Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups
The notion of matched pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo><mo>▹</mo&g...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3479 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850068411504132096 |
|---|---|
| author | Chenfei Mao Kangqiao Li |
| author_facet | Chenfei Mao Kangqiao Li |
| author_sort | Chenfei Mao |
| collection | DOAJ |
| description | The notion of matched pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo><mo>▹</mo><mo>,</mo><mo>◃</mo><mo>)</mo></mrow></semantics></math></inline-formula> of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⋈</mo><mi>G</mi></mrow></semantics></math></inline-formula>. We find in this paper some sufficient conditions when equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>F</mi><mo>⋈</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> imply that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⋈</mo><mi>G</mi></mrow></semantics></math></inline-formula> is the external direct product of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>×</mo><mi>G</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mi>m</mi></msub></semantics></math></inline-formula> denotes <i>m</i>-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mi>G</mi></msup><mo>#</mo><mi mathvariant="double-struck">C</mi><mi>F</mi></mrow></semantics></math></inline-formula> is also mentioned. |
| format | Article |
| id | doaj-art-ea054545ec154697bd5b79ed3fa20a49 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-ea054545ec154697bd5b79ed3fa20a492025-08-20T02:48:05ZengMDPI AGMathematics2227-73902024-11-011222347910.3390/math12223479Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of GroupsChenfei Mao0Kangqiao Li1School of Mathematics, Hangzhou Normal University, Hangzhou 311121, ChinaSchool of Mathematics, Hangzhou Normal University, Hangzhou 311121, ChinaThe notion of matched pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>G</mi><mo>,</mo><mo>▹</mo><mo>,</mo><mo>◃</mo><mo>)</mo></mrow></semantics></math></inline-formula> of finite groups was introduced by Takeuchi in 1981, which is equivalent to a factorization of the group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⋈</mo><mi>G</mi></mrow></semantics></math></inline-formula>. We find in this paper some sufficient conditions when equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>F</mi><mo>⋈</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><msub><mi>ν</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> imply that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>⋈</mo><mi>G</mi></mrow></semantics></math></inline-formula> is the external direct product of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>×</mo><mi>G</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ν</mi><mi>m</mi></msub></semantics></math></inline-formula> denotes <i>m</i>-th indicator of the regular representation of a finite group. A comparison with indicators of bismash product Hopf algebras <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">C</mi><mi>G</mi></msup><mo>#</mo><mi mathvariant="double-struck">C</mi><mi>F</mi></mrow></semantics></math></inline-formula> is also mentioned.https://www.mdpi.com/2227-7390/12/22/3479finite grouphigher Frobenius–Schur indicatormatched pair of groupsfactorizable groupdirect product |
| spellingShingle | Chenfei Mao Kangqiao Li Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups Mathematics finite group higher Frobenius–Schur indicator matched pair of groups factorizable group direct product |
| title | Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups |
| title_full | Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups |
| title_fullStr | Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups |
| title_full_unstemmed | Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups |
| title_short | Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups |
| title_sort | some notes on higher frobenius schur indicators of the regular representations for matched pairs of groups |
| topic | finite group higher Frobenius–Schur indicator matched pair of groups factorizable group direct product |
| url | https://www.mdpi.com/2227-7390/12/22/3479 |
| work_keys_str_mv | AT chenfeimao somenotesonhigherfrobeniusschurindicatorsoftheregularrepresentationsformatchedpairsofgroups AT kangqiaoli somenotesonhigherfrobeniusschurindicatorsoftheregularrepresentationsformatchedpairsofgroups |