Hook variables: Cut-and-join operators and τ-functions

Abstract

Young diagrams can be parameterized with the help of hook variables, which is well known, but never studied in big detail. We demonstrate that this is the most adequate parameterization for many physical applications: from the Schur functions, conventional, skew and shifted, which all satisfy their own kinds of determinant formulas in these coordinates, to KP/Toda integrability and related basis of cut-and-join W -operators, which are both actually expressed through the single-hook diagrams. In particular, we discuss a new type of multi-component KP τ -functions, Matisse τ -functions. We also demonstrate that the Casimir operators, which are responsible for integrability, are single-hook, with the popular basis of "completed cycles" being distinguished by especially simple coefficients in the corresponding expansion. The Casimir operators also generate the q = t Ruijsenaars Hamiltonians. However, these properties are broken by the naive Macdonald deformation, which is the reason for the loss of KP/Toda integrability and related structures in q-t matrix models. * mironov@lpi.ru; mironov@itep.ru † morozov@itep.ru also related to the skew characters and show up not-quite-expectedly in the differential-expansion coefficients for twist and double-braid knots [14]. They also provide non-trivial solutions to the Plücker relations and thus provide new types of KP τ -functions [?], somewhat closer to the partition functions of the rainbow tensor models [15][16][17]. Last but not least, the Casimir operators are essentially single-hook, and they generate the ordinary τ -functions, this is what makes these latter distinguished among the Hurwitz τ -functions, which do not yet possess any clear description in terms of Hirota-like equations.This broad variety of applications makes hook representations for the Schur functions an interesting subject, which should be further explored. This paper is just the first step on this way.The paper is organized as follows. Sections 2 and 3 provide a preliminary preparation for the main part of the text, we discuss determinant representations of the Schur functions (sec.2) and of the skew Schur functions (sec.3) in terms of the hook Schur functions. Section 4 contains a review of the relations between KP τfunctions and linear and bilinear combinations of the Schur functions. In section 5, we discuss τ -functions that are multilinear combinations of the Schur functions χ R . In section 6, we discuss determinant representations of the shifted Schur functions [13] in terms of the hook Schur and shifted Schur functions. We also notice that the shifted Schur functions satisfy the Hirota bilinear equations. In sections 7 and 8, we discuss the generalized cut-and-joinŴ -operators [6,18] and their representation in terms of hooks [19]. In particular, we discuss that, at each level, there is only one (up to a linear combination of lower level operators) cut-and-joinŴ -operator that involves only single-hook Schur functions. At last, in section 9, we explicitly construct a simple basis of these ope...

Connection between cut-and-join and Casimir operators

“…> β l > 0 so that, e.g., µ = [3, 2, 1] = (3, 1|3, 1). In these variables, the shifted Schur polynomials are given [29] by…”