Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals

Combinatorics Seminar
Friday, November 18, 2011 - 15:05
1 hour (actually 50 minutes)
Skiles 005
North Carolina State University, Raleigh, NC
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of Kashiwara's sl_2 lowering operator in the theory of crystal bases.   The talk will include a survey of related past work on symmetric chain decomposition and unimodality by Greene-Kleitman, Griggs-Killian-Savage, Proctor, Stanley and others as well as a discussion of open questions that still remain.  This is joint work with Anne Schilling.