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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Notes
- 1.
See Sect. 15.1.5 of Algebra I.
- 2.
- 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) = aφ(x) for all a, x ∈ A.
- 28.
The unit elements e λ ∈ I λ are called irreducible idempotents of A.
- 29.
- 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 λ ≠ ϱ) or a scalar homothety (for λ = ϱ); apply this to φ = E ij and use the trace to evaluate the coefficient of the homothety.
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Basic Notions of Representation Theory. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_5
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