Certain classes of polynomials associated with column-strict, rectangular Young tableaux, that we call (fused) Specht polynomials,
give explicit formulas for conformal blocks (partition functions) describing a general class of
conformally invariant boundary conditions and crossing probabilities for models building on the Gaussian free field (GFF):
e.g. double-dimers contours and multi-dimer webs of higher degree.
Also, analogous determinantal formulas describe the geometry of uniform spanning tree (UST) branches (loop-erased walks).
These objects also display rich algebraic content: they carry irreducible representations of certain
diagram algebras: e.g., the (fused) Hecke algebra, Temperley-Lieb algebra,
or the Kuperberg algebra defined from $\mathfrak{sl}_n$ webs.
I shall discuss the general framework and results towards the connection to random geometry.
While the precise connection is partly conjectural, it has been verified in special cases.
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