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.

Reviewer: Naihuan Jing (Raleigh)

##### 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 |