The ordered exponential representation of GKM using the $W_{1+\mathrm{\infty }}$ operator

The generalized Kontsevich model (GKM) is a one-matrix model with arbitrary potential. Its partition function belongs to the KP hierarchy. When the potential is monomial, it is an $r$-reduced tau-function that governs the $r$-spin intersection numbers. In this paper, we present an ordered exponential representation of monomial GKM in terms of the $W_{1+\mathrm{\infty }}$ operators that preserves the KP integrability. In fact, this representation is naturally the solution of a $W_{1+\mathrm{\infty }}$ constraint that uniquely determines the tau-function. Furthermore, we show that, for the cases of Kontsevich-Witten and generalized BGW tau-functions, their $W_{1+\mathrm{\infty }}$ representations can be reduced to their cut-and-join representations under the reduction of the even time independence and Virasoro constraints.