Title: Quasi-particles in the principal picture and Rogers-Ramanujan-type identities

Abstract: Rogers-Ramanujan identities first appeared in 1894, and “classical” Rogers-Ramanujan-type identities were discovered by Gordon, Andrews and Bressoud in 1960s and 1970s. In 1980s Lepowsky, Milne and Wilson found the connection of these identities with representations of affine Lie algebra of the type A_1^(1) and vertex operators in the principal picture. In fact, by using relations in terms of quasi-particles, monomial bases of representation spaces were constructed, parametrized by partitions satisfying difference 2 conditions; the analytic identities then follow by using q-series methods. With Slaven Kožić we obtained all Andrews-Gordon and Andrews-Bressoud sums in analytic identities by constructing quasi-particle bases of representation spaces for A_1^(1), and recently, with Marijana Butorac and Kožić, we obtained the Kanade-Russel and (independently) Kursungoz sum in the Capparelli analytic identity by constructing quasi-particle bases of level 3 representation space for A_2^(2) (arXiv:2603.23120). We expect that there are similar character formulas for other representations of affine Lie algebras related to quasi-particle bases.

Acknowledgement: This lecture is supported by a grant from the Royal Physiographic Society of Lund.