Degenerate Affine Flag Varieties and Quiver Grassmannians

We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and that their irreducible components are normal Cohen-Macaulay varieties with rational singularities.


Methods and Structure.
To utilise the approximations via quiver Grassmannians, it is necessary to understand the quiver Grassmannians for the oriented cycle and their geometric properties. The link between representations of quivers and modules over finite dimensional algebras is the foundation of the realisation of quiver Grassmannians as framed module spaces. This interpretation of quiver Grassmannians allows us to translate properties between the variety of quiver representations and the quiver Grassmannian.
The equioriented cycle, and the class of quiver Grassmannians which we want to examine, are introduced Section 3. Based on word combinatorics we prove a dimension formula for the space of morphisms between nilpotent indecomposable representations of the cycle. This is applied to the parametrisation of irreducible components. Moreover it reveals certain favourable geometric properties of the quiver Grassmannians for the cycle and of the approximations of the degenerate affine flag variety, as claimed in Theorem 2.4. The proof of the geometric properties utilises the construction of the quiver Grassmannian as framed moduli space. Hence they lift from the variety of quiver representations, which was studied by G. Kempken in [21].
In Section 3.3, we introduce a C * -action on the quiver Grassmannians for the equioriented cycle. This action provides us with a combinatorial tool to compute the Euler characteristic of these quiver Grassmannians, which was introduced by G. Cerulli Irelli in [3]. It induces a cellular decomposition of the quiver Grassmannians, whose cells are in bijection with affine Dellac configurations. For the proof that such a decomposition exists, it is crucial to find the right grading of the vector spaces basis of the spaces belonging to a quiver representation. This grading defines a torus action on the indecomposable summands of the quiver representation. The action on the quiver representation induces an action on the quiver Grassmannian. The identification of cells and affine Dellac configurations is based on the parametrisation of cells by successor closed subquivers [3].
The class of quiver Grassmannians studied in this paper has some rather strong restrictions. But deviating from the Dynkin setting, these restrictions are necessary to obtain the results expected from the classical setting. If we drop the restrictions which keep up the analogy, it is hard to say anything about the quiver Grassmannians. Only the existence of a cellular decomposition is rather general among the quiver Grassmannians for the cycle. It would be interesting to find out more about the geometry of the intermediate degenerations but the methods introduced in this paper are not sufficient since the identification of the approximations with a framed moduli space fails for all intermediate degenerations.

Representations of Quivers
In this section we recall some definitions and results about quiver representations which are required to examine properties of approximations of the degenerate affine flag variety. Fix an algebraically closed field k. Let Q be a finite quiver with a finite set of vertices Q 0 , a finite set of edges Q 1 between the vertices and two maps s, t : Q 1 → Q 0 providing an orientation of the edges with source s α and target t α for all α ∈ Q 1 . A Q-representation R is a pair of tuples R = (V, M ), with a tuple of k-vector spaces over the vertices V = (V i ) i∈Q0 , and a tuple of linear maps between the vector spaces along the arrows of the underlying quiver M = (M α ) α∈Q1 .
The category of finite dimensional Q-representations over the field k is rep k (Q). A morphism ψ of Q-representations R = (V, M ) and S = (U, N ) is a collection of linear maps ψ i : V i → U i such that ψ j • M α = N α • ψ j for all edges α : i → j. The set of all Q-morphisms from R to S is denoted by Hom Q (R, S). Its dimension is abbreviated as [R, S] := dim Hom Q (R, S).
1.1. Quiver Grassmannians. The entries of the dimension vector dim R ∈ Z Q0 of a quiver representation R are given by the dimension of the vector spaces V i over the vertices of the quiver. A subrepresentation S ⊆ R is parametrised by a tuple of vector subspaces U i ⊂ V i , which is compatible with the maps between the vector spaces of the representation R, i.e. for all arrows α : i → j of Q we have M α (U i ) ⊆ U j . In [6] the stratum of U ∈ Gr ∆n e (M ) is defined as the isomorphism class of U in the quiver Grassmannian. By [6,Lemma 2.4], it is irreducible, locally closed and of dimension Hence the irreducible components of the quiver Grassmannian are given by the closures of the strata which are not contained in the closure of any other stratum.

Path Algebras and Bounded Representations.
A path in a quiver is the concatenation of successive arrows. The path algebra A := kQ of the quiver Q is the k-algebra with all paths in Q as basis and multiplication of paths is defined via concatenation [28,Definition 4.5]. A quiver representation is called indecomposable if it can not be written as direct sum of two non-zero quiver representations. Every finite dimensional quiver representation has a decomposition into indecomposable representations which is unique up to the order of the summands [22,Theorem 1.11]. Q has finitely many indecomposable representations if and only if the underlying graph of Q is a simply-laced Dynkin diagram [13]. This implies that the path algebra is finite dimensional.
For arbitrary quivers we have to restrict to representations of bound quivers, i.e. representations of Q such that the maps of the representation satisfy relations from an admissible ideal I in the path algebra kQ. Then A I := kQ/I is finite dimensional and in particular there are only finitely many indecomposable representations of the bound quiver (Q, I) [28,Theorem 5.4]. 1.4. Simple, Projective and Injective Representations. The simple representation S i of Q has a one dimensional k-vector space over the i-th vertex. All other vector spaces and the maps along the arrows are zero. The projective representation P k has a vector space with a basis indexed by paths from k to i over the i-th vertex. The maps along the arrows are determined by concatenation of the arrows with the paths labelling the basis. Analogously, the injective representation I k has a vector space with a basis indexed by paths from j to k over the j-th vertex. The maps along the arrows are determined by the factoring of paths through the arrows. For bound quivers, one has to work with the equivalence classes of paths in the bounded path algebra [28,Definition 5.3].
1.5. Coefficient Quivers. Let R = (V, M ) be a finite dimensional representation of a finite quiver Q and d its dimension vector. The coefficient quiver Q(R) has one vertex for each basis element v is called successor closed if for all vertices in the subquiver, their image along the arrows of Q(R) is also contained in the subquiver.
and M s e,d (Q, I) is the geometric quotient of R s e,d (Q, I) by the group G e . The existence of this geometric quotient is part of the statement and for its proof it is important that the stability as defined above is a stability in the sense of geometric invariant theory. Here d is a tuple with multiplicities of injective bounded quiver representations and not the dimension vector of the quiver representation J. This theorem was first proven by M. Reineke for Dynkin quivers in [26,Proposition 3.9]. S. Fedotov used the same methods to derive the statement in the generality of modules over finite dimensional algebras in [10,Theorem 3.5].
The following theorem establishes a bijection between orbits in the variety of quiver representations and strata in the corresponding quiver Grassmannian, which preserves geometric properties. It allows us to lift certain properties of the variety of quiver representation, studied by G. Kempken, to the quiver Grassmannians, which we use for the finite approximations of the affine flag variety and its degeneration. In this generality, it was already known by K. Bongartz [2] and in the case of Dynkin quivers it is proven by M. Reineke in [26,Theorem 6.4].
Define R  The proof of this statement is obtained with the same arguments as used by M. Reineke, for the Dynkin case in [25,26].
2. The Degenerate Affine Flag Variety 2.1. Alternative Parametrisations of the Affine Flag Variety. The first step in the direction of approximations by quiver Grassmannians is an alternative description of the affine flag variety, which is similar to the identification with the set of vector space chains in the classical setting. The affine flag variety is infinite dimensional such that we have to replace the finite dimensional vector space by some infinite dimensional objects.
There are two approaches to this problem. The more common construction is via lattice chains [1,14,15]. It is possible to define approximations and degenerations of the affine flag variety in this setting, but we want to take a different path, where the analogy to the classical setting is more visible. The second construction is based on Sato Grassmannians [12,19].
For ℓ ∈ Z let V ℓ be the vector space which is a subspace of the infinite dimensional C-vector space V with basis vectors v i for i ∈ Z. The Sato Grassmannian SGr k for k ∈ Z is defined as The vector spaces in the chains for the classical flag variety are elements of the Grassmannians Gr k (n). Analogously, we obtain a description of the affine flag variety as a set of cyclic chains where the vector spaces are elements of the Sato Grassmannians SGr k .
Proposition 2.1 ([12, 19]). The affine flag variety F l gl n as subset in the product of Sato Grassmannians is parametrised as Here we use a slightly different parametrisation for the affine flag variety than the version by E. Feigin, M. Finkelberg and M. Reineke in [12]. It turns out that this parametrisation fits better for the identification of finite approximations with quiver Grassmannians which we have in mind. It is shown in [11] that the degenerate SL n+1 -flag variety admits a description via vector space chains, where the spaces are related by projections instead of inclusions. In the same way, we define degenerations of the affine flag variety.
The different parametrisation of the affine flag also leads to a different definition of the degeneration, but the version as defined above is isomorphic to the degenerate affine flag variety as introduced in [12].

Finite Approximations by Quiver Grassmannians and Geometric
Properties. E. Feigin, M. Finkelberg and M. Reineke studied quiver Grassmannians for the loop quiver, to model finite approximations of the affine Grassmannian [12]. Analogously, in this paper we restrict us to quiver Grassmannians for the equioriented cycle: Both, the set of vertices, and arrows, of ∆ n are in bijection with the set Z n := Z/nZ.
For a positive integer ω, the finite approximation of the degenerate affine flag variety is defined as There is an analogous version of this statement for certain, more general, degenerations of the affine flag variety. This construction allows us to obtain statements about geometric properties of the approximations from the corresponding quiver Grassmannians.

Theorem 2.4.
For ω ∈ N, the approximation F l a ω gl n of the degenerate affine flag variety satisfies: (1) It is a projective variety of dimension ωn 2 .
(3) It admits a cellular decomposition. (4) The irreducible components are normal, Cohen-Macaulay and have rational singularities. (5) There is a bijection between irreducible components and grand Motzkin paths of length n.
Grand Motzkin paths of length n are lattice paths from (0, 0) to (n, 0) with steps (1, 1), (1, 0) and (1, −1), without the requirement that the path is not allowed to cross the x-axis. Accordingly the number of irreducible components is independent of the parameter ω and the same in every approximation. The identification of cells and affine Dellac configurations is proven in Section 4.

Isomorphism of Finite Approximations and Quiver Grassmannians.
Proof of Theorem 2.3. In the approximation to the parameter ω, the cyclic relations of the vector spaces, describing a point (U i ) i∈Zn in the degenerate affine flag variety, induce the restrictions Accordingly the corresponding approximations of the Sato Grassmannians are isomorphic to the Grassmannian of vector subspaces Gr ωn (2ωn). Here we identify the standard basis of C 2ωn with basis vectors of V as Moreover, the Sato Grassmannian SGr k is just the k-th shift of the Sato Grassmannian SGr 0 , i.e. SGr k ∼ = s k SGr 0 [12, Section 1.2]. Instead of one index shift by n we can take the Sato Grassmannians SGr 0 in the place of SGr k and have a shift by one along each arrow of the quiver and for the maps pr i+1 we have to write pr 1 . If we want to apply these shifts on the left hand side of the containment relations, they become a shift by minus one. This identification works in the same way on the level of approximations. We obtain the projection pr ωn : C 2ωn → C 2ωn if we apply the above base change for i = 0 to the restriction of the projection pr 1 : V → V to the finite support induced by the approximation. Due to this base change, the index shift by minus one is turned into a shift by plus one. Combining these properties, we obtain the following chain of isomorphisms: Gr ωn (2ωn) :

Quiver Grassmannians for the Equioriented Cycle
For every i ∈ Z n , we define the path with ℓ arrows starting at vertex i as The path algebra k∆ n is denoted by A n . This algebra is not finite dimensional because there are paths p i (ℓ) of arbitrary length around the cycle. Let be the ideal of the path algebra generated by all paths of length N . For N ∈ N, we define the bounded path algebra A n,N := k∆ n /I N . The following result is a special case of [28,Theorem 5.4].
Proposition 3.1. The category rep k (∆ n , I N ) of bounded quiver representations is equivalent to the category A n,N -mod of (right) modules over the bounded path algebra.
Let P i ∈ rep k (∆ n , I N ) be the projective bounded representation of ∆ n at vertex i ∈ Z n . Define the projective representation where x i ∈ Z ≥0 for all i ∈ Z n . Analogously, let I j ∈ rep k (∆ n , I N ) be the injective bounded representation of ∆ n at vertex j ∈ Z n and define the injective representation with y j ∈ Z ≥0 for all j ∈ Z n . Throughout this section, we study quiver Grassmannians Gr ∆n e (X ⊕ Y ), where e := dim X is the dimension vector of X. A representation U of the cycle is called nilpotent if there exists an integer N such that U satisfies the relations in I N , i.e. cyclic concatenations of the maps corresponding to U are zero after a certain length. The indecomposable nilpotent representations of ∆ n are parametrised by some starting vertex i ∈ Z n and a length parameter ℓ ∈ Z ≥0 [22, Theorem 7.6]. We denote them by U i (ℓ) and the simple representations over the vertices have length one, i.e. S i = U i (1). It turns out that we can identify bounded projective and bounded injective representations of the cycle, via indecomposable nilpotent representations.

Proposition 3.2.
For n, N ∈ N and all i, j ∈ Z n the projective and injective representations P i and I j of the bound quiver (∆ n , I N ) satisfy Proof. It follows from the definition of projective and injective representations of bound quivers as in [28,Definition 5.3] that they are of the form as described in [22,Theorem 7.6].
This identification allows us to apply Theorem 1.3 to the quiver Grassmannians as introduced above. Accordingly it is possible to realise them as framed module space and we can use Theorem 1.4 to study their geometry.

Word Combinatorics.
Definition 3.3. For a nilpotent representation U i (ℓ) of the equioriented cycle on n vertices, the corresponding word w i (ℓ) is defined as To each nilpotent quiver representation X we assign a diagram ϑ X , consisting of the words corresponding to the indecomposable direct summands of X. Define r j (w) as the number of repetitions of the letter j in a word w. These numbers can be used to compute the dimension of the space of morphisms between two indecomposable nilpotent representations of the cycle. The linearity of the morphisms allows us to generalise this formula to compute the dimension of the morphism space for all nilpotent representations of the cycle. Here we use the notation [ . , . ] as abbreviation for the dimension of the space of morphisms Hom ∆n ( . , . ).

Proposition 3.4.
For two indecomposable nilpotent representations U i (ℓ) and U j (k) of ∆ n , let w i (ℓ) and w j (k) be the corresponding words. Then the dimension of the space of morphisms from U i (ℓ) to U j (k) equals where m := min{ℓ, k}. This proposition is just a different formulation of a result by A. Hubery [18,Theorem 16 (1)]. It is also possible to compute the dimension of the space of morphisms by counting certain repetitions of the letter i in the word corresponding to U j (k). Here we have to exclude the repetitions coming before max{0, m − ℓ} such that the parametrisation of the word, wherein we have to count the repetitions, becomes more complicated. Hence we exclude this case from the proposition.
We can use word combinatorics to compute the dimension of the space of morphisms from an arbitrary representation in R e (∆ n , I N ) to an indecomposable representation of maximal length.
because the morphisms of quiver representations are linear maps between finite dimensional vector spaces.
. We want to apply Proposition 3.4. By assumption we know that ℓ ≤ N , and hence we have to count the repetitions of the vertex i + N − 1 (which is the end point of U i (N )) in the word w j (ℓ) corresponding to U j (ℓ). This value is given by the (i + N − 1)-th entry of the dimension vector of U j (ℓ).
By [6,Lemma 2.4], the dimension of the stratum of U ∈ Gr ∆n For the first part, we can apply Proposition 3.5, and obtain [U, , because X and Y consist of summands of the form U i (N ), and the morphisms are linear. To compute the dimension of the quiver Grassmannian, we are interested in the value of [U, U ], and want to find the elements of the quiver Grassmannian minimising it. On the variety of quiver representations R e (∆ n , I N ), we have an action of the group G e . The dimension of the G e -orbit of Thus we can compute maximisers for the dimension of the G e -orbit in R e (∆ n , I N ), among the elements of the Grassmannian Gr ∆n e (X ⊕ Y ), in order to find the strata of highest dimension in this quiver Grassmannian. Unfortunately, this does not work in the same way for the equioriented cycle. But in some special cases at least the dimension formula holds.
, which is computed with the word combinatorics as above. The indecomposable representation U i (ωn) has length ωn, which means that it is winding around the cycle on n points exactly ω times. For this reason we refer to ω as winding number. The proof of this statement is based on the characterisation of minimal degenerations of orbits in the variety of quiver representations as given by G. Kempken in [21].
Proof of Proposition 3.8. The idea of the proof is to show that for every element U ∈ Gr ∆n e (X ⊕ Y ), its G e -orbit in the variety of quiver representations is the degeneration of an orbit with the same codimension as the G e -orbit of X, or already has the same codimension. In the first case, if the G e -orbit of U is contained in the closure of a different G e -orbit, the codimension of O U := G e .U is strictly bigger. The following property of a subrepresentation helps us to decide how far the representation is degenerate from X.
This set includes all direct summands of U , which are not of maximal length. If the = ω for all i, j ∈ Z n and N = ω · n. This follows from Proposition 3.5 if we set M = U i (N ). Now let U ∈ Gr ∆n e (X ⊕ Y ) be given such that S(U ) = ∅. Since all entries of the dimension vector dim U are equal, and U consists of indecomposable summands of lengths at most N , the set S(U ) has to contain at least two elements.
We can find U i (ℓ), U j (k) ∈ S(U ) and can assume without loss of generality that j is contained in the word w i (ℓ + 1). This pair has to exist because otherwise the dimension vector of U could not be homogeneous, i.e. all entries being equal. By changing the labelling of the two representations, we can ensure that they satisfy the relation we want. Let w 2 be the overlap of the words w i (ℓ) and w j (k). We can write them as w i (ℓ) = w 1 w 2 and w j (k) = w 2 w 3 , where it is possible that w 2 is the empty word. We define the representation Here \ U i (ℓ) means that we do not take the direct summand U i (ℓ) of U for the definition ofÛ. We have to show that U is a degeneration ofÛ and thatÛ is contained in the quiver Grassmannian.
In the terminology of words the representation U i (i − j + ℓ + k) corresponds to w 1 w 2 w 3 and U j (j − i) corresponds to w 2 . We can assume that the word w 1 w 2 w 3 has not more than N letters, because there have to exist w i (ℓ) = w 1 w 2 and w j (k) = w 2 w 3 such that this is satisfied. Without such words it would not be possible that the dimension vector of U is homogeneous and that all words corresponding to it have at most N letters.
By construction U andÛ have the same dimension vector. It is sufficient to find an arbitrary embedding ofÛ into X ⊕ Y , to show thatÛ is contained in the quiver Grassmannian. The easiest way to do this is to identify a segment wise embedding, i.e. ι : U (i; ℓ) ֒→ U (i; k) for k ≥ ℓ. For nilpotent representations of the equioriented cycle, the existence of segment wise embeddings is equivalent to the existence of arbitrary embeddings. This follows from the structure of the Auslander Reiten quiver for the indecomposable nilpotent representations of the equioriented cycle.
For example, this is computed using the knitting algorithm [28, Chapter 3.1.1]. Given a fixed nilpotence parameter N , we take an equioriented type A quiver of length 2N n and identify the vertex i with all repetitions i+kn. Similarly we identify vertices in the Auslander Reiten quiver of type A and obtain the Auslander Reiten quiver for the equioriented cycle.
Hence there exists a segment wise embedding of U into X ⊕ Y . From the structure of segment wise embeddings of indecomposable representations of the cycle as introduced above, we know that U i (i − j + ℓ + k) embeds into the same U p (N ) as U j (k), and U j (j − i) embeds into the same U q (N ) as U i (ℓ). It follows that the representationÛ embeds into the same summands of X ⊕ Y as U .
Following [21,Satz 5.5] by G. Kempken, the orbit of U is a degeneration of the orbit OÛ , i.e. O U ⊂ OÛ . Hence we obtain dim O U < dim OÛ , since U andÛ are not isomorphic because their diagrams of words are constructed such that they do not contain the same words. This degeneration might not be minimal, but here it is not of interest to find minimal degenerations. Thus we do not have to satisfy the restrictions on the words in [21,Satz 5.5].
Since all the vector spaces U i for i ∈ Z n corresponding to the subrepresentation U are equidimensional, we can apply this procedure starting from any U in the quiver Grassmannian, until we arrive at anÛ ∈ Gr ∆n For the proof it is crucial that the dimension vector of the subrepresentations is homogeneous, and that the length of the cycle divides the length of the indecomposable projective and injective representations. This is guaranteed by the condition N = ωn. Otherwise we can not assure that the gluing procedure of the words ends in a representation with S(U ) = ∅. In the setting where N = ωn, it is not possible to control the minimal codimension of the G e -orbits.

Dimension Formula and Parametrisation of Irreducible Components.
For this subsection, we restrict us to the case N = ωn. The bounded projective and injective representations in rep k (∆ n , I ωn ) will be denoted by P ω i and I ω j . Based on Proposition 3.8 we can compute the dimension of Gr ∆n e (X ⊕ Y ). In its proof the subrepresentations U , with the same codimension as X, are characterised. This allows to determine the irreducible components of the quiver Grassmannian.
Based on the characterisation of the strata with the same codimension as the stratum of X ω , from the previous section, we obtain the following parametrisation of the irreducible components of the quiver Grassmannians.
where d i := y i + x i+1 and they all have dimension ωk(m − k).
Remark 3.11. In particular, the number of irreducible components is independent of the winding number ω.
Proof. We use the interpretation of the Grassmannian as framed moduli space Here the i-th entry of d is given by the multiplicity of the injective representation I ω i as summand of X ω ⊕ Y ω , and these numbers are independent of the winding number ω.
The irreducible components of Gr ∆n eω (X ω ⊕ Y ω ) are given by the closures of the strata which are maximal in the partial order S U ≤ S V :⇔ S U ⊆ S V . We apply Theorem 1.4 to this setting such that the irreducible components are in bijection with the maximal elements of R (d) eω (∆ n , I ωn )/G eω , with respect to the partial order induced by the inclusion of orbits in orbit closures.
In the proof of Proposition 3.8, we have seen that the maximal elements for this order are parametrised by the U ∈ Gr ∆n eω (X ω ⊕ Y ω ) such that S(U ) = ∅, i.e. all summands of U have the same dimension vector, and are of the form U i (ωn) for some i ∈ Z n . Accordingly, these subrepresentations are representatives for strata whose closures give the irreducible components. The set of all such subrepresentations of X ω ⊕ Y ω , with dimension vector e ω , is parametrised by the set C k (d). For every tuple p ∈ C k (d), define the representation The assumption i∈Zn p i = k ensures that the dimension vector of U (p) is e, since the dimension vector of U (i; N ) has all entries equal to ω. The restriction is necessary to guarantee that the representation U (p) corresponding to the tuple p embeds into X ω ⊕ Y ω . The dimension of the irreducible components is computed in the same way as done for the stratum S Xω in Lemma 3.9.
For arbitrary N , we would have d i := y i + x i−N +1 but for N = ω · n the shift by N does not change the index, because it is considered as a number in Z n . The number of irreducible components is bounded by n+k−1 The set C k is the set of all partitions of the number k into at most n parts. Partitioning the number k into at most n parts, is equivalent to choosing n − 1 points, out of n + k − 1 points, to be the separators between the n parts of a partition of the remaining k points. The number of these choices is given by For the case N = n, this parametrisation of the irreducible components, together with a precise count, is proven in the thesis of N. Haupt [16, Proposition 3.6.16]. We can use his formula for the case ω = 1, to compute the number of irreducible components of the quiver Grassmannian Gr ∆n eω X ω ⊕ Y ω , because the number of irreducible components is independent of ω.

Geometric Properties.
Back in the setting where the indecomposable summands of X and Y have arbitrary but all the same length N , we do not have a parametrisation of the irreducible components, but nevertheless they have the following properties. We can apply it to orbit closures of nilpotent representations in R e (∆ n , I N ), because by [21, Korollar 2.10] there are no non-nilpotent representations inside these orbit closures. Combining this with Theorem 1.4, we get that the closures of the strata in the quiver Grassmannian have rational singularities, which again combined with [20, p. 50] yields that they are normal and Cohen-Macaulay. Applying it to the closures of the strata, which are maximal for the partial order as introduced in Lemma 3.10, we obtain the desired result.
Moreover, G. Kempken gives a description of the types of singularities which can occur, and she also describes the structure of the orbit closures and the codimension of the minimal degenerations of orbits.

Torus Action and Cellular Decomposition.
From now on, we restrict us to the setting where k = C. For every nilpotent representation U ∈ rep C (∆ n ), there exists a nilpotence parameter N ∈ N such that U ∈ rep C (∆ n , I N ). Hence by [22,Theorem 1.11] it is conjugated to a direct sum of indecomposable nilpotent representations, i.e.
where d i,ℓ ∈ Z ≥0 for all i ∈ Z n and ℓ ∈ [N ].
Let B i := {v To simplify notation we also use the index i for the arrow α : i → i + 1. The segments of M represent its direct summands U i (ℓ) = U (i + ℓ − 1; ℓ).
The number of indecomposable direct summands of U (d), ending over the vertex i ∈ Z n , is given by We rearrange the segments of M such that they end in the d i last basis vectors over each vertex. In each package of segments ending over some vertex we order them from long to short, i.e. the shortest segment ending over the i-th vertex ends in the basis vector v (i) mi . We continue the segments, to the last free basis vectors over the i − 1-th vertex of of ∆ n , such that the order by length is preserved. This way of arranging the segments ensures that they do not cross.
The condition d(α) := d sα for all α ∈ Z n induces a grading of the vertices in the coefficient quiver of U (d). We give weight one to the starting point of the longest segment. For the other vertices on this segment, their weight increases by the weight of the arrows connecting them. The weight of the end point of the longest segment is given to all other endpoints of segments, containing more than two vertices. For the other vertices on these segments, we compute the weight along the arrows in the opposite direction. The weight of segments consisting of single vertices is chosen increasingly from the weight of the vertex above them, or starting from one if there is no vertex above them.
Using this grading, we can define a torus action on the quiver representation M . An element λ ∈ T := C * acts on every element b ∈ B • of the new ordered basis as By linearity, this action extends to all elements of M , and also to the quiver Grassmannian [3,Lemma 1.1]. This implies that the Euler-Poincaré characteristic of the quiver Grassmannian is given by the number of its torus fixed points [3,Theorem 1], and that they are parametrised by successor closed subquivers [3, Proposition 1]. Analogous to [5,Section 6.4], this action induces a cellular decomposition of the quiver Grassmannian. In Particular, these quiver Grassmannians have property (S) [8,Definition 1.7]. This explicit parametrisation of the cells allows us to use the combinatorics of coefficient quivers to study geometric properties of the quiver Grassmannians. The existence of a cellular decomposition is proven in [4].
Proof. We use the action of the torus T on the quiver Grassmannian Gr ∆n e (M ) as introduced above, to define a cellular decomposition of the product of subspace Grassmannians Gr e (m) := i∈Zn Gr ei (m i ). We parametrise each torus fixed point by a collection of subspaces L ∈ Gr e (m), and define the cells for every i ∈ Z n . This induces a decomposition of the subspace Grassmannians Here e i is the i-th entry of the dimension vector e of the subrepresentations in the quiver Grassmannian. Thus a point V i ∈ C(L i ) is spanned by vectors j,s ∈ C, because these vectors parametrise all spaces V i with limit L i . Hence the C(L i ) are affine spaces for all i ∈ Z n .
In order to prove that the attracting sets in the quiver Grassmannian are affine, we have to describe the coordinates in the intersection of the cells in the subspace Grassmannians with the quiver Grassmannian. Let V ∈ C(L) be a point in the some attracting set. Like above, it corresponds to a collection of spaces V i ∈ C(L i ) for i ∈ Z n , which is parametrised by the collection of coefficients {µ To simplify the notation we replace the indices by the weight of the corresponding basis vector. This is possible since all basis vectors over a vertex have distinct weights and the weights are increasing with increasing indices. Accordingly the sums now run over the weights corresponding to the index set [m i ] instead of the index set itself. If M i w (i) ks is non-zero, this leads to the equation , and this is satisfied if and only if Combined with the above description of M i w (i) ks , this leads to equality of coefficients µ showing that the attracting sets in the quiver Grassmannian are affine spaces.

Proof of Theorem 2.4.
The representation M ω as defined in Theorem 2.3 is nilpotent, and every nilpotent representation of ∆ n has a decomposition into the indecomposable representations U (i; ℓ) [22,Theorem 1.11]. The following observation is the key to examine geometric properties of the degenerate affine flag, because it allows us to apply the results about quiver Grassmannians from this section. Lemma 3.14. For ω ∈ N there is an isomorphism of ∆ n -representations Proof. The coefficient quiver of M ω consists of 2n segments of length ωn, and over each vertex there start and end exactly two segments. Each of these segments corresponds to a bounded injective representation I ω j ∼ = U (j; ωn).
With this identification of quiver representations, Theorem 2.4 is a specialisation of the results about quiver Grassmannians for the cycle as developed above.
Proof of Theorem 2.4. Part (1) follows from Lemma 3.9 and the identification with a quiver Grassmannian as in Theorem 2.3, i.e. dim F l a ω gl n = ωn(2n − n) = ωn 2 . From the C * -action and the induced cellular decomposition as in Theorem 3.13, we obtain (3). With x i = y i = 1, we obtain (2) and (5) as special case of Lemma 3.10, i.e. the irreducible components are equidimensional and parametrised by p ∈ Z n ≥0 : p i ≤ 2 for all i ∈ Z n , i∈Zn p i = n .
With b i := p i − 1, the tuples b describe the steps in grand Motzkin paths, and we obtain a bijection between the above set parametrising the irreducible components and the set of grand Motzkin paths of length n. The geometric properties in part (4) are obtained as application of Lemma 3.12.

Affine Dellac Configurations
For the Feigin degeneration of the classical flag variety of type A n , the Poincaré polynomial can be computed using Dellac configurations, which are counted by the median Genocchi numbers. This description was develloped by E. Feigin in [11]. The torus fixed points of the symplectic degenerated flag variety are identified with symplectic Dellac configurations by X. Fang and G. Fourier in [9]. In this section, we identify affine Dellac configurations with the cells of the degenerate affine flag variety, based on the parametrisation of the cells via successor closed subquivers. Definition 4.1. For n ∈ N an affine Dellac configurationD to the parameter ω ∈ N consists of a rectangle of 2n × n boxes, with 2n entries k j ∈ {0, 1, 2, . . . , ω} such that: (1) There is one number in each row.
(2) There are two numbers in each column. (3) 2n j=1 (p j + nr j ) = ωn 2 , where r j := max{k j − 1, 0}. The left hand side and the right hand side of the rectangle are identified to obtain boxes on a cylinder. There is a staircase around the cylinder from left to right as separator. In the planar picture, we draw it from the lower left corner to the upper right corner of the rectangle of boxes. With respect to the staircase, p j is the number of steps from the separator to the entry going left. If the entry is zero, the position is zero as well. The set of affine Dellac configurations to the parameter ω is denoted by DC n (ω).   (1) each row contains exactly one marked box, (2) each column contains exactly two marked boxes, The set of all Dellac configurations for a fixed parameter n will be denoted by DC n and its cardinality is given by the normalised median Genocchi number h n . The degenerate flag variety F l a (sl n+1 ) is isomorphic to the quiver Grassmannian where Q is the equioriented type A quiver on n vertices and A = CQ is its path algebra [6, Proposition 2.7]. Now we want to look at the relation of cells and configurations for the degenerate flag variety F l a (sl 5 ). In this setting, the coefficient quiver of the representation A ⊕ A * is of the from as in the example below.
By [3, Proposition 1] we obtain that the cells of the degenerate flag variety are in bijection with successor closed subquivers of this coefficient quiver, which have j marked points in the j-th column. From such a subquiver we obtain a Dellac configuration by marking the boxes corresponding to the starting points of the segments of the subquiver, and marking the only possible boxes in the first and the last row. If a segment contains no marked point, we have to mark the box corresponding to the point to the right of the end point of this segment.
The other way around, we can start with a Dellac configuration and transfer the marked points to the coefficient quiver of A ⊕ A * . Then we mark all necessary points to make this a successor closed subquiver. Hence the cells of F l a (sl n+1 ) are in bijection with the Dellac configurations in DC n+1 , as proven in [11].  Proof. The approximation F l a ω gl n is isomorphic to the quiver Grassmannian Gr ∆n eω M a ω , where e ω := (e i = ωn) i∈Zn and M a ω = i∈Zn U (i; ωn) ⊗ C 2 .
By [3, Proposition 1] we obtain that the cells in the approximation of the degenerate flag variety are in bijection with successor closed subquivers in the coefficient quiver of M a ω , which have ωn marked points over each vertex i ∈ Z n . Each of the subquivers consists of 2n segments with length between zero and ωn. There are exactly two segments ending over each vertex and they correspond to the indecomposable representations U (i; ℓ i,1 ) and U (i; ℓ i,2 ), where U (i; 0) is the zero representation independent of the indexing vertex. The marked points of the subquiver are encoded in the dimension vector of the representation U (l). Hence the set C a n, ω is in bijection with the set of successor closed subquivers, which parametrise the cells of F l a ω gl n .
Now we want to study how every row of an affine Dellac configuration encodes an indecomposable representation U (i; ℓ i,k ) for k ∈ {1, 2}. Given an entry k j of a configuration, and its relative position p j to the separator, we set ℓ i,k = p j + nr j , where i = j, k = 1 for j ≤ n, and i = j − n, k = 2 for j > n. Vice versa, we compute k j := ⌈ℓ i,k /n⌉ and p j := ℓ i,k − nr j , where j = i for k = 1 and j = i + n for k = 2. These maps are inverse to each other. It remains to show that the image of a cell is an affine Dellac configuration to the parameter ω, and that each of these configurations is mapped to a cell.
Given a length tuple parametrising the fixed point U (l), which describes a cell in the quiver Grassmannian, we compute the numbers k j for j ∈ [2n] as described above. By construction of the map, we obtain exactly 2n parameters k j ∈ [ω] 0 and have a unique way to write them in the 2n rows of a configuration. These parameters satisfy since the entries in the dimension vector of U (l) are equal to ωn. It follows from the parametrisation of cells by successor closed subquivers, that one cell is obtained from each other cell by moving parts of subsegments in the coefficient quiver. These movements preserve the number of segments starting and ending over each vertex of the underlying quiver, where the empty segment over the vertex i is considered as segment starting over the vertex i + 1. By Lemma 3.10 the top dimensional cells have two segments starting and ending over each vertex of the quiver ∆ n . Hence this is true for all other cells such that in the configuration, which is assigned to the length tuple l, there are exactly two entries in each column. Accordingly, the image of a cell is an affine Dellac configuration to the parameter ω.
Starting with an affine Dellac configuration D(k) to the parameter ω, we compute the numbers ℓ i,k as described above. We obtain 2n numbers parametrising the length of the subsegments in the coefficient quiver of M a ω , and by construction all of these numbers live in the set [ωn] 0 . It remains to show that the subquiver, which is parametrised by this length tuple, has dimension vector e ω = (e i = ωn) i∈Zn .
From an affine Dellac configuration, we can compute the dimension vector of the corresponding successor closed subquiver as follows. In the j-th row we fill the boxes on the right of the box with the entry k j and on the left of the separator with the number k j , and in all remaining boxes of this row we write r j . The row vector as obtained by this procedure equals the dimension vector of the quiver representation U (i, ℓ i,k ), where i and k are obtained as above. Summing all row vectors computed from the configuration, we obtain the dimension vector of the associated quiver representation.
Property (1) and (2) of affine Dellac configurations imply that over each vertex of ∆ n there are starting and ending exactly two segments of the associated subquiver. Hence the entries of the dimension vector of the subquiver are all the same and equal to ωn, since their sum is equal to ωn 2 by Property (3) of affine Dellac configurations. Thus the image of an affine Dellac configuration is a torus fixed point, which parametrises a cell in the quiver Grassmannian.

Linear Degenerations of the Affine Flag Variety
In this section, we study linear degenerations of the affine flag variety, following the approach of G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier and M. Reineke as introduced in [5]. This generalises the definition of the degenerate affine flag variety as introduced above.

Rotation Invariant Parametrisation of the Affine Flag Variety.
In Proposition 2.1 we obtained an alternative parametrisation of the affine flag variety, which based on Sato Grassmannians and is similar to the chains of vector spaces description of the SL n+1 -flag variety. Using the fact that s k SGr 0 ∼ = SGr k , we get a rotation invariant parametrisation of the affine flag variety and its linear degenerations. This is useful to simplify the study of the isomorphism classes for the linear degenerations.
Proposition 5.1. The affine flag variety F l gl n , as subset in the product of n copies of the Sato Grassmannian SGr 0 , is parametrised as If we replace the shifts s −1 by arbitrary linear maps, we obtain linear degenerations of the affine flag variety. The degeneration as in Definition 2.2 corresponds to f k = s −1 • pr 1 .

Isomorphism Classes of Linear Degenerations and Corank Tuples.
On tuples of linear maps f ∈ End ×n (V ), we have an action of g ∈ G := i∈Zn GL(V ) via base change, i.e.
If the corank tuples of two map tuples coincide for some finite approximation V (ℓ) , they coincide for all larger approximations.
An endomorphism f ∈ End ×n (V ) is called nilpotent if there exist k, ℓ ∈ N such that the concatenation of the restrictions of f i to V (ℓ) , along the arrows of the cycle, vanish for each stating vertex i ∈ Z n and paths of length k. The set of nilpotent endomorphism is denoted by End ×n nil (V ). The G-orbits of the nilpotent endomorphisms can be studied with the methods from the thesis by G. Kempken, and the corresponding degenerations of the affine flag variety admit finite approximations by quiver Grassmannians for the equioriented cycle.
We introduce an entry wise partial order on the corank tuples, and restrict to the intermediate degenerations between the non-degenerate affine flag variety and the degeneration as in Definition 2.2. These tuples correspond to the G-orbits of the maps f ∈ End ×n nil (V ), where each f i is either the shifted projection s −1 • pr 1 or the index shift s −1 . They are completely determined by coranks of the maps f k . Hence it is sufficient to view co-rank tuples in Z n . The co-rank tuple c 1 = (1, . . . , 1) ∈ Z n corresponds to the degeneration as in Definition 2.2, and the tuple c 0 = (0, . . . , 0) ∈ Z n parametrises the non-degenerate affine flag variety.
Let F l c gl n be the linear degenerate affine flag variety corresponding to the co-rank tuple c. The degenerate flag varieties, for tuples with c 1 ≤ c ≤ c 0 , are representatives, for the isomorphism classes of intermediate degenerations, between the non-degenerate affine flag F l gl n and the degenerate affine flag variety F l a gl n .

Finite Approximations of the Linear Degenerations.
In the same way as for F l a gl n , we define finite approximations of the linear degenerations, and identify them with quiver Grassmannians. Proof. For the representation M ω , the vector space over each vertex i ∈ Z n has dimension 2ωn. In the coefficient quiver of M c ω , there are 1 + c i segments starting over the vertex i ∈ Z n .
The first segment is starting in the fist point over the vertex i, and in the k-th step its arrow goes from the k-th point over the vertex i + k − 1 to the k + 1-th point over the vertex i + k. If c i = 1, this segment has length ωn and there has to be a second segment starting over the same vertex. If c i = 0, this segment has length 2ωn and there is no second segment starting over the vertex i ∈ Z n . Now assume that c i = 1. The first segment ends in the ωn-th point over the vertex i − 1, and it is not possible that there exists an arrow pointing to the ωn + 1th point over the vertex i. We choose this point as starting point for the second segment starting over the vertex i.
In the k-th step the arrow of this segment goes from the ωn + k-th point over the vertex i + k − 1 to the ωn + k + 1-th point over the vertex i + k, and it ends in the 2ωn-th point over the vertex i + n − 1 = i − 1. With this realisation of the coefficient quiver of M c ω , we have the maps M αi := s 1 • pr ci ωn , for the arrow α i from vertex i to vertex i + 1.
This yields an alternative parametrisation for the quiver Grassmannian, which can be identified with the approximation of the partial degeneration.
Proof of Theorem 5.3. The partial degeneration is parametrised as F l c gl n = U k k∈Zn ∈ SGr ×n 0 : s −1 • pr c k 1 U k ⊆ U k+1 for all k ∈ Z n .
The cyclic relations of the vector spaces, describing a point (U k ) k∈Zn in the degenerate affine flag variety, induce the restrictions for the finite approximation to the parameter ω. Accordingly, the corresponding approximations of the Sato Grassmannian SGr 0,ω := U ∈ SGr i : V −nω ⊆ U ⊆ V nω are isomorphic to the subspace Grassmannian Gr ωn (2ωn). This induces the chain of isomorphisms F l c ω gl n ∼ = U k k∈Zn ∈ SGr ×n 0,ω : s −1 • pr c k 1 U k ⊆ U k+1 for all k ∈ Z n ∼ = U k k∈Zn ∈ Gr ωn (2ωn) ×n : s 1 • pr c k ωn U k ⊂ U k+1 for all k ∈ Z n ∼ = Gr ∆n eω M ω , where the base change from SGr 0,ω to Gr ωn (2ωn), for each of the n copies over the vertices k ∈ Z n , provides the isomorphism between the approximation of the degenerate affine flag variety and the quiver Grassmannian for the equioriented cycle, in the second row. The isomorphism in the last row is a direct consequence of the proposition above.
Remark 5.5. In Section 3, we discovered that it was crucial for certain properties of the quiver Grassmannian that the length of the projective and injective representations of ∆ n is a multiple of n. For the co-rank tuples c we restricted to, we still get approximations of the corresponding linear degenerate affine flag varieties F l c gl n , where the length of all summands of M c ω is a multiple of n. This enables us to use the methods developed in Section 3 to study their approximations.
The approximation of the linear degenerate affine flag varieties F l f gl n by quiver Grassmannians for the equioriented cycle would work for all map tuples in the G-orbit of map tuples f ∈ End ×n nil (V ), where each f i is an arbitrary finite composition of projections. But the resulting quiver Grassmannians can not be studied using the methods as introduced in this paper.

Partial Degenerations of Affine Dellac Configurations.
In this section, we introduce subsets of affine Dellac configurations, which describe the cells in the approximations of the partial degenerate affine flag varieties. For every i ∈ Z n , the representation M c ω contains the summand U (i; ωn) ⊗ C 2 if c i = 1 or U (i; 2ωn) if c i = 0.
Recall that for ω ∈ N the finite approximation is given as F l c ω gl n ∼ = Gr ∆n eω M c ω , where e ω := (e i = ωn) i∈Zn .
In Theorem 3.13, we have shown that certain quiver Grassmannians for the equioriented cycle admit a cellular decomposition induced by a C * -action. This applies to the quiver Grassmannians approximating the partial degenerations of the affine flag variety. By [3, Proposition 1], the cells in the approximations are in bijection with successor closed subquivers in the coefficient quiver of M c ω , with ωn marked points over each vertex.
Accordingly, these successor closed subquivers are parametrised by the set  The precise formula for the function h c is very complicated and only practical to be implemented for a computer program. Hence we decide to omit this detail and highlight the description of the Poincaré polynomials, which is more practical for computations in small examples. For the alternative approach, we draw the successor closed subquivers based on the parametrisation by the length of the subsegments, and count the holes below the starting points of the segments. Based on the second approach, we implemnted a program to computed the Poincaré polynomials of some approximations using SageMath [30]. The explicit dimension function h c and the code for the computer program can be found in [24, Chapter 6.10, Appendix B]. The formula based on the dimension function for the affine Dellac configurations might lead to a more efficient computer program, but the algorithmic version based on the coefficient quivers is efficient enough to compute examples for n ≤ 5 and ω ≤ 6.