Algorithmic Aspects of Semistability of Quiver Representations

Here, we present the formal definition of a quiver representation. We follow the terminologies in [14, 6]. Let $Q=(Q_0,Q_1)$ be a quiver with a vertex set $Q_0$ and an arc set $Q_1$. In this paper, we consider only acyclic quivers except otherwise stated. For each arc $a\in Q_1$, we denote the tail and head of $a$ by $ta$ and $ha$, respectively. A representation $V$ of $Q$ consists of complex vector spaces $V(i)$ for vertex $i\in Q_0$ and linear maps $V(a):V(ta)\to V(ha)$ for arc $a\in Q_1$. A subrepresentation $W$ of $V$ is a representation of the same quiver such that $W(i)\le V(i)$ for $i\in Q_0$, and $W(a)={V(a)|}_{W(ta)}$ and $\mathrm{im}W(a)\le W(ha)$ for $a\in Q_1$. The vector of dimensions of $V(i)$ is called the dimension vector of the representation, denoted by $\underset{¯}{\dim }V$. We call the vector space of all representations of $Q$ with the dimension vector $\alpha$ the representation space of $Q$ with dimension vector $\alpha$, which we denote by $\mathrm{Rep}(Q,\alpha )$. After fixing the dimension vector $\alpha$ and a basis of each $V(i)$, we can represent $V(a)$ as an $\alpha (ha)\times \alpha (ta)$ matrix. Therefore, the representation space can be identified as

where $\mathrm{Mat}(m,n)$ denotes the space of $m\times n$ complex matrices.

Fix a dimension vector $\alpha$. Let

where $\mathrm{GL}(n)$ denotes the general linear group of degree $n$. Then, $\mathrm{GL}(Q,\alpha )$ acts on the representation space by a change of basis:

Note that this is a left action, i.e., $(gh)\cdot V=g\cdot (h\cdot V)$ for $g,h\in \mathrm{GL}(Q,\alpha )$. We say that a representation $V$ is semistable under the $\mathrm{GL}(Q,\alpha )$-action if the orbit closure of $V$ does not contain the origin, i.e.,

Otherwise, $V$ is said to be unstable. The set of all unstable representations is called the null-cone of the $\mathrm{GL}(Q,\alpha )$-action. It is easy to see that any representation is unstable under the $\mathrm{GL}(Q,\alpha )$-action if $Q$ is acyclic.¹¹1The readers may wonder whether we can check the semistability (under $\mathrm{GL}(Q,\alpha )$-action) of quiver representations of cyclic quivers. We will address this point later.

However, the semistability of quivers under subgroups of $\mathrm{GL}(Q,\alpha )$ turns out to be more intricate. Let $\sigma \in {\mathbb{Z}}^{Q_0}$ be an integer vector on $Q_0$, which we call a weight. Let ${\chi }_{\sigma }$ be the corresponding multiplicative character of $\mathrm{GL}(Q,\alpha )$, i.e.,

Note that ${\chi }_{\sigma }$ is a one-dimensional representation of $\mathrm{GL}(Q,\alpha )$; $\mathrm{GL}(Q,\alpha )$ acts on $ℂ$ by $g\cdot x≔{\chi }_{\sigma }(g)x$. A representation $V$ is said to be $\sigma$-semistable if the orbit closure of $(V,1)\in \mathrm{Rep}(Q,\alpha )\oplus ℂ$ under the $\mathrm{GL}(Q,\alpha )$ action does not contain the origin, i.e.,

It turns out that checking the $\sigma$-semistability of a quiver representation includes operator scaling (noncommutative rank computation) and the membership problem of the BL polytopes. We will see these examples in the following sections.

Our first result is a deterministic algorithm that, given a quiver representation $V$ and weight $\sigma$, decides whether the representation is $\sigma$-semistable in time polynomial in the bit complexity of $V$ and absolute values of the entries of $\sigma$. Let $\alpha (Q_0)≔{\sum }_{i\in Q_0}\alpha (i)$.

This improves the previous result [30] which runs in time polynomial in the number of paths in $Q$, which can be exponential in the size of $Q$. Furthermore, if the absolute value of the entries of $\sigma$ is constant, our algorithm runs in polynomial time. This includes the known result for operator scaling [24].

King [32] showed the following characterization of $\sigma$-semistability, which is known as King’s criterion. Let $\sigma (\alpha )≔{\sum }_{i\in Q_0}\sigma (i)\alpha (i)$ for a dimension vector $\alpha$. Then, a representation $V$ is $\sigma$-semistable if and only if $\sigma (\underset{¯}{\dim }V)=0$ and $\sigma (\underset{¯}{\dim }W)\le 0$ for any subrepresentation $W$ of $V$.

King’s criterion is a common generalization of the noncommutative rank (nc-rank) computation and membership problem of BL polytopes.

We study the following optimization problem: given a quiver representation $V$ and weight $\sigma$, find a subrepresentation $W$ of $V$ that maximizes $\sigma (\underset{¯}{\dim }W)$. In the case of the nc-rank, such a subrepresentation corresponds to a subspace $U$ that maximizes $dimU-dim({\sum }_{a=1}^m{A}_{i}U)$. Such a subspace is called a shrunk subspace and can be regarded as a certificate of the nc-rank [31, 19]. In the case of the BL polytopes, the problem corresponds to separation for the BL polytope [23].

King’s criterion can be regarded as maximizing a modular function over the modular lattice of subrepresentations. For any subrepresentations $W_1,W_2$ of $V$, define the subrepresentations $W_1+W_2$ and $W_1\cap W_2$ as follows. For each $i\in Q_0$,

where the addition and intersection on the right-hand side are those of the vector spaces. Furthermore, the linear map of $a\in Q_1$ in $W_1+W_2$ (resp. $W_1\cap W_2$) is defined as the restriction of $V(a)$ to $(W_1+W_2)(ta)$ (resp. $(W_1\cap W_2)(ta)$). Then, $W_1+W_2$ and $W_1\cap W_2$ are indeed subrepresentations of $V$. Thus, the subrepresentations of $V$ form a modular lattice. Furthermore, the function $f(W)≔\sigma (\underset{¯}{\dim }W)$ is a modular function, i.e., for any subrepresentations $W_1,W_2$ of $V$,

Thus, subrepresentations maximizing $f$ form a sublattice, and there is a unique inclusion-wise minimum maximizer. Our second result is a deterministic algorithm to find such a maximizer of King’s criterion.

King’s criterion was originally proved using the Hilbert-Mumford criterion (see, e.g., [14, Sections 9.6 and 9.8]), a fundamental result in the GIT.

We use the algorithm for finding the maximizers of King’s criterion to devise an algorithm for finding the Harder-Narasimhan (HN) filtration [27, 28] of a quiver representation. Roughly speaking, the HN-filtration decomposes a quiver representation into the direct sum of smaller representations.

More precisely, let $\sigma \in {\mathbb{Z}}^{Q_0}$ be a weight and $\tau \in {\mathbb{Z}}_{+}^{Q_0}$ a strictly monotone weight, i.e., a nonnegative weight such that $\tau (\underset{¯}{\dim }W)>0$ if $W\ne \{0\}$. Here, $\{0\}$ denotes the zero representation, which is the representation whose dimension vector is the zero vector. We define the slope of a quiver nonzero representation $V$ as $\mu (V)=\sigma (\underset{¯}{\dim }V)/\tau (\underset{¯}{\dim }V)$. We say that $V$ is $\mu$-semistable²²2It is also called $(\sigma :\tau )$-semistability in the literature. if $\mu (W)\le \mu (V)$ for any nonzero subrepresentation $W$ of $V$. The HN-filtration theorem states that for any quiver representation $V$, there exists a unique filtration such that (i) $\mu (W_i/W_{i-1})>\mu (W_{i+1}/W_i)$ for $i\in [k-1]$ and (ii) $W_i/W_{i-1}$ is $\mu$-semistable. Here, means that $W_i$ is a subrepresentation of $W_{i+1}$ with $W_i\ne W_{i+1}$, and $W_i/W_{i-1}$ is a representation of $Q$ such that $(W_i/W_{i-1})(j)$ is the quotient space $W_i(j)/W_{i-1}(j)$ for $j\in Q_0$ and $(W_i/W_{i-1})(a)$ is the corresponding quotient linear map of $W_i(a)$ for $a\in Q_1$. We note that semistability with respect to a slope can be reduced to that for a weight.

Our third result is a deterministic algorithm for finding the HN-filtration.

This result improves a recent result [7] which runs in time polynomial in the number of paths in $Q$.

Recently, Hirai and Sakabe [29] introduced the coarse Dulmage-Mendelsohn (DM) decomposition of a linear matrix, generalizing the classic DM-decomposition of a bipartite graph. They showed that a natural gradient flow of operator scaling converges to the coarse DM-decomposition. However, their result did not provide an efficient algorithm to compute the coarse DM-decomposition because the gradient flow may take exponential time to converge. We show that the coarse DM-decomposition is indeed a special case of the HN-filtration for the generalized Kronecker quiver. Since the absolute values of the weights involved for this special case are polynomially bounded, our algorithm finds the coarse DM-decomposition in polynomial time.

Motivated by King’s criterion, we investigate a polyhedral cone that is the set of $\sigma \in {ℝ}^{Q_0}$ satisfying $\sigma (\underset{¯}{\dim }V)=0$ and $\sigma (\underset{¯}{\dim }W)\le 0$ for any subrepresentation $W$ of $V$. Since the number of distinct $\underset{¯}{\dim }W$ is finite, the above linear system is also finite and hence defines a polyhedral cone. We call the polyhedral cone the King cone of a quiver representation $V$. Interestingly, the King cone is related to the cone of feasible flows in network-flow problems.

Let us first consider the easiest case. If $V$ is a quiver representation with $dimV(i)=1$ for all $i\in Q_0$, then King’s criterion characterizes the existence of a nonnegative flow $\phi$ on the support quiver with the boundary condition $\partial \phi =\sigma$. To state it more precisely, we introduce notation on flows. Let $Q=(Q_0,Q_1)$ be a quiver (or a directed graph). For a vertex subset $X\subseteq Q_0$, let $\mathrm{Out}(X)$ denote the set of outgoing arcs from $X$, i.e., $\mathrm{Out}(X)≔\{a=(i,j):i\in X,j\in Q_0\setminus X\}$. Similarly, let $\mathrm{In}(X)$ denote the set of incoming arcs to $X$. If $X=\{i\}$, we abbreviate $\mathrm{Out}(\{i\})$ and $\mathrm{In}(\{i\})$ as $\mathrm{Out}(i)$ and $\mathrm{In}(i)$, respectively. If $\mathrm{Out}(X)=\mathrm{\varnothing }$, then $X$ is called a lower set of $Q$. For a flow $\phi \in {ℝ}^{Q_1}$ on $Q$, its boundary $\partial \phi \in {ℝ}^{Q_0}$ is defined by $\partial \phi (i)≔{\sum }_{a\in \mathrm{Out}(i)}\phi (a)-{\sum }_{a^{\prime }\in \mathrm{In}(i)}\phi (a^{\prime })$ for $i\in Q_0$.

Let us return to the $\sigma$-semistability of a representation $V$ with $dimV(i)=1$ for all $i\in Q_0$. In this case, $V(a)\in ℂ$ for each arc $a\in Q_1$, and a subrepresentation $W$ of $V$ can be identified with a vertex subset $X\subseteq Q_0$. By the definition of a subrepresentation, if $i\in X$ and $W(a)\ne 0$ for $a=(i,j)\in Q_1$, then $j\in X$. This implies that $X$ is a lower set in the support quiver of $V$, namely, the subquiver of $Q$ whose arcs are $a\in Q_1$ with $V(a)\ne 0$. Therefore, King’s criterion is equivalent to the purely combinatorial condition that $\sigma (Q_0)=0$ and $\sigma (X)\le 0$ for each lower set $X$ of $Q$, which characterizes the existence of a nonnegative flow $\phi$ on the support quiver with the boundary condition $\partial \phi =\sigma$ by Gale’s theorem [22] (see, e.g., [33, Theorem 9.2]).

By generalizing the above observation, we show that if $V$ is a rank-one representation of $Q$, i.e., $V(a)$ is a rank-one matrix for each $a\in Q_1$, then

• $\mathrm{■}$

  King’s criterion can be rephrased as a purely combinatorial condition with respect to the linear matroids arising from the rank-one matrices $V(a)$ of $Q$, and

• $\mathrm{■}$

  the rephrased condition above can be further viewed as the feasibility condition of a network flow-type problem called submodular flow.

That is, the King cone is representable as the feasibility of a certain instance of the submodular flow problem. This enables us to decide the $\sigma$-semistability for rank-one representations in strongly polynomial time.

This theorem recovers the following well-known results when applied to the generalized Kronecker quiver and a star quiver.

• $\mathrm{■}$

  A rank-one linear matrix ${\sum }_{k=1}^m{x}_k{v}_k{f}_k$ is (nc-)nonsingular (where $v_k$ is a column vector and $f_k$ a row vector) if and only if the linear matroids of $(f_k:k\in [m])$ and $(v_k:k\in [m])$ have a common base [34].

• $\mathrm{■}$

  If each linear operator $B_i=f_i$ is of rank-one for $i\in [m]$ (where $f_i$ is a row vector), the BL polytope coincides with the base polytope of the linear matroid of $(f_i:i\in [m])$ [2].

Thus far, we have considered the $\sigma$-semistability of acyclic quivers. As a complementary result, we show that the semistability of cyclic quivers under the $\mathrm{GL}(Q,\alpha )$-action can be efficiently reduced to noncommutative polynomial identity testing. In particular, we show that the polynomial can be represented by an algebraic branching program (ABP). This yields a deterministic algorithm for deciding the semistability of general quivers because noncommutative polynomial identity testing for ABP can be conducted in deterministic polynomial time [36]. Note that [6] devised another deterministic algorithm for the problem with their framework of noncommutative optimization, which is built upon deep results in various areas of mathematics. See also the discussion in related work.³³3After submitting the first version of this paper, an anonymous reviewer pointed out that this result for general quivers was sketched in [35]. See Theorem 10.8 and the last paragraph of Section 10.2 in [35]. At a very high level, our algorithm and Mulmuley’s results follow a similar strategy, although Mulmuley’s result requires several deep algebro-geometric backgrounds. We believe that our proof is more explicit and elementary, and hence, is worthy to be presented here for completeness.

In this subsection, we outline our techniques.

Existing studies on algorithms for quiver semistability have focused on bipartite quivers [11, 9, 10, 18]. We remark that semistability in bipartite quivers is essentially operator scaling with a block structure. In bipartite quivers, the number of paths is equal to the number of arcs; hence, a weak running time was sufficient in the previous studies. [30, 7] studied general acyclic quivers. They first used the reduction of [13] to the generalized Kronecker quiver and applied the nc-rank algorithm [31] in a black box manner. Hence, their algorithm runs in time polynomial in the number of paths, which can be exponential.

Several polyhedral cones associated with quiver representations have been studied in the literature [8, 37]. The moment cone of a quiver $Q$ and a dimension vector $\alpha$ is the polyhedral cone generated by the highest weights of the representations of $Q$ with the dimension vector $\alpha$. The membership of the moment cone can be decided in strongly polynomial time for bipartite quivers [8] and even general acyclic quivers [37]. Another polyhedral cone is the conic hull of weight $\sigma \in {\mathbb{Z}}^{Q_0}$ such that there exists a nonzero semi-invariant polynomial in $\mathrm{Rep}(Q,\alpha )$ with weight $\sigma$. The membership problem of this cone is called the generic semistability problem [8]. By definition, $\sigma$ is in this cone if and only if there exists a generic $\sigma$-semistable representation of $Q$, hence the name. To the best of our knowledge, the generic semistability problem remains open for general acyclic quivers.

The semistability of quiver representations is a special case of semistability in the GIT. In the most abstract setting, GIT studies group actions on algebraic varieties. We say that a point in the variety is semistable if its orbit closure does not contain the origin. Bürgisser et al. [6] proposed a framework of noncommutative optimization to devise algorithms for GIT problems in the general setting. Although noncommutative optimization is a broad and general framework, it does not always provide efficient algorithms for all GIT problems. Currently, most of the known tractable problems originate from a family of operator scaling problems, which are also contained in the semistability of quiver representations. Furthermore, it is built upon deep results in various areas of mathematics, such as algebraic geometry, Lie algebra, and representation theory, rendering it difficult for non-experts to understand. Another conceptual contribution of this paper is identifying the semistability of quiver representations as a useful subclass of GIT. The semistability of quiver representations is rich enough to capture various interesting problems in the literature while also supporting the design efficient algorithms using elementary techniques.

The rest of this paper is organized as follows. Section 2 introduces the necessary background and notation of quiver representations, operator scaling, and noncommutative rank. Section 3 presents our algorithms for deciding the $\sigma$-semistability and finding maximizers of King’s criterion. Due to the space limitation, the other results are placed in the full version.

A linear map $\mathrm{\Phi }:\mathrm{Mat}(n)\to \mathrm{Mat}(m)$ is said to be completely positive, or CP for short, if $\mathrm{\Phi }(X)={\sum }_{\mathrm{\ell }=1}^k{A}_{\mathrm{\ell }}X{A}_{\mathrm{\ell }}^{†}$ for some $A_{\mathrm{\ell }}\in \mathrm{Mat}(m,n)$. These $A_{\mathrm{\ell }}$ are called the Kraus operators of $\mathrm{\Phi }$. The dual map ${\mathrm{\Phi }}^{\ast }:\mathrm{Mat}(m)\to \mathrm{Mat}(n)$ of $\mathrm{\Phi }$ is defined by ${\mathrm{\Phi }}^{\ast }(X)={\sum }_{\mathrm{\ell }=1}^k{A}_{\mathrm{\ell }}^{†}X{A}_{\mathrm{\ell }}$. For $(g,h)\in \mathrm{GL}(m)\times \mathrm{GL}(n)$, we define the scaling ${\mathrm{\Phi }}_{g,h}$ of $\mathrm{\Phi }$ by

If $m=n$, the CP map is said to be square.

Let $\mathrm{\Phi }:\mathrm{Mat}(n)\to \mathrm{Mat}(n)$ be a square CP map. Let $\mathrm{𝖽𝗌}(\mathrm{\Phi })≔{\parallel \mathrm{\Phi }(I)-I\parallel }_{\mathrm{F}}^2+{\parallel {\mathrm{\Phi }}^{\ast }(I)-I\parallel }_{\mathrm{F}}^2$, where ${\parallel \cdot \parallel }_{\mathrm{F}}$ denotes the Frobenius norm. Then, $\mathrm{\Phi }$ is said to be approximately scalable if for any $\epsilon >0$, there exists $(g,h)\in \mathrm{GL}(n)\times \mathrm{GL}(n)$ such that $\mathrm{𝖽𝗌}({\mathrm{\Phi }}_{g,h})\le \epsilon$. The goal of the operator scaling problem is to decide whether a given CP map is approximately scalable or not.

Operator scaling is closely related to the noncommutative rank (nc-rank) of linear matrices. An $m\times n$ symbolic matrix $A$ is called a linear matrix if $A={\sum }_{\mathrm{\ell }=1}^k{x}_{\mathrm{\ell }}{A}_{\mathrm{\ell }}$ for indeterminates $x_{\mathrm{\ell }}$ and matrices $A_{\mathrm{\ell }}\in \mathrm{Mat}(m,n)$. Sometimes, it is more convenient to see a linear matrix as a matrix space spanned by $A_1,\mathrm{\dots },A_k$. We denote the corresponding matrix space of a linear matrix $A$ by $𝒜$. For a subspace $U\le {ℂ}^n$, let $𝒜U≔⟨\{Au:A\in 𝒜,u\in U\}⟩,$ which is a subspace of ${ℂ}^m$. The nc-rank of a linear matrix $A$ (denoted by $\mathrm{nc}-\mathrm{rank}A$) is defined by

A square linear matrix $A$ is said to be nc-nonsingular if $\mathrm{nc}-\mathrm{rank}A=n$. Informally speaking, $\mathrm{nc}-\mathrm{rank}A$ is the rank of $A$, where the indeterminates $x_i$ are pairwise noncommutative, i.e., $x_i{x}_j\ne x_j{x}_i$ for $i\ne j$. See [12, 15] for more details. A pair $(L,R)$ of subspaces $L\le {ℂ}^m$ and $R\le {ℂ}^n$ is called an independent subspace if $L\cap 𝒜R=\{0\}$. Over the complex field, $(L,R)$ is independent if and only if $\mathrm{tr}({\mathrm{\Pi }}_L\mathrm{\Phi }({\mathrm{\Pi }}_R))=0$, where ${\mathrm{\Pi }}_L$ denotes the orthogonal projection matrix onto $L$. Then,

An independent subspace $(L,R)$ is said to be maximum if $dimL+dimR$ is maximum. In particular, a square linear matrix $A$ is nc-nonsingular if and only if $dimL+dimR\le n$ for any independent subspace $(L,R)$. Gurvits’ theorem [25] states that a square CP map $\mathrm{\Phi }$ with the Kraus operator $A_{\mathrm{\ell }}$ ($\mathrm{\ell }\in [k]$) is approximately scalable if and only if the linear matrix ${\sum }_{\mathrm{\ell }=1}^k{x}_{\mathrm{\ell }}{A}_{\mathrm{\ell }}$ is nc-nonsingular.

Operator scaling with specified marginals is a generalization of operator scaling. Let $(b^{+},b^{-})\in {ℝ}_{+}^n\times {ℝ}_{+}^m$ be a pair of nonincreasing nonnegative vectors, which we call the target marginals. We say that a CP map $\mathrm{\Phi }:\mathrm{Mat}(n)\to \mathrm{Mat}(m)$ is approximately scalable to the target marginals $(b^{+},b^{-})$ if there exist nonsingular upper triangular matrices $(g,h)\in \mathrm{B}(m)\times \mathrm{B}(n)$ such that

Such target marginals are said to be feasible. We define $\mathrm{\Delta }{b}^{+}\in {ℝ}_{+}^n$ as $\mathrm{\Delta }{b}_j^{+}=b_j^{+}-b_{j+1}^{+}$ for $j\in [n]$, where we conventionally define $b_{n+1}^{+}≔0$. Similarly, we define $\mathrm{\Delta }{b}^{-}\in {ℝ}_{+}^m$. Let $F_j^{+}=⟨{𝐞}_1,\mathrm{\dots },{𝐞}_j⟩$ be the standard flag of ${ℂ}^n$ for $j\in [n]$. Similarly, we define $F_i^{-}$ for $i\in [m]$. The following theorem characterizes the set of feasible marginals by a certain linear system.

Let $(b^{+},b^{-})$ be a feasible marginal with rational entries. There is an efficient algorithm that finds a scaling of a given CP map $\mathrm{\Phi }$ whose marginal is $\epsilon$-close to $(b^{+},b^{-})$ [17, 5].

Here, we recall the reduction of the $\sigma$-semistability of an acyclic quiver to nc-nonsingularity testing [13].

Let $Q$ be an acyclic quiver and $V$ a representation of $Q$ with the dimension vector $\alpha$. Let $\sigma \in {\mathbb{Z}}^{Q_0}$ be a weight.⁶⁶6We do not assume $\sigma (\alpha )=0$ here because the reduction does not need it. This is useful for finding a maximizer in King’s criterion. Let $Q_0^{+}$ and $Q_0^{-}$ be the sets of vertices $i$ such that $\sigma (i)>0$ and , respectively. Let ${\sigma }^{+}(i)≔\max \{\sigma (i),0\}$ and ${\sigma }^{-}(i)≔\max \{-\sigma (i),0\}$ for each $i\in Q_0$. Note that $\sigma ={\sigma }^{+}-{\sigma }^{-}$. Let $N≔{\sigma }^{+}(\alpha )={\sigma }^{-}(\alpha )$. We define an $N\times N$ partitioned linear matrix $A$ as follows. As a linear map,

The $(s,p;t,q)$-block ($s\in Q_0^{+}$, $p\in [{\sigma }^{+}(s)]$, $t\in Q_0^{-}$, $q\in [{\sigma }^{-}(t)]$) of $A$ is given by a linear matrix

where $x_{P,p,q}$ is an indeterminate and $V(P)$ is the linear map corresponding to the path $P$, i.e., $V(P)≔V(a_k)\mathrm{\cdots }V(a_1)$ for $P=(a_1,\mathrm{\dots },a_k)$ as a sequence of arcs. The number of indeterminates in $A$ is equal to

where $m(s,t)$ denotes the number of $s$–$t$ paths in $Q$. Thus, the number of indeterminates may be exponential in general.

The following lemma connects the nc-rank of $A$ with King’s criterion, which is shown in [30, Theorem 3.3] using abstract algebra.

By the lemma, one can check the $\sigma$-semistability of a quiver representation by checking whether the corresponding linear matrix is nc-nonsingular or not. However, the naive reduction does not give a polynomial time algorithm, as the number of indeterminates in the linear matrix may be exponential.

We present a scaling algorithm for deciding the $\sigma$-semistability. The idea is to reduce the problem to operator scaling with specified marginals.

Let us define a CP map ${\mathrm{\Phi }}_V$ corresponding to a quiver representation $V$. As a linear map,

Let $X=⨁(X_s:s\in Q_0^{+})$ be an input block matrix. Then,

for $t\in Q_0^{-}$. The dual map