Representing Finite Groups: A Semisimple Introduction by Ambar N. Sengupta PDF

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By Ambar N. Sengupta

ISBN-10: 1461412307

ISBN-13: 9781461412304

This graduate textbook offers the fundamentals of illustration conception for finite teams from the viewpoint of semisimple algebras and modules over them. The presentation interweaves insights from particular examples with improvement of basic and robust instruments according to the concept of semisimplicity. The dependent rules of commutant duality are brought, besides an creation to representations of unitary teams. The textual content progresses systematically and the presentation is pleasant and alluring. imperative options are revisited and explored from a number of viewpoints. routines on the finish of the bankruptcy aid strengthen the material.

Representing Finite teams: A Semisimple Introduction could function a textbook for graduate and a few complex undergraduate classes in arithmetic. must haves comprise acquaintance with simple team idea and a few familiarity with jewelry and modules. a last bankruptcy offers a self-contained account of notions and ends up in algebra which are used. Researchers in arithmetic and mathematical physics also will locate this booklet useful.

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Example text

7. Here F is a field and n ≥ 2 is an integer for which n1F = 0. Recall that for σ ∈ Sn , and any standard-basis vector ej of Fn , def R(σ)ej = eσ(j) . Hence, χR (σ) = |{j ∈ [n] : σ(j) = j}| = number of fixed points of σ. 45) Now consider the restriction RD of this action to the “diagonal” subspace D = F(e1 + · · · + en ). Clearly, RD (σ) is the identity map for every σ ∈ Sn , and so the character of RD is given by χD (σ) = 1 for all σ ∈ Sn . Then the character χ0 of the representation R0 = R(·)|E0 is given by χ0 (σ) = χR (σ) − χD (σ) = |{j : σ(j) = j}| − 1.

The two-dimensional representation ρ+ 2 on R in which c is associated ◦ with rotation by 120 ; 2 3. The two-dimensional representation ρ− 2 on R in which c is associated ◦ with rotation by −120 . These are clearly all irreducible. Moreover, any irreducible representation of C3 on R2 is clearly either (2) or (3). Now consider a general real vector space V on which C3 has a representation ρ. Choose a basis B in V , and let VC be the complex vector space with B as a basis (put another way, VC is C ⊗R V , viewed as a complex vector space).

Cn−1 r}. 10) Note that there are 3 + n/2 conjugacy classes, and this exactly matches the number of inequivalent irreducible complex representations obtained earlier. To see how this plays out in practice, let us look at D4 . Our analysis shows that there are five conjugacy classes: {e}, {c, c3 }, {c2 }, {r, c2 r}, {cr, c3 r}. There are four one-dimensional complex representations θ±,± , and one irreducible two-dimensional complex representation ρ1 specified through ρ1 (c) = i 0 , 0 −i ρ1 (r) = 0 1 .

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Representing Finite Groups: A Semisimple Introduction by Ambar N. Sengupta

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