A
finiteabelianp-groupM is a direct sum of
cyclicp-power components where
is a
partition of called the type of M. Let be the number of subgroups N of M such that N has type and the quotient M/N has type . Hall proved that the functions g are
polynomial functions of p with integer coefficients. Thus we may replace p with an indeterminate q, which results in the Hall polynomials
Hall next constructs an
associative ring over , now called the Hall algebra. This ring has a basis consisting of the symbols and the structure constants of the multiplication in this basis are given by the Hall polynomials:
It turns out that H is a commutative ring, freely generated by the elements corresponding to the
elementary p-groups. The linear map from H to the algebra of
symmetric functions defined on the generators by the formula
Schiffmann, Olivier (2012), "Lectures on Hall algebras", Geometric methods in representation theory. II, Sémin. Congr., vol. 24-II, Paris: Soc. Math. France, pp. 1–141,
arXiv:math/0611617,
Bibcode:
2006math.....11617S,
MR3202707