The boundary of the irreducible components for invariant subspace varieties
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Abstract
Keywords
Nilpotent operator Invariant subspace Partial order Degeneration Littlewood–Richardson tableauMathematics Subject Classification
14L30 (primary) 16G20 47A151 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 can be written, by means of Littlewood–Richardson tableaux, as a union of components of equal dimension. They are the topic of this paper.
Throughout we assume that k is an algebraically closed field. A nilpotent klinear operator is a finite dimensional module over the localized polynomial ring \(k[T]_{(T)}\) in one indeterminate, hence it has the form \(N_\alpha =\bigoplus _{i=1}^sk[T]/(T^{\alpha _i})\) for a uniquely determined partition \(\alpha =(\alpha _1,\ldots ,\alpha _s)\) which represents the sizes of its Jordan blocks, (see Notation 2.12).
It turns out that the variety \({\mathbb {V}}_\varGamma ={\mathcal {O}}_M\) consists of a single orbit (\(M=P_1^3\oplus P_2^2\oplus P_0^1\), see Sect. 2), while \({\mathbb {V}}_{{\widetilde{\varGamma }}}={\mathcal {O}}_{M'}\cup {\mathcal {O}}_{M''}\) is the disjoint union of two orbits (\(M'=P_2^3\oplus P_0^2\oplus P_1^1,M''=P(0,2)\oplus P_1^2\)). One can deduce the following, see [11]. The orbit \({\mathcal {O}}_{M'}\) is dense. Moreover, \({\mathbb {V}}_{{\widetilde{\varGamma }}}\) is closed while \(\overline{{\mathbb {V}}}_\varGamma ={\mathcal {O}}_M\cup {\mathcal {O}}_{M''}\). Thus, \({\mathbb {V}}_{{\widetilde{\varGamma }}}\cap \overline{{\mathbb {V}}}_\varGamma \ne \emptyset \) and hence \(\varGamma \preceq _\mathsf{boundary}{\widetilde{\varGamma }}\).
Obviously, \(\preceq _\mathsf{boundary}\) is reflexive and we will see that it is antisymmetric. We denote by \(\le _\mathsf{boundary}\) the transitive closure of \(\preceq _\mathsf{boundary}\). In general, for a reflexive and antisymmetric relation \(\preceq _\mathsf{x}\) we denote its transitive closure by \(\le _\mathsf{x}\).
However, if \(\beta {\setminus }\gamma \) is a horizontal and vertical strip, then all the above relations are equal.
Since the set \(\mathcal {P}\) of irreducible components of \(\mathbb {V}_{\alpha ,\gamma }^\beta \) is in bijection with the set \(\mathcal {T}_{\alpha ,\gamma }^\beta \), we will work with posets \((\mathcal {T}_{\alpha ,\gamma }^\beta ,\le _\mathsf{x})\) instead of \((\mathcal {P},\le _\mathsf{x})\).
We are ready to present the main results of the paper.
1.1 Two algebraic tests
The algebraic group \(G=\mathrm{Aut}_{k[T]}(N_\alpha )\times \mathrm{Aut}_{k[T]}(N_\beta )\) acts on \({\mathbb {V}}_{\alpha ,\gamma }^\beta \) via \((a,b)\cdot f=bfa^{1}\). The orbits of this group action are in onetoone correspondence with the isomorphism classes of embeddings \(f:N_\alpha \rightarrow N_\beta \).
We consider the following reflexive relation for LRtableaux. We say \(\varGamma \prec _\mathsf{deg}\widetilde{\varGamma }\) if there are embeddings \(f\in {\mathbb {V}}_\varGamma , {\tilde{f}}\in {\mathbb {V}}_{\widetilde{\varGamma }}\) such that \(f\le _\mathsf{deg}{\tilde{f}}\), that is, \({\mathcal {O}}_{{\tilde{f}}}\subset \overline{{\mathcal {O}}}_f\), where \(\mathcal {O}_f\) is the orbit of f under the action of G on \({\mathbb {V}}_{\alpha ,\gamma }^\beta \).
The degeneration relation is under control algebraically as the extrelation \(\prec _\mathsf{ext}\) implies the degrelation \(\prec _\mathsf{deg}\), which in turn implies the homrelation \(\prec _\mathsf{hom}\) (see Sect. 4). We show that the boundary relation implies the restricition \(\le _\mathsf{hompicket}\) of the hom order to certain objects called pickets.
Theorem 1.2
We present proofs in Sect. 4.
1.2 Two combinatorial criteria
On the set \({\mathcal {T}}_{\alpha ,\gamma }^\beta \), there are two partial orders of combinatorial nature. The dominance relation \(\le _\mathsf{dom}\) is given by the natural partial orders of the partitions defining the tableaux. The second relation is the boxorder \(\le _\mathsf{box}\), 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 Sect. 2.1.
The following result presents a necessary and a sufficient criterion of combinatorial nature for two LRtableaux to be in boundary relation:
Theorem 1.3
 (a)
If \(\varGamma \le _\mathsf{boundary}{\widetilde{\varGamma }}\) then \(\varGamma \le _\mathsf{dom}{\widetilde{\varGamma }}\).
 (b)
Suppose \(\beta {\setminus }\gamma \) is a horizontal strip. If \(\varGamma \le _\mathsf{box}{\widetilde{\varGamma }}\) then \(\varGamma \le _\mathsf{boundary}{\widetilde{\varGamma }}\).
We show in Sect. 3.1 that the dominance relation is in fact equivalent to the restriction of the homorder to pickets. The second part follows from a result in [13].
Proposition 1.4
Suppose \(\varGamma ,{\widetilde{\varGamma }}\) are LRtableaux which have the same shape and which are horizontal strips. If \(\varGamma \le _\mathsf{box}{\widetilde{\varGamma }}\) then \(\varGamma \le _\mathsf{ext}{\widetilde{\varGamma }}\).
1.3 Horizontal and vertical strips
Of particular interest is the case where the partitions are such that \(\beta {\setminus }\gamma \) is both a horizontal and a vertical strip. In this situation, the combinatorial relations \(\le _\mathsf{box}\) and \(\le _\mathsf{dom}\) are in fact equivalent. In [14] we give two proofs for this statement; below in Sect. 2.1 we sketch the algorithmic approach in one of them. We deduce the following result.
Theorem 1.5
For comparison we note that there is a related result about the six partial orders in the case where all parts of \(\alpha \) are at most two. In this situation the orbits and the boundary relation are given combinatorially in terms of arc diagrams and of resolution of crossings, respectively [11, 12].
Theorem 1.6
Suppose \(\alpha ,\beta ,\gamma \) are partitions such that all parts of \(\alpha \) are at most two. The relations \(\le _\mathsf{dom},\le _\mathsf{hom},\le _\mathsf{boundary}, \le _\mathsf{deg},\le _\mathsf{ext},\le _\mathsf{box}\) are all partial orders which are equivalent to each other.
1.4 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, 15]. We investigate a similar problem: connections of the dominance order of LRtableaux 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 [17]. Well understood are generic extensions and their relationships with the specializations to \(q=0\) of the RingelHall algebras, see [3, 4, 10, 19, 20].
1.5 Organization of this paper
In Sect. 2, we describe how partitions and tableaux describe short exact sequences of linear operators, or equivalently of embeddings or invariant subspaces of linear operators. Moreover, we introduce pickets as special types of embeddings.
In Sect. 3, we show that the boundary relation in Formula (1.1) implies the dominance order (Part (a) of Theorem 1.3). As a consequence we obtain that the boundary relation is antisymmetric. We present an example showing that \(\preceq _\mathsf{boundary}\) may not be transitive. Example 3.6 shows that the dominance order does not imply the boundary relation in general, not even for vertical strips. But note that the two relations are equivalent when we are dealing with horizontal and vertical strips (Theorem 1.5).
In Sect. 4, we adapt the ext deg and homrelations for modules to tableaux. As for modules, the extorder implies the degeneration order, which implies the homorder. Moreover, the homrelation implies the dominance order. This completes the proof of Theorem 1.2. Using results given in [13] and in [14], we show part (b) of Theorem 1.3 and complete the proof of Theorem 1.5.
2 Littlewood–Richardson tableaux
Given three partitions, \(\alpha ,\beta ,\gamma \), we consider the set \({\mathcal {T}}_{\alpha ,\gamma }^\beta \) of all Littlewood–Richardson tableaux of shape \((\alpha ,\beta ,\gamma )\). We define the dominance order on the set \({\mathcal {T}}_{\alpha ,\gamma }^\beta \). Moreover, we introduce the LRtableau of a short exact sequence, and determine the tableaux for certain types of short exact sequences, in particular pickets. For the case where the skew diagram \(\beta {\setminus }\gamma \) is a horizontal strip, we also introduce the boxorder.
2.1 Combinatorial orders on the set of LRtableaux
Notation 2.1

in each row, the entries are weakly increasing,

in each column, the entries are strictly increasing,

for each \(\ell >1\) and for each column c: on the right hand side of c, the number of entries \(\ell 1\) is at least the number of entries \(\ell \).
Example 2.2
Notation 2.3
In the example above, the first tableau is given by the sequence of partitions \(\varGamma =[(3,2,2,1),(3,3,2,1,1),(4,3,2,2,1),(4,3,3,2,1)]\).
We introduce two partial orders on the set \({\mathcal {T}}_{\alpha ,\gamma }^\beta \) of all LRtableaux of shape \((\alpha ,\beta ,\gamma )\).
Definition 2.4
Two LRtableaux \(\varGamma =[\gamma ^{(0)},\ldots ,\gamma ^{(t)}], \widetilde{\varGamma }=[\widetilde{\gamma }^{(0)},\ldots ,\widetilde{\gamma }^{(t)}]\) of the same shape are in the dominance order, in symbols \(\varGamma \le _\mathsf{dom}\widetilde{\varGamma }\), if for each i, the corresponding partitions \(\gamma ^{(i)},\widetilde{\gamma }^{(i)}\) are in the natural partial order, i.e. \(\gamma ^{(i)}\le _\mathsf{nat}\widetilde{\gamma }^{(i)}\).
Definition 2.5
Suppose \(\varGamma ,{\widetilde{\varGamma }}\) are LRtableaux of the same shape which we assume to be a horizontal strip. We say \({\widetilde{\varGamma }}\) is obtained from \(\varGamma \) by a box move if after two entries in \(\varGamma \) have been exchanged in such a way that the smaller entry is in the higher position in \({\widetilde{\varGamma }}\), we obtain \({\widetilde{\varGamma }}\) by resorting the list of columns if necessary. We denote by \(\le _\mathsf{box}\) the partial order generated by box moves.
Remark 2.6
In [13] the boxorder is defined in a more general case: in the case when LRtableaux are unions of so called columns. For simplicity, we present definitions and results for horizontal strips.
Lemma 2.7
For LRtableaux of the same shape, the \(\le _\mathsf{box}\)order implies the \(\le _\mathsf{dom}\)order.
Proof
Suppose the LRtableau \({\widetilde{\varGamma }}=[{\widetilde{\gamma }}^{(0)},\ldots ,{\widetilde{\gamma }}^{(t)}]\) is obtained from \(\varGamma =[\gamma ^{(0)},\ldots ,\gamma ^{(t)}]\) by a box move based on entries i and j with, say, \(i<j\). The process of reordering the entries in each row will not affect entries less than i or larger than j, so the partitions \(\gamma ^{(0)},\ldots ,\gamma ^{(i1)}\), and \(\gamma ^{(j)},\ldots ,\gamma ^{(t)}\) remain unchanged. The partitions \(\gamma ^{(\ell )},{\widetilde{\gamma }}^{(\ell )}\) for \(i\le \ell <j\) are different and satisfy \(\gamma ^{(\ell )}<_\mathsf{nat}{\widetilde{\gamma }}^{(\ell )}\) (since the defining partial sums can only increase). This shows that \(\varGamma <_\mathsf{dom}{\widetilde{\varGamma }}\). \(\square \)
The converse does not always hold, not even for horizontal strips:
Example 2.8
However for horizontal and vertical strips, the two partial orders are equivalent [14]:
Theorem 2.9
Suppose \(\alpha ,\beta ,\gamma \) are partitions such that \(\beta {\setminus }\gamma \) is a horizontal and vertical strip. Then the two partial orders \(\le _\mathsf{dom},\le _\mathsf{box}\) are equivalent on \({\mathcal {T}}_{\alpha ,\gamma }^\beta \). \(\square \)
In [14] we present two proofs of the fact that \(\le _\mathsf{dom}\) implies \(\le _\mathsf{box}\) (for horizontal and vertical strips). Both are algorithmic. Below we present one of these algorithms without any proof of its correctness. The reader is referred to [14] for details and proofs.
Algorithm 2.10

Input: Two LRtableaux \(\varGamma ,{\widetilde{\varGamma }}\) of shape \((\alpha ,\beta ,\gamma )\) such that \(\beta {\setminus }\gamma \) is a horizontal and vertical strip and such that \(\varGamma <_\mathsf{dom}{\widetilde{\varGamma }}\).

Output: An LRtableau \({\widehat{\varGamma }}\) of the shape \((\alpha ,\beta ,\gamma )\) such that \(\varGamma \le _\mathsf{dom}\widehat{\varGamma }\) and \(\widehat{\varGamma }<_\mathsf{box}\widetilde{\varGamma }\).

Step 1. Find the smallest k such that \(\omega (\varGamma )_k\ne \omega ({\widetilde{\varGamma }})_k\) and put \(x=\omega (\varGamma )_k\).

Step 2. Choose the minimal \(m\ge k+1\) such that \(x=\omega ({\widetilde{\varGamma }})_m\).

Step 3. Let \(y=\min \{\omega ({\widetilde{\varGamma }})_i>x:k\le i < m\}\).

Step 4. Choose \(k\le l < m\) such that \(y=\omega ({\widetilde{\varGamma }})_l\).

Step 5. Define \({\widehat{\varGamma }}\) such that \(\omega ({\widehat{\varGamma }})_i=\omega ({\widetilde{\varGamma }})_i\), for \(i\ne l,m\), and \(\omega ({\widehat{\varGamma }})_l=x,\omega ({\widehat{\varGamma }})_m=y\).
Example 2.11
2.2 The LRtableau of a short exact sequence
Notation 2.12
Definition 2.13
Given two partitions \(\gamma ,{\widetilde{\gamma }}\), the union \(\gamma \cup {\widetilde{\gamma }}\) has as Young diagram the sorted union of the columns in the Young diagrams for \(\gamma \) and \({\widetilde{\gamma }}\), in symbols, \((\gamma \cup {\widetilde{\gamma }})'_i= \gamma '_i+{\widetilde{\gamma }}'_i\).
Lemma 2.14
Suppose the exact sequences \(E,{\widetilde{E}}\) have LRtableaux \(\varGamma ,{\widetilde{\varGamma }}\), respectively. Then the LRtableau of the direct sum \(E\oplus {\widetilde{E}}\) is \(\varGamma \cup {\widetilde{\varGamma }}\).
Proof
Suppose \(E,{\widetilde{E}}\) are given by the embeddings \(A\subset B,{\widetilde{A}}\subset {\widetilde{B}}\). The jth partition in the LRtableau for \(E\oplus {\widetilde{E}}\) is the Jordan type for \(B/T^jA \oplus {\widetilde{B}}/T^j{\widetilde{A}}\), which is \(\gamma ^{(j)}\cup {\widetilde{\gamma }}^{(j)}\). \(\square \)
Thus, the LRtableau of a direct sum is obtained by merging the rows of the LRtableaux of the summands, starting at the top, and by sorting the entries in each row.
We present a formula for the number \(\mu _{\ell ,r}\) of boxes Open image in new window in the rth row in the LRtableau \(\varGamma =[\gamma ^{(0)},\ldots ,\gamma ^{(t)}]\) of an embedding \((A\subset B)\). We refer to [23, Theorem 1] for a moduletheoretic and homological interpretation of this number.
2.3 Example 1: pickets
Definition 2.17
A short exact sequence \(E: 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\) is a picket if B is indecomposable as a k[T]module (so the partition \(\beta \) has only one part). A picket E is empty if \(A=0\).
Remark 2.18
2.4 Example 2: poles
Definition 2.19
A short exact sequence \(E: 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\) is a pole if A is indecomposable as a k[T]module and E is indecomposable as a short exact sequence.
Poles have been classified, up to isomorphy, by Kaplansky [7, Theorem 24].
Theorem 2.20
A pole with submodule generator a is determined uniquely, up to isomorphy, by the radical layers of the elements \(T^ia\). \(\square \)
For a nonempty, strictly increasing sequence \(S=(x_0,\ldots ,x_{L1})\) of nonnegative integers we construct the pole P(S) for which the submodule generator a satisfies that each \(T^ia\) occurs in the \(x_i\)st power of the radical.
The LRtableau for \(P(S)=[\gamma ^{(0)},\ldots ,\gamma ^{(L)}]\) is easily computed as \(\gamma ^{(i)}{\setminus }\gamma ^{(i1)}\) consists of a single box Open image in new window in row \(x_{i1}+1\).
3 The boundary relation and its properties
In this section we present properties of the boundary relation defined in Formula 1.1.
3.1 The boundary relation is antisymmetric
We show that the boundary relation for LRtableaux is antisymmetric by verifying that it implies the dominance order. This is Part (a) in Theorem 1.3.
Lemma 3.1
Proof
Proposition 3.2
Proof
Suppose \(f:A\rightarrow B\) is an embedding in \({\mathbb {V}}_\varGamma \). Recall that the partitions in \(\varGamma \) are given by \(B/f(T^iA)=N_{\gamma ^{(i)}}\).
We can now show that the boundary relation implies the dominance order.
Proof [of Part (a) of Theorem 1.3]
As a consequence we obtain:
Corollary 3.3
The boundary relation is reflexive and antisymmetric. \(\square \)
We conclude this section with a result for later use.
Lemma 3.4
Examples of possible invariant submodules of \(N_\beta \) are the powers of the radical \(T^rN_\beta \), powers of the socle \(T^{s}0\), and their intersections \(T^rN_\beta \cap T^{s}0\).
Proof
Let \(h_\lambda :N_\alpha \rightarrow N_\beta \) be a oneparameter family of objects in \({\mathbb {V}}_{\alpha ,\gamma }^\beta \) such that \(h_\lambda \cong g\) for \(\lambda \ne 0\) and \(h_0\cong f\). Put \(n=\dim \mathrm{Im}g\cap W\).
3.2 The boundary relation and the dominance relation
We have seen in Sect. 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 \({\mathcal {S}}(4)\) studied in [22, (6.4)] of all pairs consisting of a nilpotent linear operator with nilpotency index at most 4 and an invariant subspace.
Lemma 3.5
Recall that the LRtableau of a direct sum is obtained by merging the rows of the LRtableaux of the summands, see Lemma 2.14.
Example 3.6
3.3 The boundary relation may not be transitive
Example 3.7
The notation is such that \(M_i\) or \(M_{ix}\) has LRtableau \(\varGamma _i\).
However, \({\mathbb {V}}_{\varGamma _1}\cap \overline{{\mathbb {V}}}_{\varGamma _3}=\emptyset \). The only possible orbit in the intersection is \({\mathcal {O}}(M_{12})\), since there are only two orbits in \({\mathbb {V}}_{\varGamma _1}\), and since the other orbit \({\mathcal {O}}(M_1)\) has the same dimension as \({\mathbb {V}}_{\varGamma _3}={\mathcal {O}}(M_3)\).
Note that the module \(M_{12}=(U\subset V)\) has the property that \(\dim U\cap T^2V\cap \mathrm{soc\,}V=1\), while for the module \(M_3\), the corresponding dimension is 2. It follows from Lemma 3.4 with \(W=T^2V\cap \mathrm{soc\,}V\) that \({\mathcal {O}}(M_{12})\not \subseteq \overline{{\mathcal {O}}}(M_3)\).
This finishes the example which illustrates that in general, the condition for LRtableaux that \({\mathbb {V}}_{\widetilde{\varGamma }}\cap \overline{{\mathbb {V}}}_\varGamma \ne \emptyset \) may not define a partial order. \(\square \)
4 The algebraic orders for LRtableaux

The relation \(f \le _\mathsf{ext} g\) holds if there exist embeddings \(h_i,u_i,v_i\) of linear operators and short exact sequences \(0\rightarrow u_i\rightarrow h_i\rightarrow v_i\rightarrow 0\) of embeddings such that \(f\cong h_1,u_i\oplus v_i\cong h_{i+1}\) for \(1\le i\le s\), and \(g\cong h_{s+1}\), for some natural number s.

The relation \(f \le _\mathsf{deg} g\) holds if \(\mathcal {O}_g \subseteq \overline{\mathcal {O}_f}\) in \({\mathbb {V}}_{\alpha ,\gamma }^\beta (k)\).
 The relation \(f\le _\mathsf{hom} g\) holds iffor any embedding h, where [f, h] denotes the dimension of the linear space \(\mathrm{Hom}(f,h)\) of all homomorphisms of embeddings.$$\begin{aligned}{}[f,h]\le [g,h] \end{aligned}$$
Definition 4.1

\(\varGamma \le _\mathsf{ext}\widetilde{\varGamma }\) implies \(\varGamma \le _\mathsf{deg}\widetilde{\varGamma }\) and

\(\varGamma \le _\mathsf{deg}\widetilde{\varGamma }\) implies \(\varGamma \le _\mathsf{hom}\widetilde{\varGamma }\).

\(\varGamma \le _\mathsf{deg}\widetilde{\varGamma }\) implies \(\varGamma \le _\mathsf{boundary}\widetilde{\varGamma }\).

\(\varGamma \le _\mathsf{boundary}\widetilde{\varGamma }\) implies \(\varGamma \le _\mathsf{deg}\widetilde{\varGamma }\).
We have seen in Sect. 3.1 that the boundary relation implies the dominance order \(\le _\mathsf{dom}\). In the following section we show that also the homrelation implies the dominance order. As a consequence, each of the three relations \(\le _\mathsf{ext}, \;\le _\mathsf{deg}, \;\le _\mathsf{hom}\) is antisymmetric, hence a partial ordering.
4.1 The Homrelation implies the dominance order
We start with an abstract result.
Lemma 4.2
For each \(i\in {\mathbb {N}}\), the functors \(R_i,L_i\) form an adjoint pair.
Proof
We recognize that the objects of the form \(P_i^\ell =L_i(N_{(\ell )})\) are pickets.
Proposition 4.3
 1.
\(\varGamma \le _\mathsf{dom}\widetilde{\varGamma }\)
 2.For each picket \(P_i^\ell \) the inequality holds:$$\begin{aligned}\dim \mathrm{Hom}_{{\mathcal {S}}}((A\subset B),P_i^\ell )\le \dim \mathrm{Hom}_{{\mathcal {S}}}((\widetilde{A}\subset \widetilde{B}),P_i^\ell )\end{aligned}$$
Proof
4.2 The boxorder implies the extorder (for horizontal strips)
Remark 4.4
 1.
In [13] the implication \(\le _\mathsf{box}\;\Longrightarrow \; \le _\mathsf{ext}\) is proved in a more general case.
 2.
Example 2.8 shows that these orders are not equivalent in general (even for horizontal strips).
 3.
Results of [14] prove the equivalence of all these orders in the case \(\beta {\setminus }\gamma \) is a horizontal and vertical strip (compare Theorem 1.5).
4.3 The ext and degrelations are not equivalent
It is wellknown that for modules, the extrelation \(\le _\mathsf{ext}\) implies the degrelation \(\le _\mathsf{deg}\). In general for modules, the converse is not the case. Here we give an example for embeddings of linear operators.
Example 4.5
We claim that there are no further isomorphism types of objects in \({\mathbb {V}}_{\alpha ,\gamma }^\beta \).
For finite fields, the Hall polynomial \(g_{\alpha ,\gamma }^\beta \) counts the number of submodules of \(N_\beta \) which are isomorphic to \(N_\alpha \) and have factor \(N_\gamma \). For each of the isomorphism types of embeddings (that is, \(M_1,M_{12},M_{123}, M_2(\lambda )\) (\(\lambda \ne 0,1\)), \(M_{23},M_3\)), we can count the corresponding numbers of submodules of \(N_\beta \). It is straightforward to verify that the sum, taken over the isomorphism types, is exactly \(g_{\alpha ,\gamma }^\beta \).
For algebraically closed fields, the embeddings \(M_1,M_{123},M_3\) are sums of exceptional objects in the covering category \({\mathcal {S}}({\widetilde{6}})\) studied in [22], the others are indecomposable nonexceptional objects. The \(M_2(\lambda )\) 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 [22, (2.3)], they contain no nonexceptional objects in \({\mathbb {V}}_{\alpha ,\gamma }^\beta \). All nonexceptional objects in tubes of index different from 0 have higher dimension. (Namely, an indecomposable module of index different from zero occurs as the image under the covering functor of a regular module over the tubular algebra \(\varTheta _0\) corresponding to a tubular index \(\gamma =(p:q)\in {\mathbb {Q}}^+\) [22, (1.4),(1.1)]. Since the modules in an extended tube in \({\mathcal {T}}_\gamma \) which have distance from the mouth less than the rank of the tube are all exceptional, one deduces that each nonexceptional module in \({\mathcal {T}}_\gamma \) has dimension pair at least \((p+q)\cdot (12,6)\).)
It follows that each remaining object in \({\mathbb {V}}_{\alpha ,\gamma }^\beta \) is a direct sum of exceptional modules. Each exceptional object X is determined uniquely by its dimension vector in \({\mathcal {S}}({\widetilde{6}})\) and can be realized over any field. The dimension of the homomorphism spaces \(\mathrm{Hom}(P,X)\) where P is a picket, and hence the LRtableau for X ([23]) do not depend on the base field. Hence \(M_1,M_{123}\) and \(M_3\) are the only objects in \({\mathbb {V}}_{\alpha ,\gamma }^\beta \) which have an exceptional direct summand.
We determine the extorder and the degorder on \({\mathcal {T}}_{\alpha ,\gamma }^\beta \).
Since the extrelation implies the degrelation, it remains to show that \(\varGamma _2\ge _\mathsf{deg}\varGamma _3\). As mentioned, the modules \(M_1\) and \(M_3\) are dual to each other, so their orbits have the same dimension. As \({\mathcal {O}}_{M_3}={\mathbb {V}}_{\varGamma _3}\), and since all varieties given by LRtableaux are irreducible of the same dimension, it follows that \({\mathcal {O}}_{M_1}\) is dense in \({\mathbb {V}}_{\varGamma _1}\). In particular, \({\mathcal {O}}_{M_1}\) contains \({\mathcal {O}}_{M_{12}}\) in its closure. Applying duality again, we obtain that \({\mathcal {O}}_{M_3}\) contains \({\mathcal {O}}_{M_{23}}\) in its closure. Thus, \({\mathcal {O}}_{M_{23}}\) is in the closure of \({\mathbb {V}}_{\varGamma _3}\). \(\square \)
Notes
Acknowledgements
The authors are indebted to Hugh Thomas for discussions which led to the proof of Theorem 2.9, and for his contribution of the example in Sect. 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.
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