Let G be a finitely generated torsion free nilpotent group (T-group for short). Choose g_1, g_2, ..., g_n ∈ G such that the subgroups G_i := ⟨ g_i, ..., g_n⟩ form a central series for G with infinite cyclic factors. n is unique and is called the Hirsch-length of G. (g_1, ..., g_n) is called a T-basis for G. For every h ∈ G there exist unique e_1, ..., e_n ∈ ℤ with h = g_1^e_1 ⋯ g_n^e_n; this is called the normal form of h (w.r.t. g_1, ..., g_n). Hence there are functions F_i : ℤ^n × ℤ^n -> ℤ and K_i : ℤ^n × ℤ -> ℤ for 1 le i le n with
(g_1^{a_1} \cdots g_n^{a_n}) \cdot (g_1^{a_1} \cdots g_n^{a_n}) = g_1^{F_1(a,b)} \cdots g_n^{F_n(a,b)}
and
(g_1^{a_1} \cdots g_n^{a_n})^x = g_1^{K_1(a,x)} \cdots g_n^{K_n(a,x)}
for all a = (a_1, ..., a_n), b = (b_1, ..., b_n) ∈ ℤ^n and x ∈ ℤ. Philip Hall showed in [Hal57] that the functions F_i and K_i can be described as rational polynomials in 2n resp. n+1 indeterminates for 1 le i le n. Hence the functions F_i and K_i are called Hall-polynomials. In particular the polynomials F_i are called multiplication polynomials and the polynomials K_i are called power polynomials.
The purpose of the HallPoly Package is to compute Hall-polynomials for arbitrary T-groups of arbitrary Hirsch-length. To do so, the T-groups need to be uniformly presented. Assume G is a T-group and (g_1, ..., g_n) a T-basis for G. Then there is a unique t = (t_i,j,k | 1 le i < j < k le n) ∈ ℤ^{n choose 3} so that
g_j g_i = g_i g_j g_{j+1}^{t_{i,j,j+1}} \cdots g_n^{t_{i,j,n}}
for all 1 le i < j < n. One can show that G has the presentation
G(t) := \langle g_1, \ldots, g_n \ \vert \ g_j g_i = g_i g_j g_{j+1}^{t_{i,j,j+1}} \cdots g_n^{t_{i,j,n}} \ (1 \le i < j \le n)\rangle.
Hence it is possible to describe any T-group G of Hirsch-length n by n choose 3 parameters t_i,j,k ∈ ℤ.
Vice versa any group defined by a presentation of the form G(t) for t ∈ ℤ^{n choose 3} is finitely generated and nilpotent with a Hirsch-length less or equal n. The group is torsion free (and therefore a T-group) iff it has Hirsch-length n.
The presentation G(t) with t ∈ ℤ^{n choose 3} is called consistent, if the group defined by G(t) has Hirsch-length n. The previous arguments imply that each T-group can be described via a consistent presentation. Hence we consider only consistent presentations.
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