Principal Components Along Quiver Representations

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-022-09563-x Principal Components Along Quiver Representations Anna Seigal1,2 · Heather A. Harrington1 · Vidit Nanda1 Received: 22 April 2021 / Revised: 24 November 2021 / Accepted: 23 December 2021 © The Author(s) 2022 Abstract Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil. Keywords Principal components · Quiver representations Mathematics Subject Classification 15A22 · 16G20 · 55N30 · 62H25 Introduction A quiver representation is an arrangement of vector spaces and linear maps tethered to the vertices and edges of a directed graph [14, 43]. The quiver illustrated below will be our running example throughout the paper. Communicated by Peter Bubenik. B Vidit Nanda 1 University of Oxford, Oxford, UK 2 Harvard University, Cambridge, MA, USA 123 Foundations of Computational Mathematics Despite being relatively concrete mathematical objects, quiver representations provide a uniform framework for a host of fundamental abstract problems in linear algebra [7]. Isomorphisms of quiver representations can be used to characterise, for example, the Jordan normal form of matrices and the Kronecker normal form of matrix pencils. They also play an important role in various other fields, including the study of associative algebras [3], Gromov–Witten invariants [22], representations of Kac–Moody algebras [39], moduli stacks [48], Morse theory [35], persistent homology [41], and perverse sheaves [19], among others. In most of these contexts, the crucial property of a given quiver representation is its decomposability into a direct sum of smaller representations. Gabriel’s celebrated result [18] establishes that a quiver admits finitely many (isomorphism classes of) indecomposable representations if and only if its underlying undirected graph is a union of simply laced Dynkin diagrams (i.e. type A, D or E). Thus, most quivers have rather complicated sets of indecomposable representations and are said to be of wild type. It is a direct consequence of this trifecta—concreteness, ubiquity and generic wildness—that ideas from disparate branches of mathematics have conversely been deployed to study representations of quivers. These include algebraic geometry [34], combinatorics [13], differential geometry [23, 25], geometric representation theory [20], invariant theory [32, 33], and multilinear algebra [27]. Quiver representations have recently emerged in far more applied and computational contexts than the classical ones listed above. We are aware of three such appearances: (1) Cellular Sheaves: A vector space-valued sheaf defined over a cell complex [10, 11] constitutes a representation of the underlying Hasse diagram; here the vertices are cells and edges arise from face inclusions. The stalks of the sheaf form vector spaces over the vertices, while restriction maps are associated to edges. (2) Conley theory: Morse decompositions in computational dynamics [26, Def 9.19] are representations of Conley–Morse quivers associated to discrete dynamical systems (vertices are recurrent sets and edges represent gradient flow). The linear maps of such representations arise from connection matrices [17]; these assemble into a chain complex that allows one to recover the homology of the phase space. (3) Algebraic statistics: Matrix normal models can be studied via quiver representations [1, 12]. The sample data gives a representation of a Kronecker quiver. The stability of the representation [33] can then be used to characterise the existence and uniqueness of a maximum likelihood estimate in the model. We expect (and hope) that this influx of quiver theory into more applied and computational domains will continue. 123 Foundations of Computational Mathematics This Paper We consider a representation A• of a quiver Q. It assigns vector spaces Av to each vertex v of Q and linear maps Ae to each edge e in Q. We construct a vector space (Q; A• ) called the space of sections of the quiver representation. An element of (Q; A• ) selects one vector γv from the vector space Av assigned to each vertex v so that for every edge e : u → v the linear map  Ae sends γu to γv . As such, (Q; A• ) is a subspace of the total space Tot(A• ) := v Av . The assignment A• → (Q; A• ) can directly be seen to be a functor from the category of Q-representations to the category of vector spaces. We do not expect this functor to immediately answer any deep questions regarding (in)decomposability of quiver representations. Rather, we hope that the space of sections will become a useful and practical tool for those who encounter quiver representations in applied and computational contexts. Our first contribution is an algorithm for computing the space of sections for any finite-dimensional representation of a finite quiver. This is of some relevance even to those who have no warm feelings for quiver representations, since it is a purely categorical procedure for computing the limit (i.e. the universal cone) of a diagram in the category of vector spaces. With minor modifications, it can be made to work for diagrams valued in any abelian category that has computable products and equalisers. There are two types of restriction imposed on the space of sections: the first of these arises from directed cycles, where we are forced to restrict to a fixed point space of an endomorphism; and the second is the presence of multiple incoming edges at a vertex, where we are forced to restrict to an equaliser. None of these difficulties arise when the quiver is a directed rooted tree. Our algorithm consists of two steps—the first step removes all directed cycles and updates the representation A• accordingly; and the second step replaces this acyclic quiver with a directed rooted tree, again updating the representation. The result is a + new representation A+ • of a rooted directed tree T , which has all the same vertices as Q (plus an additional root vertex) and satisfies A+ v ⊂ Av at each vertex v. Here is our first main result. Theorem (A). The space of sections (Q; A• ) is the image of the map F : A+ ρ −→ Tot(A• ), obtained by composing the linear maps assigned by the quiver representation A+ • along the unique path in the rooted directed tree T + from the root ρ to each other vertex. Although the constructions of T + and A+ • are explicit and readily implementable on a computer, they require making several intermediate choices. Each such choice is liable to produce a diff (...truncated)