Rational Cherednik Algebras and Knot Invariants: Filtrations

In [19], the HOMFLY polynomial of the $(m,n)$ torus knot Tm,n is extracted from the hook isotypic components within the graded character of the finite-dimensional Cherednik algebra representation Lm/n. It is furthermore conjectured in loc. cit. that one can obtain the triply-graded Khovanov–Rozansky homology of Tm,n by considering a certain filtration on Lm/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.