The Kazhdan-Lusztig basis of the Hecke algebra has several important consequences and applications in geometric representation theory and adjacent fields. In this talk, I will describe a new conjectural approach to obtaining this basis at q=1 via quadratic optimisation. Focusing on the symmetric groups, we attempt to recover the Kazhdan-Lusztig basis of Specht modules as the solution to continuous optimisation problems over bases that are upper triangular in the polytabloid basis. This is subject to the constraint that the action of (1+s) is non-negative.

We prove that minimizing the trace of the Gram matrix uniquely recovers Young’s seminormal basis. Conversely, we conjecture that the Kazhdan-Lusztig left cell basis corresponds to the global maximum. I will present evidence for this conjecture, including techniques for verification for partitions of small ranks (less than 9). Geometrically, we interpret these bases as generators of invariant cones, proving that the Kazhdan-Lusztig basis generates the unique maximal invariant cone for hook shapes and two-column partitions. This is joint work with Geordie Williamson.