Nonnegative Hall Polynomials
Lynne M. Butler
and Alfred W. Hales
DOI: 10.1023/A:1022407523839
Abstract
The number of subgroups of type m v l ( p) _{μv}^λ(p) with integral coefficients. We prove g m v l ( p) _{μv}^λ(p) 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.
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.Pages: 125–135
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References
1. G.E. Andrews, "The Theory of Partitions," in Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, Reading, MA, 1976.
2. G. Birkhoff, "Subgroups of abelian groups," Proc. London Math. Soc. (2) 38 (1934/5), 385-401.
3. L.M. Butler, "Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups," Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1986.
4. L.M. Butler, "Order analogues and Betti polynomials," Advances in Math, to appear.
5. L.M. Butler and A.W. Hales, "Generalized flags in p-groups," submitted.
6. P.E. Djubjuk, or P.E. Dyubyuk, "On the number of subgroups of a finite abelian group," Izv, Akad. Nauk. SSSR. Ser. Mat. 12 (1948), 351-378 (translated in Soviet Math., 2 (1961), 298-300).
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12. A.O. Morris, " The characters of the group GL(n, q)," Math. Zeitschr. 81 (1963), 112-123.
13. A.O. Morris, "The multiplication of Hall functions," Proc. London. Math. Soc. (3) 13 (1963), 733-742.
2. G. Birkhoff, "Subgroups of abelian groups," Proc. London Math. Soc. (2) 38 (1934/5), 385-401.
3. L.M. Butler, "Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups," Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1986.
4. L.M. Butler, "Order analogues and Betti polynomials," Advances in Math, to appear.
5. L.M. Butler and A.W. Hales, "Generalized flags in p-groups," submitted.
6. P.E. Djubjuk, or P.E. Dyubyuk, "On the number of subgroups of a finite abelian group," Izv, Akad. Nauk. SSSR. Ser. Mat. 12 (1948), 351-378 (translated in Soviet Math., 2 (1961), 298-300).
7. J.A. Green, "The characters of the finite general linear groups," Thins. Amer. Math. Soc. 80 (1955), 402-447.
8. P. Hall, " The algebra of partitions," Proc. 4th Canadian Math. Congress, Banff, 1957, 147-159.
9. T. Klein, " The Hall polynomial," J. Alg. 12 (1969), 61-78.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, England, 1979.
11. G.A. Miller, "On the subgroups of an abelian group," Ann. Math. 6 (1904), 1-6.
12. A.O. Morris, " The characters of the group GL(n, q)," Math. Zeitschr. 81 (1963), 112-123.
13. A.O. Morris, "The multiplication of Hall functions," Proc. London. Math. Soc. (3) 13 (1963), 733-742.