2022/10/07 by Marcelo Hernández Caro, Caro, Marcelo Hernández, Steen Ryom-Hansen +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2210.03847
openalex publication_date 2022/10/07 · openalex created_date 2022/10/12 · openalex updated_date 2026/07/28
We study the representation theory of the Soergel calculus algebra Aw := End_\mathcal D(W,S) (\underlinew) over \mathbb C in type A1. We generalize the recent isomorphism between the nil-blob algebra \mathbbNBn and Aw to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for A1. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of \mathbbNBn, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module Δw(v) for Aw . The entries of the diagonalized matrices turn out to be products of roots for A1. We use this to study Jantzen type filtrations of Δw(v) for Aw . We show that at enriched Grothendieck group level the corresponding sum formula has terms Δw(sαv)[ l(sαv)- l(v)] , where [ ⋅ ] denotes grading shift.