WALUIGI, THE UNSUNG HERO THOMAS RUD¨ Abstract. Since his inception, Waluigi has had a lukewarm reception from the public. Seen as a character with no personality, no voice, no video-game to shine by himself in, the eternal sidekick has become a running joke on the intrawebs. As if the originality used for Mario washed away after making so many copies. However, there is much more to it than it seems, we shall see here why Waluigi is objectively closer to his red-wearing ancestor than his other relatives. \AwaWAAAAAAAAAH." -Waluigi Acknowledgement I want to thank all the people around me every day that did not stop me when I decided to do this stupid essay. I want to thank my main sponsor, Coca Cola for providing me with the adequate quantity of sugar and coffeine to keep me awake during two days, after which I ended up writing this. Thank you all from the bottom of my heart, you are all important to me. 1. Preliminaries To understand how great Waluigi is, let us analyze what he is accused of by those mean internet people. This has a historical reason. Indeed after Mario got tired to fighting a horny Monkey, gave up Pauline and decided to fight turtles instead for the hot new babe on the block, aka Peach. For this purpose, in 1985, he teamed up with this brother, who was just a good guy all around, went adventuring with his fatter smaller brother, with no reward in sight, aside from \watching". Yes, he was weird from the get-go, but he remained a simple color swap for some time, so this weird man had very little importance. He was given more and more importance with such games as the Mario RPG games, the Mario kart/tennis/etc games, the super smash bros games, but his role was minimal. He had to wait for Luigi Mansion to really make a breakthrough as a strong and independent green man. People tend to like him, 2013 was even declared to be the year of Luigi. One still has to observe that both Mario and his defective brother have very little charisma and personality on their own. 1 2 THOMAS RUD¨ Then came Wario, who appeared as late as 1992, most of us were almost born, or being conceived, or at least thought about. He is meant to be the opposite of Mario, he is an antihero, he is evil, greedy, that's already more character traits than Mario himself, but that is for another time. Wario is nothing but a success story as far as I am concerned, this character found his place immediately and turned people to the dark side even before the highly superior Star Wars prequel movies (by the way, have I written an essay about how much better the prequel is ?). So, I have been talking for a bit, but so far so good right ? Everyone is happy ? All is about to change in the year 2000. The y2k bug might have been a hoax, but this has been much worse according to the internet, Waluigi appeared. Why you ask ? Because Wario needed a partner for Mario tennis, that's it. Nothing more, nothing less, they made a character that cannot really speak, that get no importance, no solo game, he is Wario's partner, nothing more, nothing less. Then, Kulthavy, are you saying those internet trolls are right ? He really is a placeholder, nothing more than a random extra bland mustache holder ? NO, here is where you are wrong, he is more than a copy of a copy, I shall explain now why he is in my honest opinion, closer (and to my appreciation, superior) to the original red plumber than both Luigi and Wario. 2. Wario's copy or Luigi's inverse ? This is the crucial question to ask. He is made a partner for Wario in Mario tennis, so Waluigi should be a copy of Wario, but at the same time, Mario is opposed to his inverse Wario on the tennis court, and so Waluigi appears as Luigi's opponent. WALUIGI, THE UNSUNG HERO 3 Figure 1. Waluigi's Creation The whole point of this is that it is the same, geometrically, and more generally mathematically speaking, the inverse of the reflection is the reflection of the inverse. Now we are ready to observe why Waluigi is closer. 3. A geometric approach Firstly, here is Mario : 4 THOMAS RUD¨ Figure 2. Mario Now we create Luigi, its copy, by simplfy taking is reflection through the y-axis. For Wario, the inverse, that will be reflection through the x-axis. Figure 3. Luigi and Wario The sharp observer will notice that Wario's hat already has his W logo. Notice the big difference between mario and his copies ? Mario is facing his left, but taking reflections changes the orientation of the space, so both his copies are WALUIGI, THE UNSUNG HERO 5 facing their right ! Another way to see this is the orientation of the axis, on the first picture, the axis x; y are ordered anticlockwise, but they become clockwise once the reflection is applied ! Both of them are mirror image, so they look similar, but they are far from being THE original. HOWEVER, one can combine both transformations, in whichever order we chose, we get the same result. Ladies and gentlemen, it is my utmost pleasure to introduce to you .... Waluigi ! Figure 4. Waluigi Result. The order of the two axis is back in the anti-clockwise fashion, he is back to facing his left, why ? Because we inverted the orientation twice ! Indeed, if we did the Luigi transformation twice, or the Wario transformation twice, we would get back to Mario, but combining both, we get the closest copy to Mario we can. This is not a symmetry anymore, if you know your middle-school maths, you know we now just have a rotation, it is just Mario rotated, keeping the same orientation ! This is a general fact, it is not only true for reflections of the plane, we will see next another slightly more advanced parallel. 4. A matrix analog Sorry from the bottom of my heart to all the fans of the movie franchise Matrix, this is not going to be about it. 6 THOMAS RUD¨ Here we assume some basic knowledge of matrices, nothing more than knowl- edge of transposition and inversion of matrices will be needed. What is Mario in this world ? Here we associate Mario with some \generic" element of GLn, i.e. some generic invertible matrix. For the expert reader, think about him as generic point of the affine scheme, or without going to those ex- tremities, just thing about some element in general position in the linear algebraic group GLn, so one can think of it as some Zariski dense open subset (i.e. some nonempty Zariski open subset) of GLn. We will not specify any base field for GLn since it applies to all, but we indeed work over a field to avoid saying anything too stupid. Consider n at least 2. So, take our matrix M in general position, we can assume all the usual stuff, n distinct eigenvalues, not symmetric, ... we just need the fact that M −1 6= M 6= M T . The act of taking a \mirror image" in this world is taking the transposition, nothing simpler, just flip the entries, in a similar sense to before, we take some symmetry. For Wario, well, he is the inverse of Mario, so it is simply the inverse matrix. So Mario : M Luigi : M T Wario : M −1: And again in this case, the transposition and inversion maps commute ! We get the same result regardless of the order, we get the inverse transpose, so −1 Waluigi : M −1T = M T : Now you may ask, how is he special ? How is he more special than Luigi and Wario. Well, let me tell you, taking the transpose (taking the Luigi) is NOT a group homomorphism, it's an antihomomorphism, SAME thing for taking the inverse(Wario), but combine both, and you take the Waluigi ! (the inverse transpose) and THIS is an automorphism of GLn, Waluigi is the image of Mario by an actual automorphism, not just some involution ! What's more, the inverse transpose is not just any automorphism, it is the most famous outer automorphism of GLn ! Here is the definition if you need it. Definition 4.1 (Inner/Outer automorphism). An automorphism GLn ! GLn is −1 inner if it is of the form x 7! gxg for some g 2 GLn. An automorphism is outer if it is not inner. Of course there are other outer automorphisms when our field has automor- phisms itself, but this is really THE outer au tomorphism inherent to the structure GLn itself. It is a truly remarkable map and in this respect, Waluigi is probably the most interesting of the four mustache aficionados. Example 4.2. Here is a simple computation of some example. WALUIGI, THE UNSUNG HERO 7 0 1 0 1 1 1 1 ()T 1 0 0 Mario = @0 −1 1A @1 −1 0A = Luigi 0 0 1 1 1 1 ()−1 ()−1 0 1 0 1 1 1 −2 ()T 1 0 0 Wario = @0 −1 1 A @ 1 −1 0A = Waluigi 0 0 1 −2 1 1 Note that we did not take a matrix with distinct eigenvalues, but we already mentioned that this was unnecessary. 5. Main Theorem Our two examples are general enough so that we trust the reader to condense it into a proof of the following.