Graded Sum Formula for ~ A 1-Soergel Calculus and the Nil-Blob Algebra

We study the representation theory of the Soergel calculus algebra $ {{ \tilde {A}_w^{{\mathbb {C}}}:= \mbox {End}_{ {{\mathcal {D}}}_{(W,S)}} (\underline {w}) $ over $ {\mathbb {C}}$ in type $ \tilde {A}_1$. We generalize the recent isomorphism between the nil-blob algebra $ {{\mathbb {N}\mathbb {B}}_n}$ and a diagrammatically defined subalgebra $ {A}_w^{{\mathbb {C}}}$ of ${{ \tilde {A}_w^{{\mathbb {C}}}$ to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for $ \tilde {A}_1$. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of $ {{\mathbb {N}\mathbb {B}}_n}$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module $ \Delta _w(v) $ for $ {{ \tilde {A}_w^{{\mathbb {C}}} $. The entries of the diagonalized matrices turn out to be products of roots for $ \tilde {A}_1$. We use this to study Jantzen-type filtrations of $ \Delta _w(v) $ for $ {{ \tilde {A}_w^{{\mathbb {C}}}$. We show that, at an enriched Grothendieck group level, the corresponding sum formula has terms $ \Delta _w(s_{\alpha }v)[ l(s_{\alpha }v)- l(v)] $, where $ [ \cdot ] $ denotes grading shift.