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Séminaire Lotharingien de Combinatoire, 89B.38 (2023), 12 pp.

# Shashank Kanade

# On the *A*_{2} Andrews-Schilling-Warnaar Identities

**Abstract.**
In a groundbreaking work, Andrews-Schilling-Warnaar invented an A_{2} generalization of the A_{1} Bailey machinery and discovered many identities related to the principal characters of standard modules for **sl**^{^}_{3}, or equivalently, for the vertex operator algebra **W**_{3}(3,*p*'). Jointly with Russell, we have given conjectures for completing this set of identities and proved these conjectures for small values of *p*'. In another direction, character of **W**_{r}(*p*,*p*') has been related to an appropriate limit of certain **sl**^{^}_{r} coloured Jones polynomials of torus knots T(*p*,*p*') under some restrictions on *r*,*p*,*p*'. This note summarizes these developments.

Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.

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