## University of Sydney Algebra Seminar

# Donald Taylor (University of Sydney)

## Friday 28th June, 12:05-12:55pm, Carslaw 373

### Gaussian elimination in groups of Lie type

The factorisation of a matrix as a product of a lower triangular by an upper triangular matrix (up to a permutation of the rows) is a form of 'Gauss elimination'. The lower and upper triangular matrices are themselves products of elementary matrices representing row operations. This is described in connection with machine computation in a 1948 paper of Alan Turing but the history of elimination methods goes back at least another 2000 years.

The generalisation of the LU-decomposition to classical groups by Bruhat (1954) and to semisimple algebraic groups by Chevalley (1955) led to the introduction of the general notion of a BN-pair by Jacques Tits.

In joint work with Arjeh Cohen and Scott Murray a general reduction algorithm has been developed -- and implemented in Magma -- to write a matrix acting on a highest weight representation of a group of Lie type as a word in the Steinberg generators of the group.

This talk will focus on examples and on the application of the reduction algorithm in the recent work of Liebeck and O'Brien on the recognition of finite groups of Lie type.