# Soergel bimodules

The next ingredient in my paper with Geordie is understanding a bit about Soergel bimodules.

Soergel bimodules (or “Soergelsche Bimoduln” auf Deutsch) are a remarkable category of bimodules over a polynomial ring. The main thing that’s remarkable about them is that they categorify the Hecke algebra of $S_n$ (for those of you who don’t know any other Hecke algebras, pretend I just said “Hecke algebra”).

Theorem (Soergel). The split Grothendieck group of the category of Soergel bimodules for $S_n$ is the Hecke algebra $\mathcal{H}_n$ of $S_n$.

We’ll unpack that a bit later. For the moment, bear with me. Of course, one of the most important things about Grothendieck groups is that they have a natural basis. If one uses a split Grothendieck group (like a normal Grothendieck group, except we only have a relation that says $[M]+[N]=[M\oplus N]$, not for non-trivial extensions), then this basis will be the classes of the indecomposable elements.

Theorem (Soergel). The basis of indecomposable objects is the famous basis of Kazhdan and Lusztig.

Now that I’ve told you why people like Soergel bimodules, I guess I had better define them.

For the sake of notation, let $V$ be the reflection representation of $S_n$ (that is the complement to the obvious invariant in the obvious action of $S_n$ on $\mathbb{C}^n$). Let $R=\mathbb{C}[V]$ be polynomials on $V$, which we can also think of as $\mathbb{C}[x_1,\ldots,x_n]/(x_1+\cdots +x_n)$. This is the ring that the Soergel bimodules for $S_n$ will be modules over.

In a previous post, I gave a very combinatorial definition of these modules:

Definition. The category of Soergel bimodules is the smallest full subcategory of $R-R$-bimodules which is closed under tensor product and summands, which contains $R$ and $R\otimes_{R^{s}} R$ where $s$ ranges over the adjacent transpositions of $S_n$ (Coxeter groups people would say “the simple reflections”).

Now, I suspect the crowd around this blog is most familiar with the Hecke algebra via its connection with quantum invariants. That is, they’ll agree with me, when I say one of the most interesting thing about the Hecke algebra is that there is a natural map $\Phi$ from the braid group $B_n$ to $\mathcal{H}_n$ and a bunch of traces on the Hecke algebra which when composed with this map give knot invariants.

So, of course, of the most interesting things we could do with Soergel bimodules is give categorifications of these. The map $\Phi$ is relatively easy. Its answer has been known for a long time in a slightly different form, as the shuffling action of the braid group on category $\mathcal{O}$.

The avatar of this action on the Soergel bimodules is a complex of Soergel bimodules attached to each braid. This is, slightly unfairly, called the “Rouquier complex” by knot theorists, though perhaps they should consider switching to the more neutral “shuffling complex.” Those who stare at how one writes the action of braid generators in terms of the Kazhdan-Lusztig generators will hopefully conclude that the right answer matches that dreamed up by representation theorists.

Define the shuffling complex $F(\sigma)$ of a braid $\sigma$ by the rules

1. $F(\sigma_i)=R[-2]\longrightarrow R\otimes_{R^{s_i}} R$, the complex given by the unique map of lowest degree (up to scalar).

2. $F(\sigma_i^{-1})= R\otimes_{R^{s_i}} R\longrightarrow R$, similarly.

3. $F(\sigma\tau) \simeq F(\sigma)\otimes_R F(\tau)$.

This complex is well defined up to homotopy (its very important that we use homotopy, rather than expecting equality on the nose, or only paying attention to quasi-isomorphism).

The interested reader can check that this decategorifies to the usual map of the braid group to $\mathcal{H}_n$.

Now, what’s the trace? Well, as I discussed before, the trace should be Hochschild homology. This should decategorify to a trace on $\mathcal{H}_n$ which lands in Laurent polynomials in two variables, which is, of course, the celebrated trace of Ocneanu-Jones. (The traces coming from the quantum trace of a tensor power of the standard representation of $U_q(\mathfrak{sl}_m)$, all of which are specializations of the Ocneanu-Jones trace, can be categorified by slightly more complicated functors involving matrix factorizations.

Those of you who have been following, and know your knot theory well, may well have beaten me the punchline here: if we compose the natural map from the braid group to the Hecke algebra with the Ocneanu-Jones trace, then we end up with the HOMFLYPT polynomial of the closure of that braid (up to a normalization). Thus, if we compose the categorifications of these objects, we end up with a bigraded chain complex, which we’ll call $G(\sigma)$ (Hochschild homology applied termwise to the shuffling complex) whose Euler characteristic is a knot invariant. The obvious guess is that the homology of this complex is in fact a knot invariant.

Theorem. (Khovanov-Rozansky) The homology of $G(\sigma)$ is a knot invariant (after taking care of a grading shift), categorifying the HOMFLYPT polynomial (sometimes called “triply-graded homology” or “HOMFLY homology”).

Now, this is all great, except that these Soergel bimodules are really mysterious objects. All sorts of obvious questions about them turn out to be equivalent to hard problems in representation theory (for example, if one really understood indecomposable Soergel bimodules, one would know every Kazhdan-Lusztig polynomial, not something likely to happen to any mere mortals in our lifetime). Their Hochschild homology is not even one of the many questions about these modules that people had gotten around to attacking yet. There are probably strong limits to the amount that we will ever know in general about Hochschild homology of Soergel bimodules, but it’s not as mysterious as one might fear at first, and in particular has a nice geometrical interpretation, which will have to wait for another post.

OK, this concludes part two. Next time, I promise there actually will be geometry (and content from my research, instead of just other people’s).