In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions.
Specializing q to 1, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.
They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials.
Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions.
They can be expanded in terms of Schur functions, and the coefficients Kλμ(q,t) of these expansions are called Kostka–Macdonald coefficients.
In fact, we can obtain in this manner the Schur functions, the Hall–Littlewood symmetric functions, the Jack symmetric functions, the zonal symmetric functions, the zonal spherical functions, and the elementary and monomial symmetric functions.
Various ways of counting Young tableaux have been explored and lead to the definition of and identities for Schur functions.
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