Kimberly Harry
\nUniversity of Wisconsin-Milwaukee<\/p>\n
We let L(\u03bb) denote the irreducible highest weight representation of the classical simple Lie algebra g with highest weight \u03bb. Kostant\u2019s weight multiplicity formula gives a way to compute the multiplicity of a weight \u00b5 in L(\u03bb), denoted m(\u03bb, \u00b5), via an alternating sum over the Weyl group whose terms involve the Kostant partition function. The Weyl alternation set A(\u03bb, \u00b5) is the set of Weyl group elements that contribute nontrivially to the multiplicity m(\u03bb, \u00b5). We prove that Weyl alternation sets are order ideals in the weak Bruhat order of the Weyl group. Specializing to the Lie algebra of type A, we prove that the Weyl alternation sets A(\u02dc\u03b1, \u00b5), where \u02dc\u03b1 is the highest root of sl_{r+1}(C) and \u00b5 is a positive root is a product of Fibonacci numbers. Using this result, we show that the q-multiplicity of the positive root in the representation L(\u02dc\u03b1) is precisely a power of q. We give a complete characterization of the Weyl alternation sets A(\u02dc\u03b1, \u00b5), where \u00b5 is now a negative root of sl_{r+1}(C). We also show that the cardinality of these Weyl alternation sets satisfies a two-term recurrence relation involving Fibonacci numbers. Time permitting I will present further results related to collaborative projects I have contributed to during my years at 51ÁÔÆæ.<\/p>\n
Advisor:<\/strong> Pamela E. Harris<\/p>\n Committee Members: Kostant’s Formula and Parking Functions: Combinatorial Explorations Kimberly Harry University of Wisconsin-Milwaukee We let L(\u03bb) denote the irreducible highest weight representation of the classical simple Lie algebra g with highest weight \u03bb. Kostant\u2019s weight multiplicity formula gives a way to …<\/p>\n","protected":false},"author":4217,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","uwm_wg_additional_authors":[]},"tags":[],"tribe_events_cat":[64],"class_list":["post-14861","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-graduate-student-defenses","cat_graduate-student-defenses"],"yoast_head":"\n
\n<\/strong>Profs. Jeb Willenbring, Kevin McLeod, Gabriella Pinter, and Jonah Gaster<\/p>\n","protected":false},"excerpt":{"rendered":"