The Bott–Samelson bimodule of a word

Let $R$ and the generators $B_i=R{\otimes }_{R^{s_i}}R(1)$ be as in The Soergel bimodule $B_i$ of a simple reflection. For a word $\underset{‾}{i}=(i_1,\dots ,i_r)$ in the simple reflections, i.e. a sequence with $1\le i_k\le n-1$, put $B_{\underset{‾}{i}}:=B_{i_1}{\otimes }_R{B}_{i_2}{\otimes }_R\cdots {\otimes }_R{B}_{i_r},$ the iterated balanced tensor product of graded $(R,R)$-bimodules, carrying the total internal grading, the outer left action on the first factor and the outer right action on the last factor. For the empty word $r=0$ we put $B_{\mathrm{\varnothing }}:=R$, the regular graded $(R,R)$-bimodule; this is the unit of the tensor product, so the two conventions agree and $B_{(i)}=B_i$. The word $\underset{‾}{i}$ is reduced when $s_{i_1}\cdots s_{i_r}\in S_n$ has length $r$. The notation $B_{\underset{‾}{i}}$ retains the chosen word even when that word is reduced: two reduced words for the same $w$ can give nonisomorphic Bott–Samelson bimodules. The word-independent indecomposable summand indexed by $w$ is constructed later; it is not the whole $B_{\underset{‾}{i}}$ in general. The freeness of each $B_{\underset{‾}{i}}$ on both sides is proved in Soergel generators and Bott–Samelson products are finite free on both sides. Every element of a Bott–Samelson product is a finite sum of tensors of homogeneous elements, with degrees adding.

Trivial conventions. If $n\le 1$ the empty word is the only word and only $B_{\mathrm{\varnothing }}=R$ is present. The word itself is never claimed to be visible from the isomorphism type of $B_{\underset{‾}{i}}$: two words related by a Coxeter braid move need not give isomorphic bimodules, and the rank-two decompositions of the adjacent triple products are the point where this is computed on this page.