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PDF(411 KB)
PDF(411 KB)
二面体群的Grothendieck环结构
The Structure of Grothendieck Rings of Dihedral Groups
二面体群的表示范畴为对称半单monoidal范畴，因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元，并且进一步证明了该Grothendieck环与某一多项式环的商环同构.
The representation category of a dihedral group is a symmetric semisimple monoidal category, so the Grothendieck ring of such a category is a commutative ring generated by finitely many elements. In this paper, the minimal generators of the Grothendieck ring are determined. Moreover, it is shown that the Grothendieck ring is isomorphic to a quotient of a polynomial ring.
二面体群 / Grothendieck环 / 表示范畴 {{custom_keyword}} /
dihedral group / Grothendieck ring / representation category {{custom_keyword}} /
[1] Chen H., Oystaeyen F. V., Zhang Y., The Green rings of Taft algebras, Proc. Amer. Math. Soc., 2014, 142:765-775.
[2] Green J. A., The modular representation algebra of a finite group, Ill. J. Math., 1962, 6(4):607-619.
[3] Green J. A., A transfer theorem for modular representations, J. Algebra, 1964, 1:71-84.
[4] Huang H., Oystaeyen F. V., Yang Y., Zhang Y., The Green rings of pointed tensor categories of finite type, J. Pure Appl. Algebra, 2014, 218:333-342.
[5] Li L., Zhang Y., The Green rings of the generalized Taft Hopf algebras, Contemp. Math., 2013, 585:275-288.
[6] Li Y., Hu N., The Green rings of the 2-rank Taft algebra and its two relatives twisted, J. Algebra, 2014, 410:1-35.
[7] Lidl R., Mullen G. L., Turnwald G., Dickson polynomials, Harlow:Longman Scientific & Technical, 1993.
[8] Naidu D., Rowell E. C., A finiteness property for braided fusion categories, Algebr. Represent. Theory, 2011, 14(5):837-855.
[9] Ostrik V., Pivotal fusion categories of rank 3, Mosc. Math. J., 2015, 15(2):373-396.
[10] Serre J. P., Linear Representations of Finite Groups, Springer-Verlag, New Yark, 1995.
[11] Siehler J., Near-group categories, Algebr. Geom. Topol., 2003, 3(2):719-775.
[12] Wang Z., Li L., Zhang Y., Green rings of pointed rank one Hopf algebras of nilpotent type, Algebr. Represent. Theory, 2014, 17:1901-1924.
[13] Wang Z., Li L., Zhang Y., Green rings of pointed rank one Hopf algebras of non-nilpotent type, J. Algebra, 2016, 449:108-137.
[14] Wang Z., Li L. B., The Casimir Number of a Verlinde Modular Category and Its Applications, Acta Math. Sin., Chin. Series, 2018, 61(1):59-66.
国家自然科学基金资助项目（11871063）；江苏省自然科学基金项目（BK20170589）
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