Geometric realization of $W$-operators

Abstract: Certain integrable hierarchies appearing in random matrix theory, enumerative geometry, and conformal field theory are governed by Virasoro/$W$-algebra constraints and their $W$-representations.Motivated by the Gaussian Hermitian $β$-ensemble and recent studies of superintegrable partition function hierarchies, we build an explicit bridge from symmetric group class algebras to bosonic Fock spaces and further to geometry. On the algebraic side, we decompose the transposition class sum into cut and join channels and recover the classical cut-and-join operator on the ring of symmetric functions. On the geometric side, we use the Grojnowski-Nakajima Fock space identification to realize the ladder operator $E_1=[W_0,p_1]$ as the Hecke correspondence on $\mathrm{Hilb}n(\mathbb C^2)$, and we interpret the cubic generator $W_0$ as a normal ordered triple incidence correspondence. We then explain how the $β$-deformed cubic generator $W_0^{(β)}$ arises from the Ward identities/Virasoro constraints of the Gaussian $β$-ensemble via a background charge parametrization, clarifying its conformal field theoretic meaning. Finally, using the Grojnowski-Nakajima Heisenberg-Fock isomorphism $Φ{\mathrm{Hilb}}:Λ\xrightarrow{\sim}\bigoplus_{n\ge0}H_T^*(\Hilb^n(\mathbb C^2))$, we transport the resulting commutator hierarchy to Hilbert schemes, where $E_1$ is realised by the Hecke correspondence (adding one point) and the diagonal correction terms are computed by equivariant localization from the $T$-weights of the tangent bundle $T\Hilb^n(\mathbb C^2)$ and the tautological bundle $\mathcal V$. This provides a geometric realization framework that unifies $β$-deformed integrable structures and offers new tools for studying quiver gauge theory partition functions.