geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Classical groups
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Lie groups
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Cohomology and Extensions
Related concepts
Given a finite group of cardinality , then the complex valued class functions have an inner product – the Schur inner product – given by
The Schur orthogonality relation is the statement that the complex irreducible group characters form an orthonormal basis of the space of class functions under this inner product:
(e.g. Fulton & Harris 1991 Thm. 2.12, tom Dieck 2009 Prop. 2.2.1 & (2.3.3), Steinberg 2012 Thm. 4.3.9)
In fact, this holds also at the level of complex irreducible representations (“grand Schur orthogonality”), regarded for each as unitary matrices with matrix entries :
(e.g. tom Dieck 09 (2.2.7))
Conversely, the sum over all (isomorphism classes) of irreducible characters yields:
(e.g. Fulton & Harris 1991 Ex. 2.21, tom Dieck 2009 Prop. 2.2.1 & (2.3.3))
Textbook accounts:
William Fulton, Joe Harris, Representation Theory: A First Course, Springer (1991) [doi:10.1007/978-1-4612-0979-9]
Benjamin Steinberg: Character Theory and the Orthogonality Relations, chapter 4 in: Representation Theory of Finite Groups – An Introductory Approach, Springer (2012) [doi:10.1007/978-1-4614-0776-8_4]
Lecture notes:
Tammo tom Dieck, Representation theory, 2009 (pdf, pdf)
Andrei Yafaev, Characters of finite groups (pdf)
See also:
Last revised on March 31, 2025 at 17:00:04. See the history of this page for a list of all contributions to it.