Intrinsic characterization of Soergel bimodules? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:34:34Z http://mathoverflow.net/feeds/question/92221 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92221/intrinsic-characterization-of-soergel-bimodules Intrinsic characterization of Soergel bimodules? Dylan Thurston 2012-03-26T02:34:05Z 2012-06-18T08:20:14Z <p>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:</p> <p>(1) $M$ is free as a left module or as a right module, although not necessarily as a bimodule.</p> <p>(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 $$pm = mp.$$</p> <p>I think they also have the following property:</p> <p>(3) There is an invariant vector, an element $m_0 \in M$ so that $$x_i m_0 = m_0 x_i$$ for every $i=1,\dots,n$.</p> <p>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 $\sigma$.</p> <p><strong>Edit:</strong> 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).</p> <p>Any references are welcome. If it's not known, I'll try to prove it.</p> <p><strong>Edit:</strong> Ben Webster gave a counterexample below. More generally, I'm still interested in some sort of intrinsic, elementary characterization.</p> http://mathoverflow.net/questions/92221/intrinsic-characterization-of-soergel-bimodules/92314#92314 Answer by Ben Webster for Intrinsic characterization of Soergel bimodules? Ben Webster 2012-03-26T22:10:19Z 2012-03-28T16:31:15Z <p>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. </p> <p>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.</p> <p>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.</p> <p>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.</p> <p><strong>EDIT:</strong> Just as an extra comment; if $\overline{BwB}\subset GL_n$ isn't smooth, then <code>$IH_{B\times B}^*(\overline{BwB})$</code> (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.</p> http://mathoverflow.net/questions/92221/intrinsic-characterization-of-soergel-bimodules/99880#99880 Answer by Geordie Williamson for Intrinsic characterization of Soergel bimodules? Geordie Williamson 2012-06-18T08:20:14Z 2012-06-18T08:20:14Z <p>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 ...</p> <p>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:</p> <p><code>$Gr_w = \{ (wv, v) \;| \;v \in V \}$</code></p> <p>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 $\Gamma_I M \subset M$.</p> <p>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</p> <p>$\Gamma_{\le w / &lt; w} (M) := \Gamma_{\le w} M / \Gamma_{&lt; w} M$</p> <p>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".</p> <p>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 $\Gamma_{\le w / &lt; w}$ one obtains a split exact sequence of $R$-bimodules (necessarily isomorphic to direct sums of shifts of $R_w$'s).</p> <p>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:</p> <p>arxiv.org/abs/math.RT/0505108</p> <p>(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.)</p> <p>By the way, the condition that the subsequent quotients in this filtration be split has other applications. In this paper</p> <p><a href="http://arxiv.org/abs/1205.4206" rel="nofollow">http://arxiv.org/abs/1205.4206</a></p> <p>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.</p>