The rank-two longest type-A Soergel bimodule

The parabolic. Keep $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$, the simple reflections $s_i=(ii+1)$ and the invariants $R^{s_i}$ of The standard type-A reflection realization and its polynomial ring, and let $1\le i\le n-2$. Put $W_{i,i+1}:=⟨s_i,s_{i+1}⟩\le S_n,R^{W_{i,i+1}}:=R^{s_i}\cap R^{s_{i+1}},$ the parabolic subgroup generated by the two adjacent simple reflections and its invariant ring. The two generators act as adjacent transpositions on the three indicated coordinates and fix every other coordinate, so their restriction is an isomorphism $W_{i,i+1}\cong S_3$ by the generation clause of Type-A reduced words and the Coxeter presentation; the subgroup permutes the three coordinates $x_i,x_{i+1},x_{i+2}$ and fixes every other coordinate, so $R^{W_{i,i+1}}$ is the ring of polynomials in $x_i,x_{i+1},x_{i+2}$ that are invariant under all permutations of the three, with all other $x_j$ adjoined as invariants. The element $w_{i,i+1}:=s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1}\in W_{i,i+1}$ is the longest element of $W_{i,i+1}$, of length $3$.

The bimodule. Define $B_{i,i+1,i}:=R{\otimes }_{R^{W_{i,i+1}}}R(3),$ the balanced tensor product of the $(R,R^{W_{i,i+1}})$-bimodule $R$ with the $(R^{W_{i,i+1}},R)$-bimodule $R$, with the total grading of the tensor product shifted by the external shift $(3)$ of The Soergel bimodule $B_i$ of a simple reflection, that is $(3)=\{-3\},(B_{i,i+1,i}{)}_d=(R{\otimes }_{R^{W_{i,i+1}}}R{)}_{d+3}\text{ in the convention }M(r{)}_d=M_{d+r}\mathrm{.}$ It is the rank-two longest type-A Soergel bimodule attached to $i$. Equivalently $B_{i,i+1,i}=R{\otimes }_{R^{W_{i,i+1}}}R\{-3\}$ in the internal notation $M\{r{\}}_d=M_{d-r}$ of the library, and when $n=3$ and $i=1$ it is $B_{1,2,1}=R{\otimes }_{R^{S_3}}R(3)$. The Bott–Samelson bimodule of the word $(i,i+1,i)$ is $B_{\underset{‾}{i,i+1,i}}=B_i{\otimes }_R{B}_{i+1}{\otimes }_R{B}_i=R{\otimes }_{R^{s_i}}R{\otimes }_{R^{s_{i+1}}}R{\otimes }_{R^{s_i}}R(3)$ in the notation of The Bott–Samelson bimodule of a word.

(a) Why "longest". For adjacent $i,i+1$ the rank-two parabolic is finite of type $A_2$, so by the description of the $2m_{st}$-valent vertex of Elias–Williamson (p. 6) the indecomposable bimodule $B_{w_{i,i+1}}$ indexed by the longest element of the parabolic occurs as a direct summand with multiplicity one in each of the two Bott–Samelson bimodules $B_i{B}_{i+1}{B}_i$ and $B_{i+1}{B}_i{B}_{i+1}$; the rank-two decomposition theorem of this page identifies that summand explicitly as $B_{i,i+1,i}$, so that $B_i{B}_{i+1}{B}_i\cong B_{i,i+1,i}\oplus B_i$. The reader should not read this remark as a proof: the summand identification is the content of the next item on this page.

(b) The underlying rank. The ring $R$ is free as an $R^{W_{i,i+1}}$-module of rank $\mathrm{\mid }{W}_{i,i+1}\mathrm{\mid }=6$; this is the classical Chevalley theorem for the rank-two parabolic, and in the case $n=3$ it is the isomorphism $R\cong {⨁}_{w\in S_3}{R}^{S_3}(-2\mathrm{\ell }(w))$ of $R^{S_3}$-modules used in the source. Consequently $B_{i,i+1,i}$, whose grading is shifted by $(3)$, is a free graded $R$-module of rank six on each side, with homogeneous basis degrees $2\mathrm{\ell }(w)-3$ in the convention $(r)$, $w\in S_3$; as a graded left $R$-module $B_{i,i+1,i}\cong R(-3)\oplus R(-1{)}^{\oplus 2}\oplus R(1{)}^{\oplus 2}\oplus R(3)\mathrm{.}$

(c) The rank-one analogue. For $n=2$ the parabolic of the single simple reflection is $W_1=⟨s_1⟩$ with longest element $s_1$ of length $1$. Its analogous invariant-ring formula is $B_{W_1}:=R{\otimes }_{R^{s_1}}R(1)=B_1$ of The Soergel bimodule $B_i$ of a simple reflection. This is the rank-one generator, not an instance of the rank-two notation $B_{i,i+1,i}$, since two adjacent simple reflections do not exist for $n=2$.

(d) Degenerate indices. For $n\le 2$ there is no pair of adjacent simple reflections, $W_{i,i+1}$ is undefined, and the notation $B_{i,i+1,i}$ is not used; the objects of the type-A Soergel categories of this page are then generated by the single $B_1$ (or by $R$ alone when $n\le 1$).

(e) Imported facts of Libedinsky used by the decomposition theorem. The rank-two decomposition theorem of this page imports the computation of Libedinsky's §4.4 for the triple product $B_s{B}_t{B}_s$ of the rank-two parabolic generated by $s\mathrm{\ne }t$; in the type-A notation of this item, with $s=s_i$, $t=s_{i+1}$ and the three coordinates $x_i,x_{i+1},x_{i+2}$, its content is the following. For the idempotent $e$ constructed from that computation, the image $\mathrm{im}(1-e)$ is generated as an $R$-bimodule by the $1$-tensor $1^{\otimes }:=1\otimes 1\otimes 1\otimes 1\in B_{\underset{‾}{i,i+1,i}}$, and $B_i{B}_{i+1}{B}_i$ is generated as an $R$-bimodule by $1^{\otimes }$ together with $1\otimes x_i\otimes 1\otimes 1$. Moreover there is a graded homomorphism of $R$-bimodules $R{\otimes }_{R^{W_{i,i+1}}}R(3)⟶B_{\underset{‾}{i,i+1,i}},p\otimes q⟼p\otimes 1\otimes 1\otimes q,$ whose image is the sub-bimodule $⟨1^{\otimes }⟩$ generated by the $1$-tensor; since both sides are isomorphic to $R(-3)\oplus R(-1{)}^{\oplus 2}\oplus R(1{)}^{\oplus 2}\oplus R(3)$ as graded left $R$-modules, this homomorphism is an isomorphism onto $\mathrm{im}(1-e)$. These four statements are quoted from Libedinsky, §4.4.1 and §4.4.2, and are used in the decomposition theorem of this page, not proved here.