论文标题
带有足够循环的Quivers的KAC多项式的改进
A refinement of the Kac polynomials for quivers with enough loops
论文作者
论文摘要
KAC的一个猜想现在是一个定理认为，现在称为KAC多项式的多项式，它计算具有给定维度矢量的有限磁场上绝对不可分解的代表的同构类别，仅具有非负整数系数。在本文中，我们表明，对于具有足够循环的颤动，每个KAC多项式都可以表示为精制的KAC多项式的总和，这些kAC多项式由分区的分组进行参数，并且仅具有非阴性整数系数。给出了精制KAC多项式的封闭公式。我们进一步介绍了一种称为块的新一类表示形式，并对砂块的精制KAC多项式进行了猜想的解释，并在块表示的数量上进行了足够的循环。
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.
