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Ribbon Operators and Hall–Littlewood Symmetric Functions

✍ Scribed by Mike Zabrocki

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
185 KB
Volume
156
Category
Article
ISSN
0001-8708

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✦ Synopsis

Given a partition *=(* 1 , * 2 , ..., * k ), let * rc =(* 2 &1, * 3 &1, ..., * k &1). It is easily seen that the diagram *Â* rc is connected and has no 2_2 subdiagrams, we shall call it 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 q 1 k = R q maj(R) S R where the sum is over all 2 k&1 ribbons of size k. We show here that H q 1 k has truly remarkable properties, in particular, 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 for establishing further symmetric function identities.

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