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DTSTART;TZID=America/Chicago:20241101T123000
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DTSTAMP:20260614T221841
CREATED:20241022T141444Z
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SUMMARY:Graduate Student Colloquium: Kim Harry
DESCRIPTION:A q-analog of Kostant’s Weight Multiplicity Formula and a Product of Fibonacci Numbers\nKim Harry\nPhD Graduate Student\nUniversity of Wisconsin-Milwaukee \nUsing Kostant’s weight multiplicity formula\, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root µ in the adjoint representation of sl_{r+1}(C)\, which we denote L(˜α)\, where ˜α is the highest root of sl_{r+1}(C). We prove that the number of terms contributing a nonzero value to the multiplicity of the positive root µ = α_i + α_i+1 + · · · + α_j with 1 ≤ i ≤ j ≤ r in L(˜α) is given by the product F_i · F_(r−j+1)\, where F_n is the nth Fibonacci number. Using this result\, we show that the q-multiplicity of the positive root µ = α_i + α_i+1 + · · · + α_j with 1 ≤ i ≤ j ≤ r in the representation L(˜α) is precisely q^{r−h(µ)}\, where h(µ) = j − i + 1 is the height of the positive root µ. Setting q = 1 recovers the known result that the multiplicity of a positive root in the adjoint representation of sl_{r+1}(C).
URL:https://uwm.edu/math/event/graduate-student-colloquium-kim-harry/
LOCATION:EMS Building\, Room E495\, E495; 3200 N Cramer St.\, Milwaukee\, WI\, 53211\, United States
CATEGORIES:Graduate Student Colloquia
ORGANIZER;CN="The Department of Mathematical Sciences":MAILTO:math-staff@uwm.edu
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LAST-MODIFIED:20241025T170950Z
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SUMMARY:Colloquium: aBa Mbirika & Morgan Fiebig
DESCRIPTION:A graphical approach to the Fibonacci sequence (Fn) n≥0 modulo m extended to the Lucas sequences (Un(p\,q))n≥0 and (Vn(P\,q))n≥0\naBa Mbirika & Morgan Fiebig\nUniversity of Wisconsin – Eau Claire \nThe goal of this talk is twofold: (1) extend theory on statistics in the Fibonacci and Lucas sequences modulo m to the Lucas sequences U :=(Un(p\,q))n≥0 and V :=(Vn(p\,q)n 0\, and (2) apply this theory to a novel graphical approach of U and V modulo m. The statistics we explore are the period π(m)\, entry point e(m)\, and order ω(m) := pi(m)/e(m). We generalize a wealth of known Fibonacci and Lucas statistical results to the U and V setting. Based on ω(m)\, we describe behaviors shared by infinite families of nondegenerate U and V sequences with parameters q = ± 1. In our graphical approach we place the cycle of repeating terms of the periods of U and V in a circle\, and patterns which would otherwise be overlooked emerge. In particular\, we exhibit some tantalizing examples in the following three sequence pairs: Fibonacci and Lucas\, Pell and associated Pell\, and\, balancing and Lucas-balancing. Our proofs utilize results from primary sources ranging from the ground-breaking papers of Lucas in 1878 and Carmichael in 1913\, to the seminal works of Wall in 1960 and Vinson in 1963\, amongst others.
URL:https://uwm.edu/math/event/colloquium-aba-mbirika/
LOCATION:WI
CATEGORIES:Colloquia
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