A-D-E Polynomial and Rogers--Ramanujan Identities

TL;DR

This paper proposes polynomial identities that imply Rogers--Ramanujan type identities for certain algebraic cosets, supporting their conjectures with duality behavior and asymptotic analysis, and discusses potential generalizations.

Contribution

It introduces conjectured polynomial identities for A-D-E type cosets and verifies their properties, advancing understanding of Rogers--Ramanujan identities in algebraic contexts.

Findings

Confirmed correct behavior under level-rank duality for A-type algebras.

Established expected q→1 asymptotics via dilogarithm identities.

Discussed potential extensions to more general cosets.

Abstract

We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal G}$=A$_{n-1}$ \mbox{$(\ell\geq 2)$}, D$_{n-1}$ $(\ell\geq 2)$, E$_{6,7,8}$ $(\ell=2)$. In support of our conjectures we establish the correct behaviour under level-rank duality for $\cal G$=A$_{n-1}$ and show that the A-D-E Rogers--Ramanujan identities have the expected $q\to 1^{-}$ asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary cosets are also discussed briefly.