Related papers: On Maximal Green Sequences
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism.
Exceptional sequences are important sequences of quiver representations in the study of representation theory of algebras. They are also closely related to the theory of cluster algebras and the combinatorics of Coxeter groups. We…
An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the…
For modules over an artin algebra a linear stability condition is given by a "central charge" and a nonlinear stability condition is given by the wall-crossing sequence of a "green path". Finite Harder-Narasimhan stratifications of the…
Let Q be a connected directed quiver with n vertices. We show that Q is representation-infinite if and only if there do exist n isomorphism classes of exceptional modules of some fixed length at least 2.
Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in…
We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual…
Maximum-length sequences (m-sequences for short) over finite fields are generated by linear feedback shift registers with primitive characteristic polynomials. These sequences have nice mathematical structures and good randomness properties…
The classical construction of representations of quivers enables us to consider linear maps between several vector spaces. The mixed representations of quivers helps us to work with linear maps as well as bilinear forms on several vector…
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite…
The unrestricted red size of a quiver is the maximal number of red vertices in its framed quiver after any given mutation sequence. In a 2023 paper by E. Bucher and J. Machacek, it was shown that connected, mutation-finite quivers either…
The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic,…
The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied…
Let $\Lambda$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of…
We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.
Quantum gauge theories with finite-dimensional representation spaces are constructed that can have canonical gauge field theories as singular limits. They describe nature as a recursive quantum assembly by iterating Fermi-Dirac…
In this paper, we study a maximization problem on real sequences. More precisely, for a given sequence, we are interested in computing the supremum of the sequence and an index for which the associated term is maximal. We propose a general…