Abstract
By utilizing the vertex operator formulation of the polynomial Lie algebra \(gl_\infty^{(n)}\) , we develop a representation-theoretic framework for the coupled Toda hierarchy, which is a special matrix-type Toda hierarchy. This approach not only provides a deeper understanding of its algebraic structure but also allows us to derive the Hirota bilinear form for the coupled \(2\) -Toda hierarchy. Starting from the bilinear relation of the wave-function matrix, we construct the associated dressing operator matrix, which in turn enables us to define the Lax operator matrix and derive the corresponding Lax equation.
