[Paper Review] Poincaré series and rational cohomology rings of Kac-Moody groups and their flag manifolds

Xuan Zhao, Jin Chunhua+1 more|arXiv (Cornell University)|Jan 12, 2013

Algebraic structures and combinatorial models25 references3 citations

TL;DR

This paper determines the rational cohomology rings of indefinite Kac-Moody groups and their flag manifolds by analyzing their Poincaré series via the Leray-Serre spectral sequence and Cartan matrix structure. The key result is a complete classification: if the Cartan matrix is symmetrizable, the rational cohomology ring of the group is an exterior algebra on a single degree-3 generator tensored with a polynomial algebra; otherwise, it is purely polynomial. This fully determines their rational homotopy types and rational homotopy groups.

ABSTRACT

In this paper, we study the rational cohomology rings of indefinite Kac-Moody groups and their flag manifolds. By extracting the information of cohomology from the Poincaré series, we are able to determine the rational cohomology rings of Kac-Moody groups and their flag manifolds. Since Kac-Moody groups and their flag manifolds are rational formal, we also determine their rational homotopy groups and rational homotopy types.

Motivation & Objective

• To compute the rational cohomology rings of indefinite Kac-Moody groups and their flag manifolds, a long-standing open problem.
• To determine the rational homotopy types and rational homotopy groups of these groups by leveraging rational formality and cohomological data.
• To classify the cohomology structure based on the symmetrizability of the Cartan matrix, distinguishing between symmetrizable and non-symmetrizable cases.
• To provide a complete algebraic description of $ H^*(G(A)) $ and $ H^*(F(A)) $ in terms of polynomial and exterior algebras, with degrees determined by the Poincaré series.

Proposed method

• Use the Leray-Serre spectral sequence for the fibration $ G(A) \to F(A) \to BB(A) $ to relate the Poincaré series of $ G(A) $ and $ F(A) $.
• Apply an inductive method from prior work [13][14] to compute the Poincaré series $ P_A(q) $ of the flag manifold $ F(A) $ directly from the Cartan matrix $ A $.
• Compare the Poincaré series of $ F(A) $ from two sources: the spectral sequence and direct computation, to extract the Betti numbers $ i_k $.
• Use the Hopf algebra structure of rational cohomology to express $ H^*(G(A)) $ as a tensor product of polynomial and exterior algebras, with generators determined by the $ i_k $.
• Leverage Kac's result on regular sequences in $ H^*(BB(A)) $ to derive a closed-form expression for $ P_A(q) $ in terms of the $ i_k $.
• Use the symmetrizability of the Cartan matrix $ A $ as a key invariant: if symmetrizable, $ i_3 = 1 $, else $ i_3 = 0 $, and all odd-degree generators vanish beyond degree 3.

Experimental results

Research questions

• RQ1 What is the structure of the rational cohomology ring $ H^*(G(A)) $ for indefinite Kac-Moody groups?
• RQ2 How does the rational cohomology of the flag manifold $ F(A) $ depend on the Cartan matrix $ A $, particularly its symmetrizability?
• RQ3 Can the rational homotopy type of $ G(A) $ and $ F(A) $ be fully determined from their Poincaré series and Cartan matrix data?
• RQ4 What is the role of the symmetrizability condition in classifying the rational cohomology generators of $ G(A) $?
• RQ5 How do the degrees of rational cohomology generators relate to the Poincaré series and the structure of the Weyl group?

Key findings

• For an indecomposable indefinite Cartan matrix $ A $, if $ A $ is symmetrizable, then $ H^*(G(A)) \cong \Lambda_{\mathbb{Q}}(y_3) \otimes \mathbb{Q}[z_1, \dots, z_k, \dots] $, with $ \deg z_k \geq 4 $ even.
• If $ A $ is non-symmetrizable, then $ H^*(G(A)) \cong \mathbb{Q}[z_1, \dots, z_k, \dots] $, a purely polynomial algebra.
• The rational cohomology of the flag manifold $ F(A) $ is $ \mathbb{Q}[\omega_1, \dots, \omega_n]/\langle \psi \rangle \otimes \mathbb{Q}[z_1, \dots, z_k, \dots] $ when $ A $ is symmetrizable, and $ \mathbb{Q}[\omega_1, \dots, \omega_n] \otimes \mathbb{Q}[z_1, \dots, z_k, \dots] $ when non-symmetrizable.
• The degrees of the generators $ z_k $ are determined by the Poincaré series $ P_A(q) $ and the symmetrizability indicator $ \epsilon(A) $.
• For the example with $ a_{ij}a_{ji} \geq 4 $ for all $ i \neq j $, the Poincaré series is $ P_A(q) = \frac{1 - q^4}{(1 - q^2)^n} \cdot \frac{1}{\prod_{k=2}^\infty (1 - q^{2k})^{\dim L^{k}_{n-1}}} $.
• In this example, if $ A $ is symmetrizable, $ i_1 = i_2 = 0 $, $ i_3 = 1 $, $ i_{2k} = \dim L^{k}_{n-1} $ for $ k \geq 2 $, and all odd-degree generators beyond degree 3 vanish.