The boundary of the irreducible components for invariant subspace varieties

Given partitions $\alpha$, $\beta$, $\gamma$, the short exact sequences $0\to N_\alpha \to N_\beta \to N_\gamma \to 0$ of nilpotent linear operators of Jordan types $\alpha$, $\beta$, $\gamma$, respectively, define a constructible subset $\mathbb V_{\alpha,\gamma}^\beta$ of an affine variety. Geometrically, the varieties $\mathbb V_{\alpha,\gamma}^\beta$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $\Gamma$ of shape $(\alpha,\beta,\gamma)$ contributes one irreducible component $\overline{\mathbb V}_\Gamma$. We consider the partial order $\Gamma\leq_{\sf bound}^*\widetilde{\Gamma}$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetilde{\Gamma}}\cap \overline{\mathbb V}_\Gamma\neq \emptyset$. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of $\alpha$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $\beta\setminus\gamma$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.


Introduction
Often in geometry, naturally occuring conditions define subsets of varieties which are either very big in size or tiny. For example, among all linear operators acting on a given finite dimensional vector space, the invertible ones form an open and dense subset. And so do, among all nilpotent operators, those which have only one Jordan block. A notable exception to this rule occurs in the variety of short exact sequences of nilpotent linear operators; it is partitioned, by means of Littlewood-Richardson tableaux, into components of equal dimension. They are the topic of this paper.
Throughout we assume that k is an algebraically closed field. Each nilpotent k-linear operator is given uniquely, up to isomorphy, as a k[T ]-module N α = s i=1 k[T ]/(T α i ) for some partition α = (α 1 , . . . , α s ) which represents the sizes of its Jordan blocks.
The Theorem of Green and Klein [8] states that for given partitions α, β, γ, there exists a short exact sequence of nilpotent linear operators if and only if there is a Littlewood-Richardson (LR-) tableau of shape (α, β, γ). The collection of all such short exact sequences forms a variety V β α,γ (k) which can be partitioned using LR-tableaux, as follows. Consider the affine variety Hom k (N α , N β ) of k-linear maps endowed with the Zariski topology, and assume that all subsets carry the induced topology. Define The irreducible components of V β α,γ (k) are counted by the Littlewood-Richardson coefficient. Namely, to each monomorphism in V β α,γ one can associate an LR-tableau Γ of shape (α, β, γ), as we will see in Section 2. The subset V Γ of Hom k (N α , N β ) of all such monomorphisms is constructible and irreducible. All the V Γ have the same dimension. We denote by V Γ the closure of V Γ in V β α,γ ; the sets V Γ define the irreducible components of V β α,γ , they are indexed by the set T β α,γ of all LR-tableaux of shape (α, β, γ) (see [15,Theorem 4.3] and [17]).
Our aim in this paper is to shed light on the geometry in the variety by studying the boundary relation given as follows.
We are ready to present the main results of the paper.

Two combinatorial criteria
On the set T β α,γ , there are two partial orders of combinatorial nature. The dominance relation ≤ dom is given by the natural partial orders of the partitions defining the tableaux. The second relation is the box-order, it is given by repeatedly exchanging two entries in the tableau in such a way that the smaller entry moves up and such that the lattice permutation condition is preserved. We introduce the two orders formally in Section 2.1.
The following result presents a necessary and a sufficient criterion for two LR-tableaux to be in boundary relation (see Section 3.1 for a proof of (a); Sections 4 and 5 for a proof of (b)).
(b) Suppose β \ γ is a horizontal strip. If Γ ≤ box Γ then Γ ≤ * bound Γ. As a consequence of (a), the relation ≤ bound is antisymmetric. Hence, the transitive closure ≤ * bound is a partial order on T β α,γ .

Two algebraic tests
The algebraic group G = Aut k[T ] (N α )×Aut k [T ] (N β ) acts on V β α,γ via (a, b)·f = bf a −1 . The orbits of this group action are in one-to-one correspondence with the isomorphism classes of embeddings f : N α → N β .
We consider the following preorder for LR-tableaux. We say Γ ≤ deg Γ if there are embeddings f ∈ V Γ ,f ∈ V Γ such that f ≤ degf , that is, Of ⊂ O f , where O f is the orbit of f under the action of G on V β α,γ . The degeneration relation is under control algebraically as the ext-relation implies the deg-relation, which in turn implies the hom-relation (see Section 4).
As the deg-relation, also the hom-and ext-relations give rise to preorders for LR-tableaux. In the diagram below, the relations introduced so far on the set T β α,γ are ordered vertically by containment, with the dominance order the weakest of the relations pictured.
We show that the dominance order is in fact equivalent to the hom-order restricted to certain objects called pickets. Thus we have also algebraic tests both for the validity and for the failure of the boundary relation. We present proofs in Section 4.

Horizontal and vertical strips
We can say more for tableaux with additional properties. If the partitions α, β, γ are such that β \ γ is a horizontal strip, the box-order implies the ext-relation, as we will show in Section 5.
Proposition 1.4. Suppose Γ, Γ are LR-tableaux which have the same shape and which are horizontal strips.
In the special case where the partitions are such that β \ γ is a horizontal and vertical strip, the combinatorial relations ≤ box and ≤ dom are equivalent.
In [13] we give two proofs for this statement; in Section 2.1 we sketch the algorithmic approach in one of them. We deduce the following result.
Theorem 1.5. Suppose α, β, γ are partitions such that β \ γ is a horizontal and vertical strip. The following relations are partial orders which are equivalent to each other.
In Section 3.2 we provide an example showing that the assumtion that β \ γ is a vertical strips is necessary in Theorem 1.5. For comparison we recall the following result from [11,12]. Theorem 1.6. Suppose α, β, γ are partitions such that all parts of α are at most two. The relations ≤ dom , ≤ hom , ≤ bound , ≤ deg , ≤ ext , ≤ box are all partial orders which are equivalent to each other.
Related results. The Theorem of Gerstenhaber and Hesselink shows that the natural partial order of partitions is equivalent to the degeneration order of nilpotent linear operators, see [5,6,14]. We investigate a similar problem: connections of the dominance order of LR-tableaux with the boundary order defined below. Also extensions of nilpotent linear operators are of interest as they are connected with the classical Hall algebras and Hall polynomials, see [16]. Well understood are generic extensions and their relationships with the specializations to q = 0 of the Ringel-Hall algebras, see [3,4,10,18,19].
Organization of this paper. In Section 2, we describe how partitions and tableaux describe short exact sequences of linear operators, or equivalently of embeddings or invariant subspaces of linear operators. We introduce special types of embeddings, namely pickets and poles, and show that every LRtableau which is a horizontal strip can be realized by a direct sum of poles and empty pickets. In Section 3, we show that the boundary relation in Formula 1.1 is a preorder and present an example showing that ≤ bound may not be transitive. Moreover, we prove that the boundary relation implies the dom-order (Part (a) of Theorem 1.2). In Section 4, we adapt the ext-deg-and hom-relations for modules to tableaux. As for modules, the ext-order implies the degeneration order, which implies the hom-order. Moreover, the hom-relation implies the dominance order. This completes the proof of Theorem 1.3. In Section 5, we consider LR-tableaux which are horizontal strips and show that the box-order implies the ext-relation (Proposition 1.4), and hence the boundary relation (Part (b) of Theorem 1.2). We complete the proof of Theorem 1.5.
Acknowledgements. The authors would like to thank Hugh Thomas for discussions which led to the proof of Theorem 2.2, and for contributing the example in Section 3.2. This research project was started when the authors visited the Mathematische Forschungsinstitut Oberwolfach in a Research in Pairs project. They would like to thank the members of the Forschungsinstitut for creating an outstanding environment for their mathematical research.

Littlewood-Richardson tableaux
Given three partitions, α, β, γ, we consider the set T β α,γ of all Littlewood-Richardson tableaux of shape (α, β, γ). We define the domination order on the set T β α,γ . Moreover, we introduce the LR-tableau of a short exact sequence, and determine the tableaux for certain types of short exact sequences, namely pickets and poles. For the case where the skew diagram β \ γ is a horizontal strip, we introduce the box-order and show that any LR-tableau of shape (α, β, γ) can be realized as the LR-tableau of a direct sum of poles and empty pickets.

Combinatorial orders on the set of LR-tableaux
Notation: Recall that a partition α = (α 1 , . . . , α s ) is a finite non-increasing sequence of natural numbers; we picture α by its Young diagram which consists of s columns of length given by the parts of α. The transpose α ′ of α is given by the formula α ′ j = #{i : α i ≥ j}, it is pictured by the transpose of the Young diagram for α. Two partitions α, α are in the natural partial order, in symbols α ≤ nat α, if the inequality Given three partitions α, β, γ, an LR-tableau of shape (α, β, γ) is a Young diagram of shape β in which the region β \ γ contains α ′ 1 entries 1 , . . ., α ′ s entries s , where s = α 1 is the largest entry, such that • in each row, the entries are weakly increasing, • in each column, the entries are strictly increasing, • for each ℓ > 1 and for each column c: on the right hand side of c, the number of entries ℓ − 1 is at least the number of entries ℓ.
The skew diagram β \ γ is said to be a horizontal strip if β i ≤ γ i + 1 holds for all i, and a vertical strip if β ′ \ γ ′ is a horizontal strip.
Definition: Two LR-tableaux Γ = [γ (0) , . . . , γ (s) ], Γ = [ γ (0) , . . . , γ (s) ] of the same shape are in the dominance order, in symbols Γ ≤ dom Γ, if for each i, the corresponding partitions γ (i) , γ (i) are in the natural partial order, i.e. γ (i) ≤ nat γ (i) . Definition: Suppose Γ, Γ are LR-tableaux of the same shape which we assume to be a horizontal strip. We say Γ is obtained from Γ by a box move if after two entries in Γ have been exchanged in such a way that the smaller entry is in the higher position in Γ, we obtain Γ by re-sorting the list of columns if necessary. We denote by ≤ box the partial order generated by box moves.
Here is an example: Γ : Lemma 2.1. For LR-tableaux of the same shape, the ≤ box -order implies the ≤ dom -order.
The converse does not always hold, not even for horizontal strips: Example: Let β = (4, 3, 3, 2, 1), γ = (3, 2, 2, 1) and α = (3, 2). We have seen that there are two LR-tableaux of type (α, β, γ). They are incomparable in ≤ box -relation, but However for horizontal and vertical strips, the two partial orders are equivalent [13]: Theorem 2.2. Suppose α, β, γ are partitions such that β \ γ is a horizontal and vertical strip. Then the two partial orders ≤ dom , ≤ box are equivalent on T β α,γ . In [13] we present two proofs of the fact that ≤ dom implies ≤ box (for horizontal and vertical strips). One of them is algorithmic. Below we present this algorithm without any proof of its correctness. The reader is referred to [13] for details and proofs. Algorithm: For an LR-tableau Γ we denote by ω(Γ) the list of entries when read from left to right, and each column from the bottom up. Clearly, Γ is determined uniquely by its shape and by the list of entries. Input: LR-tableaux Γ, Γ of the shape (α, β, γ) such that β \ γ is a vertical and a horizontal strip and Γ < dom Γ.

The LR-tableau of a short exact sequence
Notation: By a nilpotent operator we understand a pair (V, T ) where V is a finite dimensional k-vector space and T : V → V a k-linear nilpotent operator. Each such pair is determined uniquely, up to isomorphy, by the partition α = (α 1 , . . . , α s ) which records the sizes of the Jordan blocks. We consider (V, T ) as the module over the polynomial ring Conversely, given a k[T ]-module M on which the variable T acts nilpotently, the transpose of the partition β such that M ∼ = N β is given by Given three partitions α, β, γ, there is a short exact sequence E : 0 → N α → N β → N γ → 0 if and only if there is an LR-tableau of shape (α, β, γ) [8].
The tableau Γ corresponding to the sequence E is obtained as follows. Let Definition: Given two partitions γ, γ, the union γ ∪ γ has as Young diagram the sorted union of the columns in the Young diagrams for γ and γ, in , the union of the tableaux is given rowwise: Thus, the LR-tableau of a direct sum is obtained by merging the rows of the LR-tableaux of the summands, starting at the top, and by sorting the entries in each row.
We present a formula for the number µ ℓ,r of boxes ℓ in the r-th row in the LR-tableau Γ = [γ (0) , . . . , γ (t) ] of an embedding (A ⊂ B). We refer to [22, Theorem 1] for a module-theoretic and homological interpretation of this number. Denote by γ ≤r = (γ ′ 1 , . . . , γ ′ r ) ′ the partition which consists of the first r rows of γ. Thus, if a k[T ]-module C has type γ, then C/T r C has type γ ≤r . In particular, the first r rows of the partitions γ (ℓ) are given as follows.
As an immediate consequence, the number of boxes ℓ in the first r rows of Γ is given by and the formula for µ ℓ,r is as follows.
In the remainder of this section we study several types of examples.

Example 1: Pickets
Remark: Recall that the invariant subspaces of a linear operator with only one Jordan block are determined uniquely by their dimension. As a consequence, a picket E as above is determined uniquely, up to isomorphy, by the dimensions n = dim B and m = dim A. We write We picture pickets as follows. In the diagram, the column represents the Jordan block of B and the dot in the (n − m + 1)-st box the submodule generator T n−m in B.
To determine the LR-tableau Γ = [γ (0) , . . . , γ (t) ] of a picket, note that t = m, Theorem 2.6. A pole with submodule generator a is determined uniquely, up to isomorphy, by the radical layers of the elements T i a.
Proof. Consider the pole P as an embedding (A ⊂ B) and apply Formula (2.5). The number µ ℓ,r is either 0 or 1 since A is cyclic. It takes value 1 if ℓ = i for some 1 ≤ i ≤ k and r = x i + 1.
X : To verify, we compute the sequence of radical layers given by the T -powers T i a of the submodule generator a, and the corresponding quotients B/(T i a).
Clearly, each extended pole is a direct sum of a pole and a sum of empty pickets. Using the above results, they can be characterized as follows. Proof. The extended pole is the direct sum of a pole and a sum of empty pickets. The radical layers of the elements T i a, which can be read off from the positions of the entries in the LR-tableau (Lemma 2.7), determine the pole uniquely, up to isomorphy (Theorem 2.6). The ambient space of the pole is a direct summand of B. The isomorphism type of a direct complement determines the empty pickets.

Horizontal strips
We consider short exact sequences for which the LR-tableau is a horizontal strip. Such a sequence can be decomposed as a direct sum of extended poles. Moreover, the decomposition can be chosen such that it is compatible with a given box move.
Lemma 2.9. Suppose Γ is an LR-tableau and a horizontal strip which is not empty. Then Γ = Γ e ∪ Γ ′ where: • Γ e is the LR-tableau of an extended pole (i.e., there are no multiple entries), it is a horizontal strip and has no empty columns; • and Γ ′ is an LR-tableau and a horizontal strip.
Proof. Let Γ e consist of the following columns of Γ: Start with the first occurance of a column which contains the largest entry, say ℓ, in Γ. Then, for each i = ℓ − 1, ℓ − 2, . . . , 1, take the first column containing an i on the right hand side of the previously selected column. The remaining columns form Γ ′ .
Corollary 2.10. Every LR-tableau which is a horizontal strip can be realized as the LR-tableau of a direct sum of poles and empty pickets.
Proof. Use Lemma 2.9 repeatedly to decompose the given LR-tableau Γ into LR-tableaux of poles and empty pickets. According to Lemma 2.3, Γ is the LR-tableau of the direct sum of those poles and empty pickets.
Lemma 2.11. Suppose Γ, Γ are LR-tableaux of the same shape, both horizontal strips, such that Γ is obtained from Γ by a single box move which exchanges the entries u and v with u < v in such a way that Γ < box Γ. Then the columns of Γ, Γ can be partitioned into LR-tableaux with the following properties: 1. Γ ′ , Γ ′′ , Γ ′ , Γ ′′ are LR-tableaux of extended poles, they are horizontal strips and have no empty columns; 2. Γ ′′′ is an LR-tableau which is a horizontal strip; 3. Γ ′ and Γ ′ differ only in the length of one column containing an u , and neither tableau has a column of length in between; 4. Γ ′′ and Γ ′′ differ only in the length of one column containing a v , and neither tableau has a column of length in between; and 5. Γ ′ ∪ Γ ′′ is obtained from Γ ′ ∪ Γ ′′ by a single box move which exchanges the entries u and v.
Proof. Suppose the entries involved in the box move are u and v with u < v.
Denote by c u , c v , c u , c v the columns in Γ and Γ, respectively, which contain those entries. Since Γ < box Γ, column c u occurs on the left of column c v in Γ and column c u occurs on the right of c v in Γ. Let Γ be obtained from Γ by replacing column c v by c v . (So Γ has two long columns.) By arranging the columns in order, Γ becomes an LR-tableau, because Γ and Γ are LRtableaux. Using the algorithm in Lemma 2.9, repeatedly split off the LRtableau Γ e of an extended pole from Γ. Three cases are possible.
• If Γ e does contain neither column c u nor column c v , take Γ e as part of Γ ′′′ .
• If the algorithm encounters c u , disregard in Γ e all columns on the right hand side of c u . Instead, take as next column the first occurance of u − 1 in Γ on the right hand side of the (missing) column c v , and then continue the algorithm. This yields the LR-tableau Γ ′ which is a horizontal strip, has no multiple entries and no empty columns. By construction, there is no column of length between the lengths of c u and c v . Let Γ ′ be obtained from Γ ′ by replacing c u by c u .
• Similarly, if the algorithm encounters c v , disregard in Γ e all columns on the right hand side of c v . Instead, take as next column the first occurance of v − 1 in Γ on the right hand side of the (missing) column c v , and continue. This extended pole is Γ ′′ . Let Γ ′′ be obtained from Γ ′′ by replacing c v by c v .
In each case, put Γ := Γ \ Γ e . By construction, the new Γ is an LR-tableau which is a horizontal strip. Repeat, starting with the algorithm which computes the new Γ e , until there are no entries left in Γ.
Finally, add the remaining empty columns in Γ to Γ ′′′ . The verification of the above properties is easy.
Example: We will revisit the following example in Section 5. The tableau Γ is obtained from Γ via a box move.
Γ : The partition into poles is given by the LR-tableaux below. Here, Γ ′′′ is empty.

The boundary relation and its properties
In this section we present properties of the boundary relation defined in Formula 1.1.

The boundary relation is a preorder
We show that the boundary relation for LR-tableaux is a preorder which implies the dominance order. For the proof of Part (a) in Theorem 1.2 we give a lemma.

Proof. Denote by P ℓ the k[T ]-module k[T ]/T ℓ with only one Jordan block, so
where B = N β , A = N α and f ∈ V β α,γ . Recall that dim Hom k[T ] (P ℓ , P m ) = min{ℓ, m} = dim P ℓ T m P ℓ . Thus: We can now show that the boundary relation implies the dominance order.
Proof of Part (a) of Theorem 1.2. We assume that Γ ≤ dom Γ and show that V Γ ∩ V Γ = ∅. By assumption, there exist i, ℓ such that As a consequence we obtain: The boundary relation is a preorder (i.e. reflexive and antisymmetric).
We conclude this section with a result for later use.
Examples of possible invariant submodules of N β are the powers of the radical T r N β , powers of the socle T −s 0, and their intersections T r N β ∩ T −s 0.

The boundary relation and dominance
We have seen in Section 3.1 that the boundary relation implies the dominance relation. Here we give an example that in general, the boundary relation is strictly stronger than the dominance relation.
In this and in the following section, we determine all isomorphism types of objects which realize a given tableau that has at most 4 rows. Such objects occur in the category S(4) studied in [21, (6.4)] of all pairs consisting of a nilpotent linear operator with nilpotency index at most 4 and an invariant subspace.
Lemma 3.5. Each object in the category S(4) is a direct sum of indecomposables. There are 20 indecomposable objects, up to isomorphy: Four empty pickets P 1 0 , . . ., P 4 0 , fifteen poles P (S), where S is a non-empty subset of {0, 1, 2, 3, 4}, and a remaining object X which has the property that the invariant subspace has two Jordan blocks: For the computation of tableaux recall that the tableau of a pole P (x 1 , . . . , x k ) has entries 1, . . ., k in rows x 1 +1, . . ., x k +1, see Lemma 2.7; and that the LRtableau of a direct sum is obtained by merging the rows of the LR-tableaux of the summands, see Lemma 2.3. Example: For α = (3, 1), β = (4, 3, 1), γ = (3, 1), there are two LR-tableaux of shape (α, β, γ): We determine the possible isomorphism types of embeddings which have LR-tableaux Γ 1 and Γ 2 , respectively. For each tableau, there is only one realization, up to isomorphy.
There are no other realizations: Any such embedding occurs in the category S(4), so Lemma 3.5 can be used. Considering the LR-tableau for X, this module cannot occur as a summand. Hence any realization is a direct sum of poles and empty pickets. Note that the pole P (0, 2, 3) cannot occur in a decomposition for Γ 1 since this would require that P 2 1 is a summand, which is not possible since there is no column of length 2 in Γ 1 . Since poles are determined by the rows in which their entries occur, there are no other choices.
As a consequence, the varieties V Γ 1 and V Γ 2 have the same dimension, and each consists of only one orbit. Hence Thus, Γ 1 and Γ 2 are not in boundary relation, but clearly Γ 1 > dom Γ 2 .
The notation is such that M i or M ix has LR-tableau Γ i . For the convenience of the reader, we picture the poles P (0, 2, 3) and P (0, 1, 3) and their LR-tableaux. We show that the containment relation of orbit closures is as follows.
The short exact sequence shows that O(M 12 ) ⊂ O(M 2 ) (since the ext-order implies the degeneration order, see Section 4). Hence V Γ 1 ∩ V Γ 2 = ∅ and Γ 1 > * bound Γ 2 . Similarly, the short exact sequence Note that the module M 12 = (U ⊂ V ) has the property that dim U ∩ T 2 V ∩ soc V = 1, while for the module M 3 , the corresponding dimension is 2. It follows from Lemma 3.
This finishes the example which illustrates that in general, the condition for LR-tableaux that V Γ ∩ V Γ = ∅ may not define a partial order.

The algebraic orders for LR-tableaux
For modules of a fixed dimension over a finite dimensional algebra the three partial orders ≤ ext , ≤ deg , ≤ hom have been studied extensively, see for example [1,2,20,9,23]. In particular, the partial orders are available for invariant subspaces in V β α,γ , see [11,Section 3.2]. For the convenience of the reader we recall these definitions. Let • The relation f ≤ hom g holds if for any embedding h in V β α,γ , where [f, h] denotes the dimension of the linear space Hom(f, h) of all homomorphisms of embeddings.
They induce three preorders on the set T β α,γ : ≤ * ext , ≤ * deg , ≤ * hom which, as we will see, are in fact partial orders. Namely, each of the relations listed implies the dominance order. Definition: Suppose Γ, Γ are two LR-tableaux of shape (α, β, γ). We write It follows from the corresponding properties for modules that: We have seen in Section 3.1 that the boundary relation implies the dominance order ≤ dom . In the following section we show that also the hom-relation implies the dominance order.

Hom-relation implies dominance order
We start with an abstract result.  Proof. Given an operator X ∈ N and an invariant subspace (A ⊂ B) ∈ S, we need to show that there is a natural isomorphism A morphism in S is given by a commutative diagram: It gives rise to the commutative diagram: Hence we obtain a morphism in N : Conversely, the morphism in N gives rise to a commutative diagram and hence to a morphism in S. Clearly, the two constructions are inverse to each other.
We recognize that the objects of the form P ℓ i = L i (P ℓ ) are pickets. Proof. By the definition given in Section 2.1, the condition Γ ≤ dom Γ is equivalent to for each i and ℓ.
Let i and ℓ be natural numbers. We obtain from Lemma 4.1 and from the equality in the proof of Proposition 3.2 that The claim follows from this and from the corresponding equality for ( A ⊂ B).
It follows that the hom-relation implies the dominance order. Here is what we have. So far, we have not imposed any conditions on the triple (α, β, γ).

The ext-and deg-relations are not equivalent
It is well-known that for modules, the ext-relation ≤ ext implies the degrelation ≤ deg . In general for modules, the converse is not the case. Here we give an example for embeddings of linear operators. Example: For α = (4, 2), β = (6, 4, 2), γ = (4, 2), there are three LRtableaux: We show that the partial orders given by ≤ ext and ≤ deg are as follows: ext: (In each case, Γ 1 is the largest element in the poset.) First we describe the embeddings which realize the tableaux. From [21] we know that there is a one-parameter family of indecomposable embeddings M 2 (λ) occurring on the mouths of the homogeneous tubes with tubular index 0; they all have type Γ 2 . There are two additional indecomposables, they occur in the tube of circumference 2 at index 0; the modules are dual to each other and have type Γ 1 and Γ 2 , respectively. We sketch the modules, using the conventions as in [21]. The modules M 1 = P 6 4 ⊕ P 4 0 ⊕ P 2 2 and M 123 = P (0, 1, 4, 5) ⊕ P 4 2 have type Γ 1 , and M 3 = P 6 2 ⊕ P 4 4 ⊕ P 2 0 has type Γ 3 . We claim that there are no further isomorphism types of objects in V β α,γ . For finite fields, the Hall polynomial g β α,γ counts the number of submodules of N β which are isomorphic to N α and have factor N γ . For each of the isomorphism types of embeddings (that is, M 1 , M 12 , M 123 , M 2 (λ) (λ = 0, 1), M 23 , M 3 ), we can count the corresponding numbers of submodules of N β . It is straightforward to verify that the sum, taken over the isomorphism types, is exactly g β α,γ . For algebraically closed fields, the embeddings M 1 , M 123 , M 3 are sums of exceptional objects in the covering category S( 6) studied in [21], the others are indecomposable non-exceptional objects. The M 2 (λ) occur in the homogeneous tubes, M 12 and M 23 in the tube of circumference 2 in the tubular family of index 0; the remaining tubes of index 0 are pictured in [21, (2.3)], they contain no non-exceptional objects in V β α,γ . All non-exceptional objects in tubes of index different from 0 have higher dimension. It follows that each remaining object in V β α,γ is a direct sum of exceptional modules. Each exceptional object X is determined uniquely by its dimension vector in S( 6) and can be realized over any field. The dimension of the homomorphism spaces Hom(P, X) where P is a picket, and hence the LR-tableau for X ( [22]) do not depend on the base field. Hence M 1 , M 123 and M 3 are the only objects in V β α,γ which have an exceptional direct summand. We determine the ext-order and the deg-order on T β α,γ . Consider the short exact sequences In each, the sum of the end terms is M 123 . It follows that Γ 1 ≥ ext Γ 2 and Γ 1 ≥ ext Γ 3 , respectively. Note that Γ 2 > ext Γ 3 since there is no decomposable module of type Γ 2 .
Since the ext-relation implies the deg-relation, it remains to show that Γ 2 ≥ deg Γ 3 . As mentioned, the modules M 1 and M 3 are dual to each other, so their orbits have the same dimension. As O M 3 = V Γ 3 , and since all varieties given by LR-tableaux are irreducible of the same dimension, it follows that O M 1 is dense in V Γ 1 . In particular, O M 1 contains O M 12 in its closure. Applying duality again, we obtain that O M 3 contains O M 23 in its closure. Thus, O M 23 is in the closure of V Γ 3 .

Box-relation and ext-relation
In this section we assume that α, β, γ are partitions such that β \ γ is a horizontal strip. We show that the box-order implies the ext-relation (Proposition 1.4). As a consequence, we can complete the proof of Theorem 1.5.

Examples
We present two examples where the box-relation implies the ext-relation. Example: We first consider the LR-tableaux from the Example at the end of Section 2.5. Γ : Here, Γ is obtained from Γ by exchanging a box 3 in row 5 with a box 2 in row 7. Since in Γ, the box with the smaller entry is in the higher position, we have Γ < box Γ. Our goal is to show that Γ < ext Γ.
For this, we use Lemma 2.11 to decompose Γ and Γ as the union of two LR-tableaux of poles, and possibly another tableau. This has been done in the example following the lemma.
We picture the four poles.
X : Let Y be given by the following diagram. (Clearly, it is an extension of Z by X.) Y : The statement Γ < ext Γ is a consequence of the following two facts which we will show in a more general set-up in Section 5.3: 1. There is a short exact sequence 0 → X → Y → Z → 0.
2. The LR-tableau for Y is Γ.
Note that in the above example, the modules X, Z, X, Z are all indecomposable. We present a second example in which X, X and Z are poles, hence indecomposable, but Z is the direct sum of a pole and an empty picket. Example: Γ : In this example we can put Y = X ⊕ Z. One can check that Y occurs as the middle term of a short exact sequence 0 → X → Y → Z → 0.

Gradings
To present a general construction for the exact sequence which witnesses the implication from ≤ box to ≤ ext , we will work in the category of graded embeddings. Assume that the type of Γ is (α, β, γ) where β = (β 1 , . . . , β t ). We specify an extended pole P (Γ) which is given by the following graded embedding (A ⊂ B). The grading is such that the submodule generator has degree 0. Thus, the i-th column of Γ (which contains the entry t − i + 1) will give rise to the summand P Here, the elements g β i are the generators of the summands of B. The notation is defined since β has no multiple parts.
We obtain as the proof of Lemma 2.8: Lemma 5.1. The extended pole P (Γ) has LR-tableau Γ.
Here is an example of a graded pole. Example: Consider the following LR-tableau Γ corresponding to the pole P (0, 2, 5). Γ : • • • · · · · · · · · · · · · · · · · · · Remark: Note that the above pole is a horizontal strip. Also poles which are not horizontal strips can be graded, but the above formula has to be modified. To keep the notation simple, we work with extended poles which are horizontal strips. In this case in general, the grading is not unique, up to the shift, since the indecomposable summands can be moved against each other.

Exact sequences for horizontal strips
In order to prove Theorem 1.4, we assume the set-up from Lemma 2.11. Let α, β, γ be partitions such that β \ γ is a horizontal strip. Let Γ, Γ be LR-tableaux of type (α, β, γ).
Assume that Γ is obtained from Γ by a single box move so that Γ < box Γ. Hence the LR-tableau Γ has two boxes u and v in rows r and s where u < v and r > s, and the tableau Γ is obtained from Γ by replacing the boxes u and v .
Our goal is to construct a short exact sequence such that the LR-type of Y is Γ and the LR-type of X ⊕ Z is Γ. This shows Γ ≤ ext Γ.
According to Property (2), the common part Γ ′′′ is an LR-tableau, so Γ ′′′ can be realized as the tableau of an embedding U; in fact, since Γ ′′′ is a horizontal strip, U can be taken as a direct sum of poles and empty pickets (Corollary 2.10). For the purpose of constructing the short exact sequence, we may assume that Γ ′′′ is empty since the embedding U can be added later to both X and Y , say.
According to Property (1), the LR-tableaux Γ ′ , Γ ′′ , Γ ′ and Γ ′′ are horizontal strips with no multiple entries and no empty columns. Hence they can be realized as LR-tableaux of graded extended poles (Section 5.2), which we denote as follows: To refer to the subspace and to the ambient space of such an embedding, we write X : (X sub ⊂ X amb ) etc.
Using Property (3), we see that the embedding X embeds into X as follows. Namely, Γ ′ and Γ ′ differ only in one column which gives X a column that is by r − s units shorter than the corresponding column in X. On the ambient spaces, the inclusion map is as follows.
ι X,X : X amb → X amb , g ℓ X → g r X T r−s if ℓ = s g ℓ X otherwise Note that cok(ι X,X ) is an empty picket of height r−s. Recalling the definition of the subspace generators in Section 5.2, ι X,X maps a X to a X , and hence is a morphism in S Z .
Dually, we can use Property (4) to embed Z into Z with cokernel also an empty picket of height r − s.
Similarly, ι Z, Z maps a Z to a Z and hence is a morphism in S Z .
We can now introduce the module Y . Define the ambient space as the sum Y amb = X amb ⊕ Z amb of graded modules, and let Y sub = (a X , g s Z T s−u ) + (0, a Z )