 in [11], but we will simply take it as a definition of F w . One should be aware, however, that the F w defined in [20] is our F w −1 .
Now we discuss a subset of positroid varieties whose cohomology classes will turn out, in Section 4, to be represented by ordinary Stanley symmetric functions.    Being defined by rank conditions on intersections with interval subspaces, rank varieties should be special instances of positroid varieties.
Let f M be the affine permutation mapping b i to a i + n and d i to c i . Then f M is bounded because a i ≤ b i , which implies d i ≤ c i . This provides a bijection between rank sets for Gr(k, n) and members f of Bound(k, n) such that the subsequence of f (1) · · · f (n) with entries in [n] is increasing.  The following are equivalent: That is, V is in Σ M . Both Π fM and Σ M are irreducible, so equality will follow if we show they have the same codimension ℓ(f M ). By Theorem 3.8, Inversions of f M come in three types: In particular, there are no inversions just among the entries f M (d i ).
For fixed i, inversions of type (1) correspond to elements of Hence the total number of inversions of this type is S∈M (#S(M ) − 1).
It remains to show that Say g ∈ S n is the ordinary permutation with g(d p ) = f M (d p ) and g(

Cohomology classes of rank varieties
Let Λ be the ring of symmetric functions over Z, and φ : Λ → H * (Gr(k, n), Z) the ring homomorphism sending the Schur function s λ to the Schubert class σ λ , or to 0 if λ is not contained in a k × (n − k) rectangle.
Coskun gives a recursive rule to calculate the cohomology class of a rank variety [3]. Since rank varieties are positroid varieties, Theorem 3.6 gives a more direct answer, namely that [Σ M ] is φ(F fM ). The goal of this section is to show that [Σ M ] is actually represented by an ordinary Stanley symmetric function. We will not prove this theorem by showing thatF fM is an ordinary Stanley symmetric function, since this is not true in general, as the next example shows.  Proof. We first check that the entries of f M lying in [n] come in increasing order, so Let w be the restriction of τ −y+1 f M τ b−2 to [n]. By definition, The isomorphismsS r n →S 0 n are given by left multiplying by τ r , soF τ f =F f for any f . On the other hand, τ −1 s i τ = s i−1 , and so conjugation by τ gives a bijection between the cyclically decreasing factorizations of f and those of τ −1 f τ which preserves the lengths of the factors. This meansF τ −1 f τ =F f , sõ  Before proving Theorem 4.4, we explicitly describe an algorithm which, given a rank variety Σ M ⊆ Gr(k, n), produces a permutation w M such that [Σ M ] = φ(F wM ).
Proof. We must show that for all S, T ∈ M is equivalent to for all S, T ∈ M . In fact, it will turn out that for each i, j, the left (resp. right) side of the first inequality is equal to the left (resp. right) side of the second. It is not hard to check that [a, b] ⊆ S ∩ T (with a ≤ b) if and only if [a, b + 1] ⊆ κS ∩ κT , and so #(S ∩ T )(M ) = #(κS ∩ κT )(κM ). Since h is injective, Suppose p ∈ S ∩ T , and write S = [a i , b i ], T = [a j , b j ]. If p = b q + 1 for any q < k, then h(e p ) = e p ∈ κS ∩ κT , so assume p = b q + 1.
Write p for the space of linear maps C n → C n sending e 1 , . . . , e k into itself. The tangent space to Gr(k, n) at e 1 , . . . , e k is gl n /p ≃ Hom( e 1 , . . . , e k , C n / e 1 , . . . , e k ). More generally, the tangent space to Gr(k, n) at V is Hom(V, C n /V ). With this identification, the differential of the quotient map q : GL n (C) ։ Gr(k, n) sending A to the span of its first k rows is where φ ∈ gl n and π V : C n ։ C n /V is the quotient map. Because of our convention of writing members of Gr(k, n) as row spans, A −1 should be interpreted as the linear transformation sending e i to the ith row of A −1 .
Say ψ ij ∈ T A Z is the map sending e i to e j and all other e p 's to 0, for i ≤ k and j ∈ [a i , b i +1]. Using equation (3), If j is not equal to any b p + 1, thenαφ ij = 0, so φ ij ∈ T V Gr(k, ker α). If j = b p + 1, write The first summand is in T V Gr(k, ker α), and the second is in T V Σ κM . Thus T V Gr(k, ker α) + T V Σ κM = T V Gr(k, n + 1), and so Σ κM and Gr(k, ker α) intersect transversely on U .
Proof of Theorem 4.4. Since ι maps distinct Schubert varieties in Gr(k, n) onto distinct Schubert varieties in Gr(k, n + 1), ι * is injective. Therefore it suffices to show that The right side here is [ι(Σ M )] since ι is an embedding. By the projection formula, the left side is [Σ κM ][ι(Gr(k, n))]. Lemmas 4.6 and 4.7 below show that for a suitable choice of ι, Σ κM and ι(Gr(k, n)) intersect generically transversely in ι(Σ M ), so we are done.

Degenerations of rank varieties
Let φ t,i→j be the linear transformation sending e i to te i + (1 − t)e j . For t = 0, the varieties φ t,i→j Σ M are all isomorphic, so they form a flat family [4, Proposition III-56]. The flat limit lim t→0 φ t,i→j Σ M then exists as a scheme [7,Proposition 9.8]. The key fact for us is that Σ M and lim t→0 φ t,i→j Σ M have the same Chow ring class, hence the same cohomology class. In this section we will show that for an appropriate choice of M , lim t→0 φ t,i→j Σ M contains the diagram variety X D(w) as an irreducible component. Other authors have used these degenerations or similar ones to calculate cohomology classes or K-theory classes of subvarieties of Grassmannian, including Coskun [3] and Vakil [21].
Define an operator C i→j on matrices of a fixed size as follows: For example, Sometimes we also apply C i→j to k-planes, or sets of k-planes. Strictly speaking this is illdefined, since it can happen that rowspan A = rowspan A ′ but rowspan C i→j A = rowspan C i→j A ′ , so we only use this notation when the k-planes are represented by specific matrices.
Lemma 5.1. Say F is any subset of [k]×[n−k] and U is the set of k-planes rowspan(A) where A ij = 0 whenever (i, j) ∈ F . If rowspan A ∈ U and C i→j A has rank k, then rowspan C i→j A ∈ lim t→0 φ t,i→j U .
Proof. Given i, j, define a matrix A(t) by Then rowspan A(t) ∈ U for t = 0, and lim t→0 φ t,i→j rowspan A(t) = rowspan C i→j A.
Rank varieties and diagram varieties both have dense open subsets to which Lemma 5.1 applies. We will apply this lemma to rank varieties, but it has an interesting interpretation for diagram varieties as well. Define C i→j on diagrams as on matrices (e.g. by viewing diagrams as 0,1-matrices). Lemma 5.1 shows that X Ci→jD ⊆ lim t→0 φ t,i→j X D , which implies that [X D ] − [X Ci→jD ] is a nonnegative linear combination of Schubert classes. On the other hand, James and Peel [8] showed that for the diagram Specht modules S D , there is always an S |D| -equivariant injection S Ci→jD ֒→ S D . Equivalently, s D − s Ci→jD is a nonnegative linear combination of Schur functions. A more powerful version of this connection is important in Liu's proofs of several cases of Conjectures 2.5 and 2.6 in [13].
Here, for w ∈ S n and v ∈ S m , w × v is the permutation in S n+m sending i to w(i) if i ≤ n and to v(i − n) + n otherwise.
Define (2) • · · · • φ t,2n→w(n) . We will show that lim t→0 φ t,w Σ M(w) contains the diagram variety X D(w) as an irreducible component. First we give an explicit example of this degeneration.