Generating functions for the powers in $\text{GL}(n,q)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866910396570402816 |
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| author | Kundu, Rijubrata Singh, Anupam |
| author_facet | Kundu, Rijubrata Singh, Anupam |
| contents | Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set $\text{GL}(n, q)^M$, i.e., the elements or classes of these kinds which are $M^{th}$ powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when $(q,M)=1$, (ii) for semisimple elements and all elements (and classes) when $M$ is a prime power and $(q,M)=1$, and (iii) for all kinds when $M$ is a prime and $q$ is a power of $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_14057 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Generating functions for the powers in $\text{GL}(n,q)$ Kundu, Rijubrata Singh, Anupam Group Theory Combinatorics 20G40 Consider the set of all powers $\text{GL}(n ,q)^M = \{x^M \mid x\in \text{GL}(n, q)\}$ for an integer $M\geq 2$. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set $\text{GL}(n, q)^M$, i.e., the elements or classes of these kinds which are $M^{th}$ powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when $(q,M)=1$, (ii) for semisimple elements and all elements (and classes) when $M$ is a prime power and $(q,M)=1$, and (iii) for all kinds when $M$ is a prime and $q$ is a power of $M$. |
| title | Generating functions for the powers in $\text{GL}(n,q)$ |
| topic | Group Theory Combinatorics 20G40 |
| url | https://arxiv.org/abs/2003.14057 |