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.

Based on the work by G. Kempken and A. Hubery we derive statements about the geometry of quiver Grassmannians for the equioriented cycle and generalise a result by N. Haupt about the parametrisation of their irreducible components.
The main goal of this paper is the study of degenerate affine flag varieties of type gl n . Analogous to the classical setting the affine flag variety is defined as the quotient of the affine Kac-Moody group by its standard Iwahori subgroup [25,Chapter XIII]. Based on the identification of the finite approximations with quiver Grassmannians for the equioriented cycle, we examine geometric properties of the degenerations.

Main Results
Finite approximations of the degenerate affine flag variety are isomorphic to certain quiver Grassmannians for the cyclic quiver, as shown in Theorem 3.7. The approximations have equidimensional irreducible components, which are parametrised by grand Motzkin paths, and the components are normal, Cohen-Macaulay and have rational singularities (Theorem 3.8). Moreover, certain quiver Grassmannians for the cycle admit a cellular decomposition, into attracting sets of the fixed points of an action by a one-dimensional torus (Theorem 4.13). The cells in the approximations of the degenerate affine flag are parametrised by affine Dellac configurations, as shown in Theorem 5.3.

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. Some basic definitions and required constructions are recalled in Section 2.
In Section 3, we define the affine flag variety of type gl n and its degeneration and state the main theorems of this article. The rest of the article is devoted to prove and generalise these statements.
The equioriented cycle, and the class of quiver Grassmannians which we want to examine, are introduced Section 4. 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 3.8. 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 [23].
In Section 4.4, 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 [4,Theorem 1]. Moreover, it induces a cellular decomposition of the quiver Grassmannians. For the proof that such a decomposition exists, it is crucial to find the right grading for the basis of the vector spaces belonging to a quiver representation. This grading defines a torus action on the quiver representation and induces a torus action on the quiver Grassmannian.
For the degenerate affine flag variety, the cells are in bijection with affine Dellac configurations. This bijection is established in Section 5 and is based on the parametrisation of torus fixed points by successor closed subquivers [4,Proposition 1]. Finally, in Section 6, we generalise the constructions from the previous sections to certain linear degenerations of the affine flag variety and examine their geometry.
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 geometry of 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 linear degenerations, as introduced in Section 6, but this requires new methods, since the identification of the approximations with a framed moduli space fails in this setting.

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∈Q 0 , and a tuple of linear maps between the vector spaces along the arrows of the underlying quiver M = (M α ) α∈Q 1 .
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 i 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).

Quiver Grassmannians
The entries of the dimension vector dim R ∈ Z Q 0 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 : In [7] the stratum of U ∈ Gr Δ n e (M) is defined as the isomorphism class of U in the quiver Grassmannian. By [7,Lemma 2.4], it is irreducible, locally closed and of dimension

Definition 2.1 The quiver Grassmannian Gr
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 [30,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 [24,Theorem 1.11]. Q has finitely many indecomposable representations if and only if the underlying graph of Q is a simply-laced Dynkin diagram [15]. 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) [30,Theorem 5.4].

Variety of Quiver Representations
For a dimension vector e ∈ Z Q 0 , define the variety of Q-representations as This generalises to representations of bound quivers if we assume that the maps satisfy admissible relations I and we write R e (Q, I). On points M in both varieties an element g of the group

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 [30,Definition 5.3].

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 (i) k with k ∈ [d i ] of the vector spaces V i for i ∈ Q 0 . We draw an arrow from a vertex 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.

Framed Moduli Interpretation of Quiver Grassmannians
The extended representation variety is defined as R e,d (Q, I) := R e (Q, I) × Hom k (e, d) 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 [28,Proposition 3.9]. S. Fedotov used the same methods to derive the statement in the generality of modules over finite dimensional algebras in [12,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 [28,Theorem 6.4].
Define R  The proof of this theorem is obtained with the same arguments as used by M. Reineke, for the Dynkin case in [27,28].

The Degenerate Affine Flag Variety
In this section we recall the definition of the affine flag variety, define its degeneration and state some geometric properties of the degeneration, which are based on the identification with quiver Grassmannians. Let C[[t]] be the ring of formal power series over C and let C((t)) be the field of Laurent series over C. Then is the loop group associated to GL n (C). Let B ⊂ GL n (C) be a Borel subgroup. For example, it can be chosen as the subgroup of upper triangular matrices. Let be the projection, which evaluates every formal power series at the point t = 0. The preimage I := p −1 (B) is a subgroup of GL n and is called Iwahori subgroup.   [25, Chapter XIII] and a brief overview can be found in [14, Section 1.1]. For our purposes it is sufficient to have the above version of the definition, since this is the definition used in [21], where the parametrisation as introduced in Proposition 3.3 is constructed.
Before we describe the alternative parametrisation of the affine flag variety, we recall it for the SL n -flag variety. Let B n denote a Borel subgroup of SL n (C). Then With the parametrisation on the right hand side, it can be realised as quiver Grassmannian [7, Remark 2.8].

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 [1,16,17]. 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 [14,21].
For ∈ Z let V 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 3.3 [14,21] The affine flag variety Fl gl n as subset in the product of Sato Grassmannians is parametrised as Remark 3.4 Here we use a slightly different parametrisation for the affine flag variety than the version by E. Feigin, M. Finkelberg and M. Reineke in [14]. 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 by E. Feigin in [13, Theorem 0.1] that the degenerate SL n -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.

Definition 3.5
The degenerate affine flag variety Fl a gl n is defined as The isomorphism from Remark 3.6 is described below. Translated to our notation, Definition 1.3 in [14] reads as SGr k : pr k+1 U k ⊂ U k+1 and U n = s n U 0 .

This follows from their identification of basis of
and k ∈ Z as in [14,Equation 1.1]. Here e 1 , . . . , e n is the standard basis of C n . Since U n is required to be equal to s n U 0 , we loose no information, if we remove it from the parametrisation and impose the condition pr n U n−1 ⊂ s n U 0 .

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 [14]. Analogously, in this paper we restrict us to quiver Grassmannians for the equioriented cycle: Both, the set of vertices, and arrows, of Δ 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 Fl a ω gl n := (U k ) n−1 k=0 ∈ Fl a gl n : This construction allows us to obtain statements about geometric properties of the approximations from the corresponding quiver Grassmannians.

Theorem 3.8
For ω ∈ N, the approximation Fl a ω gl n of the degenerate affine flag variety satisfies: It is a projective variety of dimension ωn 2 .
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.
In Section 4, we study geometric properties of quiver Grassmannians for the equioriented cycle. This specialises to the proof of Theorem 3.8 as given in Section 4.5. The identification of cells and affine Dellac configurations is proven in Section 5. In Section 6, we generalise the constructions from the previous sections to certain linear degenerations of the affine flag variety.

Isomorphism of Finite Approximations and Quiver Grassmannians
Proof of Theorem 3.7 In the approximation to the parameter ω, the cyclic relations of the vector spaces, describing a point (U k ) n−1 k=0 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 [14, 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 k+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 k = 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:

Quiver Grassmannians for the Equioriented Cycle
For every i ∈ Z n , we define the path with arrows starting at vertex i as The path algebra kΔ n is denoted by A n . This algebra is not finite dimensional because there are paths p i ( ) 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 := kΔ n /I N . The following result is a special case of [30,Theorem 5.4].

Proposition 4.1
The category rep k (Δ 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 (Δ n , I N ) be the projective bounded representation of Δ n at vertex i ∈ Z n . Define the projective representation be the injective bounded representation of Δ 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 Δ 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. By [24,Theorem 7.6], the indecomposable nilpotent representations of Δ n are parametrised by some starting vertex i ∈ Z n and a length parameter ∈ Z ≥0 . We denote them by U i ( ) 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 4.2
For n, N ∈ N and all i, j ∈ Z n the projective and injective representations P i and I j of the bound quiver (Δ n , I N ) satisfy Proof It follows from the definition of projective and injective representations of bound quivers as in [30,Definition 5.3] that they are of the form as in [24,Theorem 7.6].
This identification allows us to apply Theorem 2.3 to the quiver Grassmannians as introduced above. Accordingly it is possible to realise them as framed module space and we can use Theorem 2.4 to study their geometry.

Definition 4.3 For a nilpotent representation U i ( ) of the equioriented cycle on n vertices, the corresponding word w i ( ) is defined as
where we view each number in Z n .
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 Δ n ( . , . ).

Proposition 4.4 For two indecomposable nilpotent representations U i ( ) and U j (k) of Δ n , let w i ( ) and w j (k) be the corresponding words. Then the dimension of the space of morphisms from
This proposition is just a different formulation of [20, Theorem 16 (1)] by A. Hubery. 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 − } 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 (Δ n , I N ) to an indecomposable representation of maximal length. Proof of Proposition 4.5 Every representation M ∈ R e (Δ n , I N ) is nilpotent and can be written as a direct sum of indecomposable nilpotent representations, namely because the morphisms of quiver representations are linear maps between finite dimensional vector spaces.
We want to apply Proposition 4.4. By assumption we know that ≤ 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 ( ) corresponding to U j ( ). This value is given by the (i + N − 1)-th entry of the dimension vector of U j ( ).
For the first part, we can apply Proposition 4.5, and obtain [U,   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 [23].
Proof of Proposition 4.8 The idea of the proof is to show that for U ∈ Gr Δ 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. For U ∈ Gr Δ n e (X ⊕ Y ) define This set includes all direct summands of U , which are not of maximal length. If the set S(U ) is empty, we directly obtain = ω for all i, j ∈ Z n and N = ω · n. This follows from Proposition 4.5 if we set M = U i (N ). Now let U ∈ Gr Δ 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 ( ), U j (k) ∈ S(U ) and can assume without loss of generality that j is contained in the word w i ( + 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 ( ) and w j (k). We can write them as w i ( ) = 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 ( ) means that we do not take the direct summand U i ( ) 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 + + 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 ( ) = 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 is to identify a segment wise embedding, i.e. ι : U(i; ) → U(i; k) for k ≥ . 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 [30, Chapter 3.1.1]. Given a fixed nilpotence parameter N , we take an equioriented type A quiver of length 2Nn 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 + + 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 ( ). It follows that the representationÛ embeds into the same summands of X ⊕ Y as U .
Following [23, 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 [23,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 Δ n e (X ⊕ Y ) with S(Û) = ∅. Thus we obtain 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 (Δ n , I ωn ) will be denoted by P ω i and I ω j . Based on Proposition 4.8 we can compute the dimension of Gr Δ 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. . It follows that dim S U ≤ dim S X holds for all U ∈ Gr Δ n e (X ⊕ Y ). It remains to compute the dimension of the stratum of X, which is given by 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.

Lemma 4.10 The irreducible components of Gr Δ n e ω (X ω ⊕ Y ω ) are in bijection with the set
where d i := y i + x i+1 and they all have dimension ωk(m − k).

Remark 4.11
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 Gr Δ n e ω (X ω ⊕ Y ω ) ∼ = R s e ω ,d (Δ n , I ωn )/G e ω as in Theorem 2.3 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 Δ n e ω (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 2.4 to this setting such that the irreducible components are in bijection with the maximal elements of R (d) e ω (Δ n , I ωn )/G e ω , with respect to the partial order induced by the inclusion of orbits in orbit closures.
In the proof of Proposition 4.8, we have seen that the maximal elements for this order are parametrised by the U ∈ Gr Δ n e ω (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∈Z n 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 X ω in Lemma 4.9.
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 k since C k (d) ⊆ C k := p ∈ Z n ≥0 : p i ≤ k for all i ∈ Z n , i∈Z n 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 [18,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 Δ n e ω 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.

Torus Action and Cellular Decomposition
From now on, we restrict us to the setting where k = C. For every nilpotent representation U ∈ rep C (Δ n ), there exists a nilpotence parameter N ∈ N such that U ∈ rep C (Δ n , I N ). Hence by [24,Theorem 1.11] it is conjugated to a direct sum of indecomposable nilpotent representations, i.e.
where d i, ∈ Z ≥0 for all i ∈ Z n and ∈ [N ]. Let 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 ( ) = U(i + − 1; ). 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) m i . We continue the segments, to the last free basis vectors over the (i − 1)-th vertex of of Δ n , such that their initial order by length is preserved. This way of arranging the segments ensures that they do not cross.
The condition d(α) := d s α for all α ∈ Z n induces a grading of the vertices in the coefficient quiver of U(d). Choose the nilpotence parameter N such that there exists an i ∈ Z n such that d i,N = 0 and define q i := m i − d i for all i ∈ Z n . Take i 0 ∈ Z n such that Give weight one to v (j 0 ) 1 . Compute the weight of the other points on the segment starting in v (j 0 ) 1 using the grading of the arrows as defined above. The end point of this segment is v (i 0 ) q i 0 +1 and its weight is 1 be the largest index such that v (i) r i lives on a non-trivial segment and let k be the weight of v (i 0 ) r i 0 . If r i 0 = 0, we set k = 0, since it implies that r i = 0 for all i ∈ Z n .
For the remaining end points v (i) q i +j with i ∈ Z n and j ∈ [d i ] their weight is given by In Particular, these quiver Grassmannians have property (S) [9, Definition 1.7]. For Dynkin qiuvers and acyclic orientations of affine Dynkin diagrams, the existence of a cellular decomposition is proven in [5]. Using similar methods, it might also be possible to prove it for the equioriented cycle. But this explicit parametrisation of the cells allows us to use the combinatorics of coefficient quivers to study geometric properties of the quiver Grassmannians, like Poincaré polynomials or moment graphs for the action of a bigger torus. It follows from the definition of quiver Grassmannians that where Gr e (m) is the Grassmannian of e-dimensional subspaces in C m . First we show that the attractive sets C(L (i) ) := C(L) ∩ Gr e i (m i ) are isomorphic to affine spaces. By the structure of the C * -fixed points as described in [4,Theorem 1], there exists an index set K i ∈ [m i ] e i such that L (i) is the span of the v (i) k for k ∈ K i . From the structure of the C * -action we deduce that a point V i in the attracting set C(L (i) ) has generators for k ∈ K i and μ (i) ,k ∈ C. Hence C(L (i) ) is an affine space. Observe that for a representation V in an attracting set of a C * -fixed point L, it holds that With this identification of quiver representations, Theorem 3.8 is a specialisation of the results about quiver Grassmannians for the cycle as developed above.
Proof of Theorem 3.8 Part (1) follows from Lemma 4.9 and the identification with a quiver Grassmannian as in Theorem 3.7, i.e.
From the C * -action and the induced cellular decomposition as in Theorem 4.13, we obtain part (3). With x i = y i = 1, we obtain (2) and (5)

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 [13]. The torus fixed points of the symplectic degenerated flag variety are identified with symplectic Dellac configurations by X. Fang and G. Fourier in [10]. In this section, we identify affine Dellac configurations with the cells of the degenerate affine flag variety, based on the parametrisation of its torus fixed points via successor closed subquivers.

Definition 5.1
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.
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).

Proof of Theorem 5.3
For k ∈ N, let [k] 0 denote the set {0, 1, . . . , k}. The combinatorics of successor closed subquivers yields the following parametrisation of the cells. Proof The approximation Fl a ω gl n is isomorphic to the quiver Grassmannian Gr Δ n e ω M a ω , where e ω := (e i = ωn) i∈Z n and M a ω = i∈Z n U(i; ωn) ⊗ C 2 . By [4, Proposition 1] and Theorem 4.13 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; i,1 ) and U(i; 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 Fl a ω gl n .
Now we want to study how every row of an affine Dellac configuration encodes an indecomposable representation U(i; 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 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 := i,k /n and p j := 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 2n j =1 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 4.10 the top-dimensional cells have two segments starting and ending over each vertex of the quiver Δ 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 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∈Z n .
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, 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 Δ 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 [6]. In contrast to toric degenerations, where methods from toric geometry are used to study the degenerations and deduce properties of the original variety, linear degenerations are tailored to be studied as quiver Grassmannians. For more details we refer to the overview [11].

Rotation Invariant Parametrisation of the Affine Flag Variety
In Proposition 3.3 we obtained an alternative parametrisation of the affine flag variety, which is based on Sato Grassmannians. Using the fact that s k SGr 0 ∼ = SGr k , we get a rotation invariant parametrisation of the affine flag variety.

Proposition 6.1
The affine flag variety Fl 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 3.5 corresponds to f i = s −1 • pr 1 . In the rest of this section we study the geometry of the linear degenerations via quiver Grassmannians for the equioriented cycle and parametrise the isomorphism classes of linear degenerations. The rotation invariance of the affine flag as in Proposition 6.1 simplifies the study of isomorphism classes for the linear degenerations.

Finite Approximations of the Linear Degenerations
For a positive integer ω, the finite approximation of the f -linear degenerate affine flag variety is defined as In order to view this finite approximation as quiver Grassmannian, define the finite dimensional vector space Proof Analogous to the proof of Theorem 3.7 we obtain Fl f ω gl n = U i i∈Z n ∈ SGr ×n 0,ω : Remark 6.4 This theorem implies that all approximations of linear degenerations of the affine flag variety can be studied using quiver Grassmannians for the equioriented cycle.
However it is only possible to apply the methods introduced in the present paper if the quiver representation M ω as in Theorem 6.3 is nilpotent.

Nilpotent Linear Degenerations
In this subsection we characterise the degenerations which can be studied with the methods introduced in this article. An endomorphism f ∈ End ×n (V ) is called nilpotent if for all ∈ N there exists a k ∈ N such that the concatenation of the restrictions of f i to V ( ) , along the arrows of the cycle, vanish for each stating vertex i ∈ Z n and paths of length k, i.e.: The set of nilpotent endomorphism is denoted by End ×n nil (V ).
Remark 6.5 If f ∈ End ×n (V ) is nilpotent, the quiver representation M ω as in Theorem 6.3 is nilpotent for every finite approximation and we can apply the results from Section 4 to the corresponding quiver Grassmannians.
In particular, we obtain a cellular decomposition of each approximation. Theorem 6.6 Let f ∈ End ×n nil (V ) and ω ∈ N, then Fl f ω gl n admits a cellular decomposition into the attracting sets of its C * -fixed points.
Proof By Remark 6.5 the quiver representation M ω is nilpotent if f ∈ End ×n nil (V ). Hence we can apply Theorem 4.13 to the corresponding approximation of the f -linear degenerate flag variety, by Theorem 6.3. Remark 6.7 Analogous to [6, Remark 6], Theorem 6.3 allows to compute the Euler characteristic and the Poincaré polynomial of each finite approximation, corresponding to an f ∈ End ×n nil (V ).
Remark 6.8 For the linear degenerations, which do not correspond to nilpotent endomorphisms, it is not possible to use the methods in the present paper to deduce results about their geometry. Hence it would be of interest to understand quiver Grassmannians for the cycle, with not necessarily nilpotent representations.

Isomorphism Classes of Linear Degenerations
On tuples of linear maps f ∈ End ×n (V ), we have an action of g = (g i ) i∈Z n in the group G := i∈Z n GL(V ) via base change, i.e.
Two linear degenerations are isomorphic if and only if their defining tuples of maps live in the same G-orbit. This follows directly from the parametrisation of the degenerations via Sato Grassmannians as in Definition 6.2. Hence it is sufficient to study one representative for each G-orbit.
For the restrictions of the map tuples f = (f i ) i∈Z n to V ( ) , these G-orbits where studied by G. Kempken in her thesis [23]. In each finite approximation End ×n (V ( ) ), the number of G-orbits is finite, whereas there are infinitely many different G-orbits in End ×n (V ). This makes it more complicated to characterise the isomorphism classes of degenerations in general.
The G-action does not effect the corank of the maps in the tuples and the corank of their concatenations, since the matrices g i are invertible and by definition their action cancels out for the middle terms of concatenations along paths in the quiver. Hence, each G-orbit is characterised by these coranks for one representing map tuple. By Remark 6.8, the corresponding degenerations can be studied with the methods from the present paper if and only if the map tuple is nilpotent. In the next subsection we give one special type of degenerations, where our methods apply and which is independent of the approximation parameter ω.

Partial Degenerations and their Corank Tuples
In this subsection, we restrict us to intermediate degenerations between the non-degenerate affine flag variety and its degeneration as in Definition 3.5. These degenerations correspond to map tuples in the G-orbits of f ∈ End ×n nil (V ), where each f i is either the shifted projection s −1 • pr 1 or the index shift s −1 .
Since degenerations corresponding to map tuples in one G-orbit are isomorphic, the intermediate degenerations are completely determined by the corank of the maps f i , which is independent of the approximation parameter ω. Hence it is sufficient to view corank tuples c ∈ {1, 2} Z n . The corank tuple c (2) = (2, . . . , 2) corresponds to the degeneration from Definition 3.5, and the tuple c (1) = (1, . . . , 1) parametrises the non-degenerate affine flag variety as in Proposition 6.1.
Let Fl c gl n be a representative, for the isomorphism class of linear degenerations of the affine flag variety, corresponding to the corank tuple c ∈ {1, 2} Z n . In Section 6.6, it is shown that these degenerations play a special role among the linear degenerations of the affine flag variety. Namely, they are the maximal subclass of degenerations, where it is possible to adapt the parametrisation of cells via affine Dellac configurations.

Theorem 6.9
For ω ∈ N and c ∈ {1, 2} Z n the finite approximation is given as  The precise formula for the function h c is very complicated and only practical for computer programs. Hence we decide to omit this detail and highlight the description of the Poincaré polynomials, which is easier to handle in small examples. For the alternative approach, we draw the successor closed subquivers as described in Section 4.4, and count the holes below the starting points of the segments as in [6,Remark 6]. By Theorem 4.13, this even works for all quiver Grassmannians associated to nilpotent representations of the cycle. Based on the second approach, we implemented a program to compute the Poincaré polynomials of some approximations using SageMath [32]. The explicit dimension function h c and the code for the computer program can be found in [26, 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.