论文标题
$ \ tilde {a} _1 $ -soergel calculus和nil-blob代数
Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra
论文作者
论文摘要
我们研究soergel calculus algebra $ a_w：= \ mbox {end} _ {{\ Mathcal d} _ {（w，s）}}（\ useverline {w}）$ over $ \ mathbb c $ in Type $ \ \ tilde {a a a} _1 $。我们概括了nil-blob代数$ {\ mathbb {nb}} _ n $和$ a_w $之间的最新同构，以处理两参数blob代数。在此概括下，两个参数对应于$ \ tilde {a} _1 $的两个简单根。使用此功能，加上涉及$ \ mathbb {nb} _n $的jones-wenzl iDempotents的temperley-lieb subegra，我们获得了bil $Δ_W（v）$ a_w $ a_w $ by bilinear形式的矩阵的混凝土对角度化。对角矩阵的条目被证明是$ \ tilde {a} _1 $的根源。我们用它来研究$Δ_W（v）$的Jantzen类型过滤，$ a_w $。我们表明，在富集的grothendieck组级别，相应的总和公式具有$Δ_W（s_ {α} v）[l（s_ {α} v）-l（v）] $，其中$ [\ cdot] $表示分级移位。
We study the representation theory of the Soergel calculus algebra $ A_w := \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 ${\mathbb{NB}}_n$ and $ A_w $ 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{NB}_n$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module $Δ_w(v) $ for $ A_w $. 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 $ Δ_w(v)$ for $A_w $. We show that at enriched Grothendieck group level the corresponding sum formula has terms $ Δ_w(s_{α}v)[ l(s_{α}v)- l(v)] $, where $[ \cdot ] $ denotes grading shift.
