Related papers: On quiver representations over $\mathbb{F}_1$

We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.

We classify the irreducible representations of a family of finite-dimensional pointed liftings $H_\lambda$ of the Nichols algebra associated with the diagram $A_2$ with parameter $q=-1$. We show that these algebras have infinite…

Let G be a connected reductive group defined over a finite field F_q. We give a parametrization of the irreducible representations of G(F_q) in terms of (twisted) categorical centres of various monoidal categories associated to G. (Results…

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a…

We consider quantum group representations Rep(G_q) for a semisimple algebraic group G at a complex root of unity q. Here we allow q to be of any order. We first show that the Tannakian center in Rep(G_q) is calculated via a twisting of…

By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the…

The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with…

It is well known that there exists a significant equivalence between the vector space $\mathbb{F}_{q}^n$ and the finite fields $\mathbb{F}_{q^n}$, and many scholars often view them as the same in most contexts. However, the precise…

Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $\mathbb Z Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. The stable $n$-translation quiver $\mathbb Z|_{n-1} Q$ is…

Let G be a Lie group and Q a quiver with relations. In this paper, we define G-valued representations of Q which directly generalize G-valued representations of finitely generated groups. Although as G-spaces, the G-valued quiver…

We consider quiver representations respecting a quiver automorphism and show that the dimension vectors of the indecomposables are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algebra. Moreover, every such Lie…

We show that the category of partial comodules over a Hopf algebra $H$ is comonadic over ${\sf Vect}_k$ and provide an explicit construction of this comonad using topological vector spaces. The case when $H$ is finite dimensional is treated…

Let \hat G be the semidirect product of a connected reductive group G over F_q with a finite cyclic group generated by a quasisemisimple automorphism of G defined over F_q. In this paper we prove a conjecture of G. Malle concerning the…

We construct from a finitary exact category with duality a module over its Hall algebra, called the Hall module, encoding the first order self-dual extension structure of the category. We study in detail Hall modules arising from the…

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability…

In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation…

In this paper we prove an identity in terms of generating functions which enables us to calculate the numbers of isomorphism classes of absolutely indecomposable semistable representations of quivers over finite fields.

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.

We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category. When $\Bbbk$ is complex field…

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of…