Representations of skew braces

In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace \((A, \cdot, \circ)\) induces an action \(\lambda^{\op}: (A, \circ)...

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Bibliographic Details
Published inarXiv.org
Main Authors Rathee, Nishant, Singh, Mahender, Udeep, Ayush
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 22.08.2024
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Summary:In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace \((A, \cdot, \circ)\) induces an action \(\lambda^{\op}: (A, \circ) \to \Aut (A, \cdot)\), which gives rise to the group \(\Lambda_{A^{\op}} = (A, \cdot) \rtimes_{\lambda^{\op}} (A, \circ)\). We prove that if \(A\) and \(B\) are isoclinic skew left braces, then \(\Lambda_{A^{\op}}\) and \(\Lambda_{B^{\op}}\) are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of \((A, \cdot, \circ)\) and that of the group \(\Lambda_{A^{\op}}\). We obtain a decomposition of the induced representation of the additive group \((A, \cdot)\) and of the multiplicative group \((A, \circ)\) corresponding to the regular representation of the group \(\Lambda_{A^{\op}}\). As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.
ISSN:2331-8422