We study the relations between Blot–Shadrin–Singh tautological relation and the DR formula for Hodge character class \({{\,\textrm{ch}\,}}_{2g-1}(\mathbb {E})\) on \(\overline{\mathcal {M}}_{g}\) . In particular, we prove the equivalence between Blot–Shadrin–Singh tautological relation on \(\overline{\mathcal {M}}_{g,1}\) and the DR formula for Hodge character class \({{\,\textrm{ch}\,}}_{2g-1}(\mathbb {E})\) on \(\overline{\mathcal {M}}_{g}\) . As two applications, we first find a new push-forward tautological relation on \(\overline{\mathcal {M}}_{g+1}\) for \(g\ge 1\) , which extends the Liu-Pandharipande relation to a lower degree. Second, we obtain a new partial differential equation for higher genus descendant Gromov–Witten invariants of any smooth projective variety.
