Maximum antichains in posets of quiver representations

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type A we construct a maximum antichain in the poset. Secondly, we consider the set of subrepresentations of a given quiver representation, again ordered by inclusion. It is a finite set if we restrict to linear representations over finite fields or to representations with values in the category of pointed sets. For particular situations we prove that this poset is Sperner.


Introduction and notation 1.Maximum antichains in posets
Let (P, ≤) be a poset.Two elements a, b ∈ P are called incomparable if neither a ≤ b nor b ≤ a holds.The elements are called comparable otherwise.A subset F ⊆ P of pairwise incomparable elements is called an antichain.An antichain F ⊆ P is called maximal if there does not exist an element a ∈ P such that F ∪ {a} is an antichain.It is called maximum if there does not exist an antichain F ′ ⊆ P such that F ′ > F .Note that every maximum antichain is a maximal antichain, but the converese does not hold in general.The size of maximum antichain is sometimes called the width of the poset.Furthermore, a subset C ⊆ P of pairwise comparable elements is called a chain.Note that the elements of a chain can be reordered to form a sequence (a 1 ≤ a 2 ≤ . . .≤ a k ) and we will often use this notation to describe a chain.Maximal and maximum chains are defined in a similar way to maximal and maximum antichains.
Let n ≥ 1 be an integer.We denote by Pn set of all subsets of the finite set {1, 2, . . ., n}.Note that Pn is partially ordered by inclusion.Sperner [11] constructs a maximum antichain in (Pn, ⊆): Theorem 1.1 (Sperner).The set {A ∈ Pn∶ A = ⌊n 2⌋} is a maximum antichain in the poset (Pn, ⊆) so that width of the poset is given by the binomial coefficient n ⌊n 2⌋ .
In a later work, Stanley [12,Theorem 2.2] gives an elegant proof of Sperner's theorem using linear algebra and a grading of the poset.We give a sketch of the proof after recalling some basic notions about posets.We say that the element a ∈ P covers the element b ∈ P if a > b and there does not exist an element c ∈ P with a > c > b.Moreover, we say that a ∈ P is minimal if there does not exist an element b ∈ P with b < a. Dually, we say that a ∈ P is maximal if there does not exist an element b ∈ P with b > a.The poset (P, ≤) is called graded if there exists a map deg∶ P → N such that deg(a) = 0 for all minimal elements a ∈ P and deg(a) = deg(b) + 1 whenever a covers b.For example, the poset Pn is graded by the cardinality viewed as a map ⋅ ∶ Pn → N. A poset (P, ≤) is called bounded if it contains a miminum, i. e. an element a ∈ P such that a ≤ b for every b ∈ P, and a maximum, i. e. an element a ∈ P such that b ≤ a for every b ∈ P. Note that a finite, bounded poset is graded if and only if all maximal chains have the same cardinality.The rank of a graded poset (P, ≤) is rk(P) = max(deg(a)∶ a ∈ P).
Suppose that the poset (P, ≤) is indeed finite, bounded and graded of rank n.Then we denote by P i ⊆ P the subset of elements of degree i.It is easy to see that every P i with i ≥ 0 is an antichain.We say that P is Sperner if there exists a natural number i such that F ≤ P i for every antichain F ⊆ P. In other words, P i is a maximum antichain but there may exist other maximum antichains in P. In the case of the power set Pn we simply write P n,i instead of (Pn) i .Sperner's theorem implies that the power set Pn is Sperner.For every i we consider the Q-vector space QP i with basis P i .
Theorem 1.2 (Stanley).Let k be a natural number.Suppose that there are injective linear maps U i ∶ QP i → QP i+1 for 0 ≤ i ≤ k − 1 such that U i (a) ∈ ∑ b≥a Qb for every a ∈ P i ; suppose further that there are injective linear maps D i ∶ QP i → QP i−1 for k + 1 ≤ i ≤ n such that for D i (a) ∈ ∑ b<a Qb for every a ∈ P i , see Figure 1 for an illustration.Then the poset (P, ≤) is Sperner and P k is a maximum antichain.
Hence D i+1 U i is invertible so that U i must be injective in this case.By a similar argument we can show that D i is injective for ⌊ n 2 ⌋ < i ≤ n.As a second application, Stanley proves that the poset of vector subspaces of a finite-dimensional vector space over a finite field, again ordered by inclusion, is Sperner.
A chain decomposition of a poset (P, ≤) is a disjoint union where every C i is a chain in P. If F ⊆ P is an antichain, then every chain C i contains at most one element of F. Especially, we have F ≤ k.A chain decomposition is called a Dilworth decomposition if there does not exist a chain decomposition with a smaller number of chains.The next theorem is due to Dilworth [4, Theorem 1.1]: Theorem 1.3 (Dilworth).The cardinality of a maximum antichain in a finite poset (P, ≤) is equal to the smallest number of chains in a chain decomposition of (P, ≤).
Suppose that the poset (P, ≤) is graded with degree map deg∶ P → N and has finite rank n = rk(P).A chain We say that the graded poset (P, ≤) is a symmetric chain order if it admits a symmetric chain decomposition.Engel [5,Theorem 5.1.4]proves a relationship between symmetric chain orders and Sperner posets: Theorem 1.4.If the graded poset (P, ≤) is a symmetric chain order, then it is Sperner.
As an application of the theorem, M ühle [8] shows that certain posets of noncrossing partitions are Sperner.Noncrossing partition posets can be attached to Coxeter groups and play an important role in combinatorics and representation theory.
Suppose that (P, ≤ P ) and (Q, ≤ Q ) are two posets.The direct sum is the partial order ≤ P×Q on the set The next theorem is a product theorem for symmetric chain orders.The main idea is due to de Bruijn, van Ebbenhorst Tengbergen and Kruyswijk [3] and formal proofs are due to Aigner [1], Alekseev [2] and Griggs [7].
Theorem 1.5.If the graded posets (P, ≤ P ) and (Q, ≤ Q ) are both symmetric chain orders, then the direct product (P × Q, ≤ P×Q ) is a symmetric chain order as well.
Example 1.6.For a natural number k the poset Ch(k) = (0 ≤ 1 ≤ . . .≤ k) is called the chain poset of length k + 1.It becomes a graded poset when we define deg(a) = a for all 0 ≤ a ≤ k.By construction the chain poset Ch(k) is symmetric chain order.For natural numbers k 1 , k 2 , . . ., kr let Ch(k 1 , k 2 , . . ., kr) be the set of all sequences (a 1 , a 2 , . . ., ar) ∈ N r such that 0 ≤ a i ≤ k i for all 1 ≤ i ≤ r.We order the set by the dominance order, i. e. we say (a 1 , a 2 , . . . and the product theorem implies that the poset Ch(k 1 , k 2 , . . ., kr) is a symmetric chain order and hence Sperner.The poset is known as the chain product.

Quiver representations
is a k-vector subspace for every vertex i ∈ Q i and Uα(x) = Vα(x) for every arrow α∶ i → j in Q 1 and every element x ∈ U i .In particular, we have Uα(U i ) ⊆ U j for every arrow α.Given a subrepresentation U ⊆ V, we can define a quotient Dually, a morphism φ is called an epimorphism if every linear map φ i is surjective.A morphism φ∶ V → W is called an isomorphism if it is both a monomorphism and an epimorphism.In this case we say that V and W are isomorphic and we write V ≅ W. The representation V with V i = 0 for all i ∈ Q 0 is called the zero representation, where necessarily Vα = 0 for all α ∈ Q 1 .Suppose that V, W are two representations of the same quiver Q.The direct sum V ⊕ W is the representation with (V ⊕ W) i = V i ⊕ W i for all vertices i ∈ Q 0 and Let a path of length n be an undirected graph as in Figure 2. To unify the description of quivers which are representation finite, let us introduce star-shaped undirected graphs as graphs with a central vertex c from which r-many paths of varying lengths start.More formally, for integers r ≥ 0 and ℓ 1 , . . ., ℓr ≥ 1 let Star(ℓ 1 , . . . ,ℓr) be the graph with n = 1 + ∑ r i=1 ℓ i many vertices and edges for 1 ≤ i ≤ r and 1 ≤ j ≤ ℓ i − 1. Pictorially such a graph can be seen in Figure 3.
A star-shaped undirected graph with n vertices is said to be a Dynkin diagram of type An Gabriel [6] then classifies representation finite quivers as follows: Theorem 1.7 (Gabriel).A (non-empty) connected quiver with n vertices is representation finite if and only if its underlying undirected graph is a Dynkin diagram of type An, Dn or En.In this case, the map V ↦ dimV induces a bijection between the isomorphism classes of indecomposable representations and the positive roots in the corresponding root system.
Especially, representation finiteness does only depend on the underlying diagram but not on the orientation.We say that quivers as in Theorem 1.7 are of type An, Dn and En respectively and call them Dynkin if we do not wish to distinguish between these three families.For those readers not familiar with representation theory, let us consider one basic example to clarify the notions above.
Example 1.8.Let Q be the quiver 1 α → 2 of type A 2 .One representation is then given by V 1 = k, V 2 = 0 and the zero map; denote this representation by S 1 = (k → 0).Since its only proper subrepresentation is the zero representation, we clearly see that it is simple.Similarly, S 2 = (0 → k) is a simple and thus also an indecomposable representation.Gabriel's theorem asserts the existence of a third indecomposable representation with dimension vector (1, 1), namely the representation P 1 = (k id → k).It is an easy observation that the zero morphism is the only morphism from S 1 to P 1 , i. e. the left diagram in Figure 4 commutes if and only if φ 1 = 0 = φ 2 .
Figure 4: Morphisms between indecomposable representations of a quiver of type A 2 On the other hand, the choice ψ 1 = 0 and ψ 2 = id makes the right diagram of Figure 4 commutative, hence we obtain a nonzero morphism from S 2 to P 1 .Since the identity map is injective, the morphism (ψ 1 , ψ 2 ) is even a monomorphism of representations.By Gabriel's theorem, a general representation has the form , and Vα = 0 id 0 0 in block form.In this article, we wish to study maximum antichains in posets attached to various quivers representations.On a related note, Ringel [9] studies maximal antichains in the product ordering on the set of dimension vectors of indecomposable representations of a Dynkin quiver.

Maximum antichains in monomorphism posets of indecomposable representations for type A quivers 2.1 Poset properties
Let n ≥ 1 be a natural number and Q be a Dynkin qiver of type An.For natural numbers 1 ≤ a ≤ b ≤ n use the shorthand notation [a, b] for the representation with vector spaces V i = k for a ≤ i ≤ b and V i = 0 elsewhere, and linear maps Vα = (V i → V j ) = id k for arrows α∶ i → j with a ≤ i, j ≤ b and Vα = 0 for all others.In the case where a = b we simply write  5. To simplify the notation, whenever we draw an arrow between one-dimensional vector spaces, we assume that the associated map is the identity.For the rest of this section, let (P Q , ≤) be the poset with set

Example 2.3. Consider again the quiver as in Example 2.2. Then
The Hasse diagram of this poset is shown in Figure 6.Remark 2.4.The poset (P Q , ≤) for Q a Dynkin quiver of type An may be graded as in Example 2.3, but this is not generally the case as we observe later on in Example 2.7.What is more, for n > 1 these posets are never bounded and thus never Sperner as the simple representations supported on a source yield isolated vertices in the poset.

Linear orientation
In this and the following subsections, we will restrict to particular orientations of the path from Figure 2.For now, we consider the linear orientation with unique sink at 1 and unique source at n, see Figure 7.
for all 1 ≤ i ≤ n and we immediately obtain the following result.
Proposition 2.5.A maximum antichain of (P Q , ≤) for linearly oriented An consists of exactly n elements.

Simple zigzag
Let 1 ≤ s ≤ n and consider the orientation of the path of length n with a unique source at position s.For s = 1 or s = n the case degenerates to the linear orientation (up to reordering the vertices) of Subsection 2.2.
To simplify the notation, denote ℓ = s − 1 and r = n − s so that the quiver is of the form as shown in Figure 8.To show that F is a maximum antichain, we describe a particular chain decomposition of (P Q , ≤) and observe that F contains precisely one element of each chain.
For 1 ≤ i ≤ ℓ and 1 ≤ j ≤ r let We then obtain a chain decomposition of the entire poset P as follows: Chain Name Description Condition Cardinality Table 1: Chains depending on a choice of one or two sources Furthermore, we also consider two chains C [1,n] and C [2,n] not depending on a choice of sources, both of cardinality m + 1: By construction, all of the chains above are pairwise disjoint.Altogether these chains exhaust all elements of the poset since Thus they form a chain decomposition of (P Q , ≤) and every element of F lies in exactly one chain as the maximal element.
We can apply this construction to a particular subquiver of Q.Let Q ′ be the full subquiver of Q with the vertex n deleted, i. e. the alternating orientation of the path of length n − 1 starting with a sink and ending with a source.Then we notate n ′ = n − 1 and observe that Q ′ has indeed n ′ = 2m vertices and s sources.
Proof.The Dilworth decomposition in the proof of Theorem 2.8 degenerates to a Dilworth decomposition of P Q ′ if one removes all those elements supported at vertex n.
Example 2.10.Let us consider the case for m = 3, hence n = 7 and n ′ = 6.The Hasse diagrams of the posets (P Q , ≤) and (P Q ′ , ≤) are shown in Figure 11.Those nodes contained in P Q but not in P Q ′ are shaded gray above the downward diagonal.The elements in F are highlighted in blue below the downward diagonal and those of F ′ in red above the downward diagonal.
Remark 2.11.For the only case where the alternating orientation of this section coincides with the simple zigzag of Subsection 2.3, the maximum antichains of Theorem 2.6 and Theorem 2.8 are identical.
Remark 2.12.The cases considered in this section should be extended to arbitrary orientations of quivers of type An, and even to the types Dn and E 6,7,8 .Computational experiments suggest that the combinatorics of general Dynkin cases are much more intricate than what we have previously seen.

A Sperner theorem for subrepresentations posets in type A 2
Let V be a representation of a quiver Q = (Q 0 , Q 1 ) over a field k.We denote by P V the set of all subrepresentations U ⊆ V. Assume that U 1 , U 2 ∈ P V .We say that U 1 ≤ U 2 if and only if U 1 is a subrepresentation of U 2 .In this way, (P V , ≤) becomes a partially ordered set.Proof.The only minimal element in P V is the zero representation which satisfies dim(0) = 0. Suppose that U 2 ∈ P V covers U 1 ∈ P V , i. e. U 1 ≤ U 2 and there does not exist an element The third isomorphism theorem implies that the quotient representation U 2 U 1 is nonzero and that there does not exist a representation 0 Note that the condition of the previous proposition is satisfied if the quiver Q does not contain oriented cycles which we assume from now on.Additionally, we assume that k = Fq is a finite field Fq with q elements.For every natural number n ∈ N we define the Gaussian integer as [n]q = (q n − 1) (q − 1).Note that [n]q = 1 + q + q 2 + . . .+ q n−1 can be simplified to a polynomial in q which specializes to n when we plug in q = 1.Moreover, [n]q is equal to the number of 1-dimensional vector subspaces of F n q .The polynomial is called Gaussian binomial coefficient.Note that n d q is equal to the number of d-dimensional vector subspaces of F n q .Likewise the number of (d + 1)-dimensional k-vector spaces U such that k d ⊆ U ⊆ k n is equal to [n − d]q.Dually, the number of (n − 1)-dimensional vector spaces U such that is the quiver of type A 2 and V = P a 1 is a direct sum of copies of the indecomposable, projective representation P 1 , then the subrepresentation poset (P V , ≤) is Sperner.
Proof.Note that the rank of the poset is equal to dim k (V) = 2a.For brevity we write P i,V instead of (P V ) i for 0 ≤ i ≤ 2a.A subrepresentation X ⊆ V is given by two vector spaces 0 ⊆ X 1 ⊆ X 2 ⊆ k a .We apply Stanley's Theorem 1.2.For every integer i we define two maps for every X ∈ P i,V and extend linearly.By construction D i is adjoint to U i−1 for every i.We can generalize Stanley's commutation relation for the up and down operators from type A 1 to type A 2 .More precisely, we claim for every X ∈ P i,V with dimension vector (d 1 , d 2 ).
For a proof of the claim, let X ∈ P i,V .We compute the coefficients of the expansion of (D i+1 ○ U i − U i−1 ○ D i )(X) in the standard basis P i,V of QP i,V .We distinguish the following cases.First, suppose that X ′ ∈ P i,V is a representation such that X ′ 1 ⊆ X 1 and X 2 ⊆ X ′ 2 are both subspaces of codimension 1.In this case, (X 1 ⊆ X ′ 2 ) and (X ′ 1 ⊆ X 2 ) are both subrepresentations of V so that the coefficient corresponding to X ′ in the expansion of is a representation such that X 1 ⊆ X ′ 1 and X ′ 2 ⊆ X 2 are both subspaces of codimension 1.With the same arguments as before the coefficient corresponding to X ′ in the expansion of Moreover, the coefficient corresponding to X ′ in the expansion of (U i−1 ○ D i )(X) is equal to the number of (d 1 − 1)-dimensional vector spaces Z 1 such that 0 ⊆ Z 1 ⊂ X 1 plus the number of (d 2 − 1)-dimensional vector spaces Z 2 such that X 1 ⊆ Z 2 ⊆ X 2 , namely [d 1 ]q + [d 2 − d 1 ]q.The difference of the two terms is equal to the term [a − d 2 ]q − [d 1 ]q from equation (2).For all other cases X ′ , the coefficient corresponding to X ′ ∈ P i,V in the expansion of (D i+1 ○ U i − U i−1 ○ D i )(X) is equal to 0.
Next, we claim that if 0 ≤ i < a, then U i is injective.Dually, if a < i ≤ 2a, then D i is injective.We prove the first statement only, the second statement can be proved using the same arguments.The assumption i < a implies 0 < a − i = a − d 2 − d 1 for every X ∈ P i,V .It follows that a − d 2 > d 1 and [a − d 2 ]q > [d 1 ]q for every X ∈ P i,V .By equation (2) the matrix of the linear map D i+1 ○ U i − U i−1 ○ D i (with respect to the standard basis) is a diagonal matrix with positive diagonal entries.Especially, it is positive definite.Moreover, the adjointness of U i−1 and D i implies that the matrix corresponding to the composition U i−1 ○ D i (with respect to the standard basis) is positive semi-definite.The sum of the two matrices must be positive definite as well.Especially, D i+1 ○ U i must be invertible so that U i is injective.
Example 3.3.Figure 12 shows the poset of subrepresentations of the 4-dimensional representation P 2 1 over the finite field with 5 elements.The subrepresentations of dimension 2 form a maximum antichain of cardinality 7.

A Sperner result for subrepresentation posets over pointed sets
A pointed set (A, 0 A ) is a set A together with a distinguished element 0 A ∈ A. The set is also known as a based set with basepoint 0 A .A pointed subset of a pointed set (A, 0 A ) is a subset B ⊆ A that contains 0 A as its basepoint.Let (A, 0 A ) and (B, 0 B ) be two pointed sets.A morphism between (A, 0 A ) and (B, 0 B ) is a map f ∶ A → B with f (0 A ) = 0 B such that the restriction f A∖ f −1 (O B ) ∶ A ∖ f −1 (O B ) → B is injective.The pointed sets together with the morphisms form the category of pointed sets.Some authors also use the term vector spaces over the field with one element.Especially, we say that the morphism f ∶ (A, 0 A ) → (B, 0 B ) is an isomorphism if there exist a morphism g∶ (B, 0 B ) → (A, 0 A ) such that g ○ f = id A and f ○ g = id B .Note that in this case f and g are bijections and we say that (A, 0 A ) and (B, 0 B ) are isomorphic and we write

Figure 1 :
Figure 1: Injective maps in Stanley's theorem (a, b) ≤ P×Q (c, d) if and only if a ≤ P c and b ≤ Q d.If P and Q are graded posets with degree maps deg P and deg Q , then the direct product (P × Q, ≤ P×Q ) is graded with collated degree map deg P×Q (a, b) = deg P (a) + deg Q (b) for all a ∈ P, b ∈ Q.

Figure 2 :
Figure 2: A path with n vertices

Figure 6 :
Figure 6: Hasse diagram of indecomposable representations of alternating A 3

Figure 7 :
Figure 7: Linear orientation of type An Then [a, b] ≤ [a ′ , b ′ ] if and only if a = a ′ and b ≤ b ′ .Hence the poset (P Q , ≤) decomposes into n disjoint

Figure 8 : 2 . 6 .
Figure 8: Simple zigzag orientation of type An Theorem 2.6.The set F = [a, b]∶ a ≤ s ≤ b is a maximum antichain of size (ℓ + 1)(r + 1) in (P Q , ≤).Proof.Let [a, b] and [a ′ , b ′ ] be distinct elements from the set F and suppose there exists a monomorphism φ∶ [a, b] ↪ [a ′ , b ′ ].This implies a ′ ≤ a and b ≤ b ′ .Since these elements are distinct, a ′ < a or b < b ′ .Without loss of generality, we may constrict to a ′ < a. Then the linear map corresponding to α∶ a →(a − 1) is zero in [a, b] whereas in [a ′ , b ′ ] it is the identity.But id k ○φa ≠ φ a−1 ○ 0 yields a contradiction.Hence F is an antichain.To show that F is a maximum antichain, we describe a particular chain decomposition of (P Q , ≤) and observe that F contains precisely one element of each chain.For 1 ≤ i ≤ ℓ and 1 ≤ j ≤ r let

Proposition 3 . 1 .
If every simple representation of Q is 1-dimensional, then the dimension function U ↦ dim k (U) defines a grading of the subrepresentation poset (P V , ≤).

Figure 11 :
Figure 11: Hasse diagrams of alternating orientations of A 6 and A 7