Nonnegative Hall polynomials

Butler, Lynne M. ; Hales, Alfred W.

Journal of Algebraic Combinatorics, Tome 2 (1993) no. 2, pp. 125-135.

Résumé

Summary: The number of subgroups of type $_{ m v} ^{ l} ( p)$ _muv^$\lambda (p)$ with integral coefficients. We prove g $_{ m v} ^{ l} ( p)$ _muv^$\lambda (p)$ has nonnegative coefficients for all partitions $mgr$ and $ngr$ if and only if no two parts of $lambda$ 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 $\varphi$ that associate to each subgroup a vector dominated componentwise by $lambda$. The nonzero components of $\varphi( H)$ are the parts of $mgr$, the type of $H$; if no two parts of $lambda$ differ by more than one, the nonzero components of $lambda - \varphi( H)$ are the parts of $ngr$, 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.