By A. H. Clifford
The fabric during this quantity was once provided in a second-year graduate path at Tulane collage, throughout the educational yr 1958-1959. The publication goals at being mostly self-contained, however it is believed that the reader has a few familiarity with units, mappings, teams, and lattices. basically in bankruptcy five will extra initial wisdom be required, or even there the classical definitions and theorems at the matrix representations of algebras and teams are summarized
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Additional resources for The algebraic theory of semigroups. Vol.2
Proof. 1). Hence, ρ2 (g) Tu = Tu ρ1 (g). The operator Tu is thus an intertwining operator for ρ1 and ρ2 . 8. 4). (i) If ρ1 and ρ2 are inequivalent, then Tu = 0. (ii) If E1 = E2 = E and ρ1 = ρ2 = ρ, then Tu = Tr u IdE . dim E Proof. 15). For the second, we need only calculate λ given that Tu = λ IdE . So we obtain 1 Tr u Tr Tu = |G| g∈G Tr u = Tr u, and thus λ = dim E . 9. Let (E1 , ρ1 ) and (E2 , ρ2 ) be irreducible representations of G. We choose bases in E1 and E2 . (i) If ρ1 and ρ2 are inequivalent, then (ρ2 (g))k (ρ1 (g −1 ))ji = 0.
A representation ρ is irreducible if and only if (χρ | χρ ) = 1. 1 Deﬁnition In general, if a group G acts on a set M , then G acts linearly on the space F(M ) of functions on M taking values in C by (g, f ) ∈ G × F(M ) → g · f ∈ F(M ), where ∀x ∈ M, (g · f )(x) = f (g −1 x). We can see immediately that this gives us a representation of G on F(M ). Take M = G, the group acting on itself by left multiplication. One obtains a representation R of G on F(G) called the left regular representation (or simply regular representation) of G.
15 Representation of GL(2, C) on the polynomials of degree 2. Let G be be a group and let ρ be a representation of G on V = Cn . Let (k) P (V ) be the vector space of complex polynomials on V that are homogeneous of degree k. (a) For f ∈ P (k) (V ), we set ρ(k) (g)(f ) = f ◦ ρ(g −1 ). Show that this deﬁnes a representation ρ(k) of G on P (k) (V ). (b) Compare ρ(1) and the dual representation of ρ. (c) Suppose that G = GL(2, C), V = C2 , and ρ is the fundamental representation. Let k = 2. To the polynomial f ∈ P (2) (C2 ) deﬁned by f (x, y) = ax2 + 2bxy + cy 2 we associate the vector vf = a b c ∈ C3 .
The algebraic theory of semigroups. Vol.2 by A. H. Clifford