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\def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/2014-04/}}}

\def\green{\color{green}}

\def\diag{{\operatorname{diag}}}
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\begin{document}
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{\LARGE\bf Cheat Sheet $\beta$}\hfill
\parbox[b]{5.2in}{\tiny
  \null\hfill\sheeturl; verification at {\tt 2014-04/CheatSheetBeta-Verification.nb}
  \newline\null\hfill initiated 24/3/13; continues \href{http://drorbn.net/AcademicPensieve/2013-06/}{2013-06}; continued \href{http://drorbn.net/AcademicPensieve/2014-05/}{2014-05}; modified \today, \ampmtime
}

\vskip -10pt
\rule{\textwidth}{1pt}

\textbf{$\beta$-calculus.} With $\epsilon:=1+\alpha$, $\langle\alpha\rangle:=\sum_v\alpha_v$, and $\langle\gamma\rangle:=\sum_{v\neq u}\gamma_v$,

\vskip 1mm  $\displaystyle
  \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\end{array}
  \ast
  \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\end{array}
  \underset{\beta}{=}
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & A_1 & 0 \\
    T_2 & 0 & A_2
  \end{array}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|c}
      \omega & H \\
      \hline
      u & \alpha \\
      v & \beta \\
      T & \gamma
    \end{array}
    @>tm^{uv}_w>\beta>
    \left(\begin{array}{c|c}
      \omega & H \\
      \hline
      w & \alpha+\beta \\
      T & \gamma
    \end{array}\right)\sslash({u,v\atop\to w})
  \end{CD}
$
\hfill $\displaystyle
  \rho^\pm_{ux} \underset{\beta}{=}
  \begin{array}{c|cc}
    1 & x \\
    \hline
    u & t_u^{\pm 1}-1
  \end{array}
$

\vskip 1mm $\displaystyle
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & x & y & H \\
      \hline
      T & \alpha & \beta & \gamma
    \end{array}
    @>{hm^{xy}_z}>\beta>
    \begin{array}{c|cc}
      \omega & z & H \\
      \hline
      T & \alpha+\beta+\langle\alpha\rangle\beta & \gamma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|cc}
      \omega & x & H \\
      \hline
      u & \alpha & \beta \\
      T & \gamma & \delta
    \end{array}
    @>{sw^{ux}_{th}}>\beta>
    \begin{array}{c|cc}
      \omega\epsilon & x & H \\
      \hline
      u & \alpha(1+\langle\gamma\rangle/\epsilon)
        & \beta(1+\langle\gamma\rangle/\epsilon) \\
      T & \gamma/\epsilon & \delta-\gamma\beta/\epsilon
    \end{array}
  \end{CD}
$

\vskip 1mm Constraints.
$\bullet$ Column sums are monomials minus 1.
\quad $\bullet$ At $t_\ast=1$, $\omega=1$ and $A=0$.

\vskip -8pt
\rule{\textwidth}{1pt}
\textbf{$\sigma$ calculus.}
\hfill $\sigma_1\ast\sigma_2=\sigma_1\cup\sigma_2$,
\quad $tm^{uv}_w=({u,v\atop\to w})$,
\quad $\begin{CD}\sigma @>{hm^{xy}_z}>> (\sigma\remove\{x,y\})\cup(z\to\sigma_x\sigma_y)\end{CD},$
\quad $sw^{ux}_{th}=I,$
\quad $R^\pm_{ux}\to t_u^{\pm 1}$

\textbf{$\beta$-better calculus.}
\hfill Constraints.
$\bullet$ Sum of column $x$ is $(\sigma_x-1)w$.
\quad $\bullet$ $\omega^{k-1}\mid\Lambda^kA$.
\quad $\bullet$ At $t_\ast=1$, $\omega=1$ and $A=0$.

\vskip 1mm  $\displaystyle
  \begin{array}{c|c}\omega_1&H_1\\\hline T_1&A_1\end{array}
  \ast
  \begin{array}{c|c}\omega_2&H_2\\\hline T_2&A_2\end{array}
  \underset{\beta_b}{=}
  \begin{array}{c|cc}
    \omega_1\omega_2 & H_1 & H_2 \\
    \hline
    T_1 & \omega_2A_1 & 0 \\
    T_2 & 0 & \omega_1A_2
  \end{array}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|c}
      \omega & H \\
      \hline
      u & \alpha \\
      v & \beta \\
      T & \gamma
    \end{array}
    @>tm^{uv}_w>\beta_b>
    \left(\begin{array}{c|c}
      \omega & H \\
      \hline
      w & \alpha+\beta \\
      T & \gamma
    \end{array}\right)\sslash({u,v\atop\to w})
  \end{CD}
$
\hfill $\displaystyle \rho^\pm_{ux}
  \underset{\beta_b}{=}
  \begin{array}{c|cc}
    1 & x \\
    \hline
    u & t_u^{\pm 1}-1
  \end{array}
$

\vskip 1mm $\displaystyle
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & x & y & H \\
      \hline
      T & \alpha & \beta & \gamma
    \end{array}
    @>{hm^{xy}_z}>\beta_b>
    \begin{array}{c|cc}
      \omega & z & H \\
      \hline
      T & \alpha+\sigma_x\beta & \gamma
    \end{array}
  \end{CD}
$
\hfill $\displaystyle
  \begin{CD}
    \begin{array}{c|cc}
      \omega & x & H \\
      \hline
      u & \alpha & \beta \\
      T & \gamma & \delta
    \end{array}
    @>{sw^{ux}_{th}}>\beta_b>
    \begin{array}{c|cc}
      \omega+\alpha & x & H \\
      \hline
      u & \sigma_x\alpha & \sigma_x\beta \\
      T & \gamma & \delta+\frac{\alpha\delta-\gamma\beta}{\omega}
    \end{array}
    =: \begin{array}{c|c}
      \cdot & - \\
      \hline
      \mid & \begin{pmatrix}\sigma_x&0\\0&1\end{pmatrix}\cdot A^{ux}
    \end{array}
  \end{CD}
$

\[
  \begin{CD}
    \begin{array}{c|ccc}
      \omega & a & b & S \\
      \hline
      a & \alpha & \beta & \theta \\
      b & \gamma & \delta & \epsilon \\
      S & \phi & \psi & \Xi
    \end{array}
    @>{dm^{ab}_{c}}>\beta_b\ {\green\checkmark}>
    \left(\begin{array}{c|cc}
      \omega+\beta & c & S \\
      \hline
      c & \gamma+\sigma_a\delta+\sigma_b(\alpha+\sigma_a\beta)
          +\frac{\beta\gamma-\alpha\delta}{\omega}
        & \epsilon+\sigma_b\theta+\frac{\beta\epsilon-\delta\theta}{\omega} \\
      S & \phi+\sigma_a\psi+\frac{\beta\phi-\alpha\psi}{\omega}
        & \Xi+\frac{\beta\Xi-\psi\theta}{\omega}
    \end{array}\right)
    \sslash({t_a,t_b\atop\to t_c})
  \end{CD}
\]

The MVA (mod units):\hfill
$\displaystyle
  n\text{-component }L\underset{\beta b}{\mapsto}(\sigma,\omega,A)
  \mapsto\frac{\omega^{2-n}\det'(A-\omega\diag((\sigma_i-1))}{1-t_1}
$
\hfill${\green\checkmark}$

\rule{\textwidth}{0.5pt}

Note. $A^{ux}=\begin{pmatrix}
  \alpha & \beta \\
  \gamma & \delta+\frac{\alpha\delta-\gamma\beta}{\omega}
\end{pmatrix}
= \begin{pmatrix}
  \alpha & \beta \\
  \gamma & \frac{(\omega+\alpha)\delta-\gamma\beta}{\omega}
\end{pmatrix}
= \frac{1}{\omega}\left[
  (\omega+\alpha)\begin{pmatrix}
    \alpha & \beta \\ \gamma & \delta
  \end{pmatrix}
  - \begin{pmatrix}\alpha\\\gamma\end{pmatrix}
    \begin{pmatrix}\alpha&\beta\end{pmatrix}
  \right]
  =\frac{1}{\omega}\left[(\omega+a_{ux})A-a_{\ast x}a_{u\ast}\right].
$

\vskip 1mm
{\bf Claim.} $\omega^{k-1}\mid \Lambda^kA$ and $\omega^k\mid \Lambda^{k+1}A$ implies $(\omega+\alpha)^{k-1}\mid \Lambda^kA^{ux}$, with $\alpha=a_{ux}$.

{\bf Proof.} With $\bar{u}\in T^k$ and $\bar{x}\in H^k$, $\omega^k$ divides
$\left|\begin{array}{cc} \omega & 0 \\ 0 & a_{\bar{u}\bar{x}}\end{array}\right|$
and
$\left|\begin{array}{cc} a_{ux} & a_{u\bar{x}} \\ a_{\bar{u}x} & a_{\bar{u}\bar{x}} \end{array}\right|$
and hence their sum,
$\left|\begin{array}{cc} \omega+\alpha & a_{u\bar{x}} \\ a_{\bar{u}x} & a_{\bar{u}\bar{x}} \end{array}\right|$
$= (\omega+\alpha)\left|\begin{array}{cc} 1 & 0 \\ 0 & a_{\bar{u}\bar{x}}-\frac{1}{\omega+\alpha}a_{\bar{u}x}a_{u\bar{x}} \end{array}\right|$
$=\frac{1}{(\omega+\alpha)^{k-1}}\left|(\omega+\alpha)a_{\bar{u}\bar{x}}-a_{\bar{u}x}a_{u\bar{x}}\right|$.
So
$\frac{1}{(\omega+\alpha)^{k-1}} \left|\frac{1}{\omega}\left[(\omega+\alpha)a_{\bar{u}\bar{x}}-a_{\bar{u}x}a_{u\bar{x}}\right]\right|$
is integral. \quad$\Box$

That is, with $A_{\bar{u};\bar{x}}$ denoting minors, if $\omega^{k-1}\mu_{\bar{u};\bar{x}}=A_{\bar{u};\bar{x}}$ and $\omega^k\mu_{u\bar{u};x\bar{x}}=A_{u\bar{u};x\bar{x}}$, then $(\omega+\alpha)^{k-1} (\mu_{\bar{u};\bar{x}}+\mu_{u\bar{u};x\bar{x}}) = A^{ux}_{\bar{u};\bar{x}}$.

\vskip -8pt
\rule{\textwidth}{1pt}

{\bf Relations.} $\bullet$ $\rho^+_{ux}\rho^-_{vy}\sslash tm^{uv}_w\sslash
hm^{xy}_z=t\epsilon_w h\epsilon_z$.
\quad$\bullet$
$
  \rho^{s_1}_{ux}\rho^{s_2}_{vy}\rho^{s_2}_{wz}
    \sslash tm^{vw}_v\sslash hm^{xy}_x\sslash sw^{ux}_{th}
  = \rho^{s_2}_{vx}\rho^{s_2}_{wz}\rho^{s_1}_{uy}
    \sslash tm^{vw}_v\sslash hm^{xy}_x.
$

\vskip -8pt
\rule{\textwidth}{1pt}

\textbf{$\Lambda$-calculus.} $\Lambda(T;H)=R(T)\otimes\left(\Lambda(T)\otimes\Lambda(H)\right)_=$, with $R(T)$ Laurent polynomials in $\{t_u\}_{u\in T}$.
\hfill$\lambda_1\ast\lambda_2 = \lambda_1(\wedge\otimes\wedge)\lambda_2$

$tm^{uv}_w:\ u,v\to w,\,t_u,t_v\to t_w$
\hfill$hm^{xy}_z:\ x\to z,\, y\to\sigma_xz$
\hfill$sw^{ux}_{th}:\lambda\mapsto(1+i_u\otimes i_x)\lambda\sslash(u\to\sigma_xu)$
\hfill$\rho^\pm_{ux}=1+(t_u^{\pm 1}-1)ux$

\vskip -8pt
\rule{\textwidth}{1pt}

\vfill
\rule{\textwidth}{1pt}

{\bf To do.}
$\bullet$ A verification program.
\quad$\bullet$ Add Burau calculus.

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