Topological and Algebraic Combinatorics of Crystal Posets

LIST OF FIGURES......Page 8
Introduction......Page 9
Crystal bases......Page 10
Stembridge axioms and Sternberg relations......Page 14
Partially ordered sets......Page 18
Discrete Morse functions......Page 20
Lexicographic discrete Morse functions......Page 23
Motivation......Page 26
Introduction......Page 32
Consequences of the Stembridge axioms......Page 35
Connections between the Möbius function of a poset and relations among crystal operators......Page 40
New relations in highest weight crystals of doubly laced type......Page 56
Discussion and open questions......Page 58
Introduction......Page 60
Results towards a proof of Conjecture 3.2.2......Page 63
Stembridge and HL relations in two rowed shape crystals......Page 69
Rank three intervals in two rowed shape crystals of type An......Page 76
Analyzing arbitrary rank intervals......Page 85
Discussion and open questions......Page 88
References......Page 90