Let $\rho,V$ be a representation of $S_3$.   In my lecture notes, It says that we can deduce the characters of $V\otimes V$ and $\operatorname{Sym}^2(V)$ and $\bigwedge^2V$ only from the character table of $S_3$.  I don't see how for the symmetric and exterior algebras, as those are quotients and I am not aware of operations on character of quotient representations.   Thank you for your hints or help.  Edit  Trying to do the exercise pointed out by @darij in Alex's Answer.  Let $G=S_3=\{\sigma, \tau|\sigma^2=\tau^3=\sigma\tau\sigma\tau=e\}$.  Taking the notations of the other   We have $m=2$ and $g=\tau$ and we try to find $\chi_{Sym^2(V)}$.  Let $1,j,j^2$ be eigenvalues of $\tau$ and $1, -1$ those of $\sigma$. I don't understand the step 2. How do we get $\chi_{Sym^2(V)}$ in terms of the eigenvalues?  Thank you.
Let $V$ be a representation with character $\chi$.  I would like to have a formula for the characters of the representations $\mathrm{Sym}^m[V]$ and $\wedge ^m[V]$ in terms of $\chi$.  Fulton and Harris presents such a formula for $m=2$, but not for general $m$.  Is there a "nice" such expression?  I found a paper by , but this doesn't seem to be what I'm looking for (it doesn't seem to give a general formula in terms of $\chi$).  In case the link does not work, here is the reference:  X. Zhou, P. Pulay. Characters for Symmetric and Antisymmetric Higher Powers of Representations: Application to  the  Number of Anharmonic Force Constants in Symmetrical Molecules. Journal of Computational Chemistry, Vol 10, Issue 7, (1989).
In which Comic Book series is Batman actually Thomas Wayne? I've heard people say that there is a series of comics where Batman is actually Thomas Wayne instead of his son, Bruce.   Where can I find this? What is the series called? Space war, humans recreated, grown from plants with an alien female supervisor This was a small hardback book I read in the UK in the late eighties. No memory of the cover.  Some kind of space war and a long lived alien female escapes the conflict in her starship, her side had lost.  An evil enemy that she needs to trap so she gets (I think) human  DNA from space dust and sets up on a habitable planet. The story then gets like episodic and looks at the world every couple of generations while she hides out there.  Plants bud on the world and people pop out from big seedpods, the population listens out for babies crying and rescue them.  There are legends of the 'Earth mother’ who secretly guides them to become high tech.   In reality she is creating attractive food bait for the bad aliens.  I think an early human and her are attracted to each other and she (maybe) gives him long life or (maybe) uses reincarnation to keep seeing him.  Edit:- the name 'Rimstalker' has just now popped up in my brain! stop Ipod from asking for old Apple ID password ‘And what on earth’s a Squib?’ said Harry.   To his surprise, Ron stifled a snigger.   ‘Well – it’s not funny really – but as it’s Filch ...’ he said. ‘A Squib is someone who was born into a wizarding family but hasn’t got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual. If Filch’s trying to learn magic from a Kwikspell course, I reckon he must be a Squib. It would explain a lot. Like why he hates students so much.’ Ron gave a satisfied smile. ‘He’s bitter.’   Chamber of Secrets - pages 110-111 - British Hardcover   If a Squib is 'kind of the opposite of a Muggle-born wizards' wouldn't that make a Squib more Muggle than wizard? Does a Muggle strictly have to be born to Muggle parents to be considered a Muggle, or can the lack of magic render a person, even of wizarding parentage, a Muggle?  Hermione can say in all truthfulness that she is a 'Muggle-born witch.' The nickname (albeit derogatory) for a Muggle-born is 'Mudblood.' Filch can say in all truthfulness that he is a 'wizarding-born non-magic person,' which, according to Hagrid is a Muggle.  'Squib' may be derogatory in the same way as 'Mudblood' is. Neville laments his family thought he was a Squib when he showed no magical tendencies as a child; the Longbottoms were so rattled at the thought of Neville being a Squib that they, among other things, threw him out a second-story window to see if he would bounce. So could 'Squib' actually be a derogatory term meaning the opposite of 'Mudblood'? After all, the Ministry doesn't allow Squibs to register as magical beings, and many Squibs are schooled and trained in Muggle professions. Squibs don't get a Hogwarts letter.    Wizards can be born to Muggles. Can Muggles be born to wizards?  ETA 11.22.14: In regard to this question, I was glancing through Philosopher's Stone, and I came across this, that Neville said:     ‘Well, my gran brought me up and she’s a witch,’ said Neville, ‘but the family thought I was all Muggle for ages. My great-uncle Algie kept trying to catch me off my guard and force some magic out of me – he pushed me off the end of Blackpool pier once, I nearly drowned – but nothing happened until I was eight.’   Philosopher's Stone - page 93 - Bloomsbury - chapter seven, The Sorting Hat   Could what Neville says contribute to the theory and/or conclusion that Squibs are, or are not, Muggles? How do we apply what Neville said? Opportunity attack when moving in and out of range of a creature in the same move? Let's say I'm 10ft away from an enemy. In my turn, I choose to move 5ft to get close enough to attack, I do attack, then I use my remaining movement to move out of range of the enemy. Does this still allow them an opportunity attack? zsh scripting - How do I NOT match a pattern in filename generation? Binomial confidence interval estimation - why is it not symmetric? I have used the following r code to estimate the confidence intervals of a binomial proportion because I understand that that substitutes for a "power calculation" when designing receiver operating characteristic curve designs looking at detection of diseases in a population.  n is 150, and the disease, we believe, is 25% prevalent in the population. I have calculated the values for 75% sensitivity and 90% specificity (because that's what people seem to do).      binom.test(c(29,9), p=0.75, alternative=c("t"), conf.level=0.95)      binom.test(c(100, 12), p=0.90, alternative=c("t"), conf.level=0.95)   I have also visited this site:    Which is a java page which calculates binomial confidence intervals, and it gives the same answer.  Anyway, after that lengthy set-up, I want to ask why the confidence intervals are not symmetric, e.g. sensitivity is      95 percent confidence interval:    0.5975876 0.8855583      sample estimate probability: 0.7631579    Sorry if this is a basic question, but everywhere I look seems to suggest that they will be symmetric, and a colleague of mine seems to think they will be too. Find the limit of $\lim\limits_{(x,y)\to (0,0)} \frac{x^2y^2}{x^2y^2+(x-y)^2}$ I have the following problem illustrated by a simple example: If I cite the paper of an author T. Coven in my document, the outcome is 'Coven (2004b)'. If I cite the appendix to this paper, the outcome is 'Coven (2004a)'. Now the appendix pops up much less frequently than the original paper, therefore I would like to reverse the order (exchange 'a' and 'b' in the citations). Is this possible?  I am using Biblatex. Here is the bib-file of the above example (named: lib.bib):  @article{co:2004, author = {T. Coven}, title = {Title A}, journaltitle = {The X Journal}, date = {2004} } @article{coapp:2004, author = {T. Coven}, title = {Appendix to Title A}, journaltitle = {The X Journal}, date = {2004} }   MWE:  \documentclass[a4paper, 12pt, headsepline, headings=small,]{scrreprt} \usepackage[backend=bibtex8,     style=authoryear-icomp,     dashed=false,     autocite=footnote,     maxcitenames=3,     mincitenames=1,     maxbibnames=100,     firstinits=true,     sorting=nty     ]{biblatex} \bibliography{lib}  \begin{document} \cite{co:2004} and \cite{coapp:2004} \end{document} What does the term $\mathcal O(\epsilon^2)$ mean? Why can a discontinuous function not be differentiable? preventDefault() on an  tag Impossible to hide navigation bars in Safari iOS 7 for iPhone/iPod touch Is  structured data (in Microdata, JSON-LD, or RDFa) relevant for SEO?   I know that structured data can provide benefits in other areas, but here I’m only asking about its significance for general web search engines. Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$ Prove: $$\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}, \quad\forall x&gt;0$$  Yeah, pretty much it. I've tried all manner of rearranging and just can't seem to get it. Thanks. Difference between logit and probit models What is the reason behind the multiple Spider-Man reboots? Installing particular versions when repo has newer versions? Let's have a metapackage which depends on the following packages  Depends: A (=2),  B (=2),  C (=2)   Note the '=' - I want to install these particular versions, not the most recent ones.  Additionally:  B depends on A(>=1)  C depends on B(>=1) and A(>=1)  If the most recent versions of A,B,C in the repository are 2 then this works.  Since I have put version 3 of A and version 3 of B in the repo sudo apt-get install my-meta-package refuses to finish with the following info:  The following packages have unmet dependencies:   my-meta-package: Depends: B (= 2) but 3 is to be installed                    Depends: A (= 2) but 3 is to be installed   If I manually install versions 2 of A and B then the metapackage installs correctly, which strongly suggests that there is no hidden dependency on A3 or B3. This is not a solution for the problem as the metapackage is installed automatically on many machines.  QUESTION: What to do to make sudo apt-get install my-meta-package install versions 2,2,2?  Additionally, is there a way I could get an extended info what actually creates the need for version 3 of A and version 3 of B? How to show to the user a progression in a script? Why isn't Higgs coupling considered a fifth fundamental force? When I first learned about the four fundamental forces of nature, I assumed that they were just the only four kind of interactions there were. But after learning a little field theory, there are many other kinds of couplings, even in the standard model. So why isn't the Yukawa Higgs coupling considered a fifth fundamental force? To be a fundamental force, does there need to be a gauge boson mediating it?