By Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj

ISBN-10: 0824798155

ISBN-13: 9780824798154

Complaints of the second one foreign convention on Cumulative Ring conception held in June 1992 at Fes, Morocco. Paper.

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Additional info for Commutative ring theory: proceedings of the II international conference

Example text

The representation theoretic information about a finite group G can be encoded in a matrix known as its character table. 11 (Character table). Let G be a finite group with irreducible characters χ1 , . . , χs and conjugacy classes C1 , . . , Cs . The character table of G is the s × s matrix X with Xij = χi (Cj ). In other words, the rows of X are indexed by the characters of G, the columns by the conjugacy classes of G and the ij-entry is the value of the ith-character on the jth-conjugacy class.

The idea is the following. Let ϕ : G −→ GL(V ) be a representation. We can think of elements of G as scalars via g · v = ϕg v for v ∈ V . A morphism between ϕ : G −→ GL(V ) and ρ : G −→ GL(W ) should be a linear transformation T : V −→ W such that T gv = gT v for all g ∈ G and v ∈ V . , T ϕg = ρg T for all g ∈ G. 1 (Morphism). Let ϕ : G −→ GL(V ) and ρ : G −→ GL(W ) be representations. A morphism1 from ϕ to ρ is by definition a linear map T : V −→ W such that T ϕg = ρg T , for all g ∈ G. In other words the diagram 1 Some authors use the term intertwiner or intertwining operator for what we call morphism.

3. Show that if v ∈ V G , then 1 |G| ϕh v = v. h∈G 4. Conclude dim V G is the rank of the operator P = 1 |G| ϕh . h∈G 5. Show that P 2 = P . 6. Conclude Tr(P ) is the rank of P . 7. Conclude dim V G = 1 |G| Tr(ϕh ). 6. Let ϕ : G −→ GLn (C) be a representation. 1. Show that setting ψg = ϕg provides a representation ψ : G −→ GLn (C). It is called the conjugate representation. Give an example showing that ϕ and ψ do not have to be equivalent. 2. Let χ : G −→ C∗ be a degree 1 representation of G. Define a map ϕχ : G −→ GLn (C) by ϕχg = χ(g)ϕg .

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