By Bruce E. Sagan
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Extra resources for The Symmetric Group
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But if Y = RXR- 1 for some fixed matrix R, then the map T---+ RTR- 1 is an algebra isomorphism from Com X to Com Y. Once the commutant algebras are isomorphic, it is easy to see that their centers are too. 8 continues to hold with all set equalities replaced by isomorphisms. There is also a module version of this result. We will use the multiplicity notation for G-modules in the same way it was used for matrices. Theorem 1. 9 Let V be a G-module such that V ~ m1 V( 1) EB m2 V( 2) EB · · · EB mk V(k), where the V(i) are pairwise inequivalent irreducibles and dim V(i) = di.
B) Suppose X has character x and degree d. Prove that g E N if and only if x(g) =d. Hint: Show that x(g) is a sum of roots of unity. (c) Show that for the coset representation, N 9i are the transversal. = nigiHg;l, where the (d) For each of the following representations, under what conditions are they faithful: trivial, regular, coset, sign for Sn, defining for Sn, degree 1 for Cn? (e) Define a function Y on the group G/N by Y(gN) = X(g) for gN E GjN. i. Prove that Y is a well-defined faithful representation of G/N.
GROUP REPRESENTATIONS 22 Proof. We prove only the first assertion, leaving the second one for the reader. It is known from the theory of vector spaces that ker () is a subspace of V since () is linear. So we only need to show closure under the action of G. But if v E ker(), then for any g E G, ()(gv) g()(v) gO = (() is a G- homomorphism) (v E ker()) 0, and so gv E ker (), as desired. • It is now an easy matter to prove Schur's lemma, which characterizes Ghomomorphisms of irreducible modules. This result plays a crucial role when we discuss the commutant algebra in the next section.
The Symmetric Group by Bruce E. Sagan
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