Kostant's generating functions, Ebeling's theorem and McKay's observation relating the Poincare series

2006/08/20 by Rafael Stekolshchik, Stekolshchik, Rafael
Mathematics · #15A18 #17B67 #20F55 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:15A18 #msc:17B67 #msc:20F55

paper · pdf · doi:10.48550/arxiv.math/0608500

22 pages, 1 figure

arxiv created 2006/08/20 · openalex publication_date 2006/08/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize B. Kostant's construction of generating functions to the case of multiply-laced diagrams and we prove for this case W. Ebeling's theorem which connects the Poincare series [P_G(t)]₀ and the Coxeter transformations. According to W. Ebeling's theorem [P_G(t)]₀ = \fracX(t²)X(t²), where X is the characteristic polynomial of the Coxeter transformation and X is the characteristic polynomial of the corresponding affine Coxeter transformation. We prove McKay's observation relating the Poincare series [P_G(t)]_i: (t+t^-1)[P_G(t)]_i = ∑_{i ← j}[P_G(t)]_j, where j runs over all vertices adjacent to i.

Related

• Coxeter Transformations, the McKay correspondence, and the Slodowy correspondence
• Notes on Coxeter Transformations and the McKay correspondence
• Poincare series of Klein groups, Coxeter polynomials, the Burau representation and Milnor invariants
• Counting chambers in restricted Coxeter arrangements
• Generating Functions in Algebraic Geometry and Sums over Trees