Plethystic Murnaghan–Nakayama rule via vertex operators

Abstract Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan–Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of $$s_{\nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> </mml:math> in the plethysm product $$(p_{n}\circ h_{k})s_{\mu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>∘</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> </mml:mrow> </mml:math> . Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.