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Washington State
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Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

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September 14, Monday, 4:10 - 5:00 PM

Daryl Deford

Department of Mathematics and Statistics

Washington State University

Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

Zoom

September 14, Monday, 4:10 - 5:00 PM

Daryl Deford

Department of Mathematics and Statistics

Washington State University

Title:
Representations of $SL_2(\mathbb{Z}/p\mathbb{Z})$
and spectral properties of Bethe trees

Abstract: A nice application of Wedderburn's theorem is to compute the spectrum of a Cayley graph using the Fourier transform over the underlying group (i.e. taking the union over a complete set of irreducible representations). While the properties of these graphs for $SL_2(\mathbb{Z}/p\mathbb{Z})$ have been studied for many years, much less is known about the more complex setting of $SL_2(\mathbb{Z}/p^n\mathbb{Z})$. In this talk, I will present some experiments making use of this technique and provide a complete algorithmic description of the irreducible representations of $SL_2(\mathbb{Z}/p^2\mathbb{Z})$.

This work also motivates questions about the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we will see that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials (making use of some early work from Professor Webb) and have a singular limiting distribution as the tree depth goes to infinity. I will also discuss natural generalizations of these results including periodic branching patterns, higher order simplicial complexes, and spectrally stable graph families. This work is joint with Dan Rockmore.

Abstract: A nice application of Wedderburn's theorem is to compute the spectrum of a Cayley graph using the Fourier transform over the underlying group (i.e. taking the union over a complete set of irreducible representations). While the properties of these graphs for $SL_2(\mathbb{Z}/p\mathbb{Z})$ have been studied for many years, much less is known about the more complex setting of $SL_2(\mathbb{Z}/p^n\mathbb{Z})$. In this talk, I will present some experiments making use of this technique and provide a complete algorithmic description of the irreducible representations of $SL_2(\mathbb{Z}/p^2\mathbb{Z})$.

This work also motivates questions about the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we will see that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials (making use of some early work from Professor Webb) and have a singular limiting distribution as the tree depth goes to infinity. I will also discuss natural generalizations of these results including periodic branching patterns, higher order simplicial complexes, and spectrally stable graph families. This work is joint with Dan Rockmore.