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

TL;DR

This paper proves that cyclic quotients of Boolean algebras have symmetric chain decompositions for any order, generalizing previous prime-order results and connecting the combinatorial structure to cyclic analogues of crystal bases.

Contribution

It provides an explicit construction of symmetric chain decompositions for cyclic quotients of Boolean algebras of any order, extending prior prime-order results and linking to cyclic crystal theory.

Findings

Proved symmetric chain decomposition exists for all cyclic quotients of Boolean algebras.

Constructed an explicit combinatorial map analogous to the $ ext{sl}_2$ lowering operator.

Connected combinatorial structures to cyclic crystal bases.

Abstract

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 the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.