Gram Determinants of Type A and Generalized Type A

W. B. Raymond Lickorish introduced the Gram determinant of type A in his work on 3-manifold invariants. Bruce Westbury and Philippe Di Francesco proved the closed formula for this invariant. Later, Xuanting Cai proved the formula using a special basis involving the Jones-Wenzl idempotent. By using Cai’s special basis, the first, second, and third authors, along with Mukherjee, proved a formula for the generalized type A bilinear form. In this lecture, we present Cai’s proof of the closed formula of the Gram determinant of type A, introduce the generalized type A bilinear form, and prove the closed formula for the Gram determinant of generalized type A.