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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-Sch├╝tzenberger 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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