Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions

Abstract

An element [Φ] ∈ Gr n (H + , F) of the Grassmannian of n-dimensional subspaces of the Hardy space H + = H 2 , extended over the field F = C(x 1 , . . . , x n ), may be associated to any polynomial basis, labelled by partitions λ, provide an analog of Jacobi's bi-alternant formula, defining a generalization of Schur polynomials. Applying the recursion relations satisfied by the polynomial system φ to the analog {h (0) i } of the complete symmetric functions generates a doubly infinite matrix h (j) * Work of J.H. supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds Québecois de la recherche sur la nature et les technologies (FQRNT).

Vertex Operators Arising from Jacobi–Trudi Identities

“…It seems that this simple fact is generally overlooked in the literature, though several papers can be named, where versions of Jacobi-Trudi identities are applied: (see e.g. [1,11] where variations of Jacobi -Trudi identities are used for construction of τ -functions). Note that, as we show below, the so-called Giambelli identity in a general form is equivalent to a general Jacobi -Trudi identity statement.…”

ABJM matrix model and 2D Toda lattice hierarchy

“…if we denote the Young diagram by λ = (α 1 , · · · , α r |β 1 , · · · , β r ). These formulas were originally found for the Schur polynomial, though it turns out that they appear naturally in integrable systems [17][18][19][20]. Hence, in this context, it is natural to ask what the integrable structure for the ABJM matrix model is.…”

“…After introducing the grand canonical ensemble 18) it was found that the normalized grand canonical two-point functions…”

“…As mentioned above, these formulas are famous in the context of solvable systems. In fact, in the study of soliton systems or solvable lattice systems, similar formulas appear (see, for example, [17][18][19][20]). Also, in [21,22] in a series of variations of the Schur function, as a final one called the ninth variation, the Jacobi-Trudi formula was regarded as the definition of the Schur function and many interesting properties such as the Giambelli formula can still be derived from this simple setup.…”

Jacobi-Trudi Identity in Super Chern-Simons Matrix Model

“…with H = H M =0 . 4 In [29] the Jacobi-Trudi identity is defined also by extending the matrix size in (1.10) to be greater than L. Since this extension does not make any differences essentially, we restrict the matrix size to be L. The Jacobi-Trudi identity defined with the automorphism ϕ plays an important role in the study of integrable models and is also called the quantum Jacobi-Trudi identity (see [4,12,23,26,27,42,43,48] for examples).…”