### Abstract

In Chap. 1 it is explained that if p is a probability on a finite group G the group matrix X_{G}(p) is a transition matrix for a random walk on G. If f is an arbitrary function on G the process of transforming X_{G}(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) |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 231-270 |

Number of pages | 40 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2233 |

ISSN (Print) | 0075-8434 |

ISSN (Electronic) | 1617-9692 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*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

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*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

**Fourier Analysis on Groups, Random Walks and Markov Chains.** / Johnson, Kenneth W.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Fourier Analysis on Groups, Random Walks and Markov Chains

AU - Johnson, Kenneth W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

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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U2 - 10.1007/978-3-030-28300-1_7

DO - 10.1007/978-3-030-28300-1_7

M3 - Chapter

AN - SCOPUS:85075173111

T3 - Lecture Notes in Mathematics

SP - 231

EP - 270

BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -