
- This event has passed.
Graduate Student Colloquium: Kim Harry
November 1, 2024 @ 12:30 pm - 1:30 pm
A q-analog of Kostant’s Weight Multiplicity Formula and a Product of Fibonacci Numbers
Kim Harry
PhD Graduate Student
University of Wisconsin-Milwaukee
Using 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).