1 Introduction

For n≥1, let be the degenerate flag variety attached to the Lie algebra \(\mathfrak {sl}_{n+1}\) (see [13, 14]). This is a flat degeneration of the classical flag variety, defined using the PBW filtration on irreducible representations of \(\mathfrak {sl}_{n+1}\) (see [17]). By construction, the is acted upon by the degenerate Lie group \(\mathit{SL}_{n+1}^{a}\), which is the semi-direct product of the Borel subgroup B and the abelian group \(\mathbb {G}_{a}^{N}\), where \(\mathbb {G}_{a}\) is the additive group of the field. In particular, \(\mathbb {G}_{a}^{N}\) acts on with an open dense orbit. The degenerate flag varieties are singular normal projective algebraic varieties, sharing many nice properties with their classical analogues. In particular, they enjoy a description in terms of linear algebra as subvarieties inside a product of Grassmann varieties.

It has been observed in [9] that the degenerate flag varieties can be identified with certain quiver Grassmannians of the equioriented quiver of type A n . More precisely, is isomorphic to the quiver Grassmannian \(\mathrm{Gr}_{\operatorname {dim}A}(A\oplus A^{*})\), where A and A ∗ are the path algebra of the equioriented A n quiver, resp. its dual. This observation was used in two different ways: first, to get a deeper understanding of the geometry and combinatorics of the degenerate flag varieties, and, second, to generalize the results and constructions to a wider class of quiver Grassmannians. In this paper we continue the study of the varieties using the techniques from the theory of quiver Grassmannians. More concretely, we achieve two things: first, we describe the combinatorial structure of the moment graph of . Second, we describe explicitly the smooth and singular loci of the degenerate flag varieties. Let us give a brief description of our results.

Recall that the notion of the moment graph attached to an algebraic variety X acted upon by an algebraic torus was introduced in [6, 21]. This combinatorial object captures the structure of zero- and one-dimensional orbits of T. It turns out to be very useful for describing various geometric properties of X, such as cohomology and intersection cohomology. Our first task is to describe the moment graph Γ of . We note that the automorphism group Aut(A⊕A ∗) acts on . The maximal torus T of the automorphism group acts with a finite number of fixed points (this number is equal to the normalized median Genocchi number, see [9, 13, 15]). It is proved in [9] that there exists a codimension one subgroup \({\mathfrak{A}}\subset\operatorname{Aut}(A\oplus A^{*})\) containing the torus T such that \({\mathfrak{A}}\)-orbits through T-fixed points are affine cells that provide a cellular decomposition of . We describe \({\mathfrak{A}}\) as a quotient of the Borel subgroup of SL 2n . Using this description, we prove the following theorem (for a more precise formulation see Sect. 3):

Theorem 1.1

The number of one-dimensional T-orbits in is finite. The edges of Γ correspond to the one-parameter subgroups of \({\mathfrak{A}}\).

We note that the structure of Γ has many common features with its classical analogue (see [8, 22, 27]).

Our next goal is to describe the smooth locus of the degenerate flag varieties. Since has a cellular decomposition by \({\mathfrak{A}}\)-orbits of T-fixed points, it suffices to decide which T-fixed points are smooth. We recall that the T-fixed points are labeled by collections S=(S 1,…,S n ) of subsets of {1,…,n+1} such that #S i =i and S i ⊂S i+1∪{i+1}. We denote the corresponding T-fixed point by p S .

Theorem 1.2

A point p S is smooth if and only if for all 1≤j<i≤n, the condition i∈S j implies j+1∈S i . The number of smooth T-fixed points is given by the large Schröder number r n .

We recall (see [20, 26]) that the large Schröder number r n is equal to the number of Schröder paths, i.e. subdiagonal lattice paths starting at (0,0) and ending at (n,n) with the following steps allowed: (1,0), (0,1) and (1,1). In particular, Theorem 1.2 implies that the Euler characteristic of the smooth locus of is equal to r n . Moreover we prove the following theorem:

Theorem 1.3

The Poincaré polynomial of the smooth locus of is equal to the (scaled) q-Schröder number q n(n−1)/2 r n (q), where r n (q) is defined via the statistics on Schröder paths, counting the number of (1,1) steps in a path.

As an application, we obtain a new proof of the statement that r n (q) is divisible by 1+q. The ratio is known to give a q-analogue of the small Schröder numbers.

Let us mention two more results of the paper. First, we prove that, for a general Dynkin type quiver Q and a projective Q-module P and an injective Q-module I, the quiver Grassmannian \(\mathrm{Gr}_{\operatorname {dim}P} (P\oplus I)\) is smooth in codimension 2. Second, we prove that the smooth locus of can be described as the subvariety of points where the desingularization map (see [16]) is one-to-one.

Finally, we note that all the results of the paper can be generalized to the case of the degenerate partial (parabolic) flag varieties.

Our paper is organized as follows:

In Sect. 1 we introduce the main objects and recall the main definitions and results needed in the rest of the paper.

In Sect. 2 we describe the moment graph of the degenerate flag varieties.

In Sect. 3 we prove a criterion for smoothness of a T-fixed point and compute the Euler characteristics and Poincaré polynomials.

In Appendix A we prove the regularity in codimension 2 of certain quiver Grassmannians.

In Appendix B we describe the smooth locus in terms of the desingularization.

In Appendix C we compute the moment graph for the degenerate flag variety .

2 Quiver Grassmannians and degenerate flag varieties

In this section we recall definitions and results on the degenerate flag varieties and quiver Grassmannians to be used in the main body of the paper.

2.1 Degenerate flag varieties

Let be the complete flag variety for the group SL n+1, i.e. the quotient SL n+1/B by the Borel subgroup B. This variety has an explicit realization as the subvariety of the product of Grassmannians \(\prod_{k=1}^{n} \mathrm{Gr}_{k}(\mathbb {C}^{n+1})\) consisting of collections (V 1,…,V n ) such that V i ⊂V i+1 for all i. In [13, 14] flat degenerations of the classical flag varieties were introduced. The degenerate flag varieties are (typically singular) irreducible normal projective algebraic varieties, sharing many nice properties with their classical analogues. In particular, they also have a very explicit description in linear algebra terms. Namely, let W be an (n+1)-dimensional vector space with a basis w 1,…,w n+1. Let pr k :W→W be the projection operators defined by pr k w k =0 and pr k w i =w i if i≠k. The following lemma is proved in [13], Theorem 2.1.

Lemma 2.1

The degenerate flag variety is a subvariety of the product of Grassmannians \(\prod_{k=1}^{n} \mathrm{Gr}_{k}(W)\), consisting of collections \((V_{k})_{k=1}^{n}\) such that

$$\mathit{pr}_{k+1} V_k\subset V_{k+1}\quad \mathit{for\ all}\ k=1,\dots,n-1. $$

Another important property of the varieties is that they admit a cellular decomposition into a disjoint union of complex cells. Moreover, there exists an algebraic group \({\mathfrak{A}}\) and a torus \(T\subset{\mathfrak{A}}\) acting on such that each cell contains exactly one T-fixed point and the \({\mathfrak{A}}\)-orbit through this point coincides with the cell. Let us describe the combinatorics of the cells, postponing the description of the group action to the next subsection. So let S=(S 1,…,S n ) be a collection of subsets of the set {1,…,n+1} such that each S i contains i elements. Then the cells in are labeled by the collections satisfying the following property:

$$ S_k\subset S_{k+1}\cup\{k+1\},\quad k=1, \dots,n-1. $$
(2.1)

We call such collections admissible. The number of admissible collections (and hence the Euler characteristic of ) is equal to the normalized median Genocchi number h n+1 (see [9, 13, 15]). We note that the correspondence between the admissible collections and T-fixed points is very explicit. Namely, for a collection S we denote by a point defined by

$$p_{\mathbf {S}}=(V_1,\dots,V_n),\qquad V_k=\mathrm{span}(w_i,\ i\in S_k). $$

Clearly, such a point belongs to if and only if the collection S is admissible.

2.2 Quiver Grassmannians

The construction above can be reformulated in the language of quiver Grassmannians (see e.g. [7, 24, 25]). Let Q be the equioriented type A n quiver with vertices labeled by numbers from 1 to n and arrows i→i+1, i=1,…,n−1:

$$Q:\quad \bullet\to\bullet\to\dots\to\bullet $$

For a representation M of Q we denote by M k the subspace of M attached to the vertex k. For a pair 1≤i≤j≤n let R i,j be an indecomposable representation of Q supported on the vertices i,…,j (i.e. (R i,j ) k =ℂ for i≤k≤j and is trivial otherwise). We have the following immediate lemma.

Lemma 2.2

$$\dim{\operatorname{Hom}} (R_{i,j}, R_{k,l})= \begin{cases} 1, & \mathit{if}\ k\le i\le l\le j,\\ 0, &\mathit{otherwise}; \end{cases} $$
$$\dim{\mathrm{Ext}}^1 (R_{i,j}, R_{k,l})= \begin{cases} 1, & \mathit{if}\ i+1\leq k\leq j+1\leq l,\\ 0, & \mathit{otherwise}. \end{cases} $$

We note that the representations R 1,j are injective and the R i,n are projective (note that these are all indecomposable injective and projective representations of Q). We set

$$I_k=R_{1,k},\qquad P_k=R_{k,n}, \qquad P=\bigoplus_{k=1}^n P_k,\qquad I=\bigoplus_{k=1}^n I_k. $$

Hence, P is isomorphic to the path algebra of Q and I is isomorphic to its linear dual. For a dimension vector e=(e 1,…,e n ) and a representation M of Q, we denote by Gr e (M) the quiver Grassmannian of e-dimensional subrepresentations of M. Then by definition one gets

(2.2)

Remark 2.3

The representation P⊕I can be visualized by the following picture (here n=4). Each fat dot corresponds to a basis vector and two dots corresponding to the vectors u and v are connected by an arrow u→v if u is mapped to v. The quiver obtained in this way is called the coefficient-quiver of P⊕I.

(2.3)

The isomorphism (2.2) has many important consequences. In particular the automorphism group of the Q-module P⊕I acts on . The group \(\operatorname{Aut} (P\oplus I)\) is of the form . The part \(\operatorname{Hom} (I,P)\) is one-dimensional (\(\operatorname{Hom} (I,P)=\operatorname{Hom} (I_{n},P_{1})\)). We denote by \({\mathfrak{A}}\subset\operatorname{Aut} (P\oplus I)\) the subgroup

The group \({\mathfrak{A}}\) contains a torus T isomorphic to (ℂ∗)2n, where each factor scales the corresponding indecomposable summand in P⊕I. The importance of the group \({\mathfrak{A}}\) comes from the following lemma, proved in [9].

Lemma 2.4

The group \({\mathfrak{A}}\) acts on with a finite number of orbits. Each orbit is a complex affine cell, containing exactly one T-fixed point. The orbits are labeled by admissible collections.

For an admissible collection S we denote by C S the cell containing the T-fixed point p S .

Remark 2.5

We note that T contains a one-parameter subgroup which acts by the identity automorphism on the degenerate flag variety. Hence one gets a (2n−1)-dimensional torus acting effectively on , while the maximal torus T c acting on the classical flag variety is n-dimensional. We note that there is a natural embedding T c⊂T. In fact recall that any point of is of the form \((V_{k})_{k=1}^{n}\), V k ⊂W≃ℂn+1. Hence any diagonal (in the basis w i ) matrix in SL(W) induces an automorphism of the degenerate flag variety. Hence we obtain the embedding T c⊂T.

Finally, we note that the torus T contains a one-dimensional sub-torus T 0 with the following properties: the set of T-fixed points coincides with the set of the T 0-fixed points and the attracting set of a fixed point p coincides the orbit \({\mathfrak{A}}p\) (which is an affine cell) [9, Theorem 5.1]. The action of the one-dimensional torus can be illustrated as follows (n=4, the scalar λ∈ℂ∗ is the parameter of the torus and the power of λ corresponds to the scaling factor of the T 0 action):

(2.4)

This picture is obtained from the picture (2.3) by putting the P-part on top of the I-part.

We conclude this section by describing the action of the torus T on the tangent space at a T-fixed point p S . Recall that the tangent space at p S is isomorphic to \(\operatorname{Hom}(p_{\mathbf{S}},M/p_{\mathbf{S}})\) where M=P⊕I ([9, Lemma 2.3], [7, 25]). Let θ M be the coefficient quiver of M (see Remark 2.3) and let π:θ M →Q be the natural projection onto the A n quiver Q. The coefficient quiver of M/p S is θ M ∖S. The vector space \(\operatorname{Hom}(p_{\mathbf{S}},M/p_{\mathbf{S}})\) has a distinguished basis, denoted by \(\mathcal{B}\), parameterized by triples (A,f,B) where A is a predecessor-closed connected sub-quiver of S, B is a successor-closed connected sub-quiver of θ M ∖S and f:A→B is a quiver isomorphism compatible with π (see [11]). For example, in the left-hand side of the picture below

(2.5)

the fat dots highlight the coefficient-quiver S of a T-fixed point p S of and the frames highlight a distinguished basis vector of the tangent space at p S .

Proposition 2.6

Given a T-fixed point p S of , the torus T acts on the tangent space at p S diagonally in the basis \(\mathcal{B}\). Moreover the eigenvalues are (generically) distinct.

Proof

Given λ∈T and \(f\in\operatorname{Hom}(p_{\mathbf{S}}, M/p_{\mathbf{S}})\), (λ.f)(v)=λ.f(λ −1.v). Now, by definition of T, each connected component R of θ M has a weight wt(R) and hence a basis vector (A,f,B) receives the weight wt(B)/wt(A). □

To illustrate the previous proposition, let us consider and the action of T depicted in the right-hand side of (2.5). The tangent space at p S has dimension 7 and the torus acts in the standard basis \(\mathcal{B}\) as the diagonal matrix \(\mathrm{diag}(\frac{1}{\lambda_{3}}, \frac{\lambda_{3}}{\lambda_{4}},\frac{\lambda_{2}}{\lambda_{4}}, \frac{\lambda_{3}}{\lambda_{1}}, \frac{\lambda_{2}}{\lambda_{1}}, \frac{1}{\lambda_{2}}, \frac{\lambda_{4}}{\lambda_{5}})\). The one-dimensional torus T 0 is given by putting λ i :=λ i. In particular its action on the tangent space at p S is given by the diagonal matrix diag(λ −3,λ −1,λ −2,λ 2,λ 1,λ −2,λ −1). Notice that the eigenvalues of the T 0 action are not distinct.

Corollary 2.7

The T-fixed one-dimensional vector subspaces of \(\operatorname{Hom}(p_{\mathbf{S}}, M/p_{\mathbf{S}})\) are precisely the coordinate ones, i.e. those generated by standard basis vectors.

2.3 Partial flag varieties

The whole picture described above has a straightforward generalization to the case of partial flag varieties. Namely, given a collection d=(d 1,…,d k ), where 1≤d 1<d 2<…<d k ≤n, let be the corresponding partial flag variety for SL n+1 ( is a quotient of SL n+1 by a parabolic subgroup). Explicitly, consists of collections \((V_{d_{1}},\dots,V_{d_{k}})\) of subspaces of an (n+1)-dimensional vector space W such that dimV m =m and \(V_{d_{i}}\subset V_{d_{i+1}}\). These varieties can be degenerated in the same way as the complete flag variety (see [13, 14]). As a result one gets a variety , consisting of collections of subspaces \((V_{d_{1}},\dots,V_{d_{k}})\) of W such that dimV m =m and

$$\mathit{pr}_{d_i+1}\cdots \mathit{pr}_{d_{i+1}}V_{d_i} \subset V_{d_{i+1}},\quad i=1,\ldots,k-1. $$

These varieties are also certain quiver Grassmannians (see [9]). Namely, consider the equioriented quiver of type A k . Then the degenerate partial flag variety is isomorphic to

$$ \mathrm{Gr}_{(d_1,\dots,d_k)} \bigl(P_1^{d_1} \oplus P_2^{d_2-d_1}\oplus\cdots\oplus P_k^{d_k-d_{k-1}} \oplus I_1^{d_2-d_1}\oplus\cdots\oplus I_{k-1}^{d_k-d_{k-1}} \oplus I_k^{n+1-d_k}\bigr), $$
(2.6)

where P i and I j are projective and injective modules of the A k quiver. There is a natural surjection , sending \((V_{i})_{i=1}^{n}\) to \((V_{d_{j}})_{j=1}^{k}\). The group \(\mathfrak {A}\) thus acts on ; the orbits are affine cells containing exactly one T-fixed point. These T-fixed points are parametrized by collections \(\mathbf {S}=(S_{d_{1}},\ldots,S_{d_{k}})\) of subsets of {1,…,n+1} subject to the conditions \(\#S_{d_{i}}=d_{i}\) and

$$ S_{d_i}\subset S_{d_{i+1}}\cup\{d_i+1, \ldots,d_{i+1}\},\quad i=1,\dots,k-1. $$
(2.7)

We call such collections d-admissible. As for the complete flags, the corresponding T-fixed point \(p_{\mathbf {S}}=(V_{d_{1}},\dots,V_{d_{k}})\) is given by \(V_{d_{i}}=\mathrm{span}(w_{j},\ j\in S_{d_{i}})\).

3 The moment graph

In this section we study the combinatorics and geometry of the cellular decomposition of the degenerate flag varieties.

3.1 The group action

Recall the group \({\mathfrak{A}}\) acting on . The following lemma is simple, but important for us. Let B⊂GL 2n be the Borel subgroup of lower-triangular matrices and N⊂B be the subgroup of matrices (a i,j ) i≥j such that a i,i =1 and a i,j =0 unless i−j>n. For example, for n=5 the group N looks as follows:

$$\left ( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[1.5pt] 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[1.5pt] 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\[1.5pt] 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\[1.5pt] 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\[1.5pt] 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\[1.5pt] * & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\[1.5pt] * & * & 0 & 0 & 0& 0 & 0 & 1 & 0 & 0\\[1.5pt] * & * & * & 0 & 0 & 0 & 0 & 0 & 1 & 0\\[1.5pt] * & * & * & * & 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right ). $$

Lemma 3.1

The group \({\mathfrak{A}}\) is isomorphic to the quotient group B/N.

Proof

Consider the isomorphism \(\operatorname{Aut} (P\oplus I)\simeq\operatorname{Aut} (\bigoplus_{i=1}^{n} P_{i} \oplus\bigoplus_{k=1}^{n} I_{k})\). We note that for any pair of indecomposable summands of P⊕I the space of homomorphisms between them is either one-dimensional or trivial. More precisely, let us introduce the following notation for the indecomposable summands of P⊕I:

$$ R_1=P_n, R_2=P_{n-1}, \ldots, R_n=P_1,\ R_{n+1}=I_n, R_{n+2}=I_{n-1},\ldots, R_{2n}=I_1. $$
(3.1)

Then for two indecomposable summands R i and R j one has \(\dim\operatorname{Hom} (R_{i},R_{j})=1\) if and only if i≤j and j−i≤n (see Lemma 2.2). Hence we obtain a surjection of groups \(B\to \mathfrak {A}\) and the kernel coincides with N. □

Remark 3.2

Let us fix a non-zero element \(\gamma_{i,j}\in\operatorname{Hom} (R_{i},R_{j})\) for each pair i,j with i≤j, j−i≤n. Then any element \(g\in \mathfrak {A}\) can be uniquely written as a sum ∑g i,j γ i,j , defining a matrix in B. This produces a section \(\mathfrak {A}\to B\).

Remark 3.3

We note that the direct summands R i in type A 4 are visualized in (2.4). Namely, R 1 is represented by the only fat dot in the upper line, R 2 is represented by the two dots in the next to the upper line, and so on up to R 8. In general, if i≤n, then the dimension vector of R i is (0,…,0,1,…,1) with i units and each non-zero (R i ) k is spanned by w n+1−i . If i>n, then the dimension vector of R i is (1,…,1,0,…,0) with 2n−i+1 units and each non-zero (R i ) k is spanned by w 2n−i+2.

Recall that the T fixed points in are labeled by the admissible collections. For an admissible collection S let p S be the corresponding T-fixed point and C S be the cell containing p S . We know that \(C_{\mathbf {S}}={\mathfrak{A}} p_{\mathbf {S}}\). Our goal now is to describe a unipotent subgroup \(U_{\mathbf {S}}\subset{\mathfrak{A}}\) such that the map U S →C S is one-to-one. Let \({\mathfrak{a}}\) be the Lie algebra of the group \({\mathfrak{A}}\). Then

$$\mathfrak{a}=\operatorname{Hom} (P,P)\oplus\operatorname{Hom} (I,I) \oplus \operatorname{Hom} (P,I). $$

The Lie algebra \(\mathfrak{a}\) is the quotient of the Borel subalgebra \(\mathfrak {b}\subset \mathfrak {gl}_{2n}\) of lower-triangular matrices by the ideal \(\mathfrak {n}\) consisting of matrices (a j,i ) j≥i such that a i,j =0 unless j−i>n (this is exactly the Lie algebra of N). In particular, the one-dimensional hom-spaces \(\operatorname{Hom}(R_{i},R_{j})\), i≤j, j−i≤n between two indecomposable summands of P⊕I correspond to the root vectors of the form \(E_{j,i}\in \mathfrak {b}\) (E j,i are matrix units). We have

$$\mathfrak{a}=\mathfrak {t}\oplus\bigoplus_{{1\le i<j\le 2n}\atop{j-i\le n}} \mathfrak{a}_{i,j}, $$

where \(\mathfrak {t}\) is the Lie algebra of the torus T and \(\mathfrak {a}_{i,j}=\operatorname{Hom}(R_{i},R_{j})\).

Consider a pair R i ,R j of direct summands of P⊕I such that \(\dim\operatorname{Hom}(R_{i},R_{j})=1\) and fix a non-zero \(\gamma\in\operatorname{Hom}(R_{i},R_{j})\).

Definition 3.4

A pair of indices (i,j) (a pair of representations R i ,R j ) is called S-effective, if p S ∩R i ≠0 and γ(p S ∩R i ) does not sit inside p S .

Remark 3.5

S-effective pairs have the following geometric interpretation: they are in bijection with standard basis vectors of the tangent space at p S on which T 0 acts with positive weight (see the end of Sect. 2.2). Let us prove this statement. In notation (3.1), we denote by R k the coefficient-quiver of R k . Given an S-effective pair (i,j) a non-zero \(\gamma\in\operatorname{Hom}(R_{i},R_{j})\) is determined (up to scalar multiplication) by a (unique) triple (A,f,B). So A⊂R i is predecessor-closed, B⊂R j is successor closed and f:A→B is a quiver isomorphism compatible with π (see Sect. 2.2). The sub-representation γ(p S ∩R i )⊂R j determines the successor-closed sub-quiver f(S∩A) of B. Since by definition γ(p S ∩R i ) does not sit inside p S , f(S∩A) strictly contains S∩B and the difference f(S∩A)∖(S∩B) is the coefficient quiver of the non trivial quotient γ(p S ∩R i )/(γ(p S ∩R i )∩p S ). The map

$$\gamma\mapsto b_\gamma:= \bigl(\mathbf {S}\cap A\setminus f^{-1}(\mathbf {S}\cap B), f|_{\mathbf {S}\cap A\setminus f^{-1}(\mathbf {S}\cap B)}, f(\mathbf {S}\cap A)\setminus (\mathbf {S}\cap B) \bigr) $$

gives a bijection between S-effective pairs and standard basis vectors of the tangent space on which T 0 acts with a positive weight. To see this we notice that S∩A is predecessor-closed in S and S∩B is successor closed in B. Then f −1(S∩B) is successor closed in S∩A and hence S∩A∖f −1(S∩B) is predecessor closed in S∩A and hence in S. We notice that S∩B coincides with S∩R j (otherwise S∩B would not be strictly contained in f(S∩A)). Since f(S∩A) is successor closed in R j and S∩B=S∩R j , it follows that f(S∩A)∖(S∩B) is successor closed in R j ∖(S∩R j ) and hence in θ M ∖S. The quiver morphism \(f|_{\mathbf {S}\cap A\setminus f^{-1}(\mathbf {S}\cap B)}\) is a quiver isomorphism between S∩A∖f −1(S∩B) and f(S∩A)∖(S∩B) compatible with π, since f is so. The image b γ of γ is hence a standard basis vector of \(\operatorname{Hom}(p_{\mathbf{s}},M/p_{\mathbf {S}})\). The action of T 0 on b γ is given by λ.b γ =λ j−i b γ . Since γ≠0, then i≤j and hence b γ has positive weight. The map is hence well-defined and injective. Let us show that it is surjective. Let b=(A′,f′,B′) be a standard basis vector of \(\operatorname{Hom}(p_{\mathbf {S}},M/p_{\mathbf {S}})\) on which T 0 acts with a positive weight. Then there are indices i and j such that A′ is a predecessor-closed sub-quiver of S∩R i , and B′ is a successor-closed sub-quiver of R j ∖(R j ∩S). The torus T 0 acts on b as λ.b=λ j−i b and hence j>i. We claim that j−i≤n. Indeed if j−i>n then π(R j ) and π(R i ) are disjoint in Q (otherwise \(\operatorname{Hom}(R_{i},R_{j})\neq0\) against the hypothesis j−i>n) and hence the quiver isomorphism f′:A′→B′ could not exist. In view of Lemma 2.2 and the proof of Lemma 3.1, it follows that there is a non-zero standard basis vector \(\gamma\in \operatorname{Hom}(R_{i}, R_{j})\) defined by a triple (A,f,B). Notice that π(A)=π(B)=π(R i )∩π(R j )⊃π(A′)=π(B′). It follows that A′⊂A, B′⊂B and f′=f| A′. From this we conclude that p S ∩R i ≠0 and γ(p S ∩R i ) does not sit inside p S and hence (i,j) is an S-effective pair.

Let \(U_{i,j}\subset \mathfrak {A}\) be the one-parameter subgroup with the Lie algebra \(\mathfrak{a}_{i,j}\). The importance of effective pairs is explained by the following lemma:

Lemma 3.6

If a pair (i,j) is not S-effective then U i,j p S =p S . Otherwise, the map , g↦gp S is injective.

Proof

Assume that a pair R i ,R j is not S-effective and take a non trivial \(\gamma\in\operatorname{Hom}(R_{i},R_{j})\). By definition, γp S ⊂p S and hence the exponent of the (scaled) operator γ fixes p S . To prove the second claim we note that

$$\exp(c\gamma) p_{\mathbf {S}}=(\mathrm{Id} + c\gamma) p_{\mathbf {S}}. $$

Hence, if γp S does not sit inside p S , then all the points exp(cγ)p S , c∈ℂ are different. □

For an admissible S let \(\mathfrak{a}_{\mathbf {S}}\subset\mathfrak{a}\) be the subspace defined as the direct sum of one-dimensional spaces \(\operatorname{Hom}(R_{i},R_{j})\) for all S-effective pairs R i , R j .

Lemma 3.7

The subspace \(\mathfrak{a}_{\mathbf {S}}\) is a Lie subalgebra of \(\mathfrak{a}\).

Proof

Let \(\gamma_{1}\in\mathfrak{a}_{i,j}\) and \(\gamma_{2}\in\mathfrak {a}_{k,l}\), i>j, k>l be two elements such that [γ 1,γ 2]≠0. Then either j=k or i=l. We work out the first case (the second is very similar). We have \([\gamma_{1},\gamma_{2}]=\gamma_{1}\gamma_{2}\in \mathfrak{a}_{i,l}\). Since γ 2 is S-effective, we have

$$\gamma_2(p_{\mathbf {S}}\cap R_l)\supsetneq p_{\mathbf {S}}\cap R_k. $$

Now, since

$$\gamma_1(p_{\mathbf {S}}\cap R_j)\supsetneq p_{\mathbf {S}}\cap R_i $$

and j=k, we obtain

$$\gamma_1\gamma_2(p_{\mathbf {S}}\cap R_l) \supsetneq p_{\mathbf {S}}\cap R_i $$

and hence γ 1 γ 2 is S-effective. □

Let U S be the Lie group of \(\mathfrak{a}_{\mathbf {S}}\), i.e. U S is generated by all U i,j with S-effective (i,j). We note that U S is invariant with respect to the torus T action by conjugation.

Theorem 3.8

The map U S →C S , g↦gp S is bijective and T-equivariant.

Proof

First, we note that T-equivariance follows from Tp S =p S . Now let us prove that the map U S →C S is surjective. Let us write an element \(\mathfrak {g}\in \mathfrak {A}\) as g=g S g 1 h, where h∈T, g S ∈U S and g 1 belongs to the subgroup of \(\mathfrak {A}\), generated by U i,j with non S-effective (i,j). Then gp S =g S p S and hence we are done. Finally, let us prove the injectivity. Assume that there exists g∈U S such that gp S =p S . We identify g with the corresponding lower-triangular matrix in GL 2n with entries g i,j satisfying g i,i =1 and g i,j =0 if i−j>n. Our goal is to prove that g i,j =0 for all i>j. Let p(S)=(V 1,…,V n ) and assume that g i,j ≠0 for i>j. Since g∈U S , the pair (i,j) is S-effective. Consider a non-zero element \(\gamma\in\mathfrak{a}_{i,j}\) (so γ∈Hom(R i ,R j )). Let t=1,…,n be a number such that V t ∩R i ≠0 and γV t ∩V t =0. Choose a non-zero vector w∈V t ∩R i . Then gw∉V t and hence gp S ≠p S . □

Remark 3.9

We note that Theorem 3.8 is analogous to the corresponding theorem for classical flag varieties, see e.g. [27], Lemma 3.2.

Proposition 3.10

The number of S-effective pairs (i,j) is equal to the sum N PI (S)+N PP (S)+N II (S) of three numbers defined by:

  • N PI (S) is the number of pairs 1≤k<l≤n+1 such that there exists t with k≤t<l such that k∈S t , l∉S t .

  • N PP (S) is the number of pairs 1≤k<l≤n such that there exists t≥l such that l∈S t , k∉S t .

  • N II (S) is the number of pairs 2≤k<l≤n+1 such that there exists t<k such that l∈S t , k∉S t .

Proof

We divide S-effective pairs into three parts R i ,R j ⊂P, R i ,R j ⊂I and R i ⊂P,R j ⊂I. We claim that the number of S-effective pairs from the first (second, third) part is equal to N PP (S) (N II (S), N PI (S)).

  1. (i)

    The case R i ⊂P, R j ⊂I. Then 1≤i≤n<j≤2n. Since (i,j) is S-effective, there exists an index t: n+1−i≤t≤2n+1−j such that n+1−i∈S t and 2(n+1)−j∉S t . Put k=n+1−i and l=2(n+1)−j.

  2. (ii)

    The case R i ,R j ⊂P. Since (i,j) is S-effective then 1≤i<j≤n and there is an index t: t≥n+1−i>n+1−j such that n+1−i∈S t and n+1−j∉S t . Put l=n+1−i and k=n+1−j.

  3. (iii)

    The case R i ,R j ⊂I. Since (i,j) is S-effective then n+1≤i<j≤2n and there is an index t: t≤2n+1−j<2n+1−i such that 2(n+1)−i∈S t and 2(n+1)−j∉S t . Put l=2(n+1)−i and k=2(n+1)−j.

 □

Corollary 3.11

The dimension of C S is equal to the sum N PI (S)+N PP (S)+N II (S).

Proof

Thanks to Theorem 3.8 the dimension of the cell C S is equal to the number of S-effective pairs R i , R j . Now Proposition 3.10 implies the corollary. □

Corollary 3.12

The Poincaré polynomial of is equal to the sum of the terms \(q^{N_{\mathit{PI}}(\mathbf {S})+N_{\mathit{PP}}(\mathbf {S})+N_{\mathit{II}}(\mathbf {S})}\), where the sum runs over the set of admissible collections.

Remark 3.13

In [9, Theorem 5.1] it is shown that although is not smooth, the one-dimensional sub-torus T 0 of T still produces a Białynicki–Birula type cell decomposition ([3], [10, Theorem 2.4.3]). In other words, the attracting set of a T 0-fixed point p S is a cell and it has dimension equal to the dimension of the positive part of the tangent space at p S (the positive part is the vector subspace generated by vectors on which T 0 acts with positive weight). In view of Remark 3.5, this dimension is precisely the number of S-effective pairs. Theorem 3.8 provides another and more explicit proof of this fact.

Remark 3.14

From the discussion above (see Corollary 3.11 and Remark 3.5), the dimension of the cell with center p S can be easily read off from S, viewed inside the coefficient quiver of P⊕I written in the form (2.4). Indeed in this diagram let us color a vertex black if it belongs to S and white otherwise. In the ith column (counting from left to right) there are precisely i black vertices. Some of them are sources of S. For every such source t∈S i let us count the number w t of white vertices below it. Let c i be the sum of the w t ’s. Then the dimension of the cell with center p S equals the sum c 1+c 2+⋯+c n . For example let us consider the following T-fixed point of :

then c 1=2, c 2=0, c 3=2 and c 4=2. The cell has hence dimension 6.

3.2 Moment graph

We briefly recall the definition of a moment graph (see [6, 21]). Let X be a projective algebraic variety acted upon by a torus T=(ℂ∗)d with a fixed one-dimensional sub-torus ı:ℂ∗⊂T. Assume that the T action on X has finitely many fixed points and one-dimensional orbits and any ℂ∗ fixed point is T-fixed (\(X^{T}=X^{\mathbb {C}^{*}}\)). Assume further that X has a decomposition as a disjoint union of T-invariant affine cells in such a way that each cell C contains exactly one ℂ∗-fixed point p and C={x∈X: lim λ→0 ı(λ)x=p} (i.e. the cell consists of all points attracted by p, see [3]). We denote this cell by C p . The moment graph Γ has its set of vertices labeled by the T-fixed points. Two points p 1 and p 2 are connected by an edge in Γ if there exists a one-dimensional T-orbit L such that \(\bar{L}=L\sqcup p_{1}\sqcup p_{2}\) (i.e. p 1 and p 2 are the T-fixed points in the closure of L). Thus the edges of Γ are labeled by the one-dimensional T-orbits. We orient Γ by the following rule: for two vertices p 1 and p 2 we say p 1≥p 2 if \(C_{p_{2}}\subset\bar{C}_{p_{1}}\). If there is an edge connecting p 1 and p 2 in Γ then we put an arrow p 1→p 2. Finally, one defines a labeling α L of the edges L of Γ by the elements \(\alpha _{L}\in \mathfrak {t}^{*}\), where \(\mathfrak {t}\) is the Lie algebra of the torus T. Namely, for an edge L let T x ⊂T be the stabilizer of a point x∈L (obviously, T x is independent of x∈L). Then the Lie algebra \(\mathfrak {t}_{x}\subset \mathfrak {t}\) is a hyperplane. We define α L as a non-zero element in the annihilator of \(\mathfrak {t}_{x}\).

Example 3.15

Here we give an example of the moment graph for the classical flag variety . The torus T has six fixed points labeled by pairs (S 1,S 2) of subsets of {1,2,3} such that #S 1=1, #S 2=2 and S 1⊂S 2. The moment graph of looks as follows:

We note that usually the arrows in the moment graph direct from bottom to top. However, for our purposes it is more convenient to draw the vertices from top to bottom, since in the degenerate situation the dense cell corresponds to the point (1,12), see Example 3.17. This is not important in the classical situation due to the Chevalley involution, but crucial in the degenerate case.

Our goal is to describe the moment graph of the degenerate flag varieties.

Remark 3.16

We note that the moment graphs turn out to be a powerful tool for computing various cohomology groups of algebraic varieties (see [6, 18, 19, 21, 27]). A crucial role is played by the notion of sheaves on moment graphs. In this paper we do not discuss Γ-sheaves, but only describe the combinatorial structure of the graphs. Computation of the (equivariant) cohomology as well as the (equivariant) intersection cohomology of the degenerate flag varieties is an interesting open problem.

Example 3.17

Here we give a picture of the moment graph for the degenerate flag variety . Recall that the T-fixed points are labeled by pairs (S 1,S 2) of subsets of the set {1,2,3} such that #S 1=1, #S 2=2 and S 1⊂S 2∪{2}.

The moment graph for the degenerate flag variety is computed in Appendix C.

We now give an explicit combinatorial description of the moment graph. We identify the Lie algebra \(\mathfrak {t}\) of T with the diagonal traceless 2n×2n matrices. For a pair of indices i,j, 1≤i<j≤2n, we denote by \(\alpha _{i,j}\in \mathfrak {t}^{*}\) the element α i,j (diag(x 1,…,x 2n ))=x i −x j .

Theorem 3.18

The number of one-dimensional T-orbits in is finite. The orbits are of the form U i,j p S ∖p S , where S is admissible and (i,j) is S-effective. The edge in the moment graph, which corresponds to U i,j p S ∖p S is labeled by α i,j .

Proof

Thanks to Theorem 3.8, we only need to describe the one-dimensional T-orbits in U S . It is easy to see that these are non-identity elements in U i,j . □

Remark 3.19

Theorem 3.18 also follows from Corollary 2.7 and Remark 3.5. Indeed in view of Corollary 2.7, the directions around p S of the one-dimensional T-orbits containing p S are precisely the standard basis vectors of the tangent space at p S . In particular the number of such T-orbits is bigger or equal than and it is equal if and only if p S is smooth. Any such curve ℓ consists of three T-orbits ℓ={p S }∪{ℓ′}∪{p R }. The direction of ℓ is fixed also by the one-dimensional torus T 0. In particular this standard basis vector of has either positive or negative T 0 weight. If the weight is positive then {p S }∪{ℓ′} sits inside the attracting set of p S which is the cell \({\mathfrak{A}}p_{\mathbf{S}}\) and hence p R (and its attracting cell) is in the closure of this cell. It follows that in the moment graph there is an arrow p S →p R . In particular the number of arrows starting from p S in the moment graph, equals the number of standard basis vector of on which T 0 acts with positive weight. In view of Remark 3.5 this number equals the number of S-effective pairs.

Corollary 3.20

The dimension of a cell C S is equal to the number of edges in the moment graph which are directed outwards the vertex p S .

The following theorem generalizes the results as above to the case of the degenerate partial flag varieties.

Theorem 3.21

The number of one-dimensional T orbits on is finite. Each of these orbits is covered by a one-dimensional T-orbit in via the surjection . All the orbits are of the form U i,j p∖p for some i,j and a T-fixed .

4 Smooth locus and the Schröder numbers

In this section we describe the smooth locus of the degenerate flag varieties and compute Euler characteristics and Poincaré polynomials.

4.1 Smooth cells

Take a point \(N\in\mathrm{Gr}_{\operatorname {dim}P}(P\oplus I)\). Then N can be split as N=N P ⊕N I , where N P ⊂P and N I ⊂I, such that N I and P/N P are of the same dimension vector (see [9, Theorem 1.3]).

Lemma 4.1

A point N in a quiver Grassmannian \(\mathrm{Gr}_{\operatorname {dim}P}(P\oplus I)\) is smooth if and only if \(\operatorname{Ext}^{1}(N_{I},P/N_{P})=0\).

Proof

Let 〈⋅,⋅〉 be the Euler form of the quiver Q, given on a pair of dimension vectors d, e by \(\langle\mathbf{d},\mathbf{e}\rangle=\sum_{i=1}^{n} d_{i}e_{i}-\sum_{i=1}^{n-1} d_{i}e_{i+1}\). Then \(\langle \operatorname {dim}X,\operatorname {dim}Y\rangle=\dim \operatorname{Hom}(X,Y)-\dim\operatorname{Ext}^{1}(X,Y)\) for arbitrary representations X and Y of Q. By [9, Theorem 1.1], we have

$$\langle \operatorname {dim}P,\operatorname {dim}I\rangle=\dim\mathrm{Gr}_{\operatorname {dim}P}(P\oplus I). $$

By the formula [9, Lemma 2.3] for the dimension of the tangent space T N to the point \(N\in\mathrm{Gr}_{\operatorname {dim}P}(P\oplus I)\), we then have

Since N P is projective and N/N I is injective, we obtain

$$\dim\mathrm{Ext}^1 (N_I\oplus N_P, P/N_P\oplus I/N_I)=\dim\mathrm{Ext}^1 (N_I, P/N_P). $$

Hence, the dimension of the tangent space at a point N is equal to the dimension of the Grassmannian if and only if Ext1(N I ,P/N P ) vanishes. □

Recall that the quiver Grassmannian \(\mathrm{Gr}_{\operatorname {dim}P}(P\oplus I)\) can be decomposed into the disjoint union of \({\mathfrak{A}}\)-orbits of the form \({\mathfrak{A}} p_{\mathbf {S}}\). Hence all the points of the orbit are smooth or singular together with p S . So it suffices to understand what are the conditions for an admissible collection S that guarantee the smoothness of p S . We use Lemma 4.1 above.

Theorem 4.2

A point p S is smooth if and only if for all 1≤j<i≤n, the condition i∈S j implies j+1∈S i .

Proof

Given an admissible collection \(\mathbf {S}=(S_{i})_{i=1}^{n}\), we introduce the following numbers for all i=1,…,n+1:

$$k_i=\min\{1\leq k<i : i\in S_k\},\qquad l_j=\min\{j\leq l\leq n : j\in S_l\}. $$

Recall the indecomposable representations R k,l with the support on the interval [k,l]. A representation p S is isomorphic to the direct sum N I ⊕N P , where N I ⊂I and N P ⊂P. It is easy to see that

$$N_I=\bigoplus_i R_{k_i,i-1}, \qquad P/N_P=\bigoplus_j R_{j,l_j-1}. $$

The extension groups between the indecomposables are given by Lemma 2.2. Thus we obtain 0≠Ext1(N I ,P/N P ) if and only if there exist indices i and j such that k i +1≤j≤i≤l j −1. This holds (writing out the three inequalities) if and only if there exist indices j≤i such that

$$\min\{1\leq k<i : i\in S_k\}<j,\qquad \min\{j\leq l\leq n : j\in S_l\}>i. $$

This translates into the condition that there exist j≤i such that i∈S j−1, but j∉S i . Conversely, this means that the orbit is smooth if and only if for all 1≤j≤i≤n+1, if i∈S j−1, then j∈S i . Note that this condition is void in case j=1 or i=n+1, so that we can replace j by j−1, and obtain the assertion of theorem. □

In what follows we call an admissible collection S smooth iff p S is a smooth point.

4.2 The large Schröder numbers

Let r n be the nth large Schröder number, defined as the number of Schröder paths, i.e. subdiagonal lattice paths from (0,0) to (n,n) consisting of the steps (0,1), (1,0) or (1,1). The sequence r 0,r 1,r 2,… starts with 1,2,6,22,90,394. Here are the six Schröder paths for n=2:

We note that (see e.g. [1, 5, 26])

$$r_n=r_{n-1}+\sum_{k=0}^{n-1} r_kr_{n-1-k}. $$

The small Schröder numbers s n are defined as halves of the large ones.

Recall that a collection \(\mathbf {S}=(S_{a})_{a=1}^{n}\) of subsets of the set {1,…,n+1} is smooth if #S a =a, S a ⊂S a+1 and for all 1≤a<b≤n the following condition holds (see Theorem 4.2):

$$\text{if}\quad b\in S_a, \quad \text{then}\quad a+1\in S_b. $$

Let LS n be the set of length n smooth collections and \(\bar {r}_{n}\) be the cardinality of LS n .

Proposition 4.3

The numbers \(\bar{r}_{n}\) satisfy the recursion

$$\bar{r}_n=\bar{r}_{n-1}+\sum_{k=0}^{n-1} \bar{r}_k\bar{r}_{n-1-k}. $$

Proof

We divide all smooth collections according to the values of S 1. So first, let S 1={1}. Let us show that the number of such smooth collections is equal to \(\bar{r}_{n-1}\). Note that all S a contain 1. For a=1,…,n−1 we set

$$S'_a=\{i:\, i+1\in S_{a+1}\}\subset\{1, \dots,n\}. $$

We claim that the collection \((S'_{a})_{a=1}^{n-1}\) is smooth and all (length n−1) smooth collections arise in this way. First, obviously, \(\#S'_{a}=a\) and \(S'_{a}\subset S'_{a+1}\). Now the conditions (b∈S a implies a+1∈S b ), 2≤a<b≤n are equivalent to the conditions (\(b\in S'_{a}\) implies \(a+1\in S'_{b}\)), 1≤a<b≤n−1.

Let \(\mathit{LS}_{n}^{k}\subset \mathit{LS}_{n}\) be the set of smooth collections satisfying S 1={k}, 2≤k≤n+1. We want to show that the cardinality of \(\mathit{LS}^{k}_{n}\) is equal to \(\bar {r}_{k-2}\bar{r}_{n+1-k}\). To this end we construct a bijection \(F:\mathit{LS}_{n}^{k}\to \mathit{LS}_{k-2}\times \mathit{LS}_{n+1-k}\). For convenience, we write F=(f,g), where

$$f:\mathit{LS}_n^k\to \mathit{LS}_{k-2},\qquad g: \mathit{LS}_n^k\to \mathit{LS}_{n+1-k}. $$

First, since S 1⊂S a for any a, we have k∈S a , 2≤a≤n. Now the conditions k∈S a for a=1,…,k−1 imply

$$\{2,\ldots,k\}\subset S_a\quad \text{for all } a\geq k. $$

Given a collection \(\mathbf {S}\in \mathit{LS}_{n}^{k}\), we define

$$g(\mathbf {S})=\bigl(g(\mathbf {S})_1,\dots,g(\mathbf {S})_{n+1-k}\bigr) $$

as follows:

$$ g(\mathbf {S})_a= \begin{cases} \{i:\, 2\le i\le n+1-k, i+k-1\in S_{a+k-1}\}, &\text{if } 1\notin S_{a+k-1},\\ \{1\}\cup\{i:\, 2\le i\le n+1-k, i+k-1\in S_{a+k-1}\}& \text{otherwise}. \end{cases} $$
(4.1)

We note that the image depends only on the sets S a with a≥k.

Now, given a collection \(\mathbf {S}\in \mathit{LS}_{n}^{k}\), we need to define

$$f(\mathbf {S})=\bigl(f(\mathbf {S})_1,\dots,f(\mathbf {S})_{k-2}\bigr). $$

Let S k ={2,…,k}∪{i} for some i=1,k+1,…,n+1. We note that since k∈S a ⊂S k for a<k, each S a ∖k for 2≤a≤k−1 is an (a−1)-element subset of the fixed set of cardinality k−1 (this set is {2,…,k−1}∪{i}). We now define the map f as follows:

$$ f(\mathbf {S})_a= \begin{cases} \{i:\, 1\le i\le k-2, i+1\in S_{a+1}\}, &\text{if } S_{a+1}\subset\{ 2,\dots,k\},\\ \{i:\, 1\le i\le k-2, i+1\in S_{a+1}\}\cup\{k-1\}, &\text{otherwise}. \end{cases} $$
(4.2)

We note (this is important) that f 1(S) depends only on S 2,…,S k−1.

Our first goal is to show that f(S)∈LS k−2 and g(S)∈LS n−k+1 for any \(\mathbf {S}\in \mathit{LS}_{n}^{k}\). By definition, g(S) a ⊂g(S) a+1 for 1≤a≤n−k and

$$g(\mathbf {S})_a\in\{1,\ldots,n-k+2\}, \qquad \#g(\mathbf {S})_a=a\quad \text{for } 1\le a\le n-k+1. $$

Let us show that for 1≤a<b≤n−k+1 the inclusion b∈g(S) a implies a+1∈g(S) b . Since b>1, b∈g(S) a implies b+k−1∈S a+k−1. Since S is smooth, we obtain a+k∈S b+k−1, which gives a+1∈g(S) b and we are done. Similarly, one proves that f(S)∈LS k−2.

Finally, we have to prove that the map \(F=(f,g):\mathit{LS}_{n}^{k}\to \mathit{LS}_{k-2}\times \mathit{LS}_{n-k+1}\) is one-to-one. Given an element (S′,S″)∈LS k−2×LS n−k+1, we use formulas (4.1) and (4.1) to reconstruct S such that F(S)=(S′,S″). □

Corollary 4.4

The Euler characteristic of the smooth locus of is equal to the nth Schröder number r n .

Finally, let us formulate the analogue of Theorem 4.2 for the degenerate partial flag varieties. We omit the proof since it is very close to the proof of Theorem 4.2. Recall that the T-fixed points in are labeled by d-admissible collections \(\mathbf {S}=(S_{d_{1}},\dots,S_{d_{k}})\) (see (2.7)).

Theorem 4.5

A T-fixed point is smooth if and only if the following conditions hold: if \(b\in S_{d_{i}}\) and d j+1≥b>d j for some j≥i, then \(\{d_{i}+1,\dots,d_{i+1}\}\subset S_{d_{j+1}}\).

4.3 Poincaré polynomials

There are several ways to define q-analogues of the Schröder numbers (see [1, 2, 5]). We will need the simplest one (see [5], page 37, polynomials d n (q)). They are called Narayana polynomials there, but in other papers the same polynomials are also referred to as Schröder polynomials, see e.g. [20]). For a Schröder path P let diag(P) be the number of the diagonal steps in P. Define r n (q) as the sum of the terms q diag(P) over the set of Schröder paths P. Here are the first several polynomials

Clearly, r n (0) is the nth Catalan number. Let \(P_{n}^{\mathrm{sm}}(q)\) be the Poincaré polynomial of the smooth locus of . Our goal here is to prove the following theorem:

Theorem 4.6

\(P_{n}^{\mathrm{sm}}(q)=q^{n(n-1)/2}r_{n}(q)\).

Recall (see [1, 5]) that

$$ r_n(q)=qr_{n-1}(q)+\sum _{k=0}^{n-1} r_k(q)r_{n-1-k}(q). $$
(4.3)

Proposition 4.7

The Poincaré polynomials of the smooth locus satisfy the following recursion:

$$ P_n^{\mathrm{sm}}(q)=q^nP_{n-1}^{\mathrm{sm}}(q) + \sum_{l=0}^{n-1} q^{(l+1)(n-l)-1} P_l^{\mathrm{sm}}(q)P_{n-1-l}^{\mathrm{sm}}(q). $$
(4.4)

Proof

First, let us consider smooth collections (S 1,…,S n ) with S 1=1. Then the cells labeling such collections are in one-to-one correspondence with smooth collections S′ of length n−1: \(S_{i}'=S_{i+1}\setminus\{1\}\). We claim that

$$ \dim C_{\mathbf {S}}= \dim C_{\mathbf {S}'} + n. $$
(4.5)

We use Proposition 3.10. Clearly, the terms N PP and N II for p S and p(S′) do coincide and the difference of the terms N PI is equal to n (since S 1={1}, in the definition of N PI we can take t=1, i=1, j=2,…,n+1). Now (4.5) produces the first term of the right hand side of (4.4).

Recall the bijection \(F=(f,g):\mathit{LS}_{n}^{k}\to \mathit{LS}_{k-2}\times \mathit{LS}_{n-k+1}\), k≥2, from the set of smooth collections with S 1={k} to the product LS k−2×LS n−k+1. Our goal is to prove that

$$ \dim C_{\mathbf {S}}= \dim C_{f(\mathbf {S})} + \dim C_{g(\mathbf {S})} + (k-1) (n+2-k)-1 $$
(4.6)

(after the shift l=k−2 one gets the corresponding term in (4.4)). Recall that since k∈S 1 we have

$$\{2,\dots,k\}\subset S_m \quad \text{for all } m\geq k. $$

In particular, S k ={2,…,k}∪{r} for some number r=1,k+1,…,n+1. We claim that

and

$$N_{\mathit{II}}(\mathbf {S})=N_{\mathit{II}}\bigl(f(\mathbf {S})\bigr) + N_{\mathit{II}} \bigl(g(\mathbf {S})\bigr) + k-2. $$

First, let us prove the first formula. Assume that 1=r=S k ∖{2,…,k}. Then

Here the term (k−1)(n+1−k) comes from the fact that in the definition of N PI (S) one can take i=2,…,k, j=k+1,…,n+1 and t=k. These possibilities are not counted in N PI (f(S))+N PI (g(S)). Now assume that r>k. Then one has

Here the term (k−1)(n−k) comes from the fact that in the definition of N PI (S) one can take i=2,…,k, j∈{k+1,…,n+1}∖r and t=k. The term k−1 in the right hand side of the second equality comes from the fact that in the definition of N PP (S) one can take i=1, j=2,…,k and t=k. All these possibilities are lost when computing N PI (f(S)), N PI (g(S)), N PP (f(S)) and N PP (g(S)).

Now let us prove that

$$N_{\mathit{II}}(\mathbf {S})=N_{\mathit{II}}\bigl(f(\mathbf {S})\bigr) + N_{\mathit{II}} \bigl(g(\mathbf {S})\bigr) + k-2. $$

Here the argument is even simpler: the missing k−2 comes from the following possibilities for N II (S) missing in N II (f(S))+N II (g(S)): i=2,…,k−1, j=k, t=1.

We thus obtain

$$\dim C_{\mathbf {S}}= \dim C_{f(\mathbf {S})} + \dim C_{g(\mathbf {S})} + (k-1) (n+1-k) + (k-2), $$

which implies (4.6) as well as the proposition. □

Corollary 4.8

Theorem 4.6 holds.

Proof

We note that \(P_{1}^{\mathrm{sm}}(q)=1+q=r_{1}(q)\). Now the induction procedure combined with (4.3) and Proposition 4.7 gives the desired result. □

Remark 4.9

It is natural to define a q,t-version h n+1(q,t) of the normalized median Genocchi numbers as the sum over admissible collections S of the terms

Then the value h n (1,1) is exactly the normalized median Genocchi number and h n+1(q,0)=q n(n−1)/2 r n (q) is the (scaled) nth Schröder polynomial. Here are first few q,t-Genocchi polynomials:

4.4 Schröder numbers: from large to small

Recall the polynomials \(P_{n}^{\mathrm{sm}}(q)\), which are equal to q n(n−1)/2 r n (q), r n (q) being the q-Schröder polynomials. Recall (see [12, 20]) that the polynomials r n (q) are divisible by 1+q. The ratios are denoted by s n (q) (thus r n (q)=s n (q)(1+q)). These are the small q-Schröder polynomials. (In particular, s n (1) are the small Schröder numbers.) Our goal here is to show that the divisibility of r n (q) by 1+q has a very simple and concrete explanation within our approach. We give two proofs: one is due to the referee and uses the result from Appendix B. The second proof is based on the existence of a certain involution on the set of smooth cells.

Theorem 4.10

The polynomials \(P_{n}^{\mathrm{sm}}(q)\) and thus r n (q) are divisible by 1+q.

Proof

According to Theorem B.1 there exists an embedding of the smooth locus of into the desingularization R n (see [16] and Appendix B for more details). Recall that a point of R n is represented by a collection of subspaces V i,j and the map (V i,j )1≤i≤j≤n−1↦V 1,n−1 is a fibration R n →ℙ1 (recall that V 1,n−1 is a subspace of the two-dimensional space span(w 1,w n )). We thus obtain a composition map ρ from the smooth locus of onto ℙ1, which is SL 2-equivariant, where the group SL 2 acts naturally on the two-dimensional space span(w 1,w n ). Therefore, the map ρ is a cellular fibration and \(P_{n}^{\mathrm{sm}}(q)\) is divisible by the Poincaré polynomial of ℙ1, which equals 1+q. □

We now give the second proof of the theorem above.

Theorem 4.11

There exists a fixed-point free involution σ on the set of smooth collections. For any smooth collection S and the corresponding cell C S one has

$$\dim C_{\mathbf {S}} = \dim C_{\sigma \mathbf {S}} \pm1. $$

Proof

Consider the map w:{1,…,n+1}→{1,…,n+1}, which interchanges 1 and n+1 and stabilizes all other elements. Define a map σ by the formula

$$\sigma(S_1,\dots,S_n)=(wS_1,\dots,w S_n). $$

First, we note that σ maps each smooth S to a smooth collection. Second, since w 2 is the identity map, σ 2=Id. Third, let us show that σ is fixed-point free. In fact, a smooth S is fixed by σ if and only if for all k=1,…,n the set S k either contains both 1 and n+1 or does not contain any of these elements. We note that #S n =n and hence S n contains at least one of the elements 1, n+1. If σ S=S, then S n ⊃{1,n+1}. Now let 1≤k<n be a number such that {1,n+1} is contained in S k+1 but not in S k+1 (since #S 1=1 such k does exist). If σ S=S, then we have 1,n+1∉S k . Since S is smooth, S k ⊂S k+1 and therefore S k+1 contains two non-intersecting sets S k and {1,n+1}. This contradicts with #S k+1=k+1.

Now let S be a smooth collection. Let k be a number such that 1∈S k ∖S k−1 and, similarly, let l be a number such that n+1∈S l ∖S l−1. As we proved above, k≠l. Assume that k<l. We claim that

$$\dim C_{\mathbf {S}} = \dim C_{\sigma \mathbf {S}} + 1. $$

Recall that dimC S is the sum of three numbers N PI (S)+N PP (S)+N II (S) (see Proposition 3.10). First, we note that a pair i=1,j=n+1 adds one to N PI (S), but not to N PI (σ S). Second, each pair i,j with 1<i,j<n+1, either shows up for both S and σ S in the dimension counting as in Proposition 3.10 or does not show up for both cells. Now let us look at other pairs and compute the difference between the dimensions of C S and that of C σ S .

Take m satisfying k≤m<l and consider j such that j>m, j∉S m . Then a pair i=1,j adds one to N PI (S), but not to N PI (σ S) (since 1∈S m , but 1∉(σS) m ). However, let us look at a pair i=m, j=n+1. Since n+1∈(σS) m ∖S m , the pair (m,n+1) adds one to N II (σ S), but not to N II (S).

Now take m satisfying k≤m<l and consider i such that i≤m, i∈S m . Then a pair i,j=n+1 adds one to N PI (S), but not to N PI (σ S) (since n+1∉S m , but n+1∈(σS) m ). However, let us look at a pair i=1, j=m. Since 1∈S m ∖(σS) m , the pair (1,m) adds one to N PP (σ S), but not to N PP (S).

Summarizing, the difference

$$N_{\mathit{PI}}(\mathbf {S})+N_{\mathit{PP}}(\mathbf {S})+N_{\mathit{II}}(\mathbf {S}) - N_{\mathit{PI}}(\sigma \mathbf {S}) - N_{\mathit{PP}}(\sigma \mathbf {S}) - N_{\mathit{II}}( \sigma \mathbf {S}) $$

is equal to one (coming from the pair i=1, j=n+1). This implies the second statement of the theorem. □

Corollary 4.12

The polynomials \(P_{n}^{\mathrm{sm}}(q)\) and r n (q) are divisible by 1+q. The ratio \(P_{n}^{\mathrm{sm}}(q)/(1+q)\) is equal to the sum of the terms \(q^{\dim C_{\mathbf {S}}}\) taken over smooth S satisfying the following conditions for all m=1,…,n: if 1∈S m then n+1∈S m .

Proof

The Theorem above states that \(P_{n}^{\mathrm{sm}}(q)\) is equal to the sum over the orbits of the involution σ of the terms q d(1+q), where d is the minimum of the dimensions of the cells corresponding to the collections in the orbit. But we know that dimC S =dimC σ S −1 if there exists m such that n+1∈S m , but 1∉S m . This implies the corollary. □

Let us relabel the smooth collections as follows. To a smooth collection S we attach a permutation π∈S n+1 by the formula π(m)=S m ∖S m−1. Then S is smooth if and only if the corresponding permutation satisfies the following conditions for all 1≤a<b≤n:

$$\text{if}\quad \pi^{-1}(b)\le a\quad \text{then}\quad \pi^{-1} (a+1) \le b. $$

Corollary 4.13

The number of permutations, corresponding to smooth collections, is equal to the large Schröder number. The number of such permutations satisfying π −1(n+1)<π −1(1) is equal to the small Schröder number.