DIMACS Discrete Mathematics Seminar

Title:

On the Mobius number of subgroup lattices

Speaker:

John Shareshian

Rutgers University

Place:

Room 431, CoRE Building,

Busch Campus, Rutgers University.

Time:

4:30 PM

Tuesday, January 31, 1995

Abstract:

If L is a finite lattice, the characteristic matrix of L is the L x L matrix with A(x,y)=1 if x<=y and 0 otherwise. The Mobius function of the lattice is the function mu defined on L x L, where mu(x,y) is the (x,y) entry of the inverse of A.

Consider the lattice of subgroups of the symmetric group S_n. Stanley has asked whether mu(1,S_n)=(-1)^(n-1)|Aut(S_n)|/2 for all n. This has been shown to be true for n<12. I will give a formula for mu(1,S_n) which holds whenever n has at most two (not necessarily distinct) prime factors or n=2^a. This formula involves mu(1,H) for certain primitive subgroups H of S_n. I will discuss the O'Nan-Scott theorem on primitive permutation groups, which has applications in several areas of combinatorics and graph theory. Using the formula and the O'Nan-Scott theorem, I will give a negative answer to Stanley's question.