Mathematics Colloquium — Vector vs. Scalar Sup Norms
April 15 @ 3:00 pm – 4:00 pm
MAT Colloquium
Wednesday, April 15, 2026
Hill Auditorium, Barrows Hall
Refreshments at 3:00pm, Talk 3:15-4:05pm
Speaker: Jack Buttcane, University of Maine
Abstract:
Rotations in three dimensions depend on three angles, which we call the Euler angles. A function of the Euler angles can be expressed in terms of Wigner-D matrices, which are representations of the rotation group SO(3). These matrix functions are strongly tied to quantum mechanics where they measure the probability of a state transition under a rotation, but as a number theorist, I am interested in their connection to automorphic forms: Suppose we have a vector-valued function of the 3×3 invertible matrices that transforms by a Wigner-D matrix under rotations. Then an interesting quantity to study is the “sup norm” sup_A ||f(A)||, i.e. the maximum value of the length of the vector f(A) over 3×3 invertible matrices A.
We can also think of the function f as a vector of scalar-valued functions f=(f_{-d},…f_d) and for each entry f_j in the vector, we can consider sup_A |f_j(A)|. The question I want to consider is, to what extent does the ratio of these two sup norms depend exclusively on the Wigner-D matrix, independent of the particular function f? As an undergraduate summer research project Andrii Obertas did some computations on this project and I’ll report on the outcome of those, as well.
