Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra
Marcelo Hern\'andez Caro, Steen Ryom-Hansen

TL;DR
This paper explores the representation theory of the Soergel calculus algebra in type A_1, generalizes the nil-blob algebra to a two-parameter setting, and provides explicit diagonalization and sum formulas for related modules.
Contribution
It introduces a two-parameter generalization of the nil-blob algebra and applies it to explicitly diagonalize bilinear forms and derive sum formulas in type A_1.
Findings
Diagonalization of bilinear form matrices using Jones-Wenzl idempotents.
Explicit sum formulas for Jantzen filtrations in the Grothendieck group.
Connection between parameters and simple roots in A_1.
Abstract
We study the representation theory of the Soergel calculus algebra over in type . We generalize the recent isomorphism between the nil-blob algebra and to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for . Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of , we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module for . The entries of the diagonalized matrices turn out to be products of roots for . We use this to study Jantzen type filtrations of for . We show that at enriched Grothendieck group level the corresponding…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
