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Jiuzhao Hua

Publications and source records attributed to Jiuzhao Hua.

7 recordsLinked to original sources
arXiv2026

Counting Semistable Representations of Quivers over Finite Fields

In this paper, we derive a closed formula for the number of isomorphism classes of absolutely indecomposable semistable representations of an arbitrary quiver over a finite field with a fixed dimension vector. This generalises a formula for Kac polynomials given by Hua. A key step in the proof is to show that any representation of a quiver with a nilpotent endomorphism over an arbitrary field admits a structured filtration by subrepresentations compatible with the nilpotent action.

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arXiv2023

A refinement of the Kac polynomials for quivers with enough loops

A conjecture of Kac now a theorem asserts that the polynomial now known as the Kac polynomial, which counts the isomorphism classes of absolutely indecomposable representations of a quiver over a finite field with a given dimension vector, has non-negative integer coefficients only. In this paper, we show that, for quivers with enough loops, every Kac polynomial can be expressed as a sum of the refined Kac polynomials which are parametrized by tuples of partitions and have non-negative integer coefficients only. A closed formula for the refined Kac polynomials is given. We further introduce a new class of representations called blocks and make a conjectural interpretation of the refined Kac polynomials for quivers with enough loops in terms of the numbers of block representations.

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arXiv2022

Some Identities in Quantum Torus Arising from Ringel-Hall Algebras

We define two classes of representations of quivers over arbitrary fields, called monomorphic representations and epimorphic representations. We show that every representation has a unique maximal nilpotent subrepresentation and the associated quotient is always monomorphic, and every representation has a unique maximal epimorphic subrepresentation and the associated quotient is always nilpotent. The uniquenesses of such subrepresenations imply two identities in the Ringel-Hall algebra. By applying Reineke's integration map, we obtain two identities in the corresponding quantum torus.

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arXiv2021

On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation

The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This paper gives a closed formula for the number of absolutely indecomposable orbits using the same methodology as Hua [5]; those orbits are non-splittable over field extensions. As a consequence, those numbers are always polynomials in the cardinality of the base field with integral coefficients. It is conjectured that those coefficients are always non-negative.

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arXiv2021

Counting Representations of Quivers Respecting Nilpotent Relations over Finite Fields

This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely indecomposable representations with given dimension vectors is given and a $q$-deformation of Weyl-Kac denominator identity is established. In principle, if the numbers of representations are known, then the numbers of isomorphism classes of absolutely indecomposable representations are known.

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