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On the number of partition weights with Kostka multiplicity one
Gates, Zachary Goldman, Brian Vinroot, C. Ryan
Gates, Zachary
Goldman, Brian
Vinroot, C. Ryan
Abstract
Given a positive integer n, and partitions lambda and mu of n, let K-lambda mu denote the Kostka number, which is the number of semistandard Young tableaux of shape lambda and weight mu. Let J(lambda) denote the number of mu such that K lambda mu = 1. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for J(lambda) in terms of restricted partition functions, which is recursive in the number of distinct part sizes of lambda. We use this to classify all partitions lambda such that J(lambda) = 1 and all lambda such that J(lambda) = 2. We then consider signed tableaux, where a semistandard signed tableau of shape lambda has entries from the ordered set {0 < < 1 < (2) over bar < 2 < ...}, and such that i and (i) over bar contribute equally to the weight. For a weight (omega(0), mu) with mu a partition, the signed Kostka number K-lambda(omega 0, mu)(+/-) is defined as the number of semistandard signed tableaux of shape lambda and weight (omega(0), mu), and J(+/-)(lambda) is then defined to be the number of weights (omega(0), mu) such that K-lambda(omega 0,mu)(+/-) = 1. Using different methods than in the unsigned case, we find that the only nonzero value which J((lambda))(+/-) can take is 1, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.
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2012-01-01
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Mathematics