Shuffle theorem for torus link homology

We prove that the symmetric function $e_{(1^k)}[-MX^{m,n}] \cdot 1$, arising from the elliptic Hall algebra, equals the generating function for $k$-tuples of cyclic $(m,n)$-parking functions. This result resolves a conjecture of Gorsky–Mazin–Vazirani and Wilson, establishing that the elliptic Hall algebra governs the Khovanov–Rozansky homology of torus links $T(km,kn)$. Consequently, this provides an affirmative answer to a question of Galashin and Lam in the torus link case. As a key step in the proof, we develop a rational analogue of the Shareshian–Wachs involution originally introduced to prove the symmetry property of the chromatic quasisymmetric functions.

💡 Research Summary

The paper “Shuffle theorem for torus link homology” by Donghyun Kim and Jaeseong Oh establishes a deep connection between the elliptic Hall algebra (EHA) and the Khovanov–Rozansky (KR) homology of torus links. The central object of study is the symmetric function
\