P.<x, y, z> = PolynomialRing(QQ)
rho_123 = matrix([[ 0, 0, 1],
[ 1, 0, 0],
[ 0, 1, 0]])
rho_12_34 = matrix([[-1, 0, 0],
[ 0,-1, 0],
[ 0, 0, 1]])
rep_image = MatrixGroup([rho_123, rho_12_34])
group_elements = [g.matrix() for g in rep_image]
def act(g, f):
ginv = g.inverse()
new_coord = ginv * vector([x, y, z])
return f(x=new_coord[0], y=new_coord[1], z=new_coord[2])
def homogeneous_monomials(d):
return P.monomials_of_degree(d)[::-1]Appendix E — Homework 4
Exercise E.1 Here we verify, for a particular example, that the induced dual action (written in terms of the dual basis) of a representation \(\rho\) is given by the matrices \(g \mapsto \rho(g^{-1})^\top\).
Write down the dual representation to the standard representation \(\rho: \mathcal{S}_{3} \to \operatorname{GL}(V)\). To make this concrete, use the basis vectors \(v_1 \coloneqq e_1-e_2\) and \(v_2 \coloneqq e_2-e_3\) for \(V\), so that \[ \rho((1\;2) = \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix} \quad \text{ and } \quad \rho((1\;2\;3)) = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}. \] Consider the action \(\mathcal{S}_{3} ⟳V^*\) in terms of the dual basis \(\{f_1, f_2\}\), where \(f_j(v_i) = 1\) if \(i=j\) and \(0\) otherwise, then write down the matrices corresponding to \((1\ 2)\) and \((1\ 2\ 3)\).
Remember: This is only a sanity check—you should get \(\rho(\sigma^{-1})^\top\)!
Is \(V\) isomorphic to \(V^*\) as an \(\mathcal{S}_{3}\)-representation? Prove or disprove.
Exercise E.2 Here we study an induced representation on spaces of linear transformations.
Consider the representations \(\sigma, \tau: \mathbb{Z}/{4}\mathbb{Z} \to \operatorname{GL}_2(\mathbb{C})\) given by \[ \sigma(1) = \begin{pmatrix} i & 0 \\ 0 & -1 \end{pmatrix} \quad \text{ and } \quad \tau(1) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. \] These representations induce a \(\mathbb{Z}/{4}\mathbb{Z}\)-action on \(\operatorname{Hom}(\mathbb{C}^2_{\sigma},\mathbb{C}^2_{\tau})\) via \[ g \cdot L \coloneqq \tau(g) \circ L \circ \sigma(g^{-1}). \]
If \(L = \begin{psmallmatrix} \alpha & \beta \\ \gamma & \delta \end{psmallmatrix} \in \operatorname{Hom}(\mathbb{C}^2_{\sigma},\mathbb{C}^2_{\tau})\), compute \(1 \cdot L\). Use this to write the action of \(1\) on \(\operatorname{Hom}(\mathbb{C}^2_{\sigma},\mathbb{C}^2_{\tau})\) as a \(4 \times 4\) matrix using the ordered basis \[ (E_{11},E_{12},E_{21},E_{22}), \] where \(E_{ij}\) is the matrix with \(1\) in the \(i\)th row and \(j\)th column and \(0\) elsewhere
Find a basis for the space of invariants \(\operatorname{Hom}(\mathbb{C}^2_{\sigma},\mathbb{C}^2_{\tau})^{\mathbb{Z}/{4}\mathbb{Z}}\).
Hint: Recall Exercise D.5.
Conclude that \(\mathbb{C}^2_{\sigma}\) and \(\mathbb{C}^2_{\tau}\), each of which decompose into two subrepresentations, share one irreducible subrepresentation in common.
Exercise E.3 In this problem we will compute some invariant polynomials. We urge the use of SageMath or other comparable tools for these calculations!
Consider the representation \(\mathcal{A}_{4} ⟳V \coloneqq \mathbb{R}^3\) given in terms of the tetrahedral symmetries (see Example 3.50 and Example 5.4): \[ (1\ 2\ 3) \mapsto \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad \text{ and } \quad (1\ 2)(3\ 4) \mapsto \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \] Since all matrices in this representation are orthogonal, \(V\) is isomorphic to its dual representation \(V^*\).
If we write \(x, y, z\) for the usual coordinate functions on \(\mathbb{R}^3\), we can think of \(V^* = \operatorname{Span}\{x,y,z\}\): each of these coordinate functions is a linear functional on \(V\)! Indeed, we can study the symmetric powers of \(V^*\) to make sense of non-linear polynomial functions on \(\mathbb{R}^3\). For example, we might consider \[ \operatorname{Sym}^{2}({V^*}) = \operatorname{Span}\{x^2,x y,y^2,x z,y z, z^2\}. \] More generally, \(\operatorname{Sym}^{d}({V^*})\) consists of the formal span of all degree \(d\) monomials: \[ \operatorname{Sym}^{d}({V^*}) = \operatorname{Span}\{ x^a y^b z^c \mid a, b, c \in \mathbb{N}\text{ and } a+b+c=d \}. \] There is a natural action of \(G\) via \((g \cdot f)(v) = f(g^{-1} \cdot v)\) for all \(v \in V\).
To facilitate the computations required in this problem, we offer Listing E.1 which implements the above group action. This block of code creates a polynomial ring (over \(\mathbb{Q}\), which is all we need here) for the formal variables \(x, y, z\); generates a group from the image of our generators and stores a list of all the resulting matrices in group_elements; and provides the act() function to compute what a given group element does to a particular polynomial.
For example, we can see the action on each quadratic monomial as in Listing E.2.
quadrics = homogeneous_monomials(2)
for f in quadrics:
f_123 = act(rho_123, f)
f_12_34 = act(rho_12_34, f)
print(f"(1 2 3) * {f} = {f_123} and (1 2)(3 4) * {f} = {f_12_34}")Given a representation \(G ⟳V\), a polynomial \(f \in \operatorname{Sym}^{d}({V^*})\) is \(G\)-invariant if \(f \in \operatorname{Sym}^{d}({V^*})^G\).
Let \(f \in \operatorname{Sym}^{d}({V^*})^G\) and \(c \in \mathbb{R}\). Show that \(G ⟳V\) restricts to an action on the level surface \[ f^{-1}(\{c\}) \coloneqq \{ v \in \mathbb{R}^3: f(v) = c \}. \]
Describe the space of all \(\mathcal{A}_{4}\)-invariant polynomials for \(d \in \{1, \dots, 4\}\).
Hint: Use Exercise D.5. There is no need to include your code, but you should indicate what calculations you carried out!
Using your favorite graphing software, plot some level surfaces of the polynomials from part (b). What sorts of surfaces do you find? Do they seem to enjoy any relevant symmetries?
Exercise E.4 Consider the following operators on \(V = \mathbb{C}^3\) and \(W = \mathbb{C}^2\), respectively: \[ A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad \text{ and } \quad B = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}. \] Write down a basis for \(V \otimes W\), use it to express \(A \otimes B\) as a matrix, then verify that \(\operatorname{Tr}(A \otimes B) = \operatorname{Tr}(A) \operatorname{Tr}(B)\).
Exercise E.5 (Bonus) Compute the eigenvalues of \(A = \begin{psmallmatrix} 0 & 0 & -1 \\ 1 & 0 & -1 \\ 0 & 1 & -1 \end{psmallmatrix}\) without calculating any determinants.
Hint: \(A^4 = \mathbb{I}\).