Nonnegative Hall Polynomials |
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Authors: | Lynne M Butler Alfred W Hales |
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Institution: | (1) Department of Mathematics, Haverford College, Haverford, PA, 19041;(2) Department of Mathematics, University of California, Los Angeles, 90024 |
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Abstract: | The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg
with integral coefficients. We prove g
has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients. |
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