Abstract
The semisimple bismash product Hopf algebra \(J_n=k^{S_{n-1}}\#kC_n\) for an algebraically closed field k is constructed using the matched pair actions of \(C_n\) and \(S_{n-1}\) on each other. In this work, we reinterpret these actions and use an understanding of the involutions of \(S_{n-1}\) to derive a new Froebnius-Schur indicator formula for irreps of \(J_n\) and show that for n odd, all indicators of \(J_n\) are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all 2-dimensional irreps of \(J_n\) and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of \(J_n\) and use these formulas to show that nonzero indicators become rare for large n.
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The Python notebook used to generate examples and graphs which aided in developing the formulas in this work. The entire notebook is available on GitHub. https://github.com/KaylaOrlinsk/USCProject
References
Chowla, S., Herstein, I.N., Moore, W.K.: On recursions connected with symmetric groups I. Canadian J. Math 3, 328–334 (1951)
Georg Frobenius, and Issai Schur. Über die reellen Darstellungen der endlichen Gruppen. Reimer, (1906)
Fuchs, J., Ganchev, ACh., Szlachányi, K., Vecsernyés, P.: \(S_4\) symmetry of \(6j\) symbols and Frobenius-Schur indicators in rigid monoidal \(C^*\) categories. J. Math. Phys. 40(1), 408–426 (1999)
Jedwab, A., Montgomery, S.: Representations of some Hopf algebras associated to the symmetric group \(S_n\). Algebras and representation theory 12(1), 1–17 (2009)
Kashina, Yevgenia: Classification of semisimple Hopf algebras of dimension 16. Journal of Algebra 232(2), 617–663 (2000)
Kashina, Yevgenia, Mason, Geoffrey, Montgomery, Susan: Computing the Frobenius-Schur indicator for abelian extensions of Hopf algebras. J. Algebra 251, 888–913 (2002)
Kashina, Yevgenia, Sommerhauser, Yorck, Zhu, Yongchang: Self-dual modules of semisimple Hopf algebras. J. Algebra 257, 88–96 (2002)
Kashina, Y., Sommerhauser, Y., Yongchang, Z.: On higher Frobenius-Schur indicators. AMS Memoirs 181(855), (2006)
Linchenko, Vitaly, Montgomery, Susan: A Frobenius-Schur Theorem for Hopf algebras. Algebras and Representation Theory 3(4), 347–355 (2000)
Mason, Geoffrey, Ng, Siu-Hung.: Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras. Adv. Math. 190, 161–195 (2005)
Masuoka, Akira: Extensions of Hopf algebras, Trabajos de Matemática 41/99. Universdad Nacional de Córdoba, Argentina, FaMAF (1999)
Montgomery, Susan: Hopf Algebras and their Actions on Rings, CBMS Lectures, vol. 82. AMS, Providence, RI (1993)
Natale, Sonia: Frobenius-Schur indicators for a class of fusion categories. Pacific journal of mathematics 221(2), 353–377 (2005)
Orlinsky, K.:Additional Materials for: An Indicator Formula for the Hopf Algebra \(k^{S_{n-1}}\#kC_n\) (2022)
Robin, G.: Estimation de la fonction de Tchebychef \(\theta \) sur le \(k\)-iéme nombre premier et grandes valeurs de la fonction \(\omega (n)\) nombre de diviseurs premiers de \(n\). Acta Arithmetica 424, 367–389 (1983)
Schauenburg, Peter: Frobenius-Schur indicators for some fusion categories associated to symmetric and alternating groups. Algebras and Representation Theory 19(3), 645–656 (2016)
Takeuchi, Mitsuhiro: Matched pairs of groups and bismash products of Hopf algebras. Communications in Algebra 9(8), 841–882 (1981)
Timmer, J.: Indicators of bismash products from exact symmetric group factorizations. Communications in Algebra 45(10), 4444–4465 (2017)
Acknowledgements
Many thanks to my advisor Susan Montgomery for being a patient and thoughtful advisor. Also thank you to Patrick M. Reardon for teaching me the basics of Python code. This work is part of a PhD thesis at the University of Southern California.
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Open access funding provided by SCELC, Statewide California Electronic Library Consortium. This work is part of a PhD thesis at University of Southern California.
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This work is part of Ph.D thesis project to advance the study of Hopf Algebras and Representation Theory.
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Presented by: Milen Yakimov
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Orlinsky, K. An Indicator Formula for the Hopf Algebra \(k^{S_{n-1}}\#kC_n\). Algebr Represent Theor 27, 495–545 (2024). https://doi.org/10.1007/s10468-023-10230-0
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DOI: https://doi.org/10.1007/s10468-023-10230-0
Keywords
- Hopf algebras
- Frobenius schur indicator
- Representations of hopf algebras
- Bismash product
- Symmetric group