Some Notes on Higher Frobenius–Schur Indicators of the Regular Representations for Matched Pairs of Groups

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...

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Bibliographic Details
Main Authors:	Chenfei Mao, Kangqiao Li
Format:	Article
Language:	English
Published:	MDPI AG 2024-11-01
Series:	Mathematics
Subjects:	finite group higher Frobenius–Schur indicator matched pair of groups factorizable group direct product
Online Access:	https://www.mdpi.com/2227-7390/12/22/3479
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Summary:	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.
ISSN:	2227-7390