Rogers-Ramanujan Identities as Specialised Weyl Characters in Affine A₂ₙ⁽²⁾ — E8 Intelligence Research

FINDING: The Rogers-Ramanujan identities are not isolated partition identities but are embedded in a doubly-infinite family labelled by the affine Kac-Moody algebra \(A_{2n}^{(2)}\), with the product side being a specialised Weyl character. | MATH: Classical Rogers-Ramanujan: \[ \sum_{k=0}^\infty \frac{q^{k^2}}{(q;q)_k} = \frac{1}{(q;q^5)_\infty (q^4;q^5)_\infty}, \quad \sum_{k=0}^\infty \frac{q^{k^2+k}}{(q;q)_k} = \frac{1}{(q^2;q^5)_\infty (q^3;q^5)_\infty} \] where \((a;q)_k = \prod_{j=0}^{k-1}(1-aq^j)\). The generalised family (arXiv:1309.5216) gives product sides as specialised characters of \(A_{2n}^{(2)}\) at level \(m\). The Weyl character formula for a highest weight \(\lambda\): \[ \chi_\lambda = \frac{\sum_{w \in W} \mathrm{sgn}(w) e^{w(\lambda+\rho)-\rho}}{\prod_{\alpha>0}(1-e^{-\alpha})} \] with \(W\) the Weyl group of \(A_{2n}^{(2)}\), \(\rho\) the half-sum of positive roots. The connection is that the sum side of Rogers-Ramanujan corresponds to the numerator (alt Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com