## ABSTRACT

For a partition *β*, denote by *N*_{β} the nilpotent linear operator of Jordan type *β*. Given partitions *β*, *γ*, we investigate the representation space ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ of all short exact sequences

*α*is any partition with each part at most 2. Due to the condition on

*α*, the isomorphism type of a sequence

*ℰ*is given by an arc diagram

*Δ*; denote by

*𝕍*

_{Δ}the subset of ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ of all sequences isomorphic to

*ℰ*. Thus, the variety ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ carries a stratification given by the subsets of type

*𝕍*

_{Δ}. We compute the dimension of each stratum and show that the boundary of a stratum

*𝕍*

_{Δ}consists exactly of those ${\mathbb{V}}_{{\Delta}^{\prime}}$ where

*Δ*

^{′}is obtained from

*Δ*by a non-empty sequence of arc moves of five possible types (A)–(E). The case where all three partitions are fixed has been studied in [Citation3, Citation5]. There, arc moves of types (A)–(D)suﬃce to describe the boundary of a

*𝕍*

_{Δ}in ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}$. Our fifth move

**(E)**, “explosion,” is needed to break up an arc into two poles to allow for changes in the partition

*α*.

2010 MATHEMATICS SUBJECT CLASSIFICATION:

## 1. Introduction

In this paper, we generalize results presented in [Citation5, Citation4] where arc diagrams were applied to investigate the degeneration order for a certain class of invariant subspaces of nilpotent linear operators. We enlarge the set of arc moves introduced in [Citation5] and compare the partial order induced by these moves with the partial order given by degenerations in a variety associated with invariant subspaces of nilpotent linear operators.

### 1.1. Arc diagrams and the arc order

Two diagrams of arcs and poles are said to be in *arc order* if the first is obtained from the second by a sequence of moves of type

If the arc diagrams *Δ* and *Δ*^{′} are in relation, we write $\Delta {\le}_{arc}{\Delta}^{\prime}$. Since each move either decreases the number of crossings or the number of poles, ≤_{arc} is a partial order. The arc moves **(A)**, **(B)**, **(C)** and **(D)** were introduced in [Citation5] and the arc move (**E**) is new.

Formally, an arc diagram is a finite set of arcs and poles in the Poincaré half plane. We assume that all end points are natural numbers (arranged from right to left) and admit multiple arcs and poles. In addition, each natural number can be marked by one or several circles (which are not affected by any of the arc moves).

Example.

The diagram *Δ* is obtained form *Δ* by a single move of type (**B**).

### 1.2. Invariant subspaces of nilpotent linear operators

Let *k* be a field. We call a *k*[*T*]-module

*N*

_{α}is a representation of the quiver subject to the relation ${\phi}^{{\alpha}_{1}}=0$. By

*𝒮*we denote the category of all representations

*M*of the quiver subject to the following conditions: The endomorphisms

*M*(

*φ*

^{′}) and

*M*(

*φ*) are nilpotent; the linear map

*M*(

*ψ*) is a monomorphism; and the relation

*φψ*=

*ψ*

*φ*

^{′}is satisfied. Equivalently,

*𝒮*is the category of all monomorphisms between nilpotent linear operators, which can be identified with the category of all short exact sequences of nilpotent linear operators.

Given three partitions *α*,*β*,*γ*, we denote by ${\mathcal{\mathcal{S}}}_{\alpha ,\gamma}^{\beta}$ the full subcategory of *𝒮* consisting of all short exact sequences $0\to {N}_{\alpha}\to {N}_{\beta}\to {N}_{\gamma}\to 0$ and consider the full subcategories of *𝒮* with objects:

We assign to each object *M* in ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ an arc diagram *Δ*(*M*), which depends only on the isomorphism type of *M*. By ${}_{2}{\mathcal{\mathcal{D}}}_{\gamma}^{\beta}$, we denote the set of all arc diagrams of type (*β*,*γ*). Given $\Delta {\in}_{2}{\mathcal{\mathcal{D}}}_{\gamma}^{\beta}$, we denote by *α*(*Δ*) the partition which contains a part 2 for each arc, and a part 1 for each pole. A short exact sequence with arc diagram *Δ* has the form $0\to {N}_{\alpha \left(\Delta \right)}\to {N}_{\beta}\to {N}_{\gamma}\to 0$, see Section 2.2.

Let *k* be an algebraically closed field, let *d* = (*a*,*b*)∈*ℕ*^{2} be a dimension vector with *b*≥*a*, and let *β* and *γ* be partitions of length *b* and *b*−*a*, respectively.

In this paper, we are interested in the subspace ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ of the representation space *rep*_{d}(*Q*) consisting of all representations which are isomorphic to an object in ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$.

The space $${}_{2}{\mathbb{V}}_{\gamma}^{\eta}$$ is invariant under the operation of the algebraic group $$G=G{l}_{a}(k)\times G{l}_{b}(k)$$ acting on *rep*_{d}(*Q*); the *G*-orbits in ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ are in one-to-one correspondence with the isomorphism classes in $${}_{2}{\mathcal{S}}_{\gamma}^{\eta}$$. For $$\mathrm{\Delta}\in {{}_{2}\mathcal{D}}_{\gamma}^{\eta}$$, we denote by *𝕍*_{Δ} the *G*-orbit of all representations which have arc diagram *Δ*.

### 1.3. Main results

The arc diagrams of invariant subspaces give a stratification of ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$:

More precisely, we have:

Proposition 1.3.

The set {*𝕍*_{Δ}:*Δ*∈*𝒟*} where $$\mathcal{D}={}_{2}{\mathcal{D}}_{\gamma}^{\eta}$$ forms a stratification for $$\mathbb{V}={{}_{2}\mathbb{V}}_{\gamma}^{\eta}$$ in the sense that

each

*𝕍*_{Δ}is locally closed in*𝕍*;*𝕍*is the disjoint union ${\bigcup}_{\Delta \in \mathcal{\mathcal{D}}}^{\bullet}{\mathbb{V}}_{\Delta}$;for each

*Δ*, there is a finite subset*U*_{Δ}⊂*𝒟*such that the closure ${\overline{\mathbb{V}}}_{\Delta}$ is just the union ${\bigcup}_{\Gamma \in {U}_{\Delta}}{\mathbb{V}}_{\Gamma}$.

Moreover an arc diagram determines the dimension of its stratum as follows.

Theorem 1.4.

Let *k* be an algebraically closed field, and let *β*,*γ* be partitions. Suppose the arc diagram $\Delta {\in}_{2}{\mathcal{\mathcal{D}}}_{\gamma}^{\beta}$ has *x*(*Δ*) crossings. Then

*α*=

*α*(

*Δ*) and where $n\left(\lambda \right)={\sum}_{i}{\lambda}_{i}(i-1)$ denotes the

**moment**and $\left|\lambda \right|={\lambda}_{1}+{\lambda}_{2}+\dots $ - the length of a partition λ.

The second main result of this paper describes the boundary of each stratum.

Theorem 1.5.

Suppose that *k* is an algebraically closed field and that *β*,*γ* are partitions. For arc diagrams $\Gamma ,\Delta {\in}_{2}{\mathcal{\mathcal{D}}}_{\gamma}^{\beta}$ we have

In the proof of the “only if” part, we use a version of an algorithm from [Citation5] which delivers a sequence of arc moves that convert *Γ* to *Δ*.

Remark.

In [Citation5] the authors investigated arc and deg orders in sets ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$, where

*α*_{1}≤2. In this paper, we work in a bigger set ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ and prove similar results without fixing the partition*α*. Moreover, in Section 9, we present examples showing that we have to fix the partitions*β*and*γ*.Working with arc diagrams, we have to assume that

*α*_{1}≤2, because otherwise we do not have a natural bijection between arc diagrams and orbits of our group action.In the paper we investigate the degeneration order induced by this group action and compare this order with the arc order and with two algebraic orders (hom-order and ext-order), see Section 3.1 for definitions.

### 1.4. Organization of this paper

The paper is organized as follows.

In Section 2, we introduce terminology and describe a bijection between arc diagrams and isomorphism types of objects in the category ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$.

In Section 3, we recall the definition of the partial orders ≤

_{arc}, ≤_{ext}, ≤_{deg}and ≤_{hom}. Moreover, in Theorem 3.1, we discuss relations between them.In Section 4, we give a brief exposition of the category

*𝒮*_{2}(*k*).Some technical facts for later use are presented in Section 5.

Section 6 contains the proof of the implication

$\begin{array}{c}\multicolumn{1}{c}{{\le}_{hom}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{\le}_{arc}}\end{array}$that is the crucial part of the proof of Theorem 1.5. Our proof uses methods derived from [Citation5].In Section 8, some combinatorial properties of the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$ are presented.

In Theorems 1.5 and 3.1, we assume that two partitions

*β*and*γ*are fixed. Some relevant counterexamples indicated in Section 9 show that this assumption is necessary.Section 10 contains a proof of Theorem 1.4.

## 2. Arc diagrams and invariant subspaces

For fixed partitions *γ*⊆*β* (i.e. ${\gamma}_{i}\le {\beta}_{i}$ for all *i*), there is a bijection between the set of arc diagrams and the set of isomorphism classes of objects in the category ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ (1.2). In this section, we describe this bijection.

In [Citation5], Littlewood–Richardson tableaux and Klein tableaux were applied to associate an arc diagram with an object in *𝒮*_{2}, where ${\mathcal{\mathcal{S}}}_{2}={\bigcup {}_{\gamma ,\beta}}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$. For the definition of a Klein tableau for an object in *𝒮*_{2}, we refer to [Citation3] or [Citation8].

### 2.1. Invariant subspaces

Let *k* be an arbitrary field. By *𝒩* or *𝒩*(*k*), we denote the category of all nilpotent linear operators *N*_{α} (1.1). We write the objects of *𝒩* as pairs (*V*,*φ*) where *V* is the underlying *k*-vector space and *φ*:*V*→*V* the nilpotent *k*-linear endomorphism given by multiplication by *T*. If (*V*,*φ*), $({V}^{\prime},{\phi}^{\prime})$ are objects in *𝒩*, then a morphism $f:(V,\phi )\to ({V}^{\prime},{\phi}^{\prime})$ in *𝒩* is a linear map *f*:*V*→*V*^{′} such that *φ*^{′}*f* = *fφ*. Let

*φ*

_{α}is given by the nilpotent block matrix consisting of Jordan blocks of type

*α*.

Denote by *𝒮* the category of all systems $f=({N}_{\alpha},{N}_{\beta},f)$, where *f* is a monomorphism. For $f=({N}_{\alpha},{N}_{\beta},f)$ and $g=({N}_{{\alpha}^{\prime}},{N}_{{\beta}^{\prime}},g)$, a morphism *H*:*f*→*g* is a pair $({h}_{1},{h}_{2})$ of homomorphisms ${h}_{1}:{N}_{\alpha}\to {N}_{{\alpha}^{\prime}}$ and ${h}_{2}:{N}_{\beta}\to {N}_{{\beta}^{\prime}}$ such that

For a natural number *n*, we write *𝒮*_{n} or *𝒮*_{n}(*k*) for the full subcategory of *𝒮* of all systems, where the operator acts on the subspace with nilpotency index at most *n*. Thus, the objects in *𝒮*_{2} are the systems $({N}_{\alpha},{N}_{\beta},f)$ where *α*_{1}≤2.

### 2.2. Pickets and bipickets

The category *𝒮*_{2}(*k*) is of particular interest for us in this paper. By Beers et al. [Citation1], each indecomposable object is either isomorphic to a *picket*, that is, it has the form

*ℓ*≤min{2,

*m*} (so the ambient space

*N*

_{(m)}has only one Jordan block, and

*N*

_{(ℓ)}is the unique

*T*-invariant subspace of dimension

*ℓ*), or to a

*bipicket*

*r*≤

*m*−2, ${N}_{(m,r)}=k\left[T\right]\u2215\left({T}^{m}\right)\oplus k\left[T\right]\u2215\left({T}^{r}\right)$ and the morphism $\delta :{N}_{\left(2\right)}\to {N}_{(m,r)}$ is given by $\delta \left(1\right)=({T}^{m-2},{T}^{r-1})$. Whenever we want to emphasize the dependence on the field

*k*, we will write ${P}_{\ell}^{m}={P}_{\ell}^{m}\left(k\right)$ and ${B}_{2}^{m,r}={B}_{2}^{m,r}\left(k\right)$.

Thanks to this classification we can associate with any object in *𝒮*_{2}(*k*) an arc diagram and partition (*β*,*γ*) type as follows. First we list some arc diagrams in the following table.

The arc diagram for an object *X* in *𝒮*_{2} is created in the following way. We decompose

*Y*does have neither summands of type ${P}_{2}^{m}$ nor of type ${P}_{0}^{m}$, and ${w}_{1},\dots ,{w}_{l}\notin \{{m}_{1}-1,\dots ,{m}_{k}-1\}$. For any indecomposable summand of

*Y*and any summand $({P}_{2}^{{s}_{1}}\oplus {P}_{0}^{{s}_{1}-1}),\dots ,({P}_{2}^{{s}_{r}}\oplus {P}_{0}^{{s}_{r}-1}),{P}_{2}^{{m}_{1}},\dots ,{P}_{2}^{{m}_{k}},{P}_{0}^{{w}_{1}},\dots ,{P}_{0}^{{w}_{l}}$ we create the arc diagram using the above table. The arc diagram

*Δ*(

*X*) of

*X*is the union of all these arc diagrams.

Similarly we define the *type* of *X* as the union of partitions *β* and *γ* described in the table above.

*Example:* For the object

*X*is (

*β*,

*γ*) = ((5,5,4,3,3,3,2,2,1,1),(4,3,3,2,2,2,1,1,1)).

Remark.

Note that two non-isomorphic object in *𝒮*_{2} can have the same arc diagram, e.g. ${B}_{2}^{5,2}$ and ${B}_{2}^{5,2}\oplus {P}_{0}^{3}$. However, if we fix partitions *γ* and *β*, the arc diagram uniquely determines the object it is associated with.

We obtain as a consequence:

Theorem 2.2.

For any field *k* and for fixed partitions *γ*⊆*β*, there is a one-to-one correspondence between the set of isomorphism classes of objects in ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ and the set of arc diagrams in ${}_{2}{\mathcal{\mathcal{D}}}_{\gamma}^{\beta}$.

Definition.

We say two invariant subspaces $Y,Z{\in}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ are in *arc order,* in symbols *Y*≤_{arc}*Z*, if *Δ*(*Y*)≤_{arc}*Δ*(*Z*) holds.

## 3. The degeneration order

In this section, following [Citation2, Citation6], we recall classical definitions of three partial orders that are investigated in varieties associated with modules and representations.

Assume that *k* is an algebraically closed field. For natural numbers *a*≤*b*, the representation space of the quiver *Q* corresponding to the dimension vector (*a*,*b*) is the aﬃne variety

*𝕄*

_{a×b}(

*k*) is the set of

*a*×

*b*matrices with coeﬃcients in

*k*and ${\mathbb{M}}_{a}\left(k\right)={\mathbb{M}}_{a\times a}\left(k\right)$. We work with the Zariski topology and with the induced topology for all subsets of ${}^{a}{\mathbb{V}}^{b}$. Fix partitions

*γ*⊆

*β*such that |

*β*| =

*b*and |

*γ*| =

*b*−

*a*. We define ${\mathbb{V}}_{\gamma}^{\beta}$ as the subset of ${}^{a}{\mathbb{V}}^{b}$ consisting of all points $F=({f}_{1},f,{f}_{2})$, such that ${f}_{1}^{a}=0,\phantom{\rule{0.3em}{0ex}}{f}_{2}^{b}=0,\phantom{\rule{0.3em}{0ex}}f{f}_{1}-{f}_{2}f=0$,

*f*

_{2}has the Jordan type

*β*,

*f*has maximal rank and

*Coker f*has Jordan–type

*γ*. Moreover, by ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ denote the subset of ${\mathbb{V}}_{\gamma}^{\beta}$ consisting of all points $({f}_{1},f,{f}_{2})$ such that

*f*

_{1}has nilpotency index less than or equal to 2, that is, the Jordan blocks in the Jordan normal form for

*f*

_{1}have size at most 2. On ${\mathbb{V}}_{\gamma}^{\beta}$ (resp. ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$), there acts the algebraic group

*G*=

*Gl*(

*a*,

*b*) =

*Gl*(

*a*)×

*Gl*(

*b*) via $(g,h)\cdot ({f}_{1},f,{f}_{2})=(g{f}_{1}{g}^{-1},hf{g}^{-1},h{f}_{2}{h}^{-1})$.

For a point $F\in {\mathbb{V}}_{\gamma}^{\beta}$, denote by *𝒪*_{F} the orbit of *F* under action of *G*.

Remark.

Note that the *G*—orbits in ${\mathbb{V}}_{\gamma}^{\beta}$ (resp. in ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$) are in 1−1—correspondence with the isomorphism classes of objects in ${\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ (resp. ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$).

Definition.

Suppose that the points $F,H{\in}_{2}{\mathbb{V}}_{\gamma}^{\beta}\left(k\right)$ correspond to objects $Y=({N}_{\alpha},{N}_{\beta},f)$ and $Z=({N}_{\stackrel{\u0303}{\alpha}},{N}_{\beta},h)$ in ${}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}\left(k\right)$.

The relation *Y*≤_{deg}*Z* is defined to hold if and only if ${\mathcal{\mathcal{O}}}_{H}\subseteq \overline{{\mathcal{\mathcal{O}}}_{F}}$ where $\overline{{\mathcal{\mathcal{O}}}_{F}}$ is the closure of *𝒪*_{F} in ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}\left(k\right)$.

### 3.1. The algebraic orders

Our aim is to prove that the degeneration order and the arc order are equivalent. In the proof, we use the following classical algebraic orders. Let $Y=({N}_{\alpha},{N}_{\beta},f)$ and $Z=({N}_{\stackrel{\u0303}{\alpha}},{N}_{\beta},g)$ be objects in ${\mathcal{\mathcal{S}}}_{\gamma}^{\beta}\left(k\right)$.

The relation

*Y*≤_{ext}*Z*holds if there exist a natural number*s*, objects*M*_{i},*U*_{i},*V*_{i}in*𝒮*(*k*) and short exact sequences $0\to {U}_{i}\to {M}_{i}\to {V}_{i}\to 0$ in*𝒮*(*k*) such that*Y*≅*M*_{1}, ${U}_{i}\oplus {V}_{i}\cong {M}_{i+1}$ for 1≤*i*≤*s*, and*Z*≅*M*_{s+1}.The relation

*Y*≤_{hom}*Z*holds if$\begin{array}{c}\multicolumn{1}{c}{[X,Y]\le [X,Z]}\end{array}$for any object*X*in*𝒮*(*k*). Here we write $[X,Y]={dim}_{k}{Hom}_{\mathcal{\mathcal{S}}}(X,Y)$ for*𝒮*-modules*X*,*Y*. By Kosakowska and Schmidmeier [Citation5, Lemma 3.3], the relation*Y*≤_{hom}*Z*for*Y*,*Z*∈*𝒮*_{2}(*k*) holds if [*X*,*Y*]≤[*X*,*Z*] for every object*X*in*𝒮*_{2}(*k*).

### 3.2. The partial orders are equivalent

Now we are going to present the proof of Theorem 1.5, up to two results about the arc order which are shown in the next section.

We restate the theorem to include statements about arbitrary fields.

Theorem 3.1.

Let *k* be an arbitrary field and assume that $Y,Z{\in}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$. The following conditions are equivalent

*Y*≤_{arc}*Z*,*Y*≤_{ext}*Z*,*Y*≤_{hom}*Z*.

If in addition the field *k* is algebraically closed, then the conditions stated above are equivalent with

*Y*≤_{deg}*Z*.

Proof.

By applying the functor *Hom*_{k}(*X*,−) to the short exact sequences given in the definition of ≤_{ext}, it is easy to see that *Y*≤_{ext}*Z* implies *Y*≤_{hom}*Z*.

If *k* is an algebraically closed field, then by Bongartz [Citation2] and Riedtmann [Citation6] (see [Citation5, Section 3] for details), we have

For any field *k*, the implications

## 4. The category *𝒮*_{2}(*k*)

In this section, we briefly recall properties of the category *𝒮*_{2}(*k*). Denote by ${\mathcal{\mathcal{S}}}_{2}^{n}\left(k\right)$ the full subcategory of *𝒮*_{2}(*k*) of all objects where the operator acts with nilpotency index at most *n* on the ambient space. It is shown in [Citation7, Section 3.2] that ${\mathcal{\mathcal{S}}}_{2}^{n}\left(k\right)$ is an exact Krull–Remak–Schmidt category with Auslander–Reiten sequences.

The Auslander–Reiten quiver for each of the categories ${\mathcal{\mathcal{S}}}_{2}^{n}\left(k\right)$ is obtained by identifying the objects of type ${P}_{1}^{r}$ on the left with their counterparts on the right in the following picture, thus yielding a Moebius band.

For each pair (*X*,*Y*) of indecomposable objects in *𝒮*_{2}(*k*), we determine in the table below the dimension of the *k*-space *Hom*_{𝒮}(*X*,*Y*), see [Citation8, Lemma 4] and [Citation5].

Table

We denote by **1** the *characteristic function* corresponding to the property specified in parentheses.

For the sake of simplicity, we use the notation

*m*≥2. We observe that the notation is consistent with the formulae above.

### 4.1. How operations change the hom spaces

Throughout this section, *Y*,*Z*∈*𝒮*_{2} will be objects of the same partition type (*β*,*γ*). Following [Citation5], we introduce two matrices, the multiplicity matrix *δM* = *δM*(*Y*,*Z*) and the hom matrix *δH* = *δH*(*Y*,*Z*); in each case, the indexing set is the set of isomorphism types of indecomposable objects in *𝒮*_{2}. The matrices are defined as follows:

*μ*

_{X}(

*Z*) denotes the number of direct summands of

*Z*that are isomorphic to

*X*.

We visualize the matrices by indicating the value at *X*∈*ind 𝒮*_{2} in the position of *X* in the Auslander–Reiten quiver for ${\mathcal{\mathcal{S}}}_{2}^{n}$, with *n* large enough. We sketch this quiver as follows: The modules on the top line are the ${P}_{2}^{m}$, those on the second line are the ${P}_{0}^{r}$; the modules in the triangle have type ${B}_{2}^{m,r}$. The modules ${P}_{1}^{r}$ are repeated twice, on the diagonal at the left and on the anti-diagonal at the right.

We determine how the matrices look like for the operation **(E)**on the arc diagram.

Suppose *Z* is obtained from *Y* by a transformation

*m*>

*r*+1. Recall that the arc in the diagram for

*Y*represents the bipicket ${B}_{2}^{m,r}$ which is replaced by the summands ${P}_{1}^{m}$ and ${P}_{1}^{r}$ in

*Z*that give rise to the corresponding poles in the diagram for

*Z*.

Thus, the multiplicity matrix is as follows.

Note that the marked points correspond to a short exact sequence

**(E)**.

Next we determine the Hom matrix $\delta H=\delta H({B}_{2}^{m,r},{P}_{1}^{m}\oplus {P}_{1}^{r})$. By Lemma 5.1, we have $\delta {H}_{{P}_{0}^{\ell}}=0=\delta {H}_{{P}_{2}^{\ell}}$. We compute the remaining numbers using the table given in the first part of Section 4:

Thus, the only indecomposables *X*∈*𝒮*_{2} for which *δ**H*_{X}≠0 are the ${B}_{2}^{\ell ,t}$ where *m*<*ℓ* and *t*≤*r* and ${P}_{1}^{\ell}$ where *ℓ*≤*r*. For each such module *X*, we have *δ**H*_{X} = 1. They lie in the shaded region in the diagram below.

We have seen in [Citation5] how the matrices are changing for the operations **(A)**–**(D)** on the arc diagram. We present these matrices in the following table:

Lemma 4.1.

Let *k* be an arbitrary field and let *Y*,*Z*∈*𝒮*_{2}(*k*) have the same partition type (*β*,*γ*). If *Y*≤_{arc}*Z*, then *Y*≤_{ext}*Z*.

Proof.

Apply the short exact sequences presented above for any operation **(A)–(E)**.

## 5. Some technical facts

We recall the following result from [Citation5, Lemma 4.2].

Lemma 5.1.

Suppose *Y*,*Z*∈*𝒮*_{2} have the same partition type (*β*,*γ*).

The Hom matrix

*δH*(*Y*,*Z*) has zero entry at each position corresponding to a module ${P}_{0}^{m}$, ${P}_{2}^{m}$ where*m*∈*ℕ*and to a module ${P}_{1}^{m}$ for*m*>*β*_{1}.Along each diagonal in the Hom matrix, the entries eventually become constant:

$\begin{array}{c}\multicolumn{1}{c}{\underset{m\to \infty}{lim}\delta H{(Y,Z)}_{{B}_{2}^{m,r}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\delta H{(Y,Z)}_{{P}_{1}^{r}},}\end{array}$more precisely, for each*m*>*β*_{1}and*U*∈*𝒮*_{2}the equality ${[{B}_{2}^{m,r},U]}_{\mathcal{\mathcal{S}}}=\left|\beta \right|+{[{P}_{1}^{r},U]}_{\mathcal{\mathcal{S}}}$ holds, where $\left|\beta \right|={\sum}_{i}{\beta}_{i}$.

The following consequence, which is technical but easy to show, will be used in the proof of Proposition 6.2.

Lemma 5.2.

Suppose that both $Y,Z\in {\mathcal{\mathcal{S}}}_{2}^{n}$ have partition type (*β*,*γ*).

The Hom matrix

*δH*(*Y*,*Z*) might have a nonzero element at the position ${P}_{1}^{1}$ and the remaining nonzero part of*δH*(*Y*,*Z*) is contained in the union of the*τ*-orbits for $X={B}_{2}^{3,1},\dots ,{B}_{2}^{n,1}$; they form a Moebius band, i.e. a quiver of type*ℤ**𝔸*_{n−2}with suitable identifications.For each non-injective $A\in ind\phantom{\rule{0.3em}{0ex}}{\mathcal{\mathcal{S}}}_{2}^{n}$, the entry $\delta {M}_{A}=\delta M{(Y,Z)}_{A}$ in the multiplicity matrix can be read off from the restriction of the Hom matrix to the Moebius band by a formula of type

$\begin{array}{c}\multicolumn{1}{c}{\delta {M}_{A}={H}_{A}+{H}_{C}-\sum _{i}{H}_{{B}_{i}}}\end{array}$where $0\to A\to {\oplus}_{i}{B}_{i}\to C\to 0$ is the Auslander–Reiten sequence starting at*A*and$\begin{array}{c}\multicolumn{1}{c}{{H}_{X}=\{\begin{array}{cc}\delta H{(Y,Z)}_{X}\hfill & \text{if}X\text{isintheMoebiusbanddescribedin1.}\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}}\end{array}$The multiplicity matrix determines the Hom matrix uniquely.

Proof.

The first statement follows from Lemma 5.1.

Let

*A*be a non-injective indecomposable object with Auslander–Reiten sequence 0→*A*→⊕*B*_{i}→*C*→0 in ${\mathcal{\mathcal{S}}}_{2}^{n}$. Then the multiplicity of*A*as a direct summand of*Y*is given by the contravariant defect ${\mu}_{A}\left(Y\right)=[A,Y]+[C,Y]-{\sum}_{i}[{B}_{i},Y]$. The second statement follows from this contravariant defect formula.The Hom matrix can be easily computed using the multiplicity matrix and dimensions of homomorphism spaces given in Section 4.

In the proof of Theorem 6.2, we will use the following lemmas Kosakowska and Schmidmeier [Citation5, Lemmas 4.6 and 4.7].

Lemma 5.3.

Consider the following matrix of integers.

Suppose the following conditions are satisfied.

All entries are nonnegative

The numbers ${h}^{0,0},{h}^{0,1},\dots ,{h}^{0,v}$ are strictly positive

*h*^{0,v+1}= 0For each 0≤

*i*≤*u*, 0≤*j*≤*v*,$\begin{array}{c}\multicolumn{1}{c}{{M}^{ij}={h}^{i,j}+{h}^{i+1,j+1}-{h}^{i,j+1}-{h}^{i+1,j}\le 0}\end{array}$

Then all entries in the parallelogram are positive: *h*^{i,j}>0 for each 0≤*i*≤*u*+1, 0≤*j*≤*v*.

Also the dual version holds:

Lemma 5.4.

Consider the following matrix of integers.

Suppose that in addition to conditions 1 and 2 from the previous lemma also the following are satisfied.

3'.

*h*^{0,−1}= 04'. For each −

*u*−1≤*i*<0, −1≤*j*<*v*,$\begin{array}{c}\multicolumn{1}{c}{{M}^{ij}={h}^{i,j}+{h}^{i+1,j+1}-{h}^{i,j+1}-{h}^{i+1,j}\le 0}\end{array}$

Then all entries in the parallelogram are positive: *h*^{i,j}>0 for each −*u*−1≤*i*≤0, 0≤*j*≤*v*.

## 6. The hom-order implies the arc order

Our aim is to show that *Y*≤_{hom}*Z* implies *Y*≤_{arc}*Z*.

Theorem 6.1.

Suppose the objects *Y* and *Z* in *𝒮*_{2}(*k*) have the same partition type (*β*,*γ*). If *Y*≤_{hom}*Z* holds, then so does *Y*≤_{arc}*Z*.

This theorem follows from the following.

Proposition 6.2.

Suppose *Y* and *Z* have the same partition type (*β*,*γ*), *Y*≤_{hom}*Z* and *Y*≇*Z*. Then there is an operation on the arc diagram for *Z* of type **(A)**, **(B)**, **(C)**, **(D)** or **(E)** which yields a module *Z*^{′} such that

Proof.

The methods from the proof of [Citation5, Proposition 4.5] can be adapted. Here, we have to consider one more case, namely the move of the type **(E)**. For the convenience of the reader, we present the proof with all details. Moreover, after the proof, we will illustrate it by an example.

### 6.1. The proof of Proposition 6.2

**The setup.** We assume that the entries in the Hom-matrix *δH*(*Y*,*Z*) are all nonnegative and that at least one entry is positive.

**The goal.** We show that there is a parallelogram in the Hom-matrix (in the shape of one of the shaded regions in Section 4.1) which satisfies the following two conditions.

(P1) All entries within the parallelogram are strictly positive.

(P2) The two indecomposable modules

*R*and*L*corresponding to the right corner of the parallelogram and to the point just left of the left corner, respectively, occur with higher multiplicity as direct summands of*Z*than as direct summands of*Y*.

**Step 1.** We choose a number *n*>*β*_{1} and work in the category ${\mathcal{\mathcal{S}}}_{2}^{n}$. Recall that by Lemma 5.2 in the Hom matrix, nonzero entries may occur at the position ${P}_{1}^{1}$ and in the orbits of the modules ${B}_{2}^{3,1},\dots ,{B}_{2}^{n,1}$. The region given by those orbits forms a stripe of type *ℤ**𝔸*_{n−2}, with identifications (see Lemma 5.1). For the purpose of this algorithm, we view the Hom matrix as a stripe of type *ℤ**𝔸*_{n}, with the orbits of the ${B}_{2}^{3,1},\dots ,{B}_{2}^{n,1}$ in the center and two orbits of zeros on the boundary (i.e. zeros at the positions ${P}_{0}^{i}$ and ${P}_{2}^{j}$ for all *i*,*j*).

An anti-diagonal in a Hom matrix is any of the ordered sets of positions corresponding to modules: $[{P}_{1}^{1},{P}_{1}^{2},\dots ,{P}_{1}^{n}]$, $[{P}_{0}^{i},{B}_{2}^{i+2,i},{B}_{2}^{i+3,i},\dots ,{B}_{2}^{n,i},{P}_{1}^{i},{B}_{2}^{i,1},{B}_{2}^{i,2},\dots ,{B}_{2}^{i,i-2},{P}_{2}^{i}]$, for *i* = 1,…,*n*−2, where at least one entry at the positions ${B}_{2}^{i,1},{B}_{2}^{i,2},\dots ,{B}_{2}^{i,i-2}$ is nonzero.

Pick a sequence ${h}^{0,0},\dots ,{h}^{0,v}$ of positive entries arranged along an anti-diagonal (different from $[{P}_{1}^{1},{P}_{1}^{2},\dots ,{P}_{1}^{n}]$) in the Hom matrix, such that the neighboring entries on the anti-diagonal, *h*^{0,-1} and *h*^{0,v+1}, are both zero. This choice is possible since the entries in the matrix are nonnegative, the entries at the positions ${P}_{0}^{i}$ and ${P}_{2}^{j}$ are zero, there is at least one positive entry and by Lemma 5.1 there are the same entries at the positions ${P}_{1}^{i}$ and ${B}_{2}^{n,i}$ (this guaranties that there exists at least one nonzero entry at the positions ${B}_{2}^{i,j}$).

**Step 2.** We determine the right corner of the parallelogram. Put *u* = 0.

Consider the sequence ${M}^{u,0},\dots ,{M}^{u,v}$ of entries in the multiplicity matrix in the positions given by the ${h}^{u,0},\dots ,{h}^{u,v}$ in the Hom matrix.

If one of the entries in the sequence ${M}^{u,0},\dots ,{M}^{u,v}$ is positive, put *u*^{″} = *u* and let ${M}^{{u}^{\u2033},{w}^{\u2033}}$ be the first such entry. In this case, $({u}^{\u2033},{w}^{\u2033})$ will be the right corner of the rectangle and *R*, the object corresponding to the position $({u}^{\u2033},{w}^{\u2033})$, will be a summand occurring with higher multiplicity in *Z* than in *Y*. We are done with the second step.

If none of the entries in ${M}^{u,0},\dots ,{M}^{u,v}$ is positive, then by the statement 2 in Lemma 5.2, we obtain that the assumptions of Lemma 5.3 are satisfied, so using this lemma we obtain that all the numbers ${h}^{u+1,0},\dots ,{h}^{u+1,v}$ in the Hom matrix, on the anti-diagonal just under the previous anti-diagonal, are positive. Put *u*: = *u*+1 and proceed with the paragraph under Step 2.

Note that this process terminates: Since in the Hom-matrix, there is zero at the position ${P}_{1}^{n}$, for *u* large enough *h*^{u,0} will correspond to:

the position of ${P}_{0}^{i}$ or ${P}_{2}^{i}$ for some

*i*, orthe position of ${P}_{1}^{1}$ in the Hom matrix.

Hence *h*^{u,0} = 0, in the case (a). In the case (b) using Lemma 5.3, it is easy to deduce that there exists *v*^{′} such that ${M}^{u,{v}^{\prime}}$ is positive, because in the Hom matrix, there is zero at the position ${P}_{0}^{1}$.

**Step 3.** We determine the left corner of the parallelogram. Put *u* = 0 and *v* = *w*^{″}.

Consider the sequence ${M}^{u-1,-1},\dots ,{M}^{u-1,v-1}$ of entries in the multiplicity matrix in the positions just left of the ${h}^{u,0},\dots ,{h}^{u,v}$ in the Hom matrix.

If one of the entries in the sequence ${M}^{u-1,-1},\dots ,{M}^{u-1,v-1}$ is positive, put *u*^{′} = *u*−1 and let ${M}^{{u}^{\prime},{w}^{\prime}}$ be the last such entry. In this case, $({u}^{\prime},{w}^{\prime})$ will be the point just left of the left corner of the rectangle and *L*, the object corresponding to the position $({u}^{\prime},{w}^{\prime})$ will be a summand occurring with higher multiplicity in *Z* than in *Y*. We are done with the third step.

If none of the entries in ${M}^{u-1,-1},\dots ,{M}^{u-1,v-1}$ is positive, then by Lemma 5.2, we obtain that the assumptions of Lemma 5.4 are satisfied, so using this lemma, we obtain that all the numbers ${h}^{u-1,0},\dots ,{h}^{u-1,v}$ in the Hom matrix, on the anti-diagonal just above the previous anti-diagonal, are positive. Put *u*: = *u*−1 and proceed with the paragraph under Step 3.

Note that this process terminates: By Lemma 5.1, in the Hom matrix, the entries at the positions ${P}_{1}^{i}$ and ${B}_{2}^{n,i}$ are equal for all *i*. It follows that the entries *h*^{i,0} do not lie on the anti-diagonal $[{P}_{1}^{1},{P}_{1}^{2},\dots ,{P}_{1}^{n}]$. Therefore, for −*u* large enough, the position corresponding to *h* will be a point on the boundary of the Hom matrix, so *h*^{u,0} = 0.

## 7. Conclusion

Let us denote by *U* and *D* the vertices corresponding to points just upperleft to the top corner and lowerleft to the bottom corner of the parallelogram we constructed, respectively. All possibilities of the vertices *L*, *D*, *U* and *D* are presented in the following table:

Table

In each case, the parallelogram marked off by *L*,*R*,*U* and *D* is of one of the types **(A)**, **(B)**, **(C)**, **(D)** or **(E)** described in Section 4.1. Let *Z*≃*Z*_{0}⊕*L*⊕*R*. We put ${Z}^{\prime}={Z}_{0}\oplus U\oplus D$ (or ${Z}^{\prime}={Z}_{0}\oplus U$ if *D* is undefined). By this construction, ${Z}^{\prime}{\le}_{arc}Z$. We obtain the arc diagram of *Z*^{′} from the arc diagram of *Z* by a single move of type (*A*), (*B*), (*C*), (*D*) or (*E*), so the hom matrix was changed in the way described in Section 4.1, we get

Since the entries for *δH*(*Y*,*Z*) within the parallelogram are all positive, the matrix *δH*(*Y*,*Z*^{′}) is nonnegative and hence

Proof (Proof of Theorem 6.1).

Assume that *Y*≤_{hom}*Z*. Since each application of Proposition 6.2 reduces the number of crossings in the arc diagram, a finite number of steps suﬃces to produce a sequence of modules *Z*, *Z*^{′}, *Z*^{″}, …, *Z*^{(m)} = *Y* such that

### 7.1. An example

In this section, we present an example illustrating the proof of Proposition 6.2. The arc diagrams

represent the modules $Y={B}_{2}^{7,3}\oplus {B}_{2}^{6,2}\oplus {P}_{2}^{5}\oplus {P}_{0}^{4}\oplus {P}_{1}^{1}$ and $Z={B}_{2}^{6,3}\oplus {B}_{2}^{5,1}\oplus {P}_{1}^{7}\oplus {P}_{1}^{4}\oplus {P}_{1}^{2}$.We first compute the Hom-matrix to verify that *Y*≤_{hom}*Z*. In this matrix, the entries on the first diagonal represent the numbers $\delta {H}_{{B}_{2}^{3,1}},\delta {H}_{{B}_{2}^{4,1}},\dots $. By Lemma 5.1, there are also zeros along the two top rows, which we indicate by solid lines.

We see that *Y*≤_{hom}*Z*. We show that *Y*≤_{arc}*Z*.

Following Steps 1, 2, 3 in the proof of Proposition 6.2, we choose the smaller parallelogram in the Hom-matrix, as indicated.

The corresponding operation of type **(A)** replaces the bipickets ${B}_{2}^{5,1}$ and ${B}_{2}^{6,3}$ in *Z* (representing arcs from 5 to 1 and 6 to 3, respectively) by bipickets ${B}_{2}^{6,1}$ and ${B}_{2}^{5,3}$ in *Z*^{′}, so *Z*^{′} is given by the following arc diagram. We also indicate the new Hom-matrix.

Following Steps 1, 2, 3 in the proof of Proposition 6.2, we choose a parallelogram (alternatively, we can choose the parallelogram given by the upper diagonal in the region marked by the 1’s) and perform the operation indicated (this is the move **(B)**): Replace in *Z*^{′} the bipicket ${B}_{2}^{6,1}$ and picket ${P}_{1}^{2}$ by bipicket ${B}_{2}^{6,2}$ and picket ${P}_{1}^{1}$ in *Z*^{″}. Arc diagram and Hom-matrix are as follows.

The new parallelogram corresponds to move of a type **(E)**, so we replace the pickets ${P}_{1}^{7}$ and ${P}_{1}^{1}$ in *Z*^{″} by bipicket ${B}_{2}^{7,1}$ in *Z*^{‴}. Arc diagram and Hom-matrix are as follows.

The parallelogram gives an operation of type **(B)**, namely we have to replace the bipicket ${B}_{2}^{5,3}$ and the picket ${P}_{1}^{4}$ in *Z*^{‴} by pickets ${P}_{2}^{5}$, ${P}_{0}^{4}$ and ${P}_{1}^{3}$ in *Z*^{IV} (note that the first two pickets represent an arc from 5 to 4, the last is a pole at 3).

Finally, an operation of type **(B)** reduces the Hom-matrix to zero and yields the module *Y*: We replace the bipicket ${B}_{2}^{7,1}$ and the picket ${P}_{1}^{3}$ in *Z*^{IV} by the bipicket ${B}_{2}^{7,3}$ and the picket ${P}_{1}^{1}$ for *Y*. We are done:

## 8. The poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$

We discuss the maximal and minimal elements in the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$. First, we give an example.

### 8.1. An example

Example.

We present the Hasse diagram of the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{(3,2,1,1)}^{(4,3,3,2,1)},{\le}_{arc})$. The numbers given on the right-hand side are the dimensions of *𝕍*_{Δ} for diagrams *Δ* in the same row. These dimensions are computed using Theorem 1.4.

### 8.2. Maximal and minimal elements

We fix two partition *γ*⊆*β*, the skew diagram *β*∖*γ* (i.e. the diagram that is obtained from the Young diagram of *β* by removing boxes that are in the Young diagram of *γ*) is said to be a vertical strip if ${\beta}_{i}\le {\gamma}_{i}+1$ holds for all *i*.

Proposition 8.1.

Let *γ*,*β* be partitions.

In the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$, there exists exactly one maximal element

*M*. Its arc diagram has only poles and loops.If

*β*∖*γ*is vertical strip, then in the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$, there are exactly ${c}_{\alpha ,\gamma}^{\beta}$ minimal elements, where ${c}_{\alpha ,\gamma}^{\beta}$ is the Littlewood–Richardson coeﬃcient,*α*= (2,2,…,2), if |*β*|∖|*γ*| is even and*α*= (2,2,…,2,1), if |*β*|∖|*γ*| is odd. Their arc diagrams have no crossings, have only arcs, at most one pole, and may contain loops.

Proof.

We start with the observation that any object with an arc can be modified by the move

**(E)**. In this way, we obtain a bigger object in our poset. It follows that each maximal element in the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$ has no arc. The location of loops in the arc diagram is determined by the partitions*β*and*γ*, so there exists exactly one maximal object, it has the required properties.If the tableau is a vertical strip, then in the arc diagram, there is no double pole at any given point. Therefore, if there are at least two poles in an arc diagram, we can choose two poles in the points

*m*and*r*such that*m*≠*r*. It follows that, applying the move**(E)**, we can obtain a smaller object in our poset. We conclude that each minimal element in the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$ has at most one pole and hence correspond to the partition*α*= (2,2,…,2), if |*β*|∖|*γ*| is even and*α*= (2,2,…,2,1), if |*β*|∖|*γ*| is odd. By Kosakowska and Schmidmeier [Citation5, Theorem 5.7], in the poset ${(}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta},{\le}_{arc})$, there are exactly ${c}_{\alpha ,\gamma}^{\beta}$ minimal elements, where ${c}_{\alpha ,\gamma}^{\beta}$ is the Littlewood–Richardson coeﬃcient,*α*= (2,2,…,2), if |*β*|∖|*γ*| is even and*α*= (2,2,…,2,1), if |*β*|∖|*γ*| is odd.

The following example shows that the assumption about the vertical strip in Statement 2 of Proposition 8.1 is necessary.

Example.

The poset ${}_{2}{\mathcal{\mathcal{S}}}_{(2,2,1)}^{(3,3,2,1)}$ looks as follows.

## 9. Two counterexamples

In the paper we fixed two partitions, *β* and *γ* and considered the variety ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$. We present two examples which show that similar results cannot be obtained if we fix two other partitions, i.e., either *α* and *γ* or *α* and *β*.

### 9.1. Fixing partitions *α* and *β*

Example.

Let *α*,*β* be partitions such that *α*_{1}≤2. We consider the full subcategory ${}_{\alpha}{\mathcal{\mathcal{S}}}^{\beta}$ of *𝒮*_{2} consisting of all objects $({N}_{\alpha},{N}_{\beta},f)$. For *α* = (2) and *β* = (*m*,*r*), where *m*>*r*+1 we define two objects $X={B}_{2}^{m,r}$ and $Y={P}_{0}^{m}\oplus {P}_{2}^{r}$ in ${}_{\alpha}{\mathcal{\mathcal{S}}}^{\beta}$. Using the table from Section 6, we compute that

The object *X* is indecomposable, so it is not true that:

### 9.2. Fixing partitions *α* and *γ*

Example.

Let *α*,*γ* be partitions such that *α*_{1}≤2. We consider the full subcategory ${}_{\alpha}{\mathcal{\mathcal{S}}}_{\gamma}$ of *𝒮*_{2} consisting of all objects $({N}_{\alpha},{N}_{\beta},f)$, such that *Coker f*≃*N*_{γ}. For *α* = (2) and *γ* = (*m*−1,*r*−1), where *m*>*r*+1, we define two objects $X={B}_{2}^{m,r}$ and $Y={P}_{2}^{m+1}\oplus {P}_{0}^{r-1}$ in ${}_{\alpha}{\mathcal{\mathcal{S}}}_{\gamma}$. Using the table from section 6 we compute that

The object *X* is indecomposable, so it is not true that:

## 10. The dimensions of the orbits

In this section, we present the proof of Theorem 1.4.

### 10.1. Previous results

We briefly recall notation and a result from [Citation5].

We consider the aﬃne variety ${\mathbb{H}}_{\alpha}^{\beta}\left(k\right)={Hom}_{k}\left({N}_{\alpha}\right(k),{N}_{\beta}(k\left)\right)$ (consisting of all |*β*|×|*α*|-matrices with coeﬃcients in *k*). On ${\mathbb{H}}_{\alpha}^{\beta}\left(k\right)$, we consider the Zariski topology and on all subsets of ${\mathbb{H}}_{\alpha}^{\beta}\left(k\right)$, we work with the induced topology. Let ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$ be the subset of ${\mathbb{H}}_{\alpha}^{\beta}\left(k\right)$ consisting of all matrices that define a monomorphism $f:{N}_{\alpha}\to {N}_{\beta}$ in the category *𝒩*(*k*) with *Coker f*≃*N*_{γ}. On ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$ there acts the algebraic group ${Aut}_{\mathcal{\mathcal{N}}}\left({N}_{\alpha}\right(k\left)\right)\times {Aut}_{\mathcal{\mathcal{N}}}\left({N}_{\beta}\right(k\left)\right)$ via (*g*,*h*)⋅*f* = *hf**g*^{−1}. The orbits of this action correspond bijectively to isomorphism classes of objects in ${\mathcal{\mathcal{S}}}_{\alpha ,\gamma}^{\beta}$. For a map $f:{N}_{\alpha}\left(k\right)\to {N}_{\beta}\left(k\right)$, we denote by *G*_{f} the orbit of *f* in ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$.

We recall a theorem which yields information about the dimensions of orbits in ${\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$. This theorem was proved [Citation5, Theorem 1.3].

Theorem 10.1.

Let *k* be an algebraically closed field, and let *α*,*β*,*γ* be partitions. Suppose the arc diagram *Δ* of an invariant subspace $Y=({N}_{\alpha},{N}_{\beta},f)\in {\mathbb{V}}_{\alpha ,\gamma}^{\beta}\left(k\right)$ has *x*(*Δ*) crossings. Then

*a*

_{α}(

*q*) which counts the automorphisms of ${N}_{\alpha}\left({\mathbb{F}}_{q}\right)$.

### 10.2. The dimensions of the orbits in ${}_{2}{\mathbb{V}}_{\gamma}^{\beta}$

For a group *G* acting on a variety *X* and for *y*∈*X*, we denote by *G*.*y* the stabilizer of *y* in *G*.

Proof (Proof of Theorem 1.4).

Let $Y=({N}_{\alpha},{N}_{\beta},f){\in}_{2}{\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$ and let $F=({\phi}_{\alpha},f,{\phi}_{\beta}){\in}_{2}{\mathbb{V}}_{\gamma}^{\beta}$ correspond to *Y*. Denote by *Δ* the arc diagram corresponding to *Y* and assume that *Δ* has *x*(*Δ*) crossings.

It is well known that

*a*= |

*α*| and

*b*= |

*β*|. Moreover,

Similarly,

Substituting 10.3 into 10.2, we obtain

Combining 10.5 with 10.6 yields

Finally, we obtain from Theorem 10.1

### 10.3. The effect of a single arc move

We finish by presenting an example which shows that there is no upper bound on the change of the orbit dimension resulting from a single arc move.

Example.

Let *β* = (3,1,1,…,1),*γ* = (2). Then in the category ${\mathcal{\mathcal{S}}}_{\gamma}^{\beta}$, we have only two objects $X={B}_{2}^{3,1}\oplus {\oplus}_{i=1}^{n}{P}_{1}^{1}$ and $Y={P}_{1}^{3}\oplus {\oplus}_{i=1}^{n+1}{P}_{1}^{1}$, up to isomorphism. They have the following arc diagrams:

Note that we can obtain the diagram *Δ*(*Y*) from the diagram *Δ*(*X*) by a single move **(E)**. Using Theorem 1.4, we compute

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