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Subgroup lattices and symmetric functions. (English) Zbl 0813.05067
Mem. Am. Math. Soc. 539, 160 p. (1994).
Let $$G$$ be an Abelian group of type $$\lambda$$, i.e. $$G\simeq \mathbb Z/(p^{\lambda_ 1}\times\cdots\times \mathbb Z/(p^{\lambda_ I})$$. The famous Hall polynomial $$g^{\lambda}_{\mu,\nu}(p)$$ counts the number of exact sequences $$0\to H\to G\to G/H\to 0$$, where $$H$$ has type $$\mu$$ and cotype $$\nu$$. It is well known that the Hall polynomial is defined over $$Z$$. The paper under review shows that a necessary and sufficient condition for the Hall polynomial $$g^{\lambda}_{\mu,\nu}(p)$$ always to have nonnegative coefficients is that no two parts of $$\lambda$$ differ by more than one, i.e. $$\lambda= i^ a(i+ 1)^ b$$, $$i\in \mathbb N$$. The author first uses a generalization of Knuth’s study of subgroup lattices to obtain some combinatorial formula for the Hall polynomial $$g^{\lambda}_ {\mu,\nu}(p)$$ in this case. She then employs the theory of Hall-Littlewood polynomials (cf. I. G. Macdonald [Symmetric functions and Hall polynomials. Oxford: Clarendon Press (1979; Zbl 0487.20007)]) to attack the necessary condition, where she also studies and examines the Lascoux-Schützenberger proof of the nonnegativity for the $$p$$-Kostka polynomial. Finally the author gives a conjecture on the relation between two $$(q,t)$$-polynomials $$K_{\lambda,\mu}$$ and $$K_{\lambda,\nu}$$, which are defined by the Macdonald polynomial as in the case of the Hall-Littlewood polynomial.

##### MSC:
 05E05 Symmetric functions and generalizations 20D30 Series and lattices of subgroups 20K01 Finite abelian groups 06A11 Algebraic aspects of posets 11B65 Binomial coefficients; factorials; $$q$$-identities
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