On Variants of Specht Polynomials and Random Geometry

Explore a 52-minute lecture on variants of Specht polynomials and their connection to random geometry, presented by Eveliina Peltola from Aalto University at IPAM's Statistical Mechanics and Discrete Geometry Workshop. Delve into the world of fused Specht polynomials associated with column-strict, rectangular Young tableaux and their role in providing explicit formulas for conformal blocks and partition functions. Discover how these polynomials describe conformally invariant boundary conditions and crossing probabilities for models based on the Gaussian free field, including double-dimers contours and multi-dimer webs. Examine the determinantal formulas that characterize the geometry of uniform spanning tree branches and loop-erased walks. Investigate the rich algebraic content of these objects, including their representation of diagram algebras such as the fused Hecke algebra, Temperley-Lieb algebra, and Kuperberg algebra defined from sln webs. Gain insights into the general framework and recent results connecting these mathematical structures to random geometry, while acknowledging the partly conjectural nature of some connections.