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BIRS Workshop Lecture Videos

Bott-Samelson algebras and Watanabe’s bold conjecture McDaniel, Christopher


Watanabe’s bold conjecture states that every Artinian complete intersection algebra (generated in degree one) can be embedded in an lArtinian complete intersection cut out by quadratic forms. We verify this conjecture for coinvariant rings of (finite) complex reflection groups generated by involutory reflections, which includes all finite Coxeter 3 groups. For a Coxeter group associated to a flag variety, the quadratic complete intersection algebras that we construct correspond to the cohomology rings of certain ”resolutions” of the flag variety, due to Bott and Samelson. These so-called Bott-Samelson algebras have been studied extensively by Soergel, whose work eventually led Elias and Williamson to a purely algebraic proof of the notorious Kazhdan-Lusztig positivity conjecture. Along the way, I will try to highlight some of these remarkable results of Soergel and Elias-Williamson, and their surprising connection with the strong Lefschetz property. (Joint with Larry Smith)

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