发表机构
Indian Institute of Science（印度科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过图论方法，利用广义色多项式和权多重性，给出了实部为A型的一类Borcherds-Kac-Moody代数根重数的显式公式。
AI 中文摘要
本文的主要目的是为一类有限秩对称化 Borcherds-Kac-Moody 代数 g(A) 的根重数获得显式公式，这类代数的实部为 A 型。每个此类代数的 Dynkin 图由 A 型 Dynkin 图与一个有限简单图 G_im 的不相交并集得到，其中 G_im 的顶点对应于虚单根，方法是在每个图中选择一个顶点，并用重数 c>0 的单边连接所选顶点。更一般地，我们考虑相同的构造，但将 A 型图替换为任意有限秩对称化 Kac-Moody 代数 g_re 的 Dynkin 图。所得的对称化 Borcherds-Kac-Moody 代数 g(A) 具有实部 g_re 和由 G_im 表示的虚部。我们的出发点是 g(A) 与 g_re 的分母乘积之商的图论描述。该商是 G_im 的多元独立多项式，其中对应于连接顶点的变量由 g_re 的归一化最高权特征标加权。我们证明该商的幂的系数分解为广义色多项式与张量幂权多重性的乘积。利用这一点，我们获得了所有具有非零虚分量的根重数的显式公式，这些公式以 G_im 的广义色多项式和 g_re 的最高权模的张量积中的权多重性表示。当 G_im 是弦图时，这些公式特化为涉及二项式系数乘积的有限除数求和；路径和完全图提供了具体例子。特别地，当实部 g_re 为 A 型时，我们获得了 g(A) 的根重数的显式公式。
英文摘要
The main aim of this article is to obtain explicit formulas for the root multiplicities of a class of finite-rank symmetrizable Borcherds-Kac-Moody algebras g(A) whose real part is of type A. The Dynkin diagram of each such algebra is obtained from the disjoint union of a Dynkin diagram of type A and a finite simple graph G_im, whose vertices correspond to imaginary simple roots, by choosing one vertex in each and joining the chosen vertices by a single edge of multiplicity c>0. More generally, we consider the same construction with the type A diagram replaced by the Dynkin diagram of an arbitrary finite-rank symmetrizable Kac-Moody algebra g_re. The resulting symmetrizable Borcherds-Kac-Moody algebra g(A) has real part g_re and imaginary part represented by G_im. Our starting point is a graph-theoretic description of the quotient of the denominator products of g(A) and g_re. This quotient is a multivariate independence polynomial of G_im, with the variable corresponding to the attachment vertex weighted by normalized highest-weight characters of g_re. We prove that the coefficients of powers of this quotient factor into a generalized chromatic polynomial and a tensor-power weight multiplicity. Using this, we obtain explicit formulas for all root multiplicities with nonzero imaginary component in terms of generalized chromatic polynomials of G_im and weight multiplicities in tensor products of highest-weight modules of g_re. When G_im is chordal, these formulas specialize to finite divisor sums involving products of binomial coefficients; paths and complete graphs provide particular examples. In particular, we obtain explicit formulas for the root multiplicities of g(A) when its real part g_re is of type A.
Comments28 pages
