42 research outputs found

    Raising operators and the Littlewood-Richardson polynomials

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    We use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates hr,sh_{r,s} given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of hr,sh_{r,s}, 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

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    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
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