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A Soergel bimodule (for $S_n$) is a bimodule over $R = \mathbb{Q}[x_1,\dots,x_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules $$B_{i,i+1} = R \otimes_{i,i+1} R$$ where $\otimes_{i,i+1}$ means the tensor product over the subring of polynomials invariant under permuting $i$ and $i+1$. It follows immediately that every Soergel bimodule $M$ has the following properties:

(1) $M$ is free as a left module or as a right module, although not necessarily as a bimodule.

(2) $M$ commutes with invariant polynomials, in the sense that for every invariant polynomial $p \in \mathbb{Q}[x_1,\dots,x_n]$ and $m \in M$, we have $$ pm = mp. $$

I think they also have the following property:

(3) There is an invariant vector, an element $m_0 \in M$ so that $$ x_i m_0 = m_0 x_i $$ for every $i=1,\dots,n$.

Do these properties characterize Soergel bimodules? Without the third condition, you could have, for instance, a bimodule that just permuted the $x_i$: a one-dimensional module with a single generator $a$ as a right module, so that $ x_i a = a x_{\sigma(i)} $ for some permutation $\sigma$.

Edit: The natural generalization for a general Weyl group $W$ would be to replace the invariant polynomials in (2) by the polynomials that are invariant under $W$. Clearly all Soergel bimodules would still satisfy this generalization of (2).

Any references are welcome. If it's not known, I'll try to prove it.

Edit: Ben Webster gave a counterexample below. More generally, I'm still interested in some sort of intrinsic, elementary characterization.

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It's a challenging question. Meanwhile it's a good idea to label the third condition 3 rather than 1 for clarity, and maybe add one or two background references. My impression is that $S_n$ behaves better than other Weyl groups and their coinvariant algebras in Soergel's set-up, so I wonder if there is any hope for a general characterization along the lines you propose for this special case? –  Jim Humphreys Mar 26 '12 at 13:39
    
Sorry about the numbering; fixed now. I'm afraid I don't know background references; I only just started looking at Soergel bimodules. I learned about them from Khovanov's paper "Triply graded link homology and Hochschild homology of Soergel bimodules", but surely there are better references. –  Dylan Thurston Mar 26 '12 at 14:10

2 Answers 2

up vote 8 down vote accepted

There is an intrinsic characterisation which is probably more complicated than what you are looking for. As Ben says, Soergel bimodules are pretty subtle things ...

Because Soergel bimodules are (finitely generated) $R$-bimodules one can think about them as coherent sheaves on $V \times V$ (where $V = Spec R$). Inside $V \times V$ one has for any $w \in S_n$ its reversed graph:

$Gr_w = \{ (wv, v) \;| \;v \in V \}$

Hence, given any subset of $U \subset S_n$ one can talk about "sections of an $R$-bimodule $M$ with support in $U$": those sections of $M$ which have support in the union of the graphs of all elements of $U$. In this way, for any subset $I$ of $W$ one can consider $\Gamma_I M \subset M$.

Your point (2) means that Soergel any Soergel bimodule satisfies $\Gamma_{S_n} M = M$ (that is, every element is supported on the union of all the graphs of elements of $S_n$). It follows that any Soergel bimodule has a canonical filtration indexed by the ideals of the poset $S_n$. A basic fact is that if one considers the quotient

$\Gamma_{\le w / < w} (M) := \Gamma_{\le w} M / \Gamma_{< w} M$

this is a free left $R$-module, isomorphic as a bimodule to a direct sum of copies of $R_w$ (the ``standard'' bimodule with normal left action and right action twisted by $w$). This is proved in Soergel's "Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen" and is also discussed and generalised in my "Singular Soergel bimodules".

So now one can consider all bimodules which satisfy the above property. One this category one can put an exact structure: a sequence is exact if whenever one applies the functor $\Gamma_{\le w / < w}$ one obtains a split exact sequence of $R$-bimodules (necessarily isomorphic to direct sums of shifts of $R_w$'s).

Then the claim is that Soergel bimodules are the injective objects in this exact structure. I don't think this is written down anywhere. In the very similar language of moment graphs it is proved by Peter Fiebig in "Sheaves on moment graphs and a localization of Verma flags" here:

arxiv.org/abs/math.RT/0505108

(I might be mixing things up a bit. I think Peter considers the opposite filtration, which is why he gets projective objects. Anyway, if this is really what you're looking for then I can try to provide some more detail.)

By the way, the condition that the subsequent quotients in this filtration be split has other applications. In this paper

http://arxiv.org/abs/1205.4206

we examine when Rouquier complexes satisfy this property. It turns out that this is the case if (probably: and only if) the braid is a positive lift of an element of the Weyl group.

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This is pretty good, though of course rather more roundabout than I was hoping for. Thank you. –  Dylan Thurston Jun 26 '12 at 3:42

This is definitely too simple of a characterization. It's even false for n=3, before the really nasty stuff caused by non-smooth Schubert varieties shows up.

Counter-example: Consider the Soergel bimodule $R\otimes_{1,2} R\otimes_{2,3} R$, and twist the right action by the transposition $(1,2)$. Thought of as a coherent sheaf on the product $\mathbb C^n\times\mathbb C^n$, this is supported on the graph of the symmetric group elements $1, (12), (312), (321)$. The fact that it has support on the diagonal shows that there are invariant vectors, and your first two conditions are unchanged when you flip the action, as you noted.

On the other hand, the group elements that a Soergel bimodule is supported on the graphs of are an ideal in Bruhat order (an instant consequence of the same fact for fixed points in Schubert varieties), which the elements I listed above are not.

As a general comment, I think Soergel bimodules are really special, and you will need something much more powerful than conditions like the ones you've listed to describe them. I would actually be pretty surprised to see a clean characterization along these lines.

EDIT: Just as an extra comment; if $\overline{BwB}\subset GL_n$ isn't smooth, then $IH_{B\times B}^*(\overline{BwB})$ (equivariant intersection cohomology) is a Soergel bimodule and $H^*_{B\times B}(\overline{BwB})$ (usual equivariant cohomology) is not. I will believe that a combinatorial characterization is plausible when I can see why it allows the former and rules out the latter, but frankly I have no idea how that's going to happen.

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Thanks, I should have thought of that example. I'm only slightly discouraged by your general pessimism... –  Dylan Thurston Mar 26 '12 at 23:44

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