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The rank-one Soergel bimodule square splits

Statement

Let $s = s_{i}$ be a simple reflection and $B_{s} = R \otimes_{R^{s}} R (1)$ , so that $B_{s} = R \otimes_{R^{s}} R {- 1}$ in the internal shift. Then there is an isomorphism of graded $(R, R)$ -bimodules $B_{s} \otimes_{R} B_{s} ≅ B_{s} (1) \oplus B_{s} (- 1),$ that is $B_{s} \otimes_{R} B_{s} ≅ B_{s} {- 1} \oplus B_{s} {1}$ ; the two summands are the idempotent images of the summands $R \otimes_{R^{s}} R^{s} \otimes_{R^{s}} R$ and $R \otimes_{R^{s}} α R^{s} \otimes_{R^{s}} R$ of the middle decomposition, and the two summands are not isomorphic as graded bimodules: they differ by the internal shift ${2}$ , so each is the shift of the other, but their graded left ranks $v^{- 1 + k} + v^{1 + k}$ for $k = - 1$ and $k = 1$ , namely $v^{- 2} + v^{0}$ and $v^{0} + v^{2}$ , are different, and an isomorphism of graded bimodules preserves graded ranks.

Facts & Assumptions

Given: A simple reflection $s = s_{i}$ , the invariant ring $R^{s}$ , the element $α = α_{i}$ with $s (α) = - α$ , and the bimodule $B_{s} = R \otimes_{R^{s}} R (1) = R \otimes_{R^{s}} R {- 1}$ .

[F1]

$R = R^{s} \oplus α R^{s}$ as graded $(R^{s}, R^{s})$ -bimodules, and the inclusion $R^{s} \to R$ is a graded ring map with $2$ invertible in $k$ ; $α R^{s} ≅ R^{s} {2}$ as $(R^{s}, R^{s})$ -bimodules via multiplication by $α$ (The standard type-A reflection realization and its polynomial ring, The Soergel bimodule $B_{i}$ of a simple reflection).

[F2]

$R \otimes_{R^{s}} R^{s} \otimes_{R^{s}} R ≅ R \otimes_{R^{s}} R$ and shifts move across a balanced tensor product, so that $(R \otimes_{R^{s}} R) {k} = R \otimes_{R^{s}} (R {k})$ (The Bott–Samelson bimodule of a word).

[F3]

$B_{s}$ is free of rank two on both sides and $B_{s} \otimes_{R} B_{s}$ is free of rank four on both sides, with $Δ$ -flag quotients $R_{s} {2}$ , $R {0}$ over the piece $R_{s} {1}$ and $R_{s} {0}$ , $R {- 2}$ over the piece $R {- 1}$ ; the $\nabla$ -flag quotients are $R {2}$ , $R_{s} {0}$ , $R {0}$ , $R_{s} {- 2}$ (Bott–Samelson bimodules carry delta and nabla support filtrations, Standard graph bimodules, support filtrations and characters).

Proof

1.1

Tensoring the middle factor: by [F1] the $(R^{s}, R^{s})$ -bimodule $R$ decomposes as $R^{s} \oplus α R^{s}$ , so applying $R \otimes_{R^{s}} - \otimes_{R^{s}} R$ to it gives $R \otimes_{R^{s}} R \otimes_{R^{s}} R ≅ (R \otimes_{R^{s}} R) \oplus (R \otimes_{R^{s}} α R^{s} \otimes_{R^{s}} R)$ .

2.1

The second summand: multiplication by $α$ is an $(R^{s}, R^{s})$ -bimodule isomorphism $R^{s} {2} \to α R^{s}$ , so $R \otimes_{R^{s}} α R^{s} \otimes_{R^{s}} R ≅ (R \otimes_{R^{s}} R) {2}$ by [F2]; the first summand is $R \otimes_{R^{s}} R ≅ (R \otimes_{R^{s}} R) {0}$ .

F1F2step 1.1

3.1

Shifts: $B_{s} = R \otimes_{R^{s}} R {- 1}$ , so $B_{s} \otimes_{R} B_{s} = (R \otimes_{R^{s}} R \otimes_{R^{s}} R) {- 2}$ ; applying the decomposition of step 2.1 and distributing the shift gives $B_{s} \otimes_{R} B_{s} ≅ (R \otimes_{R^{s}} R) {- 2} \oplus (R \otimes_{R^{s}} R) {0} = B_{s} {- 1} \oplus B_{s} {1}$ .

F2step 2.1

4.1

Consistency with the support flags: the two summands $B_{s} {- 1}$ and $B_{s} {1}$ have $Δ$ -flag quotients $R_{s} {0}, R {- 2}$ and $R_{s} {2}, R {0}$ respectively, whose union $R_{s} {0}, R_{s} {2}, R {- 2}, R {0}$ is the multiset in [F3]; hence the abstract decomposition of step 3.1 realizes the flag computation, and the two summands are the $α$ -divisible and the $α$ -free part of the middle factor.

F3step 3.1

5.1

Non-isomorphism and freeness: $B_{s} {- 1}$ and $B_{s} {1}$ differ by the shift ${2}$ , and the graded rank of $B_{s} {k}$ as a left $R$ -module is $v^{- 1 + k} + v^{1 + k}$ , so the two summands have different graded ranks and are not isomorphic; both are free of rank two on each side while $B_{s} \otimes_{R} B_{s}$ is free of rank four, matching step 3.1. ∎

F3step 3.1

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Sources

Elias–Williamson, Soergel Calculus, §3.5 equation (3.6), PDF p.28 (standard reference, not scraped)
Libedinsky, Gentle Introduction to Soergel Bimodules I, §4.3, PDF pp.22–26 (standard reference, not scraped)