Clifford algebras and Littlewood-Richardson coefficients

TL;DR

This paper introduces a novel Clifford algebra approach to compute the cohomology ring of certain symmetric spaces and provides a new method for multiplying Schur polynomials via Hadamard product, simplifying Littlewood-Richardson coefficient calculations.

Contribution

It develops a Clifford algebra framework to describe the cohomology of symmetric spaces and offers a new, simplified multiplication method for Schur polynomials.

Findings

Clifford algebra techniques effectively describe the cohomology ring of symmetric spaces.

A new basis simplifies the multiplication of Schur polynomials.

Hadamard product corresponds to multiplication in the Clifford algebra setting.

Abstract

We show how to use Clifford algebra techniques to describe the de Rham cohomology ring of equal rank compact symmetric spaces $G/K$. In particular, for $G/K=U(n)/U(k)\times U(n-k)$, we obtain a new way of multiplying Schur polynomials, i.e., computing the Littlewood-Richardson coefficients. The corresponding multiplication on the Clifford algebra side is, in a convenient basis given by projections of the spin module, simply the componentwise multiplication of vectors in $\mathbb{C}^N$, also known as the Hadamard product.