GOCC 09/25/2024 "A q-analog of Kostant's Weight Multiplicity Formula and a Product of Fibonacci"

Speaker: Kim Harry (UW-Milwaukee)
Abstract: 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 slr+1(C), which we denote L(˜α), where ˜α is the highest root of slr+1(C). We prove that the number of terms contributing a nonzero value to the multiplicity of the positive root µ = αi + αi+1 + · · · + αj with 1 ≤ i ≤ j ≤ r in L(˜α) is given by the product Fi · Fr−j+1, where Fn is the nth Fibonacci number. Using this result, we show that the q-multiplicity of the positive root µ = αi + αi+1 + · · · + αj with 1 ≤ i ≤ j ≤ r in the representation L(˜α) is precisely qr−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 slr+1(C).
The Graduate Online Combinatorics Colloquium is an online combinatorics seminar organized for and run by graduate students.