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2017 | OriginalPaper | Buchkapitel

5. Basic Notions of Representation Theory

verfasst von : Alexey L. Gorodentsev

Erschienen in: Algebra II

Verlag: Springer International Publishing

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Abstract

Given a set R and a field \(\mathbb{k}\), let us write \(R \otimes \mathbb{k}\) for the vector space with basis R over \(\mathbb{k}\). It is formed by the formal linear combinations ∑ x r ⋅ r of elements r ∈ R with coefficients \(x_{r} \in \mathbb{k}\), all but a finite number of which vanish. By definition, the free associative \(\mathbb{k}\)-algebra spanned by the set R is the tensor algebra \(A_{R}\stackrel{\mathrm{def}}{=}\mathsf{T}(R \otimes \mathbb{k})\) of the vector space \(R \otimes \mathbb{k}\).

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Fußnoten
1
See Sect. 15.1.5 of Algebra I.
 
2
See Algebra I, Theorem 14.​4 in Sect. 14.​3.​1 and the discussion in Sect. 15.1.3.
 
3
That is, every totally ordered subset of \(\mathcal{S}'\) has an upper bound; see Sect. 1.4.3 of Algebra I.
 
4
See Sect. 1.4.3 of Algebra I.
 
5
This holds, for example, if the A R- orbit of every vector w∈W is finite-dimensional over \(\mathbb{k}\). In this case, an A R- invariant subspace of minimal dimension contained in the orbit has to be a simple A R- module.
 
6
Also called an intertwining map or a homomorphism of representations.
 
7
See Lemma 5.3 on p. 103.
 
8
That is, it takes every \(\xi: V \rightarrow \mathbb{k}\) to the composition ; see Sect. 7.​3 of Algebra I.
 
9
S.A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society Lecture Notes 29, Cambridge University Press (1977).
 
10
See Sect. 6.5.3 of Algebra I. Note that for \(\mathop{\mathrm{char}}\nolimits (\mathbb{k})\mid \vert G\vert\), the sum of unit masses vanishes and the barycenter is not well defined.
 
11
That is, splits as a direct sum of irreducible representations; see Sect. 5.1.2 on p. 100.
 
12
See formula (5.13) on p. 110.
 
13
Recall that the center of a ring R consists of the elements of R commuting with every element of R, \(Z(R)\stackrel{\mathrm{def}}{=}\{c \in R\mid \forall \,r \in R\;cr = rc\}\).
 
14
See Example 12.14 in Algebra I.
 
15
See Definition 5.1 on p. 109.
 
16
See Example 12.13 of Algebra I.
 
17
See Corollary 5.4 on p. 106.
 
18
For n = 2, the simplicial and sign representations coincide.
 
19
Recall that the conjugacy classes in S n are in bijection with the cyclic types of permutations, i.e., are numbered by the Young diagrams of weight n; see Sect. 12.2.3 of Algebra I.
 
20
Compare with Example 2.​3 on p. 33.
 
21
See Example 12.11 of Algebra I.
 
22
See Example 12.​10 of Algebra I.
 
23
Although Proposition 5.3 was proved under the assumption that the ground field \(\mathbb{k}\) is algebraically closed, for S n-modules it holds over the field \(\mathbb{Q}\) as well, because every complex irreducible representation of S n is actually defined over \(\mathbb{Q}\), as we will see in Chap. 7
 
24
Note that the weight n of the Young diagram λ knows nothing about the dimension d of V. However, some \(\mathbb{S}^{\lambda }V\) may turn out to be zero, as happens, say, with the exterior powers \(\Lambda ^{n}V\) for n > dimV.
 
25
W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge University Press (1997). W. Fulton and J. Harris. Representation Theory. A First Course. Graduate Texts in Mathematics, Springer (1997).
 
26
That is, for every a ≠ 0, there exists a −1 such that aa −1 = a −1a = 1. Equivalently, A satisfies all the axioms of a field except for the commutativity of multiplication (see Definition 2.1 from Algebra I).
 
27
That is, φ(ax) = (x) for all a, x ∈ A.
 
28
The unit elements e λ ∈ I λ are called irreducible idempotents of A.
 
29
Without any reference to Corollary 5.8 and Example 5.5.
 
30
Hint: use the G- linearity of s, isotypic decompositions from (a), and Schur’s lemma.
 
31
Hint: for every \(\mathbb{k}\)-linear map between irreducible representations φ: U λ → U ϱ, the average
$$\displaystyle{ \vert G\vert ^{-1}\sum _{ g\in G}g\varphi g^{-1} = \vert G\vert ^{-1}\sum _{ g\in G}g\varphi \overline{g}^{t} }$$
is G-linear, and therefore, either zero (for λϱ) or a scalar homothety (for λ = ϱ); apply this to φ = E ij and use the trace to evaluate the coefficient of the homothety.
 
Literatur
[DK]
Zurück zum Zitat Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.MathSciNetCrossRef
[Fu]
Zurück zum Zitat Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.MATH
[FH]
Zurück zum Zitat Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.MATH
[Mo]
Zurück zum Zitat Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977. Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
Metadaten
Titel
Basic Notions of Representation Theory
verfasst von
Alexey L. Gorodentsev
Copyright-Jahr
2017
Verlag
Springer International Publishing
DOI
https://doi.org/10.1007/978-3-319-50853-5_5