Kazhdan constants and isomorphic graph pairs
Let \(G\) be a finite group, and let \(\Gamma\) be a~subset of \(G\). The Kazhdan constant of the pair \((G,\Gamma)\) is defined to be the maximum distance we can guarantee that an arbitrary unit vector in an arbitrary nontrivial irreducible unitary representation space of \(G\) can be moved by some...
Збережено в:
| Видавець: | Lugansk National Taras Shevchenko University |
|---|---|
| Дата: | 2023 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2023
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1851 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
Організація
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-1851 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua:article-18512023-06-18T17:42:42Z Kazhdan constants and isomorphic graph pairs Davila, M. Hayes, T. Krebs, M. Reyes, M. Kazhdan constant, representation theory 05C25, 20C15 Let \(G\) be a finite group, and let \(\Gamma\) be a~subset of \(G\). The Kazhdan constant of the pair \((G,\Gamma)\) is defined to be the maximum distance we can guarantee that an arbitrary unit vector in an arbitrary nontrivial irreducible unitary representation space of \(G\) can be moved by some element of \(\Gamma\). The Kazhdan constant relates to the expansion properties of the Cayley graph generated by \(G\) and \(\Gamma\), and has been much studied in this context. Different pairs \((G_1,\Gamma_1)\) and \((G_2,\Gamma_2)\) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To what extent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we construct an infinite family of such pairs where the Kazhdan constants are unequal. Other relevant results are presented as well. Lugansk National Taras Shevchenko University National Science Foundation grant #HRD-1602210 2023-06-18 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1851 10.12958/adm1851 Algebra and Discrete Mathematics; Vol 35, No 1 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1851/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1851/895 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1851/896 Copyright (c) 2023 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Kazhdan constant representation theory 05C25 20C15 |
| spellingShingle |
Kazhdan constant representation theory 05C25 20C15 Davila, M. Hayes, T. Krebs, M. Reyes, M. Kazhdan constants and isomorphic graph pairs |
| topic_facet |
Kazhdan constant representation theory 05C25 20C15 |
| format |
Article |
| author |
Davila, M. Hayes, T. Krebs, M. Reyes, M. |
| author_facet |
Davila, M. Hayes, T. Krebs, M. Reyes, M. |
| author_sort |
Davila, M. |
| title |
Kazhdan constants and isomorphic graph pairs |
| title_short |
Kazhdan constants and isomorphic graph pairs |
| title_full |
Kazhdan constants and isomorphic graph pairs |
| title_fullStr |
Kazhdan constants and isomorphic graph pairs |
| title_full_unstemmed |
Kazhdan constants and isomorphic graph pairs |
| title_sort |
kazhdan constants and isomorphic graph pairs |
| description |
Let \(G\) be a finite group, and let \(\Gamma\) be a~subset of \(G\). The Kazhdan constant of the pair \((G,\Gamma)\) is defined to be the maximum distance we can guarantee that an arbitrary unit vector in an arbitrary nontrivial irreducible unitary representation space of \(G\) can be moved by some element of \(\Gamma\). The Kazhdan constant relates to the expansion properties of the Cayley graph generated by \(G\) and \(\Gamma\), and has been much studied in this context. Different pairs \((G_1,\Gamma_1)\) and \((G_2,\Gamma_2)\) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To what extent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we construct an infinite family of such pairs where the Kazhdan constants are unequal. Other relevant results are presented as well. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2023 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1851 |
| work_keys_str_mv |
AT davilam kazhdanconstantsandisomorphicgraphpairs AT hayest kazhdanconstantsandisomorphicgraphpairs AT krebsm kazhdanconstantsandisomorphicgraphpairs AT reyesm kazhdanconstantsandisomorphicgraphpairs |
| first_indexed |
2024-04-12T06:13:53Z |
| last_indexed |
2024-04-12T06:13:53Z |
| _version_ |
1804810505180676096 |