Classifications of certain SOMA(k,n)s

Soicher, in SOMA Update, has classified (up to isomorphism) all SOMA(k,n)s with n ≤ 6.

In my thesis, I have reported on classifications (up to isomorphism) of the following SOMAs:

• a classification of SOMA(k,7)s, with k ≥ 3, invariant under a non-trivial group;
• a classification of all SOMA(k,7)s with k ≥ 4;
• a classification of SOMA(k,8)s, with k ≥ 3, invariant under a group of odd order;
• a classification of SOMA(k,8)s, with k ≥ 5, invariant under a non-trivial group; and
• a partial classification of SOMA(k,9)s, with k ≥ 4, invariant under a group of odd order.

We remark here that all of these classifications are performed using a computational system GAP together with its share packages GRAPE and DESIGN. Also, we remark here that the following classifications are done in discussion with my supervisor Leonard H. Soicher.

The following tables show the possible ud-types of an arbitrary SOMA(k,n) obtained from the classifications above. Note here that the following two tables give all the ud-types of an arbitrary SOMA(k,7).

Classification of SOMA(k,7)s, with k ≥ 3, invariant under a non-trivial group

We give this classification as a GAP list of GRAPE graphs here, where exactly one SOMA from each isomorphism class is given. We remark here that this file is a compressed bz2 file. In this classification, we store the unrefinable decomposition and the ud-type of a SOMA(k,n) A under the GRAPE graph Φ(A) record components called ud and udtype respectively.

The following table shows that possible ud-types of an arbitrary SOMA(k,7) obtained from the classification.

Classification of all SOMA(k,7)s with k ≥ 4

We give this classification as a GAP list of GRAPE graphs here, where exactly one SOMA from each isomorphism class is given. In this classification, we store the unrefinable decomposition and the ud-type of a SOMA(k,n) A under the GRAPE graph Φ(A) record components called ud and udtype respectively.

The following table shows that possible ud-types of an arbitrary SOMA(k,7) obtained from the classification.

Classification of SOMA(k,8)s, with k ≥ 3, invariant under a group of odd order

We give this classification as a GAP list of GRAPE graphs here, where exactly one SOMA from each isomorphism class is given. We remark here that this file is a compressed bz2 file. In this classification, we store the unrefinable decomposition and the ud-type of a SOMA(k,n) A under the GRAPE graph Φ(A) record components called ud and udtype respectively.

The following table shows that possible ud-types of an arbitrary SOMA(k,8) obtained from the classification.

Classification of SOMA(k,8)s, with k ≥ 5, invariant under a non-trivial group

We give this classification as a GAP list of GRAPE graphs here, where exactly one SOMA from each isomorphism class is given. In this classification, we store the unrefinable decomposition and the ud-type of a SOMA(k,n) A under the GRAPE graph Φ(A) record components called ud and udtype respectively.

The following table shows that possible ud-types of an arbitrary SOMA(k,8) obtained from the classification.

Partial classification of SOMA(k,9)s, with k ≥ 4, invariant under a group of odd order

We give this classification as a GAP list of GRAPE graphs here, where exactly one SOMA from each isomorphism class is given. We remark here that this file is a compressed bz2 file. In this classification, we store the unrefinable decomposition and the ud-type of a SOMA(k,n) A under the GRAPE graph Φ(A) record components called ud and udtype respectively.

The following table shows that possible ud-types of an arbitrary SOMA(k,9) obtained from the classification.

This page is maintained by John Arhin.

Last updated: September 5th 2007.