Rational Cherednik Algebras and Knot Invariants: Filtrations

Abstract In [19], the HOMFLY polynomial of the $(m,n)$ torus knot $T_{m,n}$ is extracted from the hook isotypic components within the graded character of the finite-dimensional Cherednik algebra representation $\text{L}_{\frac{m}{n}}$. It is furthermore conjectured in loc. cit. that one can obtain the triply-graded Khovanov–Rozansky homology of $T_{m,n}$ by considering a certain filtration on $\text{L}_{\frac{m}{n}}$. In this paper, we show that two of the proposed candidates, the algebraic filtration and the inductive filtration, are equal. Our proof is combinatorial and elementary, transforming the filtration identity into a linear algebra problem on coinvariant algebras.