GAP (DeepThought) - Chapter 1: The Deep Thought algorithm

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1 The Deep Thought algorithm
1.1 Category DTObj

### 1 The Deep Thought algorithm

Polycyclic and, especially, finitely generated nilpotent groups exhibit a rich structure allowing a special approach towards multiplication using polynomials. The so-called Deep Thought algorithm introduced in [LGS98] computes these polynomials in practice for a suitable presentation of a finitely generated nilpotent group. Such a presentation is of the following form

\langle a_1, \ldots, a_n \mid a_i^{s_i} = a_{i+1}^{c_{i, i, i+1}} \cdots a_n^{c_{i, i, n}}, 1 \leq i \leq n, a_j a_i = a_i a_j a_{j+1}^{c_{i, j, j+1}} \cdots a_n^{c_{i, j, n}}, 1 \leq i < j \leq n \rangle

with s_i \in \mathbb{N} \cup \{ \infty \} and c_{i, j, k} \in \mathbb{Z}. If s_i = \infty, then the power relation a_i^{s_i} is skipped.

Let G denote the group presented by the above presentation. Then every element in G can be written uniquely in a so-called normal form. That is, if G_i := \langle a_i, \ldots, a_n \rangle and r_i := | G_i : G_{i+1}|, 1 \leq i \leq n, are the relative orders, then for certain e_i \in \mathbb{Z} each element can be written as

a_1^{e_1} \cdots a_n^{e_n}

with 0 \leq e_i < r_i if r_i < \infty. A presentation is called confluent if and only if s_i = r_i for all 1 \leq i \leq n. If a presentation is not confluent, not all functions provided in this package are applicable, see function `DTP_DTapplicability`. For more theoretical background see for example the documentation of the GAP package Polycyclic.

The Deep Thought algorithm computes rational polynomials f_1, \ldots, f_n in 2n indeterminates such that if x := a_1^{x_1} \cdots a_n^{x_n} and y := a_1^{y_1} \cdots a_n^{y_n} are two elements (in normal form or not with x_1, \ldots, x_n, y_1, \ldots, y_n \in \mathbb{Z}), then their product xy is given by

a_1^{f_1(x_1, \ldots, x_n, y_1, \ldots, y_n)} \cdots a_n^{f_n(x_1, \ldots, x_n, y_1, \ldots, y_n)}.

If the collector is confluent, also the normal form of the product can be computed. Otherwise this is not possible. Moreover, there exists a second version of the Deep Thought algorithms which computes n^2 polynomials f_{rs}, 1 \leq r, s \leq n, suitable for multiplications of the form

(a_1^{x_1} \cdots a_n^{x_n}) \cdot a_s^{y_s} = a_1^{f_{1s}(x_1, \ldots, x_n, y_s)} \cdots a_n^{f_{ns}(x_1, \ldots, x_n, y_s)}

for 1 \leq s \leq n. Each general multiplication can be expressed using these special multiplications. Depending on the presentation, polynomials of one version may be more efficient for computations than the polynomials of the other version. For a suggestion on which polynomials to use for a given presentation, see `DTP_DTapplicability`. In the following, Deep Thought type f_r refers to the Deep Thought algorithm which uses n polynomials and type f_{rs} refers to the Deep Thought algorithm using n^2 polynomials.

In order to work with the Deep Thought functions, the group presentation is expected to be given as a collector `coll`, as defined in the GAP package Polycyclic. Moreover, the Polycyclic package introduces the structure of exponent vectors `expvec`, which will be used also in this package to represent group elements. In the following text, a group element a_1^{x_1} \cdots a_n^{x_n} is identified with its exponent vector in form of the list `[x_1, ..., x_n]`. For example, if `expvec1, expvec2` are exponent vectors of elements in the same group, then `expvec1 * expvec2` describes the multiplication of the corresponding group elements, and so on. Note that generally exponent vectors are not assumed to represent normal forms.

#### 1.1 Category DTObj

This package uses the category `DTObj`. A `DTObj` is a `IsFromTheLeftCollectorRep` with certain further list entries to store the Deep Thought polynomials of a collector together with some additional information. That is, the functions `DTP_DTpols_r` and `DTP_DTpols_rs` return a `DTObj` which has entries as `IsFromTheLeftCollectorRep` and additionally:

- `DTObj![PC_DTPPolynomials]`: Deep Thought polynomials in form of (nested) lists

- `DTObj![PC_DTPOrders]`: list containing orders of group generators if the collector is confluent

- `DTObj![PC_DTPConfluent]`: boolean value indicating whether the collector is confluent or not

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