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DTSTART;TZID=America/Chicago:20241101T123000
DTEND;TZID=America/Chicago:20241101T133000
DTSTAMP:20260419T220949
CREATED:20241022T141444Z
LAST-MODIFIED:20241022T144645Z
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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:/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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DTSTART;TZID=America/Chicago:20241115T123000
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DTSTAMP:20260419T220949
CREATED:20241113T152405Z
LAST-MODIFIED:20241113T164702Z
UID:10016191-1731673800-1731677400@uwm.edu
SUMMARY:Graduate Student Colloquium: Eric Redmon
DESCRIPTION:Finite State Machines and Bounded Permutations\nEric Redmon\nGraduate Student\nMarquette University \nWe define a k-bounded permutation π of length n to be a permutation such that for each pair of adjacent entries $\pi$ and $\pi(i + 1)$ for $i = 1\, 2\, 3\, . . . \, n − 1$ we have $|\pi(i) − \pi(i + 1)| \leq k$. Previous work has shown that the generating function for this family of permutations is rational\, and has computed generating functions for small values of $k$. In this talk\, we will discuss the nature of finite state machines and how we can leverage the insertion encoding devised by Albert\, Linton\, and Ruškuc to build a finite state machine that we can use to find generating functions for larger values of $k$.
URL:/math/event/graduate-student-colloquium-eric-redmon/
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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