Intereting Posts

Solution to the second order differential equation
If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$
Proving that $\sum \limits_{d^2|n}\ \mu (d)=\mu^2(n)$, where $\mu$ is the Möbius function.
Subring of a finitely generated Noetherian ring need not be Noetherian?
Proving that $\mathbb{Z}$ is a Euclidean domain
Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$
If $\gcd( a, b ) = 1$, then is it true to say $\gcd( ac, bc ) = c$?
Let H be a proper subgroup of G of order prime $p^k$ and $N(H) = \{a \in G|aHa^{-1} = H\}.$Show that $N(H) \neq H.$
Is there a way to prove $\int {x^n e^x dx} = e^x \sum_{k = 0}^n {( – 1)^k \frac{{n!}}{{(n-k)!}}x^{n-k} } + C$ combinatorially?
Is the natural logarithm actually unique as a multiplier?
Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$
Purpose of Fermat's Little Theorem
(Unsolved) In this infinite sequence, no term is a prime: prove/disprove.
Applying Freyd-Mitchell's embedding theorem on large categories
Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules

So I’m given the following definition:

$h(g)p(z)=p(g^{-1}z)$ where g is an element of $SL(3,\mathbb{C})$, $p$ is in the vector space of homogenous complex polynomials of $3$ variables and $z$ is in $\mathbb{C}^3$.

What I’m having trouble showing is that mapping $g$ to $h(g)$ is a group homomorphism. Namely, I know that $h(ab)p(z)=p(b^{-1}a^{-1}z)$, but I can’t seem to make sense that $h(a)(h(b)(p(z))$ is not $p(a^{-1}b^{-1}z)$.

- What is $\gcd(0,a)$, where a is a positive integer?
- G is group of order pq, pq are primes
- What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
- Definition of Simple Group
- When is a divisible group a power of the multiplicative group of an algebraically closed field?
- Is the conjugation map always an isomorphism?

This is probably trivial, so condescending replies are welcome!

- Can any finite group be realized as the automorphism group of a directed acyclic graph?
- Exponential of a polynomial of the differential operator
- “Natural” example of cosets
- Nonabelian semidirect products of order $pq$?
- $\forall m,n \geq 2$, $\exists$ non-cyclic group of order $n^{m}$
- Prove or disprove: $(\mathbb{Q}, +)$ is isomorphic to $(\mathbb{Z} \times \mathbb{Z}, +)$?
- Check in detail that $\langle x,y \mid xyx^{-1}y^{-1} \rangle$ is a presentation for $\mathbb{Z} \times \mathbb{Z}$.
- The kernel of an action on the orbits of normal subgroup if group acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$
- If L is a subgroup of $\mathbb{Z}^{3}$, linearly independent, linear equations
- Finite Abelian groups: $G \times H \cong G\times K$ then $H\cong K$

You should really write $h(g) p(z)$ as $(h(g) p)(z)$. In other words, $p$ is a polynomial function on $\mathbb{C}^3$ and so is $h(g) p$, with its value at $z$ being given by your formula, i.e. $(h(g) p)(z) = p(g^{-1} z)$.

Now recompute $(h(a) h(b) p)(z)$, giving $(h(a) h(b) p)(z) = (h(b) p)(a^{-1} z) = p(b^{-1} a^{-1} z)$. For what it is worth, this may well be classified as “trivial”, but I found keeping straight what acts on what and how was a significant stumbling block for me when I first studied representation theory and invariant theory. It’s easy to get actions wrong and have calculations fall apart due to these “trivial” issues, so they are most definitely worth straightening out.

You can see also this thread for further explanation in a slightly more abstract setting.

- Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?
- How does Newton's Method work?
- What is the exact and precise definition of an ANGLE?
- Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$
- Does a countable set generate a countable group?
- Real Analysis, Folland Problem 6.1.2 $L^p$ spaces
- Equality of positive rational numbers.
- Attempting Collatz conjecture using inverse function
- How to compute the determinant of a tridiagonal matrix with constant diagonals?
- If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$
- Is the orthogonality between Associated Legendre polynomials preserved on an interval
- Combinatorics: How many ways are there to
- Identity for simple 1D random walk
- Different shapes made from particular number of squares
- How many numbers with distinct digits exist within a given range?