adding the june 23rd notes

Author: Gordie Novak

Notes/June 23rd.md

@@ -0,0 +1,24 @@
+## June 23rd Tasks
+1. The big objective for the end of the week is to have recorded the state of the different teams of the development through GitHub Issues so that people can contribute easily.  
+Which GitHub issues should be pursued depends on how this week goes. There's definitely going to be the definitino of the orbit sum part, which needs to be polished and tested, and so that will be a good beginning of the Macaulay2 week task. Then we can see the implementation and test so that beginnners in M2 can write easily because there's not much code and it's more text-based. (Playing with the code and making sure that it works).
+
+"I'm kind of implicitly thinking that Sam will be in charge of that stream."
+
+2. And then there is developing the skew algorithms, and the first one should be the weight matrix algorithm. So I think that in the sense that everyone should get a look at it, but I think that we can normally have a project manager there.  
+SAM: "We made a psuedocode that we think may work by hand"
+GANDINI: "Will it work by hand?"
+SAM: "We haven't tried yet."
+MARCUS: "It almost seems too simple"
+GANDINI: "Actually that's what I was expecting. I was wondering if someone could take the output of the commutative setting and take the powers to get the results in the skew commutative setting. I'm saying the claim that if M is an invariant in the commutative setting, M-bar (or the corresponding wedge) is an invariant in the skew setting if and only if M is power-free."
+MARCUS: "for the abelian group case?"
+GANDINI: "Yes." 
+SAM: "We should run through some tests with this."
+GANDINI: "What you could actually do today is develop as much as you can and put something on GitHub issues like, 'this is what we have and this is what we need to get done.' If you need help writing it, I can do that for you all."
+
+3. There is also the work fo the previous students, the elementary abelian groups algorithm, and we'll worry about how to get the seed from Christine tomorrow. 
+TASK: Understand the algorithms that the previous students worked on. (Look at the video).
+Let's do the easiest case: Z mod p x Z mod p acting on three variables, x, y, z. Let's say the weight matrix is three by two. 
+If the rows of the matrix are linearly independent, we'll have rank 1. Let's the the first row is 1 0 1 and second row is 0 1 1. 
+
+
+* Write down issues on the GitHub Repository.

Notes/LaTeX-notes/June 23rd.pdf

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+\documentclass[10pt,oneside]{article}
+%%%% Page Info + Commands %%%%%{
+
+%packages
+\usepackage{geometry}
+\usepackage{latexsym}
+\usepackage{amssymb}
+\usepackage{amsfonts}
+\usepackage{amstext}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{multicol}
+\usepackage{hyperref}
+\usepackage{enumerate}
+\usepackage{tikz}
+\usepackage{enumitem}
+\usepackage{xcolor}
+
+
+\setlength{\footskip}{-5mm}
+
+% a good babble textwidth is 5.75in
+\newcommand{\babblewidth}{\setlength\textwidth{5.75in}}
+
+
+% This will stretch out the page
+\newcommand{\bigpage}{  \setlength \oddsidemargin{-.25in}
+            \setlength \textwidth{6.75in}
+            \setlength \topmargin{-1in}
+            \setlength \textheight{9.75in}}
+
+
+%This will shrink the page
+\newcommand{\smallpage}{  \setlength \oddsidemargin{.5in}
+            \setlength \textwidth{5in}
+            \setlength \topmargin{0in}
+            \setlength \textheight{9in}}
+
+\newcommand{\separator}{\vglue .1in\hrule\vglue .1in}
+
+\newcommand{\pause}{\vglue .1in\hrulefill {\tiny Pause here}\hrulefill \vglue .1in}
+
+%%general stuff
+\newcommand{\caret}{\textasciicircum}
+
+%This will put a circle around something.
+\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
+            \node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
+
+
+% Commands for abstract
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\C}{\mathbb{C}}
+\newcommand{\normal}{\triangleleft}
+\newcommand{\Q}{\mathbb{Q}}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\K}{\mathbb{K}}
+\newcommand{\aut}[1]{{\rm Aut}(#1)}
+\newcommand{\Ker}{{\rm Ker}\,}
+\newcommand{\im}{{\rm Im}\,}
+\newcommand{\cyclic}[1]{\langle #1 \rangle}
+\newcommand{\isom}{\cong}
+\newcommand{\autc}[1]{{\rm Aut_c}(#1)}
+\newcommand{\autsub}[2]{{\rm Aut}_{#1}(#2)}
+
+\newcommand{\vp}{\vspace{0.15cm}\\}
+\newcommand{\vpp}{\vspace{0.25cm}\\}
+\newcommand{\vpn}{\vspace{0.05cm}\\}
+\newcommand{\rmv}[1]{\,\backslash\{#1\}}
+\newcommand{\rmvs}[1]{\,\backslash{#1}}
+\newcommand{\md}[1]{\,\text{mod } #1}
+
+%%%%%%%% command for graphics %%%%%%%%%%%%%
+%}
+\definecolor{darkgreen}{rgb}{0.0, 0.5, 0.0}
+\definecolor{sasha}{rgb}{0.0, 0.5, 0.5}
+\definecolor{marcus}{rgb}{0.7, 0.3, 0.3}
+\definecolor{sam}{rgb}{0.2, 0.2, 0.8}
+
+
+\begin{document}
+
+$$
+\begin{bmatrix}
+    1 & 0 & 1\\
+    0 & 1 & 1
+\end{bmatrix}
+$$
+
+We can imagine this as the first element leaving x and z alone, and setting y to 0. Additionally, we can imagine the second one setting x to 0 and leaving y and z alone. We need to worry about getting enough elements from the kernel so we can generate the whole ring.\vpp
+We have a set of powers that will be invariant no matter what:
+\begin{itemize}
+    \item $x^p$
+    \item $y^p$
+    \item $z^p$
+\end{itemize}
+The trick is to use this trick to lower exponents in the kernel. \\
+We also know that in the kernel we can consider:
+$$xyz^{p-1}$$
+So we have that:
+\begin{align*}
+    g_0&: xz^{p-1}\\
+    g_1&: xy^{p-1}
+\end{align*}
+If we look at our roots of unity, we see that the total powers of each group action will result in a total order of $p$, so our root of unity $\mu^p = 1$. \vpp
+Because they are all multiples of this first invariant and we can generate the entire invariant ring. \vpp
+CLAIM: If we look at the $x^2y^2z^{p-2}$, it will also be invariant because the total order will be $p$. 
+
+\end{document}

Notes/LaTeX-notes/June 23rd.tex

@@ -95,7 +95,8 @@ \subsection*{Gandini Notes on Gröbner Bases}
 $\implies F[x]$ is a PID.\vpp
 Remember Bezout's:\\
 $\implies \gcd(f_1,f_2)=af_1+bf_2$\\
-$\implies \gcd \in I = (f_1,f_2)$
-
+$\implies \gcd \in I = (f_1,f_2)$\vpp
+BIG O-PLUS:
+$${\oplus\bigoplus_{\oplus\bigoplus\oplus}^{\oplus\bigoplus\oplus} \oplus}$$
 
 \end{document}