The graded \(k\)-linear category of Bott–Samelson bimodules for \(R_n\) is the graded \(k\)-linear full subcategory \(\mathrm{BSbim}^{\mathrm{gr}}_n\) of \(R_n\)-bimodules given by all graded \(R_n\)-bimodules of the form: \[ B_{\mathbf{i}}\langle j\rangle:=R\otimes_{R^{s_{i_k}}}R\otimes_{R^{s_{i_{k-1}}}}\cdots\otimes_{R^{s_{i_1}}}R\langle j-k\rangle\] for \(R=R_n\), \(j\in\mathbb{Z}\) and some \(\mathbf{i}=(i_k,i_{k-1},\ldots, i_1)\in \{1,2,\ldots n-1\}^k\) with \(k \in \mathbb{N}_0\), including the bimodules \(R\langle j\rangle\) in case \(k=0\). In the case \(k=1\) we also abbreviate: \[B_i:= R\otimes_{R^{s_{i}}}R\langle-1\rangle\]
2.1 Review of Soergel bimodules and diagrammatics[000I]
We let \(k\) denote the rationals \(\mathbb{Q}\) or, more generally, a commutative \(\mathbb{Q}\)-algebra. We consider the \(k\)-linear monoidal categories \(\mathrm{Sbim}_n\) of Soergel bimodules for the symmetric group \(S_n\) acting on its natural representation. In this section, if not specified otherwise, categories mean ordinary categories (in contrast to \(\infty\)-categories used later) and functors mean ordinary functors. For a fixed nonnegative integer \(n\), let \(R_n=k[x_1,x_2,\ldots, x_n]\) denote the polynomial ring over \(k\) in \(n\) variables viewed as polynomial functions on \(\mathfrak{h}^*=(k^n)^*\) in the standard way. Permuting the basis vectors of \(k^n\) induces a left action of the symmetric group \(W=S_n\) on \(R_n\) such that the simple transposition \(s_i=(i,i+1)\) acts by swapping the variables \(x_i\) and \(x_{i+1}\). Denote \(\check\alpha_i=x_{i}-x_{i+1}\) for \(1\leq i\leq n-1\). Then restriction to the span of the \(\check\alpha_i\)’s gives the usual geometric representation of \(W\) viewed as the Coxeter group generated by the simple transpositions. For any subgroup \(G\) of \(W\) let \(R_n^G\) be the subalgebra of \(G\)-invariants in \(R_n\). In case \(G=\langle s_i\rangle\) for some \(1\leq i\leq n-1\) we abbreviate \(R_n^G=R_n^i\). We will view \(R_n\) as a graded (by which we mean \(\mathbb{Z}\)-graded) algebra by putting the generators \(x_i\) in degree \(2\). Note that \(R_n^i\) is a graded subalgebra and we have a canonical, grading-preserving decomposition \[ R_n= R_n^i\oplus \check\alpha_iR_n^i\simeq R_n^i\oplus R_n^i\langle 2\rangle\] as graded \(R_n^i\)-bimodules. Here and in the following we denote for \(j\in \mathbb{Z}\) and a graded (bi)module \(M=\oplus_{i\in\mathbb{Z}} M_i\) by \(M\langle j\rangle\) the graded (bi)module which equals \(M\) as (bi)module but with the grading shifted up by \(j\), i.e. \(M\langle j\rangle_i=M_{i-j}\). The grading shifting functors \(\langle j\rangle\), \(j\in\mathbb{Z}\) equip the category of graded \((R_n,R_m)\)-bimodules for fixed \(n,m\) with an action of the group \(\mathbb{Z}\).
By a graded \(k\)-linear category11 we mean a category enriched in \(\mathbb{Z}\)-graded \(k\)-modules. As an example we can take as objects graded \(R_n\)-bimodules with all \(R_n\)-bimodule maps, denoted \(\mathrm{Hom}^{\mathrm{gr}}\). In this case, the grading shift functors are compatible with the grading on morphisms as follows: \[ \mathrm{Hom}^{\mathrm{gr}}(M\langle k\rangle,N\langle l\rangle) = \mathrm{Hom}^{\mathrm{gr}}(M,N)\langle l-k\rangle\]
Using this shorthand, ([000M]) may also be expressed as: \[ B_{\mathbf{i}}\langle j\rangle \simeq B_{i_k} \otimes_{R} B_{i_{k-1}}\otimes_R \cdots\otimes_R B_{i_1}\langle j\rangle\]
There are two common variations of definition 2.1.1: 
Namely, \(\mathrm{BSbim}_n\) is the \(k\)-linear (but no longer graded \(k\)-linear) category obtained by restricting to the degree zero part of the morphism spaces. We call this the degree zero subcategory. (In the language of enriched category theory, this is the underlying category of the category enriched in graded vector spaces; it inherits the linear structure.) By remembering the \(\mathbb{Z}\)-action by grading shift functors, all other homogeneous components of morphism spaces in \(\mathrm{BSbim}^{\mathrm{gr}}_n\) can be recovered from ([000K]).
Alternatively, one can consider the graded \(k\)-linear full subcategory \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) on unshifted Bott–Samelson bimodules \(B_{\mathbf{i}}\), i.e. where \(j=0\) in ([000M]). From this category one can reconstruct the morphism spaces between shifted Bott–Samelsons, that is all objects in \(\mathrm{BSbim}^{\mathrm{gr}}_n\), again via ([000K]).
We refer to [MOS09, (2.1)] for a discussion of these essentially equivalent ways of handling graded \(k\)-linear categories.
The version \(\mathrm{BSbim}^{\mathrm{gr}}_n\) in Definition [000J] is the most flexible one, but with one caveat: when making statements about isomorphism, idempotents, and categorical constructions such as (co)products, we tacitly require that the structure morphisms are of degree zero, see e.g. ([000J]), i.e. we work in the underlying category. For this reason, we will henceforth almost exclusively work with \(\mathrm{BSbim}_n\). The only exception is subsection 2.5, where we use the version \(\overline{\mathrm{BSbim}}^{\mathrm{gr}}_n\) to connect to the diagrammatic Hecke category.
As defined, \(\mathrm{BSbim}_n\) is a monoidal full subcategory of the \(k\)-linear category of graded \(R_n\)-bimodules and grading-preserving bimodule maps, with tensor product \(-\otimes_{R_n}-\). More precisely, since each \(B_i\) is free of rank \(2\) as a graded \(R_n\)-module from the left and from the right, all objects of \(\mathrm{BSbim}_n\) are finitely generated graded-projective12 \(R_n\)-modules from both sides and the tensor product coincides with the derived tensor product.
The monoidal \(k\)-linear category \(\mathrm{Sbim}_n\) of Soergel bimodules for \(R_n\) is the Karoubian closure, that is the smallest additive idempotent-complete full subcategory of graded \(R_n\)-bimodules containing \(\mathrm{BSbim}_n\).
For later use, we also record how Bott–Samelson and Soergel bimodules for various \(n\) can be related.
Given \(a,b,c\in\mathbb{N}_0\) let \(j_{a|c}=j_{a|c}^b\colon R_b\hookrightarrow R_{a+b+c}\) be the algebra homomorphism given by \(x_i\mapsto x_{i+a}\). Given an \(R_m\)-bimodule \(M\) and an \(R_n\)-bimodule \(N\), the tensor product \(M\otimes_kN\) is an \(R_m\otimes R_n\)-bimodule, hence an \(R_{m+n}\)-bimodule via the isomorphism \(j_{0|n}\otimes j_{m|0}\). We call this functorial operation parabolic induction.
It is straightforward to see directly from Definition 2.1.1 that Bott–Samelson bimodules are sent to (bimodules isomorphic to) Bott–Samelson bimodules under parabolic induction. To distinguish the two kinds of tensor product, we will use the convention: \[\begin{aligned} \circ_1:= \otimes_{R_n} \colon \mathrm{BSbim}_n\times \mathrm{BSbim}_n&\rightarrow\mathrm{BSbim}_n\\ \boxtimes := \otimes_{k} \colon \mathrm{BSbim}_m\times \mathrm{BSbim}_n&\rightarrow\mathrm{BSbim}_{m+n} \end{aligned}\] and write \(f\circ_2g \colon M\rightarrow P\) for the composition of morphisms \(f\colon M\rightarrow N\), \(g\colon N\rightarrow P\) in \(\mathrm{BSbim}_n\). We use the same notation for \(\mathrm{Sbim}_n\).
The symbols \(\circ_2\), \(\circ_1\) and \(\boxtimes\) are meant to foreshadow that these operations should form the \(2\)- and \(1\)-morphism composition and the tensor product in a monoidal bicategory, see remark 2.3.3.
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2