Ribbon Operators and Hall-Littlewood Symmetric Functions

Details Ribbon Operators and Hall-Littlewood Symmetric Functions Ribbon Operators and Hall-Littlewood Symmetric Functions

2000-08-22

arxiv.org/abs/math/0008163v1

Advances in Mathematics, 156, No. 1 (2000), pp.33-43

Mathematics

Quantum Algebra

10 pages; to appear Advances in Mathematics

Scientific paper

Given a partition $\la = (\la_1, \la_2, ... \la_k)$, let $\la^{rc} = (\la_2-1, \la_3-1, ... \la_k-1)$. It is easily seen that the diagram $\la\slash \la^{rc}$ is connected and has no $2 \times 2$ subdiagrams which we shall refer to as a ribbon. To each ribbon $R$, we associate a symmetric function operator $S^R$. We may define the major index of a ribbon $maj(R)$ to be the major index of any permutation that fits the ribbon. This paper is concerned with the operator $H_{1^k}^q = \sum_R q^{maj(R)} S^R$ where the sum is over all $2^{k-1}$ ribbons of size $k$. We show here that $H_{1^k}^q$ has truly remarkable properties, in particular that it is a Rodriguez operator that adds a column to the Hall-Littlewood symmetric functions. We believe that some of the tools we introduce here to prove our results should also be of independent interest and may be useful to establish further symmetric function identities.

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