Fall 2003 - Section 1

Schedule: Tuesdays, Room 246 , Math East, 1 pm to 2 pm ,

Thursdays: Room 246, Math East, 12.30 pm to 2 pm.

- Office: Room 507 Mathematics
- Office Hours: by appointment.

**Text:** Algorithmic
methods in modular representation theory, by K. Lux

There are several GAP sessions accompanying the course, which you can download and experiment with.

Download

- session1. It introduces you to the concept of a group ring in GAP.
- session2 is the continuation of session1.
- session3. In this session you will calculate all the irreducible matrix representations of the sporadic Mathieu group M11 over the field with 4 elements (upto isomorphism).
- session4. Comment: This example deals with analyzing the submodule structure of the permutation representation of the sporadic Higman-Sims group of degree 100 over GF(2), see our notes. In the second part we look at the sporadic Mathieu M24 in in its permutation representation over GF(2) of degree 24. We analyze the permutation module and find the Golay-code.

Homework:

- Modify the spinning algorithm discussed in class such that it computes an echelonized basis for the space generated by the input vector. Prove the correctedness of the modified algorithm.
- Find the permutation representation of M11 on 55 points from the atlas of finite groups. Download it and convert the generators into internal MeatAxe format and call them m11.1 and m11.2 . Also download the file that constructs generators for M9:2 from the standard generators of M11. Edit the file and replace all mu's by zmu and iv's by ziv. Copy the standard generators of M11 to files named 1 and 2 and run the script. You will find generators for M9:2 as the files named 1 and 2. Now try to find the 3-Sylow subgroup of M9:2. (Hint: It is easy to find an element of order 3, now use the fact the 3-Sylow subgroup is normal in M9:2) Store the generators for the 3-Sylow subgroup in files syl3.1 and syl3.2. Copy the standard generators of M11 to files named z1 and z2. Run the script called fro. It produces new elements of M11 stored in z3 up to z11. Copy z5 to m11.3 and z6 to m11.4 . Run the script kd (condense): kd 2 m11 syl3 4 (condense the four elements of M11; use the 3-Sylow subgroup constructed above s a condensation subgroup.) Now analyze the condensed representaion called kdm11 by using chop -g 2 kdm11, then chop -g 3 kdm11, and finally chop -g 4 kdm11. Describe your observations and compare them to the structure of the permutation module of M11 on 55 points over GF(2).