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| Hall–Littlewood polynomials | |
|---|---|
| Name | Hall–Littlewood polynomials |
| Field | Algebraic combinatorics |
| Introduced | 1960s |
| Key people | Philip Hall; Dudley E. Littlewood |
Hall–Littlewood polynomials are a family of symmetric functions arising in algebraic combinatorics that interpolate between Schur functions and monomial symmetric functions via a parameter t. They were introduced in connection with the representation theory of finite groups and classical groups and have deep links to symmetric function theory, algebraic geometry, and special functions.
The standard definition uses a partition λ and variables x_1, x_2, ..., with a parameter t; the polynomial is denoted P_λ(x;t) and is normalized relative to Q_λ(x;t). The construction builds on the theory developed by Philip Hall and Dudley E. Littlewood and employs operations related to the symmetric group and Young tableau combinatorics; it also uses inner products similar to those in the work of Issai Schur and Alfred Young. Notation is compatible with bases such as the Schur functions s_λ, the monomial symmetric functions m_λ, and the power-sum symmetric functions p_λ, and normalization constants reflect connections with Hall algebra structures and the Macdonald polynomials framework developed by Ian G. Macdonald.
Hall–Littlewood polynomials satisfy triangularity and orthogonality relations with respect to a Hall inner product inspired by Philip Hall and later formalized in the context of Roger Howe-style dualities. Symmetry under the symmetric group action on variables, branching rules analogous to those for GL_n representations, and transition matrices to bases like the Schur functions are fundamental. Cauchy identities relate Hall–Littlewood generating series to bilinear forms reminiscent of identities connected to Isaac Newton-type symmetric function relations and to classical results tied to Frobenius character theory. The polynomials exhibit plethystic behavior seen in works related to Alain Lascoux and satisfy conjugation properties linked to James Arthur-style involutions in representation theory.
At t=0 the polynomials reduce to Schur functions s_λ, a cornerstone in the representation theory of GL_n and in the study of Young tableau; at t=1 they give the monomial symmetric functions m_λ (up to normalization) connecting to combinatorial bases studied by MacMahon. The Hall–Littlewood family interpolates toward Jack polynomials under certain limits explored by Henry Jack and connects to zonal polynomials that appear in the harmonic analysis on symmetric spaces considered by Harish-Chandra and Élie Cartan. Degenerations link to specialized cases in the Macdonald polynomials hierarchy influential in the work of George Lusztig and Igor Frenkel.
Combinatorial models involve fillings of Young diagrams and charge statistics developed in the literature of A. N. Kirillov and M. Kashiwara, with expansions in terms of tableaux yielding Kostka–Foulkes polynomials K_{λμ}(t). These polynomials count weighted tableau-like objects tied to the Robinson–Schensted–Knuth correspondence and to crystal bases studied by Masaki Kashiwara, with connections to Lascoux–Schützenberger charge and the work of Louis J. Billera. Combinatorial formulas relate to graded characters in contexts explored by Richard Stanley and enumerate structures appearing in enumerative studies by Gian-Carlo Rota and Paul Erdős-adjacent combinatorial frameworks.
Hall–Littlewood polynomials appear as characters of graded pieces in the cohomology of flag varieties and affine Grassmannians studied by Alexander Beilinson, Victor Ginzburg, and George Lusztig. They encode point-counting data over finite fields in settings investigated by André Weil and Pierre Deligne, linking to Hall algebras and to the representation theory of finite groups of Lie type developed by Roger Howe and Jean-Pierre Serre. Connections to geometric representation theory involve perverse sheaves and intersection cohomology in the style of Joseph Bernstein and geometric Satake equivalence as formulated by Ivan Mirković and Kathy Vilonen; they also appear in the study of affine Hecke algebras in works by I. N. Bernstein and Stephen Donkin.
Practical applications include computation of graded multiplicities in representations of GL_n and enumeration problems in algebraic combinatorics featured in the work of Richard Stanley and Bertram Kostant. Examples often show expansions of P_λ(x;t) into Schur functions with coefficients given by Kostka–Foulkes polynomials studied by D. E. Littlewood and Rosemary A. Bailey, and specializations yield character formulas relevant for modular representation theory and for counting points on varieties over finite fields in research inspired by John Tate and Alexander Grothendieck. Concrete illustrations appear in classical expositions by Ian G. Macdonald and in later surveys by scholars associated with institutions such as Cambridge University and Institute for Advanced Study.
Category:Symmetric functions