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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2001 • Question 6

Question

Math 6310 • Algebra Fall 2001 • Question 6

Let \(G\) be a finite group. Prove the following version of Schur’s Lemma: if \(\rho \colon G\to M_n(\mathbb {C})\) is a finite dimensional irreducible representation of \(G\) over the complex numbers, then every element of \(M_n(\mathbb {C})\) that commutes with \(\rho (g)\) for all \(g\in G\) is a scalar matrix.

Raw LaTeX

Let \(G\) be a finite group. Prove the following version of Schur's Lemma: if
\(\rho\colon G\to M_n(\mathbb{C})\) is a finite dimensional irreducible representation of \(G\) over
the complex numbers, then every element of \(M_n(\mathbb{C})\) that commutes with \(\rho(g)\) for all
\(g\in G\) is a scalar matrix.

Topic Tags

Finite Group Theory Irreducible Polynomials Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Representation Theory