A Strong Gram Classification of Non-negative Unit Forms of
Dynkin Type $\mathbb{A}_{r}$

An integral quadratic form $q(x_{1},\ldots,x_{n})=\sum_{1\leq i\leq j\leq n}q_{ij}x_{i}x_{j}$ is an integer homogeneous polynomial ($q_{ij}\in\mathbb{Z}$) of degree two on $n\geq 1$ integer variables $x_{1},\ldots,x_{n}$, more generally considered as a function $q:\mathbb{Z}^{n}\to\mathbb{Z}$ whose associated map

given for (column) vectors $x,y\in\mathbb{Z}^{n}$ by $\mathbbm{b}_{q}(x,y)=q(x+y)-q(x)-q(y)$, is a bilinear form, usually called polarization of $q$. The form $q$ is said to be positive (resp. non-negative) if $q(x)>0$ (resp. $q(x)\geq 0$) for any non-zero vector $x\in\mathbb{Z}^{n}$, that is, whenever the polarization $\mathbbm{b}_{q}$ is a positive (semi-) definite form, since $q(x)=\frac{1}{2}\mathbbm{b}_{q}(x,x)$ for any $x$ in $\mathbb{Z}^{n}$. Recall that two integral bilinear forms $\mathbbm{b}$ and $\mathbbm{b}^{\prime}$ are called equivalent if there is a $\mathbb{Z}$-invertible matrix $B$ such that $\mathbbm{b}^{\prime}(x,y)=\mathbbm{b}(Bx,By)$ for any $x,y\in\mathbb{Z}^{n}$.

Integral quadratic forms appear frequently, sometimes implicitly, in Lie theory, in the representation theory of groups, algebras, posets and bocses, in cluster theory, and in the spectral graph theory of signed graphs, to mention some examples. Their usefulness, in representation theory alone, which is our main motivation, has prompted extensive original research for some decades now. For instance:

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  In the early stages of the representation theory of associative algebras of finite dimension, after the work of Gabriel [23]: Bernstein, Gelfand and Ponomarev [9], Ovsienko [49] (see also [21] and [53]), Dlab and Ringel [17, 18], Ringel [53], Bongartz [12, 13], de la Peña [51, 50], Brüstle, de la Peña, Skowroński [14].

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  Usually in a graphical context, considering arithmetical properties and classification problems of quadratic forms: Barot [2, 3], Barot and de la Peña [6, 7, 8] von Höhne [28, 29, 30], Dean and de la Peña [16], Happel [27], Dräxler and de la Peña [19, 20].

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  Within Lenzing’s Coxeter formalism of bilinear forms [38]: Lenzing and Reiten [39], Mróz [44, 45], Mróz and de la Peña [46, 47].

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  In a graphical context, considering morsifications, Weyl and isotropy groups, certain mesh geometries of orbits of roots, and classification problems: Simson [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], Kosakowska [37], and Simson and collaborators: Bocian and Felisiak [10, 11], Gąsiorek and Zając [25, 26], Kasjan [35, 36], Makuracki and Zyglarski [42, 43], Zając [69, 70].

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  Within the context of quasi-Cartan matrices, defined by Barot, Geiss and Zelevinsky in [4] for the study of cluster algebras: Simson [66], Makuracki and Mróz [40, 41] and Perez, Abarca and Rivera [52].

Let us fix some of the notation and terminology used in the paper. We denote by $\mathbb{M}_{n}(\mathbb{Z})$ the set of $n\times n$ matrices with integer coefficients. The identity matrix of size $n$ is denoted by $\mathbf{Id}_{n}$, or simply by $\mathbf{Id}$ for adequate size. Recall that $M\in\mathbb{M}_{n}(\mathbb{Z})$ is $\mathbb{Z}$-invertible if and only if $\det(M)=\pm 1$. The transpose of a matrix $M$ is denoted by $M^{\mathbf{tr}}$, and if $M$ is $\mathbb{Z}$-invertible then $M^{-\mathbf{tr}}:=(M^{-1})^{\mathbf{tr}}$. Here all matrices have integer coefficients, and as usual, we identify a $m\times n$ matrix $M$ with the linear transformation $M:\mathbb{Z}^{n}\to\mathbb{Z}^{m}$ given by $x\mapsto Mx$. We denote by $\mathbf{Im}(M)$ and $\mathbf{Ker}(M)$ the column space of $M$ and the null right space of $M$, respectively. We say that the matrix $K$ is a kernel matrix of $M$ if its columns consists of a basis of $\mathbf{Ker}(M)$. The column vector with $n$ entries, all of them equal to $1$, is denoted by $\mathbbm{1}_{n}$, or simply by $\mathbbm{1}$ for appropriate size. For matrices $M_{t}$ of size $m\times n_{t}$ for $t=1,\ldots,r$ the $m\times n$ matrix with columns those of $M_{1},\ldots,M_{r}$, in that order, is denoted by $[M_{1},M_{2},\ldots,M_{r-1},M_{r}]$, where $n=\sum_{t=1}^{r}n_{t}$. For arbitrary matrices $N_{1},\ldots,N_{r}$, take

The canonical basis of $\mathbb{Z}^{n}$ is denoted by $\mathbf{e}_{1},\ldots,\mathbf{e}_{n}$. For a permutation $\rho$ of the set $\{1,\ldots,n\}$, the matrix $P(\rho)$ satisfying $P(\rho)\mathbf{e}_{t}=\mathbf{e}_{\rho(t)}$ for $t=1,\ldots,n$ is called permutation matrix of $\rho$.

The matrix with integer coefficients $G_{\mathbbm{b}}=[\mathbbm{b}(\mathbf{e}_{i},\mathbf{e}_{j})]_{i,j=1}^{n}$ is called Gram matrix of an integral bilinear form $\mathbbm{b}:\mathbb{Z}^{n}\times\mathbb{Z}^{n}\to\mathbb{Z}$, with respect to the canonical basis of $\mathbb{Z}^{n}$. By symmetric Gram matrix $G_{q}$ of an integral quadratic form $q$ we mean the Gram matrix $G_{q}=G_{\mathbbm{b}_{q}}$ of the polarization $\mathbbm{b}_{q}$ of $q$ (notice that $G_{q}$ is symmetric and has integer coefficients). We also consider the (unique) bilinear form $\widecheck{\mathbbm{b}}_{q}:\mathbb{Z}^{n}\times\mathbb{Z}^{n}\to\mathbb{Z}$ such that $q(x)=\widecheck{\mathbbm{b}}_{q}(x,x)$ for all $x\in\mathbb{Z}^{n}$, and such that its Gram matrix with respect to the canonical basis of $\mathbb{Z}^{n}$, denoted by $\widecheck{G}_{q}$, is upper triangular. Note that $G_{q}=\widecheck{G}_{q}+\widecheck{G}_{q}^{\mathbf{tr}}$. We say that $q$ is a unit form (or a unitary integral quadratic form) if $q(\mathbf{e}_{i})=1$ for $i=1,\ldots,n$. In that case, $\widecheck{G}_{q}$ is a $\mathbb{Z}$-invertible matrix (since it is upper triangular with all diagonal coefficients equal to $1$), and is called the standard morsification of $q$ in Simson’s terminology [57, 61]. Two unit forms $q$ and $q^{\prime}$ are called weakly Gram congruent if their polarizations are equivalent, that is, if there is a $\mathbb{Z}$-invertible matrix $B$ such that $G_{q^{\prime}}=B^{\mathbf{tr}}G_{q}B$ (or equivalently, $q^{\prime}=qB$). Similarly, $q$ and $q^{\prime}$ are called strongly Gram congruent if their standard morsifications are equivalent, that is, if there is a $\mathbb{Z}$-invertible matrix $B$ such that $\widecheck{G}_{q^{\prime}}=B^{\mathbf{tr}}\widecheck{G}_{q}B$. Then we write $q\sim^{B}q^{\prime}$ and $q\approx^{B}q^{\prime}$ for the weak and strong cases respectively (or simply $q\sim q^{\prime}$ and $q\approx q^{\prime}$). The weak Gram classification of connected non-negative unit forms is due to Barot and de la Peña [6] and Simson [62] (see also [70]), in terms of a unique pair $(\Delta,c)$ where $\Delta$ is a Dynkin diagram $\mathbb{A}_{r}$, $\mathbb{D}_{s}$, $\mathbb{E}_{t}$ (for $r\geq 1$, $s\geq 4$ or $t\in\{6,7,8\}$) and $c\geq 0$ is the corank of the quadratic form $q$, that is, the rank of the kernel of the symmetric Gram matrix $G_{q}$. Here we deal with the strong Gram classification problem of connected non-negative unit forms of Dynkin type $\mathbb{A}_{r}$ for $r\geq 1$.

For a unit form $q$, consider the matrix with integer coefficients $\Phi_{q}=-\widecheck{G}_{q}^{\mathbf{tr}}\widecheck{G}_{q}^{-1}$, called Coxeter matrix of $q$. The characteristic polynomial of $\Phi_{q}$, denoted by $\varphi_{q}(\lambda)$, is called Coxeter polynomial of $q$. It is well known, and can be easily shown, that if $q\approx q^{\prime}$, then $q\sim q^{\prime}$ and $\varphi_{q}=\varphi_{q^{\prime}}$ (cf. [33, Lemma 4.6]). The validity of the converse of this claim in this, or in partial or equivalent forms, is a question raised by Simson for at least a decade (see [61]). Here we give a formulation in terms of non-negative unit forms (see also [34, Problem A]), which correspond to non-negative loop-less bigraphs as in [61], or to non-negative symmetric quasi-Cartan matrices as in [4]. Generalizations of these problems may be found, for instance, in terms of Cox-regular bigraphs in [45, Problem 1.3], or of symmetrizable quasi-Cartan matrices in [66] (see also [61, 62, 63]).

Solutions to these problems for special classes of quadratic forms are known. For instance, the positive case was completed recently by Simson [64, 67, 68] (see further examples and related problems in [45]). An alternative proof for positive unit forms of Dynkin type $\mathbb{A}_{r}$ was given by the author in [33]. Here we present affirmative solutions to Problems 1 and 2 for connected non-negative unit forms of Dynkin type $\mathbb{A}_{r}$ (for $r\geq 1$) and arbitrary corank, with the combinatorial methods presented in [32], and developed to this end in [33, 34].

Recall that two connected non-negative unit forms are weakly Gram congruent if and only if they have the same Dynkin type and the same corank (see [6] or [70]), or equivalently, the same Dynkin type and the same number of variables. Simson determined in [62] representatives of the weak Gram congruence classes of connected non-negative unit forms, the so-called canonical extensions, showing that any such form having Dynkin type $\Delta$ is weakly Gram congruent to a unique canonical extension of the unit form $q_{\Delta}$, see [62, Theorems 1.12 and 2.12] and [70, Theorem 1.8] (recall that the quadratic form $q_{\Delta}$ associated to a graph $\Delta$ is determined by $G_{q_{\Delta}}:=2\mathbf{Id}-\mathrm{Adj}(\Delta)$, where $\mathrm{Adj}(\Delta)$ denotes the symmetric adjacency matrix of $\Delta$). A family of representatives for the corresponding strong Gram classes of Dynkin type $\mathbb{A}_{r}$ was proposed in [34, Definition 5.2] (see Figure 1 and 1.3 below). There they are called standard extensions of the unit form $q_{\mathbb{A}_{r}}$, and here we confirm that they are representatives of strong Gram congruence in the following classification theorem.

Alternatively, the following formulation of Theorem 1 answers directly Problem 1.

Using Theorem 2, and the results on Coxeter polynomials of [34] (based on [33, Theorem A]), we complete the following descriptive theorem of non-negative unit forms of Dynkin type $\mathbb{A}_{r}$ up to the strong Gram congruence (cf. [34, Problem B]). We need the following general notions. Let $q:\mathbb{Z}^{n}\to\mathbb{Z}$ be a unit form, with symmetric Gram matrix $G_{q}$ and Coxeter matrix $\Phi_{q}$.

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  A partition $\pi$ of an integer $m\geq 1$, denoted by $\pi\vdash m$, is a non-increasing sequence of positive integers $\pi=(\pi_{1},\ldots,\pi_{\ell(\pi)})$ such that $m=\sum_{t=1}^{\ell(\pi)}\pi_{t}$. The integer $\ell(\pi)$ is called the length of $\pi$, and the set of partitions of $m$ is denoted by $\mathcal{P}(m)$.

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  The kernel of $G_{q}$ in $\mathbb{Z}^{n}$ is called radical of $q$, denoted by $\mathbf{rad}(q)$, and its elements are called radical vectors of $q$. The rank of $\mathbf{rad}(q)$ is called corank of $q$, and is denoted by $\mathbf{cork}(q)$. The reduced corank $\mathbf{cork}_{re}(q)$ of $q$ is the rank of the kernel of the restriction $\widecheck{\mathbbm{r}}_{q}$ of $\widecheck{\mathbbm{b}}_{q}$ to its radical (see details in 2.1 below).

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  The Coxeter number $\mathbf{C}(q)$ of $q$ is the minimal $t>0$ such that $\Phi_{q}^{t}=\mathbf{Id}$, if such $t$ exists, and $\mathbf{C}(q):=\infty$ otherwise. The reduced Coxeter number $\mathbf{C}_{re}(q)$ is the minimal $t>0$ such that $\Phi_{q}^{t}-\mathbf{Id}$ is a nilpotent matrix (such $t$ always exists if $q$ is non-negative, cf. [55]). The degree of degeneracy of $q$ is the integer $\mathbbm{d}_{q}\geq 0$ such that $2\mathbbm{d}_{q}=\mathbf{cork}(q)-\mathbf{cork}_{re}(q)$, which is a non-negative even number since it is the rank of a skew-symmetric form (namely, the restriction $\widecheck{\mathbbm{r}}_{q}$ of $\widecheck{\mathbbm{b}}_{q}$ to the radical $\mathbf{rad}(q)$ of $q$, see (13) and further details in 2.1 below).

A solution for the positive and principal cases (coranks zero and one) of Theorem 2, within our combinatorial framework, was shown in [33, Theorems A and B] by means of certain (admissible) flations at the level of loop-less quivers. Here we pursue an alternative strategy which we sketch as follows (see details in Section 4). Let $q$ be a unit form in $\mathbf{UQuad}_{\mathbb{A}}^{c}(n)$. We fix a unique standard extension $\vec{q}$ such that $q\sim\vec{q}$ and $\varphi_{q}=\varphi_{\vec{q}}$ (see Definition 3.3 and Remark 1.17 below), and proceed in three main steps, for which, for a $n\times n$ matrix $B$, we consider the matrix $B^{*}:=\widecheck{G}_{\vec{q}}^{-1}B^{\mathbf{tr}}\widecheck{G}_{q}$.

• Step 1. Find a matrix $B$ such that, among other technical conditions (see Definition 3.3$(a)$), satisfies

  $$\vec{q}=qB,\quad\text{and}\quad q=\vec{q}B^{*}.$$

It can be shown that for corank zero or one, the matrix $B$ of Step 1 determines itself a strong Gram congruence $q\approx^{B}\vec{q}$ (cf. Lemma 4.22). In general, analyzing how far $B$ is from being a strong Gram congruence, we arrive to the following correction steps.

• Step 2. Find a matrix $M$ such that the matrix $B+M$ is $\mathbb{Z}$-invertible, and satisfies the same conditions of Step 1:

  $$\vec{q}=q(B+M),\quad\text{and}\quad q=\vec{q}(B+M)^{*}.$$

• Step 3. Find a matrix $C$ such that $[(B+M)C]^{*}[(B+M)C]=\mathbf{Id}$.

Clearly, the condition $N^{*}N=\mathbf{Id}$ for a square matrix $N$ implies that $N$ is $\mathbb{Z}$-invertible and $q\approx^{N}\vec{q}$. The goal of this paper is to constructively exhibit the existence of matrices $B$, $M$ and $C$. A solution to Step 1 is given in Section 1, using the specific structure of standard extensions, see Proposition1.23. Steps 2 and 3 are simple correction algorithms that work in a much general context (see Remark 4.17). However, their justification is long and technical at some points, and requires a special condition that can be easily verified for standard extension (see Lemma 2.13 and Corollary 4.15). Steps 2 and 3 are shown in Sections 2 and 3, see Propositions 2.17 and 3.13 and their implementable formulations as Algorithms 3 and 4, respectively. The proofs to our main theorems, comments on generalizations and suggestions for an implementation are collected in Section 4.

In this section we summarize the needed combinatorial notions and results introduced in [32] and developed in [33, 34] for the Coxeter analysis of non-negative unit forms of Dynkin type $\mathbb{A}_{r}$, namely, structural walks, incidence vectors, the inverse of a quiver and standard quivers.