### Abstract

An element [Φ]∈GrnH+,F of the Grassmannian of n-dimensional subspaces of the Hardy space H+=H2, extended over the field F = C(x_{1}, …, x_{n}), may be associated to any polynomial basis ϕ = {ϕ_{0}, ϕ_{1}, ⋯ } for C(x). The Plücker coordinates Sλ,nϕ(x1,…,xn) of [Φ], labeled 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 {hi(0)} of the complete symmetric functions generates a doubly infinite matrix hi(j) of symmetric polynomials that determine an element [H]∈Grn(H+,F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161-168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials Sλ,nϕ(x1,…,xn) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the x_{a}’s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑λSλ,nϕ([x])Sλ,nθ(t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.

Original language | English |
---|---|

Article number | 091411 |

Journal | Journal of Mathematical Physics |

Volume | 59 |

Issue number | 9 |

DOIs | |

Publication status | Published - Sep 1 2018 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*59*(9), [091411]. https://doi.org/10.1063/1.5051546

**Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions.** / Harnad, J.; Lee, Eunghyun.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 59, no. 9, 091411. https://doi.org/10.1063/1.5051546

}

TY - JOUR

T1 - Symmetric polynomials, generalized Jacobi-Trudi identities and τ-functions

AU - Harnad, J.

AU - Lee, Eunghyun

PY - 2018/9/1

Y1 - 2018/9/1

N2 - An element [Φ]∈GrnH+,F of the Grassmannian of n-dimensional subspaces of the Hardy space H+=H2, extended over the field F = C(x1, …, xn), may be associated to any polynomial basis ϕ = {ϕ0, ϕ1, ⋯ } for C(x). The Plücker coordinates Sλ,nϕ(x1,…,xn) of [Φ], labeled 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 {hi(0)} of the complete symmetric functions generates a doubly infinite matrix hi(j) of symmetric polynomials that determine an element [H]∈Grn(H+,F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161-168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials Sλ,nϕ(x1,…,xn) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the xa’s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑λSλ,nϕ([x])Sλ,nθ(t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.

AB - An element [Φ]∈GrnH+,F of the Grassmannian of n-dimensional subspaces of the Hardy space H+=H2, extended over the field F = C(x1, …, xn), may be associated to any polynomial basis ϕ = {ϕ0, ϕ1, ⋯ } for C(x). The Plücker coordinates Sλ,nϕ(x1,…,xn) of [Φ], labeled 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 {hi(0)} of the complete symmetric functions generates a doubly infinite matrix hi(j) of symmetric polynomials that determine an element [H]∈Grn(H+,F). This is shown to coincide with [Φ], implying a set of generalized Jacobi identities, extending a result obtained by Sergeev and Veselov [Moscow Math. J. 14, 161-168 (2014)] for the case of orthogonal polynomials. The symmetric polynomials Sλ,nϕ(x1,…,xn) are shown to be KP (Kadomtsev-Petviashvili) τ-functions in terms of the power sums [x] of the xa’s, viewed as KP flow variables. A fermionic operator representation is derived for these, as well as for the infinite sums ∑λSλ,nϕ([x])Sλ,nθ(t) associated to any pair of polynomial bases (ϕ, θ), which are shown to be 2D Toda lattice τ-functions. A number of applications are given, including classical group character expansions, matrix model partition functions, and generators for random processes.

UR - http://www.scopus.com/inward/record.url?scp=85053000728&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053000728&partnerID=8YFLogxK

U2 - 10.1063/1.5051546

DO - 10.1063/1.5051546

M3 - Article

VL - 59

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 091411

ER -