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In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials.
The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups. Ringel (1990) generalized Hall algebras to more general categories, such as the category of representations of a quiver. Construction
Mar 20, 2023 · Hall polynomials are Lie polynomials obtained from elements of a given Hall set. They furnish a basis of the free Lie algebra over a (finite or infinite) set of generators $\{a_1,a_2,\ldots\}$. Elements of a Hall set $H$ may be seen as completely bracketed words (or rooted planar binary trees with leaves labelled by generators $a_1,a_2,\ldots$.
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables.
Symmetric functions and Hall polynomials / I. G. Macdonald. - 2nd ed. (Oxford mathematical monographs) Includes bibliographical references and index. 1. Abelian groups. 2. Finite groups. 3. Hall polynomials. 4. Symmetric functions. L Title. 11. Series. QA180.M33 1995 512'.2-dc2O 94-27392 CIP ISBN 0 19 853489 2 h/b 0 19 850450 0 p/b Typeset by
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3 days ago · Also let be the number of parts of of size . Then the permutation , where is the symmetric group, acts on the variables , ..., by sending to . Letting be a complex number, the Hall-Littlewood polynomials are defined by. These polynomials interpolate between the Schur functions (with ) and the monomial symmetric functions (with ; Fulman 1999).
The Hall polynomial \(P^{ u}_{\mu,\lambda}(q)\) (in the indeterminate \(q\)) is defined as follows: Specialize \(q\) to a prime power, and consider the category of \(\GF{q}\)-vector spaces with a distinguished nilpotent endomorphism. The morphisms in this category shall be the linear maps commuting with the distinguished endomorphisms.
