## Linear Representations of GroupsThis book gives an exposition of the fundamentals of the theory of linear representations of finite and compact groups, as well as elements of the the ory of linear representations of Lie groups. As an application we derive the Laplace spherical functions. The book is based on lectures that I delivered in the framework of the experimental program at the Mathematics-Mechanics Faculty of Moscow State University and at the Faculty of Professional Skill Improvement. My aim has been to give as simple and detailed an account as possible of the problems considered. The book therefore makes no claim to completeness. Also, it can in no way give a representative picture of the modern state of the field under study as does, for example, the monograph of A. A. Kirillov [3]. For a more complete acquaintance with the theory of representations of finite groups we recommend the book of C. W. Curtis and I. Reiner [2], and for the theory of representations of Lie groups, that of M. A. Naimark [6]. Introduction The theory of linear representations of groups is one of the most widely ap plied branches of algebra. Practically every time that groups are encountered, their linear representations play an important role. In the theory of groups itself, linear representations are an irreplaceable source of examples and a tool for investigating groups. In the introduction we discuss some examples and en route we introduce a number of notions of representation theory. O. |

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action Applying arbitrary assertion basis called canonical character closed commutator compact completely reducible complex linear representation condition connected Consequently consider constructed contains coordinates Corollary corresponding decomposition defined definition denote described determined differentiable dimension direct sum equal equations equivalent Example Exercises exists expressed fact factor field finite group finite-dimensional follows formula functions given GL(V GLn(K group G Hence holds identity independent inner product integration irreducible complex representations irreducible representation isomorphic Lemma Lie group Lie group G linear operator linear representation manner matrix elements matrix representation means minimal invariant subspaces multiplication needed obtained one-dimensional representation orthogonal particular permutation polynomials positive PROOF Prove regarded relations representation of G respectively result rotations rule satisfying subgroup Suppose symmetric takes Theorem theory tions topological group transformation trivial uniquely unitary vector space verified