Representation and character theory in 2-categories

• Autores: Nora Ganter, Mikhail Kapranov
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 217, Nº 5, 2008, págs. 2268-2300
• Idioma: inglés
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• Resumen

  • We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of . Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins, Kuhn and Ravenel [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553�594 (electronic)] on equivariant Morava E-theories. We define the concept of induced categorical representation and show that the corresponding 2-character is given by the same formula as was obtained in [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553�594 (electronic)] for the transfer map in the second equivariant Morava E-theory