Abstract
Let V be a vector space over an arbitrary field \(\mathbb{k}\). We write \(V ^{\otimes n}\stackrel{\mathrm{def}}{=}V \otimes V \otimes \,\cdots \, \otimes V\) for the tensor product of n copies of V and call it the n th tensor power of V.
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Notes
- 1.
See Sect. 7.2 of Algebra I.
- 2.
See Sect. 6.6.1 of Algebra I.
- 3.
See Proposition 2.1 of Algebra I.
- 4.
See Example 15.3 in Algebra I.
- 5.
See Proposition 12.2 of Algebra I.
- 6.
See Sect. 7.1.4 of Algebra I.
- 7.
See Example 2.1 on p. 22.
- 8.
Recall that the zero set of this form in \(\mathbb{P}(V )\) is the hyperplane intersecting the quadric \(Z(f) \subset \mathbb{P}(V )\) along its apparent contour viewed from v.
- 9.
That is, of first degree.
- 10.
With respect to inclusions.
- 11.
See Sect. 2.2.3 on p. 23.
- 12.
Here we use that \(\mathbb{k}\) is algebraically closed.
- 13.
See Sect. 11.3.3 of Algebra I.
- 14.
Compare with Sect. 2.5.6 on p. 41.
- 15.
Compare with Problem 17.20 of Algebra I.
- 16.
Though the last two curves are given by their affine equations within the standard chart \(U_{0} \subset \mathbb{P}_{2}\), the points at infinity should also be taken into account.
- 17.
That is, a triple of rational functions \(x_{0}(t),x_{1}(t),x_{2}(t) \in \mathbb{k}(t)\) such that f(x 0(t), x 1(t), x 2(t)) = 0 in \(\mathbb{k}(t)\), where \(f \in \mathbb{k}[x_{0},x_{1},x_{2}]\) is the equation of the curve.
- 18.
Compare with Example 11.7 and the proof of Proposition 17.6 in Algebra I.
- 19.
See Sect. 15.3.1 of Algebra I.
- 20.
This is clear if the identity in question expresses some basis-independent properties of the linear operator but not its matrix in some specific basis.
- 21.
Even for the diagonal matrices with distinct eigenvalues, because the conjugation classes of these matrices are dense in \(\mathrm{Mat}_{n}(\mathbb{C})\) as well.
- 22.
See Example 1.3 on p. 8 and Example 17.6 from Algebra I.
- 23.
Note that the decomposition of a Grassmannian polynomial into a sum of decomposable monomials is highly nonunique.
- 24.
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). Tensor Algebras. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_2
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