Download e-book for iPad: Representation Theory: A First Course by William Fulton, Joe Harris

By William Fulton, Joe Harris

ISBN-10: 0387974954

ISBN-13: 9780387974958

ISBN-10: 146120979X

ISBN-13: 9781461209799

The first target of those lectures is to introduce a newbie to the finite­ dimensional representations of Lie teams and Lie algebras. considering that this aim is shared by way of a variety of different books, we must always clarify during this Preface how our method differs, even if the capability reader can most likely see this greater by means of a short flick through the e-book. illustration conception is straightforward to outline: it's the research of the ways that a given team may possibly act on vector areas. it really is very likely designated, besides the fact that, between such basically delineated topics, within the breadth of its curiosity to mathematicians. this isn't mind-blowing: workforce activities are ubiquitous in twentieth century arithmetic, and the place the item on which a bunch acts isn't a vector area, now we have realized to exchange it via person who is {e. g. , a cohomology crew, tangent house, and so forth. }. consequently, many mathematicians except experts within the box {or even those that imagine they could are looking to be} are available touch with the topic in quite a few methods. it's for such people who this article is designed. to place it in a different way, we intend this as a e-book for novices to benefit from and never as a reference. this concept basically determines the alternative of fabric coated the following. so simple as is the definition of illustration thought given above, it fragments significantly after we try and get extra particular.

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Example text

M}} if m is even. ven with a matrix algebra or a product of two matrix algebras give a 2n-dimensional "spin" representation S of H 2n +1, and two 2n - 1 -dimensional "spin" or "halfspin" representations S+ and S- of H 2n . (b) Compute the characters of these spin representations and verify that they are irreducible. (c) Deduce that the spin representations, together with the 2m - 1 onedimensional representations coming from the abelian group Hm/{ ± I} give a complete set of irreducible representations, and compute the character table for Hm.

Before turning to a more systematic study of symmetric and alternating groups, we will work out the next couple of cases. 1. Examples: ~s and 2ls 27 Representations of the Symmetric Group 6 5 As before, we start by listing the conjugacy classes of6 s and giving the number of elements of each: we have 10 transpositions, 20 three-cycles, 30 four-cycles and 24 five-cycles; in addition, we have 15 elements conjugate to (12)(34) and 10 elements conjugate to (12)(345). 1. Find the characters of the representations V and V'; deduce in particular that V and V' are distinct irreducible representations, The first four rows of the character table are thus ~s U U' V V' 4 4 10 20 30 24 15 20 (12) (123) (1234) (12345) (12)(34) (12)(345) -1 0 1 -1 -1 1 0 0 -1 -1 -1 2 -2 0 1 Clearly, we need three more irreducible representations.

24. 25*. Which irreducible representations of 6 d remain irreducible when restricted to ~d? Which are induced from ~d? How much does this tell you about the irreducible representations of ~d? 26*. There is a unique nonabelian group of order 21, which can be realized as the group of affine transformations X 1-+ IXX + Pof the line over the field with seven elements, with IX a cube root of unity in that field. Find the irreducible representations and character table for this group. 36 3. Examples; Induced Representations; Group Algebras; Real Representations Now that we have introduced the notion of induced representation, we can state two important theorems describing the characters of representations of a finite group.

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Representation Theory: A First Course by William Fulton, Joe Harris

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