A diagrammatic category for generalized Bott-Samelson bimodules and a diagrammatic categorification of induced trivial modules for Hecke algebras

Details A diagrammatic category for generalized Bott-Samelson bimodules and a diagrammatic categorification of induced trivial modules for Hecke algebras A diagrammatic category for generalized Bott-Samelson bimodules and a diagrammatic categorification of induced trivial modules for Hecke algebras

2010-09-10

arxiv.org/abs/1009.2120v1

Mathematics

Representation Theory

Scientific paper

Let $R$ be the polynomial ring of the reflection representation of $W=S_n$, and for any parabolic subgroup $W_J \subset W$ corresponding to a subset $J$ of the Dynkin diagram, let $R^J$ be the subring of polynomials invariant under $W_J$. When $J=\{i\}$ is a singleton, denote the ring $R^i$ and the corresponding reflection $s_i$. Previously, in joint work with Mikhail Khovanov, the subcategory of $R$-bimodules generated monoidally by $R \otimes_{R^i} R$ for arbitrary $i$ was given a diagrammatic presentation, so that morphisms may be viewed as labelled planar graphs. Here, a diagrammatic presentation is given for the subcategory generated by $R \otimes_{R^J} R$ for arbitrary $J$, extending the previous diagrammatics. In addition, a diagrammatic presentation is given for a certain subcategory of $R^J-R$-bimodules which categorifies the representation of the Hecke algebra of $W$ constructed by inducing the trivial representation of the Hecke algebra of $W_J$. Whenever $x=s_{i_1}s_{i_2}\... s_{i_d}$ is a reduced expression for the longest element of $W_J$, the $R$-bimodule $R \otimes_{R^J} R$ will be a direct summand of $M_x = R \otimes_{R^{i_1}} R \otimes_{R^{i_2}} R \otimes \... \otimes_{R^{i_d}} R$. The key result is an explicit computation of the map $M_x \to M_y$, for two reduced expressions $x,y$ of the longest element, given by projecting $M_x \to R \otimes_{R^J} R$ and including $R \otimes_{R^J} R \to M_y$. The construction of this map uses the higher Coxeter theory of Manin and Schechtman.

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