Cameron–Fon-Der-Flaass IBIS theorem

In mathematics, the Cameron–Fon-Der-Flaass IBIS theorem bridges algebraic combinatorics and group theory. The theorem was discovered in 1995 by two mathematicians Peter Cameron and Dima Von-Der-Flaass.^[1][2]

Statement

Consider the group action of a permutation group ${\displaystyle G}$ acting on a set ${\displaystyle \Omega }$. A base is a sequence of elements of ${\displaystyle \Omega }$ which, when fixed, destroys all symmetry, i.e. its pointwise stabilizer is trivial. A base is irredundant if each element further reduces symmetry, i.e. no element in the sequence is fixed by the pointwise stabiliser of its predecessors. Now the following are equivalent:

• All irredundant bases of ${\displaystyle G}$ have the same size;
• The irredundant bases of ${\displaystyle G}$ are preserved by re-ordering;
• The irredundant bases of ${\displaystyle G}$ form the bases of a matroid.

References

1. Cameron, P. J; Fon-Der-Flaass, D. G (1995-11-01). "Bases for permutation groups and matroids". European Journal of Combinatorics. 16 (6): 537–544. doi:10.1016/0195-6698(95)90035-7. ISSN 0195-6698.
2. Patrias, Rebecca; Pechenik, Oliver (2020). "Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture". Forum of Mathematics, Sigma. 8: 62. arXiv:2003.13152. doi:10.1017/fms.2020.61. ISSN 2050-5094.

Further reading

• https://www.theoremoftheday.org/GroupTheory/IBIS/TotDIBIS.pdf