Grassmannian Cohomolgy Rings and Fusion Rings from Algebraic Equations

TL;DR

This paper explores the algebraic structure of cohomology and fusion rings related to Grassmannians and Lie algebras, revealing explicit polynomial potentials and their connections to classical sequences.

Contribution

It derives explicit Chebyshev polynomial potentials for fusion rings of $su(N)_K$ and $sp(N)_K$, linking algebraic equations to geometric and combinatorial structures.

Findings

Explicit Chebyshev polynomial potentials for $su(N)_K$ and $sp(N)_K$

Connection between fusion potentials and Fibonacci/Lucas numbers

Representation of cohomology rings via algebraic equations

Abstract

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for $su(N)_K$ is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for $sp(N)_K$ from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers.