Jacobi-Trudi Identity in Super Chern-Simons Matrix Model

Abstract

It was proved by Macdonald that the Giambelli identity holds if we define the Schur functions using the Jacobi-Trudi identity. Previously for the super Chern-Simons matrix model (the spherical one-point function of the superconformal Chern-Simons theory describing the worldvolume of the M2-branes) the Giambelli identity was proved from a shifted version of it. With the same shifted Giambelli identity we can further prove the Jacobi-Trudi identity, which strongly suggests an integrable structure for this matrix model.

Hanany-Witten transition in quantum curves

“…The expression keeping the Fredholm determinant (2.17) seems more elegant which leads [16] to remove the role of the matrix models and propose a conjecture between spectral theories and topological strings. We stress however that, from the viewpoint of matrix models, the open string formalism is more efficient and allows us to compute various rank deformations (and reveal some integrable structures [36][37][38][39][40]).…”

ABJM matrix model and 2D Toda lattice hierarchy

“…Using this relation, subsequently we can prove the original Giambelli formula [14,15] and the Jacobi-Trudi formula [16]. As mentioned above, these formulas are famous in the context of solvable systems.…”

“…In this notation (see figure 1), we shift the diagonal by M to the right if M is a positive integer or by |M| to the left if M is a negative integer. Then we can define a similar Frobenius notation using the shifted diagonal [13,16]. For this shifted Frobenius notation let us use the uppercase Latin characters in the following.…”

“…The idea of introducing extra auxiliary lengths may look artificial at first sight. Using this shifted Frobenius notation, however, as we review in the next subsection, we can define the shifted Giambelli relation and with the relation we can prove the original Giambelli formula [15] and the Jacobi-Trudi formula [16].…”

“…There are some properties irrelevant to the explicit form of H ( A| B) . For example, with the expression of (2.12) and (2.13) we can prove the Jacobi-Trudi formula [16]…”