42 research outputs found
Raising operators and the Littlewood-Richardson polynomials
We use Young's raising operators to derive a Pieri rule for the ring
generated by the indeterminates given in Macdonald's 9th Variation of
the Schur functions. Under an appropriate specialisation of , we
derive the Pieri rule for the ring \La(a) of double symmetric functions,
which has a basis consisting of the double Schur functions. Together with a
suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us
to obtain a new proof of a rule to calculate the Littlewood--Richardson
polynomials, which gives a multiplication rule for the double Schur functions
Asymmetric function theory
The classical theory of symmetric functions has a central position in
algebraic combinatorics, bridging aspects of representation theory,
combinatorics, and enumerative geometry. More recently, this theory has been
fruitfully extended to the larger ring of quasisymmetric functions, with
corresponding applications. Here, we survey recent work extending this theory
further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the
Schubert calculus conference in Guangzhou, Nov. 201