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### Description

The chromatic symmetric function X_G is a multivariable generalization of the chromatic polynomial of a graph G. The goal of this project is to study the `categorification' of X_G; that is, define a homology theory whose graded Frobenius characteristic can recover the polynomial X_G. To achieve this, we use techniques from Khovanov homology for knots, which has been proven to be a powerful combinatorial tool for knot theorists.
To date, in joint work with R. Sazdanovic, we have successfully defined the construction for such a homology of graded S_n-modules, such that Frob_G(q,t)=X_G at the specialization q=t=1.
There are many open questions which remain to be investigated:
1) Does there exist non-isomorphic graphs whose chromatic symmetric homology are identical? We have candidates for such examples, but computations have been difficult to carry out. This leads to the second problem.
2) Are there computational short-cuts for $H_*^{cs}$?
3) Is the homology a complete invariant for trees? Originally, Khovanov homology was invented to attack the problem of whether the Jones polynomial can detect the unknot. While that question is still open, it is now known that Khovanov homology does detect the unknot. There is a similar parallel in graph theory. It is conjectured that the polynomial X_G is a complete invariant for trees. We would like to show that the homology is a complete invariant for trees.
4) Can the homology shed light on the e-positivity conjecture for X_G? At the level of homology, the categorified version of the elementary symmetric functions appear as the top graded modules in the chain complex.
5) What is the connection between our theory, and the t-deformation of X_G defined by Shareshian and Wachs?
Of course, there are many possible ways that one can categorify a polynomial, but we have been able to show that perhaps our way of lifting the polynomial is a `correct' one as we have been able to lift a known recursive formula for X_G, due independently to Guay-Paquet/Orellana and Scott, into a long exact sequence of homology. This formula greatly simplifies the computation of homology, which is a step towards addressing problem (2) above.
Second, in independent work, I have been able to define a cohomological version, of the categorification of X_G. Compared to the original homological construction, it is simpler (computations rely on fewer natural transformations) and lends itself better to modular representation theory.
Current ongoing work is to build yet another computational model for cohomology, based on the broken-circuit complex for G. I have obtained partial results towards this end. A successful implementation of this would be a good step towards answering problem (3).

Status | Finished |
---|---|

Effective start/end date | 9/1/16 → 8/31/23 |

### Funding

- Simons Foundation

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## Projects

- 1 Finished