Equivariant Coinvariant and Bott–Samelson Rings Watanabe's Bold Conjecture

Abstract Modeled on work of Bott–Samelson, Soergel, and Elias–Williamson we introduce the Bott–Samelson ring BS ‾ ( s 1 , s 2 , … , s k ) associated to reflections s 1 , s 2 , … , s k ∈ GL ( n , F ) where F is a field. We show that for semisimple reflections it is a complete intersection algebra with a triangular pattern of relations and has the strong Lefschetz property. For a finite nonmodular reflection group generated by reflections of order two we produce a degree one embedding of the coinvariant algebra F [ V ] G into a Bott–Samelson BS ‾ ( s 1 , s 2 , … , s k ) algebra where s 1 , s 2 , … , s k ∈ G are reflections of order two generating G answering a question of J. Watanabe. We show by example that the strong Lefschetz property of BS ‾ ( s 1 , s 2 , … , s k ) need not be inherited by the embedded F [ V ] G answering another question of J. Watanabe. Finally we investigate the decomposition map where we start with a representation and its associated equivariant coinvariant algebra F [ V ] ⊗ F [ V ] G F [ V ] , and for each w ∈ W the w-twisted multiplication map is defined by μ w ( f ′ ⊗ R W f ″ ) = f ′ ⋅ w ( f ″ ) . These fit together to define the decomposition map μ W into the direct sum of | W | copies of F [ V ] indexed by the elements of W and we show this map is a monomorphism if the representation of W is nonmodular. For a reflection representation ρ of W and a family of reflections s 1 , s 2 , … , s k generating W we also define a map and characterize for which families of reflections it is a degree one embedding.