Poincar\'{e} Polynomials and Curvature Operators of Symmetric Spaces

TL;DR

This paper provides explicit formulas for the curvature operators and Poincaré polynomials of all compact irreducible symmetric spaces, revealing bounds on eigenvalues and connections to quantum numbers.

Contribution

It introduces explicit formulas for curvature operators and Poincaré polynomials of symmetric spaces, highlighting bounds and special cases like Hermitian symmetric spaces.

Findings

Maximum eigenvalue of curvature operators is bounded by the Einstein constant.

Poincaré polynomials can be derived using quantum numbers.

Equality in eigenvalue bounds characterizes Hermitian symmetric spaces.

Abstract

We compute explicit formulas for the curvature operators and Poincar\'e polynomials of all compact irreducible symmetric spaces. We can easily derive the Poincar\'e polynomials using quantum numbers, giving a formula that mirrors the known formula for the Euler characteristic. For the curvature operators, we show that their maximum eigenvalue is always bounded above by the Einstein constant, with equality attained precisely by the Hermitian symmetric spaces.