Symmetric and Skew-Symmetric Tensors

Verify that the operators Sym and Alt in the space $\mathbf{T}_{0}^{p}$ have the following properties (where $\circ$ denotes composition of maps):

Show that $S(V)$ with the operation $\bullet$ and $\Lambda (V)$ with the operation $\wedge$ are associative algebras. Moreover:

Let $e_{1}, \ldots , e_{n}$ be a basis of the space $V$. Prove that the tensors

form a basis of the space $S^{p}(V)$. Compute its dimension.

Let $e_{1}, \ldots , e_{n}$ be a basis of the space $V$. Prove that the tensors

form a basis in $\Lambda^{p}(V)$. Compute its dimension and the dimension of the space $\Lambda (V)$. Carry out analogous reasoning for $\Omega_{q}(V)$.

Prove that the linear span of the set

equals $\mathbf{S}^{k}(V)$ (the field $\mathbb {k}$ has characteristic 0).

Prove that the natural embeddings

induce isomorphisms

Let $\operatorname {dim} V > 2$. Show that the spaces $\Lambda^{2}\left(\Lambda^{2}(V)\right)$ and $\Lambda^{4}(V)$ are not isomorphic.

Show that the exterior product $a_{1} \wedge a_{2} \wedge \ldots \wedge a_{p}$ of the vectors $a_{1}, a_{2}, \ldots , a_{p}$ equals zero if and only if these vectors are linearly dependent.

Let $T$ be a skew-symmetric covariant tensor of valence $n$ in the space $V$, with $n = \operatorname {dim} V$. Show that all its components coincide, up to sign, with $T_{12 \ldots n}$ (one says that the tensor $T$ has a single essential component). Prove that this component transforms according to the rule

where $\det C$ is the determinant of the transition matrix to the new basis. Similarly, in the contravariant case, prove the formula $\left(T^{\prime }\right)^{12 \ldots n} = (\det C)^{-1} \cdot T^{12 \ldots n}$.

Let $a_{1}, \ldots , a_{p}$ and $b_{1}, \ldots , b_{p}$ be two sets of linearly independent vectors. Prove that $a_{1} \wedge \ldots \wedge a_{p} = \lambda \cdot b_{1} \wedge \ldots \wedge b_{p}$ if and only if the linear spans of the two sets of vectors coincide. Moreover, $\lambda$ equals the determinant of the transition matrix from the second system of vectors to the first.

Show that the linear span of a system of linearly independent vectors $a_{1}, \ldots , a_{k}$ coincides with the set of those vectors $a$ for which $T \wedge a = 0$, where $T = a_{1} \wedge \ldots \wedge a_{k}$.

Let $T \in \Lambda^{p}(V)$; define

Prove that $T$ can be represented in the form $\widetilde{T} \wedge S$, where $S = a_{1} \wedge \ldots \wedge a_{k} \neq 0$, if and only if $a_{i} \in \operatorname {Ann}(T)$ for $i = 1, \ldots , k$.

Prove that the correspondence $T \mapsto \operatorname {Ann}(T)$ is a bijection from the set of nonzero decomposable $p$-vectors in $V$ (considered up to multiplication by a scalar) onto the set of $p$-dimensional subspaces of the space $V$.

Show that for $T \in \Lambda^{p}(V)$ the dimension of the space $\operatorname {Ann}(T)$ does not exceed $p$, with equality holding precisely when $T$ is the exterior product of linearly independent vectors.

Let a skew-symmetric tensor $T \in \Lambda^{n-1}(V)$ be given, where $\operatorname {dim} V = n$. Prove that there exist vectors $a_{1}, \ldots , a_{n-1}$ such that $T = a_{1} \wedge \ldots \wedge a_{n-1}$ (i.e. the $(n-1)$-vector $T$ is decomposable). Similarly, for $S \in \Omega_{n-1}(V)$ there exist covectors $\xi^{1}, \ldots , \xi^{n-1}$ such that $S = \xi^{1} \wedge \ldots \wedge \xi^{n-1}$.

Prove that a bivector $T$ can be represented in the form $a \wedge b$ for some vectors $a$ and $b$ if and only if $T \wedge T = 0$.

Let $\varphi$ be a nondegenerate bilinear function on $V$. Prove that on the space $\Lambda^{2}(V)$ one can define a nondegenerate bilinear form $\Phi$ such that for any elements $v_{i} \in V$, $i = 1, \ldots , 4$, we have

Let $f: V \rightarrow V$ be a linear operator. Show that the operator $f^{\otimes p}$ (the tensor power) commutes with the operator Alt, thereby defining a linear map $f^{\wedge p}: \Lambda^{p}(V) \rightarrow \Lambda^{p}(V)$.

Let $\operatorname {dim} V = n, f: V \rightarrow V$ be a linear operator. Prove that the operator $f^{\wedge (n-1)}$ is either equal to 0, or nondegenerate, or has rank 1.

Let $f: V \rightarrow V$ be a linear operator, $\operatorname {dim} V = n$. Since $\operatorname {dim} \Lambda^{n}(V) = 1$, the operator $f^{\wedge n}$ is multiplication by a scalar. Prove that this scalar equals $\det f$.

Prove the formulas ($\circ$ is composition):

Let $e_{1}, \ldots , e_{p}$ be linearly independent eigenvectors of the operator $f: V \rightarrow V$, corresponding respectively to the eigenvalues $\lambda_{1}, \ldots , \lambda_{p}$ (not necessarily distinct). Prove that $e_{1} \wedge \ldots \wedge e_{p}$ is an eigenvector of the operator $f^{\wedge p}$, corresponding to the eigenvalue $\lambda_{1} \cdot \ldots \cdot \lambda_{p}$.

Let the linear operator $f: V \rightarrow V$ be reducible to diagonal form. Prove that in this case $f^{\wedge p}$ is also diagonalizable. Is the converse true?

Let $\lambda_{k}(k = 1, \ldots , n)$ be all the eigenvalues of the operator $f: V \rightarrow V$, taken with multiplicity, $\operatorname {dim} V = n$. Prove that the set of all eigenvalues of the operator $f^{\wedge p}$, taken with multiplicity, consists of all possible products

Prove that $\operatorname {det}\left(f^{\wedge p}\right) = (\det f)^{\binom {n-1}{p-1}}$, where $f: V \rightarrow V$ is a linear operator, $\operatorname {dim} V = n$.

Let the characteristic polynomial of the operator $f: V \rightarrow V$, $\operatorname {dim} V = n$, equal $\sum_{p = 0}^{n} a_{p} \lambda^{n-p}$. Prove the equality $\operatorname {tr}\left(f^{\wedge p}\right) = (-1)^{n-p} a_{p}$.

Find the traces of the operators $f^{\wedge p}$ for nontrivial $\Lambda^{p}(V)$, if $f$ is given by its matrix:

Prove that if $\operatorname {tr}\left(f^{\wedge p}\right) = 0$ for every $p = 1,2, \ldots$, then the operator $f: V \rightarrow V$ is nilpotent.

Find the Jordan form of $f^{\wedge 2}$ from the known Jordan form of $f$:

Let $W$ be a $k$-dimensional subspace of $V$. Describe the natural embedding of $\Lambda^{p}(W)$ into $\Lambda^{p}(V)$. Prove that the operator $f: V \rightarrow V$ has $W$ as an invariant subspace if and only if $\Lambda^{k}(W)$ is invariant under $f^{\wedge k}$.

For a covector $\varphi \in V^{*}$, define the interior product operator as the map

where, as above, $F_{T} \in \mathbf{P}_{0}^{p}$ is the multilinear function corresponding to $T \in \mathbf{T}_{0}^{p}$. Prove that the interior product operator takes symmetric tensors to symmetric ones, and skew-symmetric ones to skew-symmetric ones. Let $\varphi_{1}$ and $\varphi_{2}$ be two covectors. Show that the operators $i\left(\varphi_{1}\right)$ and $i\left(\varphi_{2}\right)$ commute in $S(V)$ and anticommute in $\Lambda (V)$.