research papers
On the Σ classes in E^{6}
^{a}Jupiterstrasse 3, CH3015 Berne, Switzerland
^{*}Correspondence email: pengel@bluewin.ch
In E^{6}, the cone of positive definite quadratic forms is subdivided into Σ_{s} subcones and its equivalence classes are determined for s = 0–3, and 18–21.
1. Introduction
The discovery of quasicrystals, the structure of which can be viewed as projected from higherdimensional translation lattices, has greatly stimulated the investigation of lattices and parallelohedra in arbitrary dimensions. The classification of the combinatorial types of primitive parallelohedra induces a structure on the cone of positive definite quadratic forms .
In a series of papers, the shape of and its subdivision into Φ and Σ subcones were discussed (Baburin & Engel, 2013; Engel, 2015, 2019).
Ryshkov (1973) defined the S subcone which contains all parallelohedra that have the same set of facet vectors , but without characterizing its boundary. The complete subcone was determined by Engel (2015) by the halfspace intersection
The investigation of translation lattices becomes most attractive in E^{6} because many new phenomena appear for the first time in dimension 6.
In the report by Engel (2019), minimal and maximal Σ_{s} classes in E^{6}, and , were investigated. The subscript `s' is an invariant of the class and denotes the number of closed zones of [see Engel (2019), equations (12)–(13)]. This classification is continued for the Σ_{s} classes, s = 0, 1, 2, 3 and 18, 19, 20, 21.
The infinite family of Σ cones generate a facetoface tiling of the cone [see Engel (2019), equation (17) ff.]. In this tiling, for each class representative are determined all the neighbouring Σ's adjacent to by a common wall, in order to find new . Proceeding in this way, for each class can be found at least one representative along a finite path of adjacent Σ's.
As a main result we obtain by this adjacency procedure:
For s = 0, 1, 2, 3 there exist 1, 1, 6, 58 Σ_{s} classes, and for s = 18, 19, 20, 21 there exist 15, 3, 1, 1 Σ_{s} classes in .
2. Determination of the Σ_{s} classes
Most concepts used in what follows were described by Engel (2019).
Beginning with Σ_{0} as a representative of its class , and its subdivision into combinatorial Φ types, the Σ_{s} classes for s = 0, 1, 2, 3 are successively determined. Recall that Σ_{0} has 216 walls , , which all are equivalent under the group (see Engel, 2019). The neighbouring Σ_{s} adjacent to Σ_{0} are equivalent too, and were determined along the following steps:
Step S1: Within the class of equivalent walls, one wall , 1 ≤ l ≤ 216, is selected, and for any leaning on , the neighbouring Φ_{k} opposite to that wall is taken. Let . The determination of requires first the computation of the primitive parallelohedron,
Note that in E^{6} every primitive parallelohedron has 126 facet vectors. The set of facet vectors of is denoted by
This shows that has one closed zone with zone vector z* = (0, 0, 0, 0, 0, 1), and thus it belongs to . Because of symmetry, for each equivalent wall an equivalent result will be obtained.
Step S2: Next all triplets that fulfil the belt condition
are determined. Their number is N_{b} = 371, and thus,
is obtained. Because of the large number of halfspaces , the direct calculation of Σ_{1} is not practicable. Instead, the calculation of the Φ subcones inside Σ_{1} will reveal the walls . Recall that is interior to Σ_{1} if
This allows the calculation of all without explicitly knowing Σ_{1}, and for the walls of Σ_{1} it holds that:
A wall of is a wall of Σ_{1} if there exists a wall , 1 ≤ h ≤ 3N_{b}, such that
By calculating a sufficiently large number of , most of the walls of Σ_{1} can be determined. The process converts relatively quickly.
Step S3: In order to verify the result, the induced symmetry of Σ_{1} is applied:
For any the induced symmetry of Σ_{1} is defined by
The centre
is invariant under the group and lies in Σ_{1}. Applying the symmetry to the walls of Σ_{1} proves that there are 166 walls which belong to seven classes under , and these are shown in Table 1. The neighbouring Σ_{s}, s = 0, 1, 2, are given in Table 2.


Step S4: For each Σ_{s} obtained, we proceed analogously to steps S1 to S3 in order to get further Σ_{s}. For each new Σ_{s} we have to check their equivalence:
and are arithmetically equivalent and belong to the same equivalence class if there exists such that for any it holds that
Because is of infinite order, the above equation is not practicable. However, if optimal bases are admitted to the forms only, then the number of transformations that have to be taken into account becomes finite. It was discovered for every Σ_{s} at maximal path length 5 from Σ_{0} (see Fig. 1) that it is sufficient to consider only, in order to verify equivalence. If equivalence is proved for any then it holds for all .
Alternatively, the combinatorial equivalence of parallelohedra may be compared:
and are equivalent and belong to the same equivalence class if there exist and such that
The latter procedure requires a sufficiently large number of Φ cones to be determined in order to find at least one equivalent pair.
Analogously, using the procedures described in steps S1 to S4 the Σ_{s} cones, s = 21, 20, 19, 18, were successively determined starting with Σ_{21} as a representative of its class . Recall that Σ_{21} has 21 walls , , which all are equivalent under the group (see Engel, 2019).
3. Results
In Table 2 are given the Σ_{s} classes, s = 0, 1, 2, 3, under the general linear group . Each equivalence class is given by its representative . is chosen such that becomes optimal. Under the heading `s' is given the number of closed zones. Under the heading `Subcone' is stated the number of walls of . In cases where the complete Σ_{s} cone was calculated, the numbers of walls and edges are indicated as N_{w} and N_{e}, respectively. Under the heading `Order' is given the order of the induced symmetry under . Under the heading `Neighbours' are stated the neighbouring , each of them preceded, in brackets, by the number of equivalent subcones under the group . If more than one number is given, it means that they are equivalent under . Remarkably, has neighbours with s = r − 1, r, r + 1 only. Note that Σ_{4} cones were not determined and the preceding number gives an upper bound for the number of equivalent subcones only. In Fig. 1 are drawn the shortest paths from Σ_{0} to each other .
In Table 3 are given the Σ_{s} classes, s = 21, 20, 19, 18. Under the heading `Subcone' are given the numbers of walls N_{w} and edges N_{e}. Most of the cones are simple with Φ and Σ cones identical. In cases where the cone is not simple, two numbers are shown as `a/b' under the heading `Types', where `a' indicates the number of Φ types and `b' gives the total number of Φ cones in . The numbers of Φ types for s = 21, 20, 19, 18 correspond to the numbers given by Baburin & Engel (2013). All these Φ types correspond to principal primitive parallelohedra. Under the heading `Order' is given the order of the induced symmetry under . Under the heading `Neighbours' are listed the neighbouring which are preceded, in brackets, by the number of equivalent types under the group . Note that the Σ_{17} cones were not determined and the preceding number gives an upper bound for the number of equivalent types only. In Fig. 2 are drawn the shortest paths from Σ_{21} to each other .

References
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Engel, P. (2019). Acta Cryst. A75, 574–583. CrossRef IUCr Journals Google Scholar
Ryshkov, S. S. (1973). Sov. Math. Dokl. 14, 1314–1318. Google Scholar
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