# Fourier Analysis on Groups, Random Walks and Markov Chains

Research output: Chapter in Book/Report/Conference proceedingChapter

### Abstract

In Chap. 1 it is explained that if p is a probability on a finite group G the group matrix XG(p) is a transition matrix for a random walk on G. If f is an arbitrary function on G the process of transforming XG(f) into a block diagonal matrix is equivalent to the obtaining the Fourier transform of f. This chapter explains the connections with harmonic analysis and the group matrix. Most of the discussion is on probability theory and random walks. The fusion of characters discussed in Chap. 4 becomes relevant, and also the idea of fission of characters is introduced, especially those fissions which preserve diaonalizability of the corresponding group matrix. As an example of how the group matrix and group determinant can be used as tools, their application to random walks which become uniform after a finite number of steps is examined.

Original language English (US) Lecture Notes in Mathematics Springer Verlag 231-270 40 https://doi.org/10.1007/978-3-030-28300-1_7 Published - Jan 1 2019

### Publication series

Name Lecture Notes in Mathematics 2233 0075-8434 1617-9692

### Fingerprint

Matrix Groups
Fourier Analysis
Random walk
Markov chain
Block Matrix
Transition Matrix
Harmonic Analysis
Diagonal matrix
Probability Theory
Fourier transform
Fusion
Determinant
Finite Group
Arbitrary
Character

### All Science Journal Classification (ASJC) codes

• Algebra and Number Theory

### Cite this

Johnson, K. W. (2019). Fourier Analysis on Groups, Random Walks and Markov Chains. In Lecture Notes in Mathematics (pp. 231-270). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_7
Johnson, Kenneth W. / Fourier Analysis on Groups, Random Walks and Markov Chains. Lecture Notes in Mathematics. Springer Verlag, 2019. pp. 231-270 (Lecture Notes in Mathematics).
title = "Fourier Analysis on Groups, Random Walks and Markov Chains",
abstract = "In Chap. 1 it is explained that if p is a probability on a finite group G the group matrix XG(p) is a transition matrix for a random walk on G. If f is an arbitrary function on G the process of transforming XG(f) into a block diagonal matrix is equivalent to the obtaining the Fourier transform of f. This chapter explains the connections with harmonic analysis and the group matrix. Most of the discussion is on probability theory and random walks. The fusion of characters discussed in Chap. 4 becomes relevant, and also the idea of fission of characters is introduced, especially those fissions which preserve diaonalizability of the corresponding group matrix. As an example of how the group matrix and group determinant can be used as tools, their application to random walks which become uniform after a finite number of steps is examined.",
author = "Johnson, {Kenneth W.}",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/978-3-030-28300-1_7",
language = "English (US)",
series = "Lecture Notes in Mathematics",
publisher = "Springer Verlag",
pages = "231--270",
booktitle = "Lecture Notes in Mathematics",

}

Johnson, KW 2019, Fourier Analysis on Groups, Random Walks and Markov Chains. in Lecture Notes in Mathematics. Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 231-270. https://doi.org/10.1007/978-3-030-28300-1_7
Lecture Notes in Mathematics. Springer Verlag, 2019. p. 231-270 (Lecture Notes in Mathematics; Vol. 2233).

Research output: Chapter in Book/Report/Conference proceedingChapter

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AB - In Chap. 1 it is explained that if p is a probability on a finite group G the group matrix XG(p) is a transition matrix for a random walk on G. If f is an arbitrary function on G the process of transforming XG(f) into a block diagonal matrix is equivalent to the obtaining the Fourier transform of f. This chapter explains the connections with harmonic analysis and the group matrix. Most of the discussion is on probability theory and random walks. The fusion of characters discussed in Chap. 4 becomes relevant, and also the idea of fission of characters is introduced, especially those fissions which preserve diaonalizability of the corresponding group matrix. As an example of how the group matrix and group determinant can be used as tools, their application to random walks which become uniform after a finite number of steps is examined.

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Johnson KW. Fourier Analysis on Groups, Random Walks and Markov Chains. In Lecture Notes in Mathematics. Springer Verlag. 2019. p. 231-270. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-030-28300-1_7