Invariant theory

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Problem 625 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection V.1.1 "Functions of Operators in Finite-Dimensional Space"

Prove that every polynomial in the coefficients of the matrix $A$ that is invariant under the similarity transformations $A\mapsto CAC^{-1}$ is a polynomial in $\operatorname {tr} A,\operatorname {tr} A^2,\ldots ,\operatorname {tr} A^n$ .

Problem 626 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection V.1.1 "Functions of Operators in Finite-Dimensional Space"

Let $A$ and $B$ be $2\times 2$ matrices. Prove that every polynomial in the coefficients of $A$ and $B$ that is invariant under the substitutions $A\to CAC^{-1}$ , $B\to CBC^{-1}$ has the form $P(\operatorname {tr} A,\operatorname {tr} B,\operatorname {tr} A^2,\operatorname {tr} B^2,\operatorname {tr} AB)$ , where $P$ is some uniquely determined polynomial in five variables.

Problem 627 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection V.1.1 "Functions of Operators in Finite-Dimensional Space"

Let $A$ and $B$ be $n\times n$ matrices. Prove that the algebra of those polynomials in the coefficients of $A$ and $B$ that are invariant under the substitutions $A\to CAC^{-1}$ , $B\to CBC^{-1}$ contains no fewer than $n^2+1$ generators.

Problem 67.70 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 67 "Extensions of fields. Galois theory"

Find the field of invariants $K\left(x_{1}, \ldots , x_{n}\right)^{A_{n}}$ for the group $\mathbf{A}_{n}$ , acting on the field of rational functions by permuting the variables.

Problem 67.71 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 67 "Extensions of fields. Galois theory"

Let $\varepsilon$ be a primitive complex $n$ -th root of 1 and let the group $G=\langle \sigma \rangle_{n}$ act on the field $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)$ by the rule

\sigma \left(x_{1}\right)=\varepsilon ^{i} x_{i} \quad (i=1, \ldots , n)

Find the field of invariants $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)^{G}$ .

Problem 67.72 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 67 "Extensions of fields. Galois theory"

Find the field of invariants for the group $G$ acting on the field $\mathbb {C}\left(x_{1}, \ldots , x_{n}\right)$ by cyclic permutation of the variables.

Problem 73.10 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 73 "Initial information on representations of continuous groups"

Let $G=\mathbf{G L}_{2}(\mathbb {C})$ . Call a complex function on $G$ polynomial if it is a polynomial in the matrix entries.

(a)

Let $t(A)=\operatorname {tr} A, d(A)=\operatorname {det} A$ . Prove that $t$ and $d$ are central polynomial functions on $G$ .

(b)

Prove that any central polynomial function on $G$ is a polynomial in $t$ and $d$ .

(c)

Let $A=\left(a_{i j}\right) \in G$ and $R=\mathbb {C}\left[x, y\right]$ . Denote by $\Psi (A)$ the homomorphism $R \rightarrow R$ for which

\begin{aligned} & \Psi (A): x \mapsto a_{11} x+a_{12} y \\ & \Psi (A): y \mapsto a_{21} x+a_{22} y \end{aligned}

Prove that $\Psi$ is a representation of the group $G$ on the space $R$ , and that the subspaces of homogeneous polynomials of degree $n$ are invariant under the representation $\Psi$ .

(d)

Prove that, for $A \in \mathbf{S L}_{2}(\mathbb {C})$ , the restriction of $\Psi (A)$ to the subspace $R_{n}$ coincides with the operator $\Phi_{n}(A)$ from problem 73.9.

(e)

Let $\chi_{n}$ be the character of the restriction $\left.\Psi \right|_{R_{n}}$ . Prove that

\chi _{n}=t \chi _{n-1}-d \chi _{n-2}

Problem III.3.7.3 from Кострикин А.И. "Основные структуры алгебры" [Кос04]More problems in Section III.3.7 "Tensor product of representations"

The forms $x y, x^{n}+y^{n}$ are invariants of the two-dimensional linear dihedral group

\begin{gathered} \left(D_{n}, \Phi \right)= \left\langle \left\Vert \begin{smallmatrix} \varepsilon & 0 \\ 0 & \varepsilon ^{-1} \end{smallmatrix}\right\Vert , \quad \left\Vert \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right\Vert \right\rangle , \\ \varepsilon ^{n}=1 \end{gathered}

(see Exercise 9 from § 5). Show that any other (polynomial) invariant of the group $\left(D_{n}, \Phi \right)$ has the form of a polynomial in $x y, x^{n}+y^{n}$ .

Problem III.3.7.4 from Кострикин А.И. "Основные структуры алгебры" [Кос04]More problems in Section III.3.7 "Tensor product of representations"

Verify that the quaternion group, considered in its two-dimensional irreducible representation, has no quadratic or cubic invariants. What can be said about the forms $x^{2} y^{2}, x^{4}+y^{4}$ ?

Problem 18.1.38 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 18.1 "Chapter Problems"

Given two $2 \times 2$ matrices $A$ and $B$ in the Lie group $GL(2, \mathbb {C})$ , i.e. $\operatorname {det}\left(A\right) \neq 0$ , $\operatorname {det}\left(B\right) \neq 0$ . The action of an $n \times n$ matrix $M = (m_{jk})$ on a polynomial $p(\overrightarrow {x}) \in \mathbf{R}_{n} = \mathbb {C}^{x_{1}, x_{2}, \ldots , x_{n}}$ may be defined by setting

T_{M}p(\overrightarrow {x}) := p(\overrightarrow {x}M)

where $\overrightarrow {x}M$ is the multiplication of the row $n$ -vector $\overrightarrow {x}$ by the $n \times n$ matrix $M$ . Thus $\overrightarrow {x}M$ is a row vector again. We denote by

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})}

the ring of polynomials in $\mathbf{R}_{4}$ that are invariant under the action of $A \otimes B$ for all pairs $A, B \in GL(2, \mathbb {C})$ , i.e.

\mathbf{R}_{4}^{GL(2,\mathbb {C}) \otimes GL(2,\mathbb {C})} := \left\{ p \in \mathbf{R}_{4} : T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})\right\} .

The action preserves degree and homogeneity. Consider the polynomial

p(x_{1}, x_{2}, x_{3}, x_{4}) = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{1}x_{2}+x_{2}x_{3}+x_{3}x_{4}+x_{4}x_{1}

and

A = B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Show that $T_{A \otimes B}p(\overrightarrow {x}) = p(\overrightarrow {x})$ .

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[Кос15] Кострикин А.И. Сборник задач по алгебре. МЦНМО, 2015.

[Кос04] Кострикин А.И. Основные структуры алгебры. ФИЗМАТЛИТ, 2004.

[StHa17] Steeb W., Hardy Y. Problems and Solutions in Introductory and Advanced Matrix Calculus. 2nd ed., World Scientific Publishing, 2017.