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\lhead{Notes on the Schur-Weyl Functor} % put text in header (left side)
\rhead{Nico Courts} % put text in header (right side)
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\begin{document}
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\title{Notes on the Schur-Weyl Functor \vspace{-1ex}}
\author{Nico Courts}
\date{Summer 2019}
\maketitle

\begin{abstract}
	These notes are my summary of the development of the Schur-Weyl functor from representations of the Schur algebra $S(n,d)$ to
	representations of $\frakS_d$. After developing the theory that arose beginning in Schur's 1901 thesis, we will establish that this
	functor is exact and behaves nicely with respect to simple modules, as well as being a monoidal functor (under the correct monoidal structure).
\end{abstract}

\section{Background and Introduction}
The primary reference for this paper is J.A. Green's \textit{Polynomial Representations of $\GL_n$} \cite{green}. Other sources will be used as well,
and will be introduced as needed.

The basic idea for this theory begins with Schur's thesis in 1901 \cite{schur-thesis}. Here he developed the theory underlying
the representation theory of the general linear group $\GL_n$. As a tool, he recognized that the irreducible representations of
$\GL_n$ took on a particular form that made them amenable to admitting actions by symmetric groups. Then Schur could rely on
Frobenius' development of the representation theory for $\frakS_d$ to say something about $\GL_n$.

\subsection{Notation}
We always use $\frakS_d$ to denote the symmetric group on $d$ letters, and let $k$ be an infinite field (unless otherwise noted).

\subsection{Representations of Semigroups}
Recall that a \textbf{semigroup} is a group where we relax the inverse requirement. If we are interested in studying the representation
theory of a semigroup $\Gamma$, there are some natural modules to consider:
\begin{ex}
	$k\Gamma=\bigoplus_{g\in \Gamma}kg$, the \textbf{semigroup algebra} for $\Gamma$ over $k$. Notice that if $\Gamma$ is infinite, we only take
	the elements that are supported at finitely many $g$ (that is, only finitely many coefficients are nonzero).
\end{ex}
\begin{ex}\label{ex:bimod} $k^\Gamma$, the set of all maps $\Gamma\to k$, which is naturally a commutative $k$-algebra. It inherits a $k\Gamma$-bimodule structure
	by considering the left- and right-translation maps $L_g:k^\Gamma\to k$ and $R_g:k^\Gamma\to k$ defined for all $g\in\Gamma$ by
	\[L_g(f)(h)= f(gh)\qquad R_g(f)(h)=f(hg)\]
	Due to the fact that $L_g$ defines the \textit{right} action and $R_g$ the left, we write for simplicity:
	\[g\circ f=R_g(f)\quad\text{and}\quad f\circ g=L_g(f).\]
\end{ex}

It is in the context of the $k\Gamma$ module $k^\Gamma$ that we get our first definition:
\begin{defn}\label{defn-finitary}
	Let $k^\Gamma$ be as above. Then an element $f:\Gamma\to k$ is called a \textbf{finitary element} or \textbf{representative function}
	if $f$ satisfies one of the following equivalent conditions:
	\begin{itemize}
		\item $k^\Gamma f$ is finite dimensional over $k$.
		\item $f k^\Gamma$ is finite dimensional over $k$.
		\item $\Delta f\in k^\Gamma\otimes k^\Gamma$ -- in other words, the comultiplication $\Delta f$ is finitely supported.
	\end{itemize}
\end{defn}
\begin{rmk}
	The collection of all finitary elements of $k^\Gamma$ will be written $F=F(k^\Gamma)$ and is a $k$-bialgebra -- in particular,
	a sub-bialgebra of $k^\Gamma$.
\end{rmk}
\begin{rmk}
	The reason such elements are called representative functions is that if we fix a finite dimensional $V$ over $k$
	and a basis $\calB$ for $V$, we can write down the structure maps of a particular $\Gamma$ action by noticing
	\[g\cdot \alpha=\sum_{\beta\in\calB}r_{\alpha\beta}(g)\beta\]
	where $r_{\alpha\beta}:\Gamma\to k$ are called the \textbf{coefficient functions} of the representation.

	Then the $k$-span $\operatorname{cf}(V)=\sum_{\alpha,\beta}k\cdot r_{\alpha\beta}$ of coefficient functions for a particular (finite-dimensional) representation
	$V$ ends up being a subcoalgebra of $k^\Gamma$ and in fact every finitary function lies in $\operatorname{cf}(V)$ for some
	finite-dimensional $V$. Here we can just take $V=k^\Gamma f$ which is finite dimensional by defn~\ref{defn-finitary}.
\end{rmk}

An important idea that comes from this that one can define an \textbf{algebraic representation theory} of $\Gamma$ over $k$. Take any subcoalgebra $A$ of $F=F(k^\Gamma)$
and define the $A$\textbf{-representation theory} of $\Gamma$ to be the full subcategory $\mathbf{mod}_A(k\Gamma)$ of $\rmod (k\Gamma)$ to be the one whose
objects are the (finite-dimensional) $k\Gamma$ modules $V$ such that $\text{cf}(V)\subseteq A$.

A nice result is that the (left) $A$-rational $k\Gamma$-modules are equivalent to the category $\mathbf{com}(A)$ of finite (right) $A$-comodules.
\begin{prop}\label{prop-modcomod}
	$\mathbf{mod}_A(k\Gamma)\simeq \mathbf{com}(A)$
\end{prop}
\begin{prf}
	\textit{Sketch:} Take an $A$-rational $k\Gamma$ module $V$ with action $\tau:\Gamma\to V$. Fix a basis $\{v_i\}_1^n$
	and as before extract the structure maps $r_{\alpha\beta}$:
	\[\tau(v_\alpha)=\sum_\beta r_{\alpha\beta} v_\beta.\]

	Then define the comodule $(V,\gamma)$ with coaction $\gamma$ via
	\[\gamma(v_\alpha)=\sum_\beta v_\beta\otimes r_{\alpha\beta}.\]
	One can check this defines a comodule structure on $V$.

	The process reverses very easily, where one extracts the $r_{\alpha\beta}$ from the coaction.
	By construction we will have that $r_{\alpha\beta}\in A$, so we get the $A$-rationality for free.

	Some mopping up shows that these are equivalences of categories, which is pretty believable.
\end{prf}
\begin{rmk}
	One of the most immediate consequences of this realization (it is barely a proof) is that we can define a left $A^\ast=\Hom_k(A,k)$-module (note that $A^\ast$ is a $k$-algebra with the convolution product)
	structure on any right $A$-comodule. We do this by writing for any $f\in A^\ast$
	\[f\cdot v=(\1_V\otimes f)(\gamma(v))\]
	or in coordinates
	\[f\cdot v_\alpha=\sum_\beta f(r_{\alpha\beta})v_\beta\]
\end{rmk}

\section{Polynomial representations of \texorpdfstring{$\GL_n$}{GLn} and the Schur Algebra}
From now on, we specialize the theory in the section above to the case when $\Gamma=\GL_n$ (we can think of this as the group scheme).

\subsection{Generators and Bases}
There are very natural choices of generators (as (co)algebras and as $k$ spaces). Starting off, we can define
\[A_k(n)\cong k[\omega_{ij}]\]
for $1\le i,j\le n$, where we can think of $\omega_{ij}:\Gamma\to k$ as the map that ``extracts'' from $g\in\Gamma$ the $(i,j)^{th}$ entry.

Then $A_k(n)$ is the set of \textbf{polynomial maps} $\Gamma\to k$. Then we define $A_k(n,r)$ to be the $k$-space spanned by all homogeneous degree $r$
polynomials in $A_k(n)$. A stars-and-bars argument gets us that there are $\binom{n^2+r-1}{r}$ ways to choose a monomial of degree $r$ from among the $n^2$ generators,
so this gives us the dimension of $A_k(n,d)$.

\subsection{Index Notation}
\label{subsec:index}
We need to use multi-indicies to rigorously discuss monomials so let $I(n,r)$ denote the set of length $r$ multi-indices drawn from $\underline n$. Then
we can impose the equivalence relation: if $\alpha,\beta\in I(n,r)$,
\[\alpha\sim\beta \quad\Leftrightarrow\quad \alpha=(n_{i_1},\dots,n_{i_r})\text{ and } \beta=(n_{i_{\sigma(1)}},\dots, n_{i_{\sigma(r)}})\]
for some $\sigma\in\frakS_r$. That is, two multi-indices are equivalent if they contain the same indicies with the same multiplicities. Sometimes it is useful to think
of this as a group action of $\frakS_r$ on $I(n,r)$, and we say two elements are equivalent if they are in the same orbit.

Extend this group action (and thus relation) to $I(n,r)\times I(n,r)$ where $G$ acts diagonally:
\[g\cdot(i,j)=(g\cdot i,g\cdot j).\]

\subsection{Module Categories}
The module categories\footnote{Equivalently the appropriate comodule categories in light of prop.~\ref{prop-modcomod}} $\mathbf{mod}_{A}(k\Gamma)$ of
(homogeneous degree $r$) polynomial representations of $\GL_n(k)$ (when $A=A_k(n,r)$ and $A=A_k(n)$ respectively) are precisely what you'd think. We will sometimes
use the notation $M_k(n)$ and $M_k(n,r)$ following the text.

One of the big realizations of Schur's was that any polynomial representation of $\GL_n$ splits into a direct sum of homogeneous polynomial representations:
\begin{thm}[{\cite[p.5]{schur-thesis}}]
	Let $V\in M_k(n)$. Then
	\[V\cong \bigoplus_{r\ge 0} V_r\]
	where $V_r\in M_k(n,r)$ for all $r$.
\end{thm}
\begin{cor}
	The indecomposable modules in $M_k(n)$ are all in $M_k(n,r)$ for some $r$.
\end{cor}

But then if we are interested in the structure of polynomial representations of $\Gamma$, we can restrict our attention entirely to the homogeneous modules
of any particular degree. The upshot here is that $A_k(n,r)$ is finite dimensional, so very computationally amenable.

\subsection{The Schur Algebra}
\begin{defn}
	The \textbf{Schur algebra} $S_k(n,d)$ is the dual of $A=A_k(n,d):$
	\[S_k(n,d)=A^\ast=\Hom_k(A(n,d),k).\]
\end{defn}

We can write down an explicit basis for $S_k(n,r)$ as a dual basis for the basis $\{c_{ij}|i,j\in I(n,r)\}$ for $A_k(n,r)$. We denote this basis $\xi_{ij}$, where
\[\xi_{ij}(c_{kl})=\begin{cases}
	1,& (i,j)\sim (k,l)\\
	0,&\text{otherwise}
\end{cases}\]
and we can define multiplication via the coproduct $\Delta$ on $A_k(s,r)$:
\[\xi\eta(c)=\sum \xi(c_{(1)})\eta(c_{(2)})\]
where we recall that $\Delta(c_{ij})=\sum_k c_{ik}\otimes c_{kj}$ and $\varepsilon(c)=c(I_n)$ are the coalgebra structure inherited on $A_k(n,r)$
from $A_k(n)$ and thus from $k^\Gamma$.

\begin{prop}
	The product of two basis elements in $S_k(n,r)$ is given by
	\[\xi_{ab}\xi_{cd}=\sum_{(p,q)\in I(n,r)^2}Z_{a,b,c,d,p,q}\xi_{pq}\]
	where
	\[Z_{a,b,c,d,p,q}=\#\{s\in I(n,r)|(a,b)\sim(p,s)\text{ and }(c,d)\sim(s,q)\}\]
\end{prop}
\begin{cor}
	$\xi_{ab}\xi_{cd}=0$ if $b\not\sim c$.
\end{cor}
\begin{cor}\label{cor:idempotent}
	$\xi_{ab}\xi_{bb}=\xi_{ab}=\xi_{aa}\xi_{ab}$.
\end{cor}

The last thing here is to notice that $\varepsilon=\sum_{I(n,r)^2}\xi_{ab}$ where equality can be seen by evaluating at the basis $c_{ab}$ of $A_k(n,r)$.

\subsection{The Evaluation Map}
There is a natural map $e:k\Gamma\to S_k(n,r)$ defined in the following way: if $f\in A_k(n,r)\subset k^\Gamma$ (which can be uniquely extended $k$-linearly
to a map on $k\Gamma$),
\[e(\kappa)(f)=f(\kappa)\]
where $\kappa\in k\Gamma$.

The use of this map is that (since it can be shown to be surjective and that things in the kernel of this map must act by zero on $M_k(n,r)$), that the category
of $S_k(n,r)$ modules and the category of $A_k(n,r)$-rational $k\Gamma$ representations are equivalent. The evaluation map gives us a direct translation between
the two actions: $\kappa\cdot v=e(\kappa)\cdot v$ and if $k\Gamma$ acts on $V$ with structure maps $(r_{ab})$, then for $\xi\in S_k(n,r)$,
\[\xi\cdot v_b=\sum_a \xi(r_{ab})v_a\]

\subsection{Modular Theory}
We can extract the $\bbZ$ forms $A_\bbZ(n)$ and $A_\bbZ(n,r)$ as the $\bbZ$ span of the basis elements $c_{ij}^\bbQ$ (the basis with respect to $k=\bbQ$). These
are closed under $\Delta$ and have the further property that for instance $\varepsilon(A_\bbZ(n,r))\subseteq\bbZ$.

Then we can extend scalars in the way we'd like: there is a $k$-colagebra isomorphism
\[A_\bbZ(n,r)\otimes_\bbZ k\cong A_k(n,r),\]
so we can recover the larger algebra from this $\bbZ$-form. But this actually transfers to the Schur algebra as well!

Define $S_\bbZ(n,r)$ to be the collection of $\xi\in S_\bbQ(n,r)$ such that $\xi(A_\bbZ(n,r))\subseteq\bbZ$. Then we get a $k$-algebra isomorphism
\[S_\bbZ(n,r)\otimes_\bbZ k\cong S_k(n,r).\]
This gives us a $\bbZ$-form for the scheme $S(n,r)$ -- or that ``the Schur scheme is defined over $\bbZ$.''

We can also define a $\bbZ$-form of a ($A_\bbQ(n,r)$-rational) $\bbQ\Gamma$-module $V$ to be denoted $V_\bbZ$ and to be the $\bbZ$-span of a $\bbQ$-basis for $V$
and to furthermore be closed under the $S_\bbZ(n,r)$-action (leveraging the equivalence of categories in the previous section).

This process of generating $V_k$ modules by extending scalars from a $\bbZ$-form is called \textbf{modular reduction}, and has a small caveat:
the $\bbZ$-form we pick needn't be unique and, in general, using different $\bbZ$-forms will yield non-isomorphic extensions to $k$-modules. The upshot here,
following the theory developed in \cite{brauer} and \cite{green-locFinReps}, is that the multiplicity with which simple modules occur in a composition series of $V_k$
are not dependent on the choice of $\bbZ$-form.

\subsection{The Trick}
To do some magic we notice that there is a duality at play between the $A_k(n,r)$-regular representations of $\GL_n$ and the representations of $\frakS_r$,
and to see this we consider the natural action of these groups on $E^{\otimes r}$, where $E\cong k^n$. The left action by $\Gamma$ is just the diagonal action
on each $k^n$ and $\frakS_r$ acts on $E^{\otimes r}$ on the right by permuting the tensor factors. The critical thing to notice here is that these two
actions commute with one another:
\[(g\cdot v)\cdot\sigma=g\cdot(v\cdot\sigma),\ \forall g\in\Gamma, \sigma\in\frakS_r.\]

Then we get some results from Schur's 1927 paper \textit{\"Uber the rationalen Darstellungen der allgemeinen linearen Gruppe}, a reference to which I am yet to cook up.
\begin{thm}[Schur]
	Let $\varphi:S_k(n,r)\to \End_k(E^{\otimes r})$ be the representation afforded by the above action of $S_k(n,r)$. Then
	\begin{itemize}
		\item $\Im \varphi=\End_{k\frakS_r}(E^{\otimes r})$; and
		\item $\ker\varphi=0$.
	\end{itemize}
	So $S_k(n,r)\cong \End_{k\frakS_r}(E^{\otimes r})$.
\end{thm}
The idea here is to take our $\xi_{ab}$ basis for $S_k(n,r)$ and determine where the images go. Not too hard, I think.
\begin{cor}[Schur]
	If $\ch k=0$ or $\ch k>r$, then $S_k(n,r)$ is semisimple. That is every element of $M_k(n,r)$ is completely reducible.
\end{cor}
To see this, we just use the fact that $k\frakS_r$ is semisimple as in usual representation theory. Thus $E^{\otimes r}$ is
completely reducible, but the endomorphism ring of a completely reducible module is semisimple, so by the above theorem, $S_k(n,r)$ is semisimple.

\subsection{Back to \texorpdfstring{$\bbZ$}{Z}-forms}
In fact, $E^{\otimes r}$ is a $\GL_n$-module (this time regarded as an affine group scheme over $\bbZ$). Let $\{V_k\}$ be a collection
of $k$-modules for each $k$ in some family of infinite fields. Then we say $\{V_k\}$ is \textbf{defined over $\bbZ$} if there is a $\bbZ$-form $V_\bbZ$
and a family of isomorphisms $\delta_k:V_\bbZ\otimes k\to V_k$.

In particular, the family of modules $E^{\otimes r}_k$ is defined over $\bbZ$.

\subsection{Contravariant Duality}
In group theory a usual way to define a dual representation for a $k\Gamma$-module $V$ is to take the linear
dual $V^\ast=\Hom_k(V,k)$ and define a left action of $\Gamma$ by
\[g\cdot f(v)=f(g^{-1}v)\]
which give you another (left) $k\Gamma$-module. The problem with this process in our case is that the resulting
module may have structure maps $r_{ab}$ that no longer lie in $A_k(n,r)$. To account for this, we instead define the
\textbf{contravariant dual} $V^\circ$, defined using the transpose:
\[g\cdot f=f(g^{\text{tr}}v).\]

It is clear that the map $J:S_k(n,r)\to S_k(n,r)$ sending $\xi_{ab}$ to $\xi_{ba}$ is an involutory antihomomorphism.
Notice
\[J(\xi)(c_{ab})=\xi(c_{ba})\]
and so by taking $\xi=e_g$, we get
\[J(e_g)(c_{ab})=e_g(c_{ba})=c_{ba}(g)=c_{ab}(g^{\text{tr}})\]
so $J(e_g)=e_{g^{\text{tr}}}$.

So given an $A_k(n,r)$-rational $k\Gamma$ representation $V$, the corresponding $S_k(n,r)$ action on $V^\circ$ is given by
\[\xi\cdot f (v)=f(J(\xi)v).\]

\subsection{\texorpdfstring{$A_k(n,r)$}{Ak(n,r)} as a \texorpdfstring{$k\Gamma$}{kGamma}-module}
Recall from example~\ref{ex:bimod} the left and right actions by $k\Gamma$ we put on $k^\Gamma$. The coproduct $\Delta$ on $k^\Gamma$
defining multiplication in the usual way, that is
\[\left(\sum_h f_h\otimes f_h'\right)(s,t)=\Delta(f)(s,t)=f(st)\]
gives us a convenient way to write, for any $c\in A_k(n,r)$, (since $R_tc(g)=c(gt)=\sum_h f_h(g)\otimes f_h'(t)=L_g c(t)$)
\[t\circ c= R_t c=\Delta(c)(-,t)=\sum_h f_h(-)f_h'(t)=\sum_h f'_h(t)f_h\]
and similarly
\[c\circ t=\sum_h f_h(t)f'_h.\]
These can be extended linearly from $t\in\Gamma$ to $k\Gamma$, giving us (commuting) left and right actions on $A_k(n,r)$.

Then some results in the book show that these actions on $A_k(n,r)$ by $k^\Gamma$ descends to ones by $A_k(n,r)$ itself,
and thus becomes an $A_k(n,r)$- and thus $S_k(n,r)$-bimodule.

Of course this gives us a $k\Gamma$-bimodule structure on $S_k(n,r)$ as well, and then using the following:
\begin{thm}
	Contravariant bilinear forms $(\cdot,\cdot):V\times W\to k$ where by contravariant we mean, for all $\xi\in S_k(n,r)$,
	\[(\xi\cdot v,w)=(v,J(\xi)\cdot w)\]
	are in bijection with linear maps $\Lambda:V\to W^\circ$ via the relation
	\[\Lambda(v)(w)=(v,w).\]
	Furthermore $\Lambda$ is an isomorphism if and only if its corresponding form under this bijection is non-degenerate.
\end{thm}
Then by letting $(\cdot,\cdot):S_k(n,r)\times A_k(n,r)\to k$ (a form on $S_k(n,r)$-modules!) be the form given by
\[(\xi, c)=J(\xi)(c)\]
one can check that this is indeed bilinear and contravariant, and furthermore non-degenerate, giving us an isomorphism
\begin{cor}
	\[A_k(n,r)\cong S_k(n,r)^\circ\]
	as $k\Gamma$-bimodules.
\end{cor}
\begin{rmk}
	The book is a bit scant on details, but I believe the idea here is that this whole thing can be done in $M'_k(n,r)$, the
	category of \textit{right} $S_k(n,r)$-modules, giving us a bimodule (instead of right module) isomorphism.
\end{rmk}

\section{Weights and Characters}
The structure theory of representation category $M_k(n,r)$ closely resembles the theory of weights that arises in the study of
semisimple Lie algebras. The standard reference is \cite{humphreys-liereps}, however I have recently been enjoying Fulton \& Harris'
lovely set of lectures in \cite{fulton-harris}. I am going to take some time to make sure I understand their treatment before diving into this fully.

\begin{defn}
	Let $\Lambda(n,r)$ be the $\frakS_r$ orbit space of $I(n,r)$ (cf. subsection~\ref{subsec:index}). This is called the (dimension $r$)
	collection of \textbf{weights of $\GL_n$}.

	Notice that these should be thought of a monomials in the polynomial algebra on $n$ indeterminates.
\end{defn}
\begin{rmk}
	Notice that each element $\alpha\in\Lambda(n,r)$ can be represented as an $n$-tuple where $\alpha_i$ is the number of times
	that $i$ appears in any multi-index in $\alpha$. This is essentially the multi-degree.
\end{rmk}

Then we can identify $\frakS_n=W$ as the Weyl group of $\GL_n$. This acts on the left of $I(n,r)$ not by simply permuting labels, but by applying
a permutation from the bigger $\frakS_n$:
\[\sigma\cdot(i_1,\dots,i_r)=(\sigma(i_1),\dots,\sigma(i_r)).\]
This action commutes with the right action of $\frakS_r$, so this action we just defined descends to one on $\Lambda(n,r)$.
\begin{defn}
	A \textbf{dominant weight} is a weight $\lambda=(\lambda_1,\dots,\lambda_r)\in\Lambda(n,r)$ such that
	\[\lambda_1\ge\cdots\ge\lambda_r.\]
	Denote the set of all dominant weights in $\Lambda(n,r)$ as $\Lambda^+(n,r)$.
\end{defn}
\begin{rmk}
	Notice that each $W$-orbit in $\Lambda(n,r)$ has a unique dominant weight (although it may be achieved multiple times).
\end{rmk}
\begin{rmk}
	Elements in $\Lambda^+(n,r)$ correspond bijectively to partions of $r$ into not more than $n$ parts.
\end{rmk}
\subsection{Weight Spaces}
Now notice that corollary~\ref{cor:idempotent} gave us that $\xi_{ii}^2=\xi_{ii}$ and the corollary before gave us $\xi_{ii}\xi_{jj}=\delta_{ij}$. Then the correspondence between $A_k(n,r)$-rational
representations and $S_k(n,r)$ modules gives us that (for $V\in M_k(n,r)$):
\[\varepsilon(v)=e(1_\Gamma)(v)=1_\Gamma\cdot v=v\]
and the decomposition\footnote{Note that here we are using that $\xi_{ii}=\xi_{jj}$ if and only if $i\sim j$, so we can represent $\xi_{ii}$
by orbit $\alpha$ of $i$ under $\frakS_r$.} $\varepsilon=\sum_{\alpha\in\Lambda(n,r)}\xi_\alpha$,
\[V=\varepsilon(V)=\sum_{\alpha\in\Lambda(n,r)}\xi_{\alpha}V\]
giving us our decomposition into weight spaces. To see{ this we show that
\[\xi_\alpha V= V^\alpha=\{v\in V|x(t)v=t_1^{\alpha_1}\cdots t_n^{\alpha_n}v,\; \forall x(t)\in T_n(k)\}\]
where $T_n(k)$ is the maximal split torus consisting of diagonal matrices in $\GL_n(k)$. Here the notation is that
\[x(t)=\operatorname{diag}(t_1,\dots,t_n).\]
\begin{rmk}
	Define the character $\chi^\alpha:T_n(k)\to k$ via
	\[\chi^\alpha(x(t))=t_1^{\alpha_1}\cdots t_n^{\alpha_n}\]
\end{rmk}
Then we have the result
\begin{prop}
	\[\xi_\alpha V= V^\alpha\]
\end{prop}
\begin{prf}
	To see this, we begin by computing the image of $c_{i,j}$ under the image of $e_{x(t)}$ and $\sum_{\alpha}\chi^\alpha(x(t))\xi_\alpha$:
	\[e_{x(t)}(c_{i,j})=\prod_k c_{i_kj_k}(e_{x(t)})\]
	which is zero unless $i_k=j_k$ for all $k$--in other words, if $i=j$. Then in this case,
	\[e_{x(t)}(c_{i,i})=\chi^\alpha(x(t))\]
	where $\alpha$ is the weight associated to $i$.

	Computing from the other end, $\xi_\alpha(c_{ij})=0$ unless $\alpha$ is the weight of $i$ and
	$j=i$. If $i=j$, a single term is nonzero--the summand corresponding to $\alpha$, the weight of $i$, giving us $\chi^\alpha(x(t))$.

	Since the $S_k(n,r)$ action via $e_{x(t)}$ corresponds to the $\GL_n$ action via $x(t)$. Then take any element $w\in\xi_\alpha V$ and using that the
	$\xi_{i,i}$ are orthogonal idempotents, we can compute the action of $T_k(n)$ on $\xi_\alpha V$:
	\[x(t)\cdot v=e_{x(t)}v=t_1^{\alpha_1}\cdots t_n^{\alpha_n} v\]
	so $v\in V^\alpha$.

	Now if $v\in V^\alpha\cap V^\beta$, consider the action by $A=\operatorname{diag}(2,1,1,\dots,1)$. Then since
	\[A\cdot v=2^{\alpha_1}v=2^{\beta_1}v\]
	this forces $\alpha_1=\beta_1$ and similar arguments show that $\alpha=\beta$. Thus the sum of $V^\alpha$'s are direct and since their sum is all of $V$ (since their sum
	contains the sum of the $\xi_\alpha V$), we get that these summands are precisely the $\xi_\alpha V$.
\end{prf}
\begin{rmk}
	A very useful consequence here is that we have a subalgebra $D_k(n,r)\subseteq S_k(n,r)$ spanned over $k$ by the $\xi_\alpha$ for $\alpha\in\Lambda(n,r)$
	This is the image of $T_k(n)$ under the evaluation map $e$.
\end{rmk}

Another useful fact: the weight spaces $V^\alpha$ are stable under the $\frakS_n$ action (up to isomorphism).
\begin{prop}
	Let $\omega\in\frakS_n$. Then
	\[V^\alpha\cong V^{\omega(\alpha)}.\]
\end{prop}
\begin{prf}
	The proof of this boils down to noticing that the action of $\GL_n$ on $V^{\omega(\alpha)}$ is the same of a translated
	version of $\GL_n$. Specifically, let $\eta_\omega$ be the permutation matrix taking the standard basis to $\{e_{\omega(1)},\dots,e_{\omega(n)}\}$.
	Then
	\[x(t_{\omega(1)},\dots,t_{\omega(n)})=\eta_\omega^{-1}x(t_1,\dots,t_n)\]
	so application of $n_\omega$ yields an isomorphism from $V^\alpha$ to $V^{\omega(\alpha)}$.
\end{prf}
The restriction to the $\alpha$-weight elements is an exact functor!
\begin{prop}
	If
	\[0\to A\to B\to C\to 0\]
	is a short exact sequence of elements in $M_k(n,r)$, then so is
	\[0\to A^\alpha\to B^\alpha\to C^\alpha\to 0\]
	for all $\alpha\in\Lambda(n,r)$.
\end{prop}
\begin{prf}
	Left exactness is immediate since this functor acts on morphisms by restriction to a subspace. Right exactness follows since for
	any $\xi^\alpha c\in\xi^\alpha C$, since the map $g:B\to C$ is surjective, there is a $b\in B$ mapping to $c$. But then
	$\xi^\alpha b\in\xi^\alpha B$ and its image is
	\[g(\xi^\alpha b)=\xi^\alpha g(b)=(\xi^\alpha)^2c=\xi^\alpha c\]
	since $\xi^\alpha$ is idempotent.
\end{prf}

\subsubsection{Extending Scalars}
Notice that a simple computation using linear algebra and counting degrees gets us that
for any $V\in M_k(n,r)$ and $W\in M_k(n,s)$, we can define the module $V\otimes W\in M_k(n,r+s)$
where if $\gamma\in \Lambda(n,r+s),$ the weight spaces are
\[(V\otimes W)^\gamma=\bigoplus_{\alpha,\beta}V^\alpha\otimes W^\beta\]
where the sum is over all $\alpha$ and $\beta$ whose sum is $\gamma$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%  Bibliography %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip

\printbibliography

\end{document}


\paragraph{}
In the traditional approach, geometric design and analysis are treated as separate modules requiring different methods and interpretations.
For example, the geometric design module employed non-uniform rational B-splines (NURBS) introduced in Sec.~\ref{lr_sec:NURBS} to describe the geometry, whilst the analysis module consisted of one of the following
\begin{enumerate}
    \item Mesh based discrete models, such as the finite element method (FEM) \citep{doi:10.1111/j.1467-8667.1989.tb00025.x}
    \item Boundary based methods, such as the boundary element method (BEM) \citep{book}, scaled boundary finite element method (SBFEM) \citep{Son1997}
    \item Meshless methods \citep{article123213eds}
\end{enumerate}
The approximation space employed in the analysis module to describe both the geometry and the fields is different from that used in the CAD system.
Hence it requires repetitive conversion between the CAD and the analysis and in this process errors are inevitable.
Moreover, the analysis module employs polynomials that do not lead to exact representation of the geometry, whilst the geometric module employs Bézier representations that use Bernstein polynomials or B-splines and NURBS that employ de Boor polynomials \citep{Pie1997}.
The above representations utilize basis functions and control points to represent the geometry, in addition to this, B-splines and NURBS also utilize a vector
of knots.
NURBS further use weights to control points to model intricate shapes.

\paragraph{}
As a consequence, the concept of isogeometric analysis is proposed \citep{Hug2005}, in which the conventional Lagrange polynomials are replaced with the NURBS basis functions.
The concept of isogeometric analysis (IGA) has revolutionized the analysis procedure.
The IGA provides a natural link with the CAD model.
A key feature of this framework is that the geometry is represented exactly by NURBS and the isoparametric concept is invoked to define the field variables.
Since its inception, the method has been applied to a variety of problems such as plates and shells \citep{NGUYENTHANH20113410,NGUYENXUAN2014222,HOSSEINI20141}, as
cohesive elements \citep{NGUYEN2014193}, for shape optimization \citep{WALL20082976}, fluid–structure interaction problems \citep{BAZILEVS201228}, problems with strong discontinuities and singularities \citep{doi:10.1093/imamat/hxu004, doi:10.1002/nme.4580, BAZILEVS201228}, optimization problems \citep{GHASEMI2014463} to name a few.
Jia et al. \citep{JIA2013342} by incorporating reproducing kernel approximation methods, alleviated the instabilities of the conventional triangular B-spline element.
The new approach yielded improved convergence rate and accuracy when compared to the conventional triangular B-spline element.
This seems to be a promising alternative to NURBS and T-splines where considerable effort is required for local refinements.
In the conventional IGA, the surfaces/volumes are represented by the tensor product of the corresponding knot vectors.
This requires the domain to be discretized with standard shapes and leads to a restricted number of boundary curves/surfaces.
Also, this leads to excessive overhead of control points with refinement.
This can be circumvented by adopting local refinement as proposed \citep{NGUYENTHANH20111892} or by employing T-splines \citep{Sederberg:2003:TT:882262.882295}.
Recently, Simpson et al. \citep{Sim2013, SIMPSON201287} proposed the isogeometric boundary element method (IGABEM), in which the NURBS functions were used to approximate the unknown fields.
This framework circumvents the need to discretize the domain, as required by the IGAFEM.
It was shown that the IGABEM is more accurate than the conventional BEM with polynomial interpolations.
Furthermore, Scott et al. \citep{Sco2013} and Simpson et al. \citep{SIMPSON2014265} combined the collocated IGABEM with T-splines for linear elastostatics and acoustic analysis, respectively.
The concept of IGABEM was further extended to damage tolerance assessment \citep{PengXuan;AtroshchenkoElena;Bordas2014} and shape sensitivity analysis \citep{LianHaojie;SimpsonRobert;Bordas2013}.


\section{Implementation}

\subsection{The Strategy of Disjoint Set}

\begin{frame}
	\frametitle{Cache Performance}
	\begin{itemize}
		\setlength\itemsep{1em}
		\item
			Traditional disjoint set merging techniques are {\em merge-by-rank}
			and {\em merge-by-size}.
		\item
			Usually, an algorithm with a lower time complexity, e.g.,
			will have more ``long jumps'' than an algorithm with a
			higher time complexity.
	\end{itemize}
	\begin{center}
	\scalebox{0.7} { \begin{minipage}{1.5\textwidth}
				\begin{figure}[!thb]
			  \centering \subfigure[Lower time complexity] {
			    \includegraphics[width=0.42\linewidth]{\GraphicPath/fig-rem-long-jump.pdf}
			    \label{fig:long-short-jump-disjoint-long}
			  } \subfigure[Higher time complexity] {
			    \includegraphics[width=0.42\linewidth]{\GraphicPath/fig-rem-short-jump.pdf}
			    \label{fig:long-short-jump-disjoint-short}
			  }
			  \caption{The parent jump in disjoint set}
			  \label{fig:long-short-jump-disjoint}
			\end{figure}
			\end{minipage}
		}
	\end{center}
\end{frame}

\begin{frame}
	\begin{itemize}
		\setlength\itemsep{1em}
		\item
			The Rem's algorithm achieves better cache performance by a
			{\em merge-by-index} technique.
		\item
			Experiments show that the merge-by-index tie-breaking
			technique improves by 3\% over the approach that breaks the
			tie randomly.
	\end{itemize}
\end{frame}

\subsection{Parallel Range Maximum Query}
\begin{frame}
    \frametitle{Parallel Range Maximum Query}
    We further improve the performance of range query by maintaining two
	extra tables in the blocked sparse table approach.
	\\~\\
	There are now three tables as follows:

	\begin{itemize}
		\setlength\itemsep{1em}
		\item
			A sparse table on the block maximum $T_S$,
		\item
			A {\em prefix maximum table} $P$, and
		\item
			A {\em suffix maximum table} $S$.
	\end{itemize}
\end{frame}

\begin{frame}
	\frametitle{The Order to Access these Tables}
	The {\em order} to access these maximum tables is important.  Our
	implementation improves performance by ``peeking'' into two
	neighboring elements in the $T_{S}$ table.
	\begin{itemize}
		\setlength\itemsep{1em}
		\item
			Access two elements in the sparse table $T_S$ in {\em the
  			same level} because accessing the first will bring in the
  			other by the hardware caching mechanism.
  		\item
  			Intermediate maximum value and neighboring elements can save
			unnecessary access to $P$ and $S$ tables.
		\item
			It improves the overall performance of query operations by
			up to 35\% when $n = 10^7$.
	\end{itemize}
\end{frame}

\begin{withoutheadline}
\begin{frame}
	\frametitle{Illustration for the Access these Tables}
	\begin{center}
	\scalebox{0.7} { \begin{minipage}{1.5\textwidth}
				\begin{figure}[!thb]
				  \centering \subfigure[The prefix/suffix maximum tables for blocks] {
				    \includegraphics[width=0.9\linewidth]{\GraphicPath/fig-compressed-sp-prefix-suffix.pdf}
				  } \subfigure[A Sparse Table for $T_S$] {
				    \includegraphics[width=0.3\linewidth]{\GraphicPath/fig-sparse-table-opt.pdf}
				  } \caption{Block maximum $T_S$, prefix maximum $P$, and suffix
				    maximum $S$.}
				  \label{fig:compressed-sp-opt}
				\end{figure}
				\end{minipage}
		}
	\end{center}
\end{frame}
\end{withoutheadline}

\begin{withoutheadline}
\begin{frame}
	\frametitle{Pseudocode for Order to Access these Tables}
	\begin{center}
	\scalebox{0.65} { \begin{minipage}{1.5\textwidth}
				\input{\AlgoPath/alg-rmq-access-order-2e}
				\end{minipage}
		}
	\end{center}
\end{frame}
\end{withoutheadline}


\section{Many-body definition of Chern number}
I want to talk about the many-body definition of Chern number.

For every band you can assign a Chern number.
At the face of it,
it's only a property of band structure,
of single-particle TI system.s
There's 2 pieces of evidence for something more general.

1. The Chern number dictate th Hall conductivity,
which is quantized.
So if the Hall conductivity is given by an integer invariant in the free band
theory,
as you slowly turn on interaction,
that shouldn't change,
so there should also be a quantized invariant for that
but how do you extract that from the GS wave function.
2. I had a continuum Dirac theory,
that you have a mass term,
and integrate out the other degrees of freedom.
In that field theory,
you can have higher order field theory interaction
and you get a 4-fermion interaction that is irrelevant by RG,
which you ignore,
so you still have this integer quantity that is still defined if you change
the sign of the mass.

So the Chern number should be well-defined for interacting systems that doesn't
require TI.

I will give you multiple ways of calculating the Chern number fro GS
wavefunctions.

Hall conductivity and twisted boundary conditions.
Suppose we put our system on a torus,
and we insert flux on both holes of the torus,
so consider changing the vector potential in the $x$ and $y$ directions.
\begin{align}
    \delta A_x &=
    \frac{\Phi_0}{L_x} \frac{\theta_x}{2\pi}\\
    \delta A_y &=
\end{align}
adding these fluxes changes the Hamiltonian,
and we get a term that looks like the gauge fields coupling to the current
\begin{align}
    \delta H &=
    \int d^2x \, \delta A_i J^i\\
    &=
    - \sum_{i=x,y} \frac{\Phi_0}{L_i} \frac{\theta_1}{2\pi}
    \int d^2 x\, J^i
\end{align}
and $H[\theta_x, \theta]$
Twisted boundary conditions vs flux through the hole of the torus are related by
singular gauge transformations.
Given this Hamiltonian,
we have a ground state wave function of this Hamiltonian
\begin{align}
    \ket{\Psi(\theta_x, \theta_y)}
\end{align}
from which we can construct the Berry connection,
just like before in momentum space.

So I can define
\begin{align}
    A_j(\theta_x, \theta_y) &=
    -i \bra{\psi(\theta_x, \theta_y)}
    \frac{\partial}{\partial \theta_j}
    \ket{\psi(\theta_x, \theta_y)}
\end{align}
and from this we can define a field strength
\begin{align}
    F_{ij} &= \partial_i A_j - \partial_j A_i
\end{align}
And here the closed surface is the torus,
because $\theta_x$ can go from $0$ to $2\pi$,
and once it's $2\pi$,
we can do a large gauge transformation
to relate it to the system without the flux.
And we get
\begin{align}
    C &=
    \frac{1}{2\pi}\int d^2\theta\,
    F
    \in \mathbb{Z}
\end{align}
And this is really an integral over the torus.
\begin{align}
    \theta_x $\sim \theta_x + 2\pi\\
    \theta_y $\sim \theta_y + 2\pi
\end{align}
The BZ was a torus,
but here the space is also a torus.
We integrate this field strengh over a closed surface,
it has to be an integer from thes ame argument I gave before.

If we look at hte $\theta$-dependence,
\begin{align}
    \frac{\delta H}{\delta \theta_i} &=
    \frac{\Phi_0}{L_i} \frac{1}{2\pi}
    \underbrace{%
    \int d^2x\,
    J_i
    }_{\text{0-momentum part of current}}
\end{align}
and so we will be able to relate the Chern number
to some current-current corelation function,
which shoud be related to the Hall conductivity.
In fact,
you can prove that the Hall conducivity is
\begin{align}
    \sigma_H &=
    \frac{e^2}{h} \frac{1}{2\pi} F_{xy}(\theta_x, \theta_y)
\end{align}
The proof is in the homework.
The relation between the Hall conductivity and the Chern number is that if you
average the Hall conductivity over the space of wave funtions,
the you find
\begin{align}
    \langle \sigma_H \rangle
    = C \frac{e^2}{h}
\end{align}
that is,
the average over twisted boundary conditions.
In fact,
you find
\begin{align}
    \sigma_H &= \langle \sigma_H \rangle_{BC's}
\end{align}
People understood you get thse two formulae
and you should expect the Hall conducitvity is an average over boudnar
yconditions,
and witha gapped system,
no quantity should care what the flux is thorugh these non-contractible cycles,
in the infinite system size,
because there is a finite-correlation lngh.
If hte cycle is large,
locally you need some coherenetnce thorugh the yccle,
so you need some correlation length that scales with teh size of the system.
Intuitively,
people expected
but in a tour-de-force of mathematical physics a decade,
this was proven about 2010 by Hastings and Michalakis.

If you think of inserting flux,
or you can do a singular gauge transformation,
remove the flux,
and do some twisted boundary conditions,
and the twist is the $\theta$.
You can relate
\begin{align}
    \psi\left( r_1, r_2, \ldots, r_N \right) &=
    e^{i\theta_x}\psi\left( r_1 + L_x \hat{x}, r_2,\ldots \right)
\end{align}
and the point is,
that this is related by a singular gauge transformation.
I'm not spelling out the whole story, because I assume you know already.

The main lesson,
is that this way of defining Chern number
gives a way to define an integer invariant in a many-body system.
We can define a Chern number for any $U(1)$ symmetry,
because the moment I have a U(1) symmetry,
I can turn on a background gauge field,
and insert flux through that torus,
and define a symmetry associated with it.
So every U(1) symmetry can have a  Chern number associated with it,
and that gives a way to define a U(1) symmetry in a gapped many-body system.
And there's few things I want to emphasize here.

This thing is intrinsically many-body.
Not a single free fermions,
not assuming translationally invariant.
So not band theory.

We also assumed implicitly that the system has a unique ground state.

This definition of the Chern number so far assumes you have a unique ground
state.
We're adiabatically inserting flux,
but in some system like fractional quantum hall effect,
if you insert integer flux,
you actually wind to a different ground state,
so you need many flux quanta to wind up in the same ground state.
So here with $2\pi$ flux,
you end up with the same exact ground state.
We get this Berry phase by moving around in one direction,
and if you consider may directions,
you get a single ground ate.

We can still define a Chern number with degenerate ground sates,
but there is an index here,
and this is a non-Abelian gauge field,
and this field strength because a non-Abelian gauge field,
and we need some extra fields,
and the Chern number is the trace of that field strength.
You can still define Chern number,
but it's not a U(1).
I won't say anything more.
You can do it,
but you need to consider non-Abelian Berry gauge fields.

\begin{question}
    To go to a non-Abelian symmetry,
    how would we do that?
\end{question}
So now you're saying,
suppose we have $SU(2)$ symmetry,
can you still define a Chern number?
The quick way to answer this question is,
Chern number we associated with an effective action
and the coefficient was the Chern number.o
With SU(2),
you turn on non-Abelian SU(2) gauge fields,
then write down a non-Abelian CS theory,
and that will also have a coefficient that is also a Chern number.
There will be a Chern number,
but to define it,
you're going to consider.
Then what you do is insert flux associated with some U(1) subgroup of
SU(2)
Just compute it for some U(1) subgroup.
I said that quickly,
because I didn't explain.

\begin{question}
    what about discrete groups instead of U(1)?
\end{question}
We can write down
\begin{align}
    S_{CS}[A] = \int \frac{C}{4\pi} A\, dA
\end{align}
but this doesn't make sense if you have some net flux through your manifold,
like a net magnetic monopole,
you can't globally define $A$,
and make sense.
But here is a remedy,
and it is to go to group cohomology.
I'll just stop there.
Once you view the Chern-Simons theory through that more appropriate lens,
then that framework you can easily extend to discrete groups.

\begin{question}
    Why not glue the state to $SU(2)$ state at the boundary?
\end{question}
Let me say just a bit more about this $SU(2)$ case
For $SU(2)$ symmetry
you would consider SU(2) gauge fields.
And the effective action would be some integer
\begin{align}
    S_{\text{eff}} &=
    \frac{C}{8\pi}
    \int \Tr\left(
    A\wedge dA
    + \frac{2}{3} A \wedge A \wedge A
    \right)
\end{align}
If the goal was just to know $C$,
you can pick $A$ to live in some $U(1)$ subgroup,
evaluate the action,
and figure out what this coefficient is.
You know the whole thing si $SU(2)$ invariatn,
so you can just pick $A$ to be in a particular $U(1)$ sugroup,
then evaluate to figure out what the coefficient is.

In general,
you could imagine you have some $SU(2)$ flux here,
but you can do an $SU(2)$ rotation,
so this $\theta_x$ lies in some $U(1)$ subgroup,
but then you have to wrory about thsi $\theta_y$.
It's not the most general thing to do,
to figure out Chern number,
just have $A$ lie in some subgroup,
and evaluate the subgroup frmo there.
Once you figured out hte Chern number,
there is only one possible Chern number.

I'm just giving you a trick to calculate the Chern number in the $SU(2)$ case.
It doesn't map analogously.

In general,
you could imagien in the $x$ direciton,
yocuould have a genera l$SU(2)$ matrix,
and you tyr to cnostruct something like the $A_j(\theat_x,\theta_y)$ formula.
But I haven't thought about tha procedure.

\begin{question}
$C$ here can be any integer,
but is it still integer for $SU(2)$?
\end{question}
The CS theory for $U(1)$,
we usually write as
\begin{align}
    S_{eff}^{U(1)} &=
    \frac{C_{U(1)}}{4\pi}\int A\, dA
\end{align}
but for $SU(2)$,
\begin{align}
    S_{\text{eff}} &=
    \frac{C}{8\pi}
    \int \Tr\left(
    A\wedge dA
    + \frac{2}{3} A \wedge A \wedge A
    \right)
\end{align}
Note the $4\pi$ vs $8\pi$.
For bosonic systems,
\begin{align}
    C_{U(1)} &\in 2\mathbb{Z},\\
    C_{SU(1)} &\in 2\mathbb{Z},
\end{align}
But for fermionic system,s
\begin{align}
    C_{U(1)} &\in \mathbb{Z}\\
    C_{SU(1)} &\in \mathbb{Z}
\end{align}

\section{Thouless Pond}
Let's view the Chern number from polarization in 1D.
Recall if we put our system on a cylinder,
and insert flux through the cylinder.
Let's say we have some $\theta_x = \Phi$ flux.
And the cyclinder is alnog the $x$ direciton.
This causaes an electric field in the $y$ direciton,
which auses a current $j_x$.
Charge is flooiwng alon the $x$ direciton,
and if you view htis as 1D system,
the polarization is changing with time that gives rise to the current.
Think about the Chern number in terms of the polarizaiton of this effecively 1D
system.

That's the perspective,
and see how it relates to the Chern number in 2D.

Let's start off discussing dimensional reduction of a 2D tight binding model.
Suppose we have a tight-binding model on a cylinder.
$x$ is alnog, $y$ is around the cylinder,
because $y$ is periodic, $k_y$ is a good quantum number.

It's a cylinder in the sense there are boundaries at the two ends
in the $x$ direction.
We can do a fermion operator,
with band $\alpha$.
\begin{align}
    C_{k_{y,\alpha}} &=
    \frac{1}{\sqrt{L_y}}
    \sum_y C_{\alpha}(x, y)
    e^{i k_y y}
\end{align}
the Hamiltonian is
\begin{align}
    H &= -\sum_{ij} t_{ij} c_i^\dagger c_j + \textrm{h.c.}\\
    &= \sum_{k_y} H_{1D}[k_y]
\end{align}
It's a sum of independent 1D systems where $k_y$ is just some parameter now.
You can think of your system as a bunch of $L_y$ different 1D chains.
For this 1D system,
$k_y$ is effective just some number.

Let's insert flux.
\begin{align}
    A_y &= - E_y t = \frac{\Phi(t)}{L_y}\\
    A_x &= 0
\end{align}
where $E_y$ is the electric field.
You can think of it as.
\begin{align}
    H &= \sum_{k_y} H_{1D}\left( k_y + A_y \right)
\end{align}
and this causes a current,
and remember the Hamiltonian is decoupled in $k_y$,
\begin{align}
    J_x &=
    \sum_{k_y} J_{1D}(k_y).
\end{align}
Let's calculate the charge flowing across the cylinder
\begin{align}
    \Delta Q &=
    \int_{0}^{\Delta t} dt\,
    \sum_{k_y} J_{1D}(k_y)
\end{align}
Now,
what's happening is,
you can think of this $J_{1D}(k_y)$,
as having some polarization,
and the current is the time derivative of this 1D system.
\begin{align}
    J_{1D}(k_y) &=
    \frac{d P_{1D, x}(k_y)}{dt}
\end{align}
so a changing polarization leads to a current,
charged being pumped in that direction.
And this is just
\begin{align}
    \Delta Q &=
    \int_{0}^{\Delta t} dt\,
    \sum_{k_y} J_{1D}(k_y)
    =
    \left.\sum_{k_y} \Delta P_x (k_y)\right|_{0}^{\Delta t}\\
    &=
    \left.\frac{L_y}{2\pi} \int_{0}^{2\pi} dk_y\,
    \Delta P_x(k_Y)\right|_{0}^{\Delta t}
\end{align}
And we're inserting flux in the adiabatic limit,
and say we insert one flux quanta.
\begin{align}
    E\, \Delta t &=
    \frac{2\pi}{L_y}
\end{align}
and here
I'm assuming all units are 1.
And the change in polarization is just going to be
\begin{align}
    \Delta P_x (k_y) &=
    P_x\left( k_y + \frac{2\pi}{L_y} \right)
    - P_x(k_y)
\end{align}
and in the limit,
this becomes
\begin{align}
    \Delta P_x (k_y) &=
    \frac{dP_x}{dk_y} \frac{2\pi}{k_y}
\end{align}
and what we leran is that the $2\pi/L_y$ factors cancel, so
\begin{align}
    \Delta Q &=
    \int_{0}^{2\pi} dk_y\,
    \frac{dP_x}{d k_y}
\end{align}
and we can just htink of this as a look integral in the Brilluoin zone.
\begin{align}
    \Delta Q &=
    \oint dk_y\,
    \frac{dP_x}{d k_y}
\end{align}
and you can just think of $\Delta Q$
as the winding number of the polarization,
which is equal to the charge pumped across the system.

\begin{question}
    Between the time and frequency arguemnts,
    $\Delta P$ should be the difference in polarizaiton at different times?
\end{question}
The current is the change in polarization,
but the reason it's change in time is becake $k_y(t)$ is in time.
\begin{align}
    J_{1D}(k_y) &=
    \frac{dP_x\left( k_y(t) \right)}{dt}
\end{align}

\begin{question}
    What's the physical meaning of polarization winding?
\end{question}
Physically,
this is what's going on.
Physically,
take the states in a given Chern band,
and because our band has a Chern number,
we cannot write localized Wannier functions,
but they can be paritally localized Wannier functions.
Physically, localized wannier functions for the Chern band
$\ket{W(k_y, x)}$.

As $k_y$ increases,
these Wannier functions are shifting in position,
the states get shifted in $x$,
specifically let's look at the expectation values
\begin{align}
    \bra{W(k_y,x)} \hat{x} \ket{W(k_y, x)}
    = x + P(k_y)
    = \bar{x}_{k_y, x}
\end{align}
Some extra steps of algebra area needed to show this.
we have partially.
The fact the polarization winds
means that if you tried to plot this average
$\bar{x}_{k_y, x}$ vs $k_y$,
it means that if you start at lattice site 0,
you end up at lattice site $C$ after $k_y$ goes up by $2\pi$.
There are two ways of saying it.

What it's saying is that the average positiion of this partially localized
Wannier function changes by $C$ units
\begin{align}
    \bar{x}_{k_y + 2\pi, x} &=
    \bar{x}_{k_y, x} + C
\end{align}
Alternatively,
after inserting $2\pi$ flux,
this average changes by $C$ windings.
At each particular poin,
the polizariotn has changed,
and the net change in polairziaton is just hte net charge that goes from one
side toe another.
This is why you relate $J$ and $P$.

\begin{question}
    What's the picture of the winding number of $P$ around $y$?
\end{question}
The winding
\begin{align}
    \oint dk_y\, \frac{dP}{dk_y}
    &=
    p(k_y + 2\pi) - P(k_y)
\end{align} shift by $2\pi$ and see how much polarization changes.

\begin{question}
    If $C$ is zero,
    we expect this to be locallized?
\end{question}
No,
if $C=0$,
there's no obstruction to write down localied Wannier funcfions,
but we can still write paritally lcoalized Wannier functions.
As we change $k_y$,
these positions would be shifted over by $C$,
which means they don't shift over at all.

\begin{question}
    These Wannier functions are not deinite combinatinos of Bloch states?
\end{question}
You cna think of these nsulators as filling some single-articulare stares,
but which basis you think in is up to you.
You alwasy have the right ot think in the basis they are paritally localized.
winding literally means graduatlly

A nice exercise is to write down the Wannier functino in terms of the Bloch
states of the bands,
and tune the superpsoitiosn so you maximize the lcoalizeation o the wannier
function.

\begin{question}
    Do we have a known quantity for the lcoalization?
\end{question}
For the moment forget about Chern numbers entirely,
and just think about 1D systesm.
More generally,
consider some 1D system that depends on some apramter $\theta$.
\begin{align}
    H_{1D}(\theta)
\end{align}
If $\theta$ varies with time,
we could get some current in this system.
Because $\theta$ is the only thing changing in this setup,
the current is going to be some response linear function $G(\theta)$ times the
derivative.
\begin{align}
    J(t) &= G(\theta) \frac{d\theta}{dt}
\end{align}
and this is varying adiabatically in time,
so we can apply linear response,
as the sytem is always in equilibrim.
Now the current,
is almost by definition
\begin{align}
    J(t) &= \frac{dP}{dt},
\end{align}
but we can also think of it in terms of
\begin{align}
    J(t) &= \frac{dP}{dt}
    = \frac{\partial P}{\partial \theta} \frac{d\theta}{dt}
\end{align}
which means
\begin{align}
    G(\theta) = \frac{\partial P}{\partial \theta}.
\end{align}
what you find for the tight-binding model is that you get a clean expression
\begin{align}
    G(\theta) &=
    \oint \frac{dk_x}{2\pi}\left(
    \frac{\partial A_x}{\partial \theta} - \frac{\partial A_\theta}{\partial k_x}
    \right)
\end{align}
where this $A_\theta$ is the Berry connection,
with
\begin{align}
    A_x &=
    -i \bra{u(k,\theta)} \frac{\partial}{\partial k_x}
    \ket{u(k,\theta)}\\
    A_\theta &=
    -i \bra{u(k,\theta)} \frac{\partial}{\partial \theta}
    \ket{u(k,\theta)}\\
\end{align}
and these $u$ are Bloch wave functions.
It should look familiar to you.
This is just the field strength of this Berry connection in this
$(k_x,\theta)$-space.
Now interestingly,
if $\theta$ is a periodic parameter so that
\begin{align}
    H(\theta + 2\pi) = H(\theta)
\end{align}
then ifwe take the inetegral,
\begin{align}
    \frac{1}{2\pi}\int d\theta\, G(\theta) &=
    C
\end{align}
where $C$ is some integer Chern number.
This is an amazing cool thing.
You have a family of 1D systems,
you try to look at the current floiwng through the system,
and that current is dteermined by a response function,
and that respnose fucntion
is just het integral of the field strength of the Berry connection.

So to do one mofre thing.
If we pick a gauge where $A_\theta$ is single-valued,
\begin{align}
    G(\theta) &=
    \frac{\partial}{\partial\theta}
    \oint \frac{dk_x}{2\pi} A_x
    = \frac{\partial P}{\partial \theta},
\end{align}
so we learnt something cool:
$P$ itself is equal to the Berry phase.
\begin{align}
    P &=
    \oint \frac{dk_x}{2\pi}A_x
\end{align}
The poliaziaton is taking the Berry conntion,
and looking at hte holonomy of the Berry gauge field in momentum space.
This is the so-called
\emph{Berry phase theory of 1D polarization}.
Once caveat is that this is only well-defined mod integer.
Because,
I can always do a gauge transformation of my insgle-aprticle gauge funtions,
which chagnes the flux thorugh the BZ by $2\pi$.
That is,
this is only well-defined mod $\mathbb{Z}$,
adn we can do large gauge transfomrations to chagne
$\oint_{k_x} A$ by $2\pi$.

This miplies that when $\theta$ is periodic,
\begin{align}
    \Delta Q &=
    \int J\, dt\\
    &= \int G(\theta) \frac{d\hteta}{dt}dt\\
    &=
    \int G(\theta)\, d\theta\\
    &= C \in \mathbb{Z}
\end{align}
So if we do a loop in paratere psace,
we pump charge,
and that can be related to the Chern number,
arising frmo the 1 extra dimension $\theta$.

So topological pumps in 1D are tightly connected to Chern numbers in 2D.

The cna also relate the charge density to the polarization.
The current is the time derivative of the charge,
but if you havet he gradient of the polarization,
that means you have some charge density somehwere.

By the continuity equation,
\begin{align}
    \frac{d\rho}{dt} &=
    -\frac{dJ}{dx}
    =
    -\frac{\partial^2 P(\theta)}{\partial x\, \partial t}
\end{align}
and that tells us the charge density is
\begin{align}
    \rho &=
    - \frac{\partial P(\theta)}{\partial x}
\end{align}
and combining iwth
\begin{align}
    J &= \frac{\partial P}{\partial t},
\end{align}
we get
\begin{align}
    i_{\mu} &=
    \epsilon_{\mu\nu} \frac{\partial P}{\partial x_\nu}
\end{align}
from which we get an effective action
\begin{align}
    S_{\text{eff}} &=
    \int dx\, dt\,
    P e^{\nu \mu} \partial_\nu A_\mu
\end{align}
which means
\begin{align}
    S_{\text{eff}} &=
    \int P\, F_{xt}.
\end{align}
Integrating by parts,
we can confirm that
\begin{align}
    j_{\mu} &=
    \frac{\partial S_{\text{eff}}}{\delta A_\mu}
    = -\epsilon^{\mu\nu} \partial_\nu P
\end{align}
So the Berry phase of the Bloch wave functions as you go around in momentum
space in 1D.
You can think of polairziaton in 1D,
where the effective action is literally just the field strength of $A$.
Finally,
there is a tight relation between pumps in 1D and chern numbers in 2D.

\begin{question}
    Does the correspondence carry in higher dimesnions?
\end{question}
There's a Simons cllaboartion caleled ultra-quantum matter.
There's an ultra quantum theory of polarizaiton.
Andy ou can generalize this to higher dimensions,
ubt it's a research topic.
What enters is you need translation gauge fields in higher dimensinos,
guage fields assocatied with tarnslation symmetry,
and then there's a natural way of witing down this higher dimnsional verios of
this effective action.

\begin{question}
    Why don't we need them now for 1D?
\end{question}
I think it's because in 2D you can go around in a loop,
but in 1D thereis no loop except for the toal loop,
so there are no small loops.
In some sense,
you can get a way with less structure in 1D,
because it just has a lot less structure.

\begin{question}
    How to generalize Chern numbers to 3D?
\end{question}
Several modern things have happend literally in the last few years.
ify ou want to geliraze the pump 1D to Chern number 2D,
therae ra few aperps about amoalies in thespace of coupling constants.
Ify ou chagne $\theta$,
it's not eactly invairatn,
and that's averison fo an anomaly.
You can recase everything that happened here more abstratly.

There's an orthoagonal set of things happend,
that is genralis the theroy of polarization to higher dimesnison.

Ther'es not really a connection betwen those two things,
but maybe that oculd e a resaerch project.

\section{Many-body definition of Chern number}
Let's go back ot the many-body defimtiion of chern number
using our knowledge of polarization in 1D.

Suppose we havea torus.
Consider $\oint A_y = \theta_y$.
Then we have the gorund state $\ket{\psi(\theta_y)}$.

Let me define the exponentiated polarization operator
\begin{align}
    R_x :=
    \prod_{x, y}
    e^{i \frac{2\pi x}{L_x} \hat{n}(x, y)}
\end{align}
and then I want to define
\begin{align}
    \mathcal{T} &=
    \frac{\bra{\psi(\theta_y)} R_x \ket{\psi(\theta_y)}}{%
    \braket{\psi(\theta_y)}{\psi(\theta_y)}
    }
\end{align}
The polarization is $\arg \mathcal{T}(\theta_y)$.
Then the Chern number is
\begin{align}
    C &= \frac{1}{2\pi} \oint d\hteta_y\,
    \frac{d}{d\hteta_y} P(\theta_y)
\end{align}
This is differnt to twisted boundary conditions defiiton.
that was twitisting in x and y,
constructing a berry connection, field strength.
But here,
I'm only inserting $\theta_y$
and seeing how the polarization winds.

The important thing to note about this formula is that before knowing the wave
function as a function of both $\theta_x$ and $\theta_y$,
we only consider the wave function has a function a function of 1 parameter,
$\theta_y$.

\begin{question}
    What's $R_x$?
    What's the meaning of it?
\end{question}
If you write down the formula for polarization in 1D,
it's just
\begin{align}
    P &\propto \sum x n(x)
\end{align}
If you have a 2D system,
\begin{align}
    P &\propto  \sum_{x, y} x n(x, y)
\end{align}
So this $R_x$ you can think of as an exponentiated polarization
\begin{align}
    P_x :\propto e^{i P}
\end{align}

\begin{question}
    What is the minimum number of ground states to find the Chern number?
\end{question}
I'm glad you asked.
The original definition uses the wave function in terms of 2 parameters.
We dropped than down to 1 parameter here.
It turns out you can drop it down to not depend on any parameters.

We can also extrat many-body Chern numbers fom a single GS wave funtion.

Suppose we have a cylinder iwth axis in $x$,
winding in $y$.
Then we have cylinder regions $R_1$, $R_2$, $R_3$, \ldots
and $l_y$ is the lenght in the $y$-direction.
This is pretty amazing
\begin{align}
    \mathcal{T}(\phi) &=
    \bra{0}
    W_{R_1}^\dagger(\phi)
    \mathrm{SWAP}_{1, 3}
    W_{R_1}(\phi)
    V_{R_1 \cup R_2}
    \ket{0}
\end{align}
where
\begin{align}
    W_R &= \prod_{(x, y)\in R} e^{i\hat{n} (x, y) \phi}\\
    V_R &=
    \prod_{(x, y)\in R} e^{i \frac{2\pi y}{l_y} \hat{n}(x, y)}
\end{align}
then the Chern number is
\begin{align}
    C &= \frac{1}{2\pi} \oint d\phi\,
    \frac{d}{d\phi} \arg\mathcal{T}(\phi)
\end{align}
We have a paper on this.
You think in terms of TQFT and cutting and gluing.
There are open questions about this.
We dropped down from 2 params, 1 param, to 0 params.
But this only works on a cylinder.o
Can we get it on justa patch of space?
Empriically,
this formula works even with patches.

The question is how much topological can you extract from a single ground state
wave function on a disc.
We don't know how to do it,
complexity,
algorithsm,
etc.

\begin{question}
    You've transfoered the parameter ot the operator?
\end{question}
Whether this is really a win depends.
I think it's a win,
it's a matter of if a single ground state wave functino contains all the
topology.

Also,
the homework is due on Monday.


% Template for ICIP-2000 paper; to be used with:
%          spconf.sty  - ICASSP/ICIP LaTeX style file, and
%          IEEEbib.bst - IEEE bibliography style file.
% --------------------------------------------------------------------------
\documentclass{article}
\usepackage{spconf,amsmath,epsfig}

% Example definitions.
% --------------------
\def\x{{\mathbf x}}
\def\L{{\cal L}}
\def\Reals{\ensuremath{\mathbf R}}
\newtheorem{theorem}{Theorem}[section]

%    Absolute value notation
\newcommand{\abs}[1]{\lvert#1\rvert}

%    Rank
\DeclareMathOperator{\rank}{rank}

%    Matroid Union
\newcommand{\munion}{\lor}

%   Set minus
\newcommand{\sminus}{\backslash}

%   Two matrices horizontally concatenated, space between can be
%   adjusted here
%
\newcommand{\hmat}[2]{[#1\;#2]}

%   Oriented Matroid Pair
\newcommand{\OMP}[2]{#1,\,#2}

\newcommand{\extra}[1]{{\small{#1}}}

% Title.
% ------
\title{\extra{REPORT:}
(\today)
AN ORIENTED MATROID PAIR MODEL FOR ELECTRICAL AND MECHANICAL NETWORKS
}
%
% Single address.
% ---------------
\name{Seth Chaiken\thanks{Part of this reseach was done during a Sabbatical
from the University at Albany in 2001.}}
\address{University at Albany\\
	Department of Computer Science\\
	Albany, NY 12222\\
	\texttt{sdc@cs.albany.edu}}
%
\begin{document}
%\ninept
%
\maketitle
%

\begin{abstract}
Both resistive electrical networks and elastic mechanical systems such
as trusses have a topological or geometric structure together with constitutive
laws for the elements prior to their interconnection.
Oriented matroids provide a common discrete mathematical model
for such structures in which relationships on the signs of element
quantities can be expressed.
Pairing of oriented matroids
enables non-linear monotone constitutive laws to be fit
into the abstraction in a way that accomodates port and nullor
insertions as well.

The resulting mathematical model clarifies some mechanical analogies for
these
circuit theory concepts, enables constraints on the signs of system response
quantities to be predicted from the structure when this is possible,
and derives topological solution formulas for linearized mechanical
systems in which the analog of a tree-sum is a sum over minimally rigid
trusses.  It also enables the theories for existance and uniqueness to
be applied to mechanical systems with small displacements.
\end{abstract}
%

\section{Introduction}
\label{sec:intro}

Our work \cite{sdcOMP} shows that the model and results of
\cite{HaslerDApplMath,HaslerNeirynck} generalize from a graph model to a
linear subspace pair model
for which
a pair of oriented
matroids (with a common ground set that generalizes the
set of resistor and other edges)
rather than a graph with designated resistor, source, nullator
and norator edges,
is the discrete structure
%in which the topological conditions are expressed
that expresses
%the
topological conditions
for existance or uniqueness of solutions.  These results, like
those of \cite{HaslerDApplMath,HaslerNeirynck} apply to
models whose non-linearities are monotone.
Each oriented matroid codes which combinations of \textit{signs} $\{+,-,0\}$
are feasible as \textit{signatures} of the real coordinate tuple  members
of each subspace.
Bachem and Kern's book \cite{BachemKern}
motivates oriented matroids from this direction.  For example, a dual
pair of oriented matroids abstracts an orthogonally complementary pair
of (finite dim. real) linear subspaces.  Minty's painting property,
most popularly known as a theorem about directed graphs, then emerges
as an excellent \textit{defining axiom} for oriented matroids.
This led us to generalize in \cite{sdcOMP}
Hasler and Neirynck's notion of a ``pair of
conjugate trees'' to a ``complementary pair of bases''; and of
a ``non-trivial uniform partial orientation of the resistors''
to a ``common (non-zero) covector''.
Theory valid for all oriented matroid pairs, not just those represented by
a pair of linear subspaces, was presented in \cite{sdcOMP}.
We will clarify below how our
generalizations of Foss\'{e}prez, Hasler and Neirynck's conditions
for unique solvability
are equivalent to the characterization of $\mathcal{W}_0$ matrix pairs
shown by Sandberg and Willson \cite{SWExistancePf,W0APPLpaper}
to be the condition for unique solvability
of a \textit{common} (not just similar)
problem with monotone non-linearities.
\extra{

In the electrical circuit theory literature, the circuit ``topology''
means the network graph (to which Kirchhoff's laws apply)
together with particular kinds of ``device elements''
such as resistors, capacitors, voltage sources (batteries),
current sources, etc., associated with single graph edges, and possibly
``multiport'' elements associated with multiple graph edges.
Kirchhoff's current law is that current, a scalar in each directed edge,
satisfies conservation of flow.  Kirchhoff's voltage law amounts to the
condition that
the voltage drop, a scalar in each directed edge, is equal to the
difference of the scalar potential values at the endpoints.
Problems with multiport elements are reduced to
those with only single edge elements, in order to apply
the theory of  \cite{HaslerDApplMath},  through the use of
``nullator'' and ``norator'' elements, discussed below.
Detailed exposition of the problems,
reductions, theory and applications is given in \cite{HaslerNeirynck}.

}We will introduce our model by using it to demonstrate
the analogy between the electrical and mechanical elastic
applications.
The  subspace
pairs of the latter
are not graphic cycle and cocycle spaces.
Unlike in the electrical case, the structural model is ``inexact'' because
it depends of the precise sizes and positions of
the bars in addition to their interconnection;
the constitutive laws describe elastic characteristics of the bars.
\extra{

Interesting purely ``discrete'' results (and open problems)
are known in rigidity theory for \textit{generic rigidity}, where
the positions are assumed to have no algebraic dependencies.  Then,
the oriented matroid of the rigidity matrix (defined below)
doesn't change with small position perturbations, and the property
of first order rigidity coincides with local rigidity:  The framework
embedding is determined up to Euclidean motions by the lengths of the
bars, when restricted to a neighborhood of the original embedding.

}We believe that our formulation
has the advantage over conventional matrix formulations
of making more explicit the discrete structure model for circuit
topology and some conditions when this model determines qualitative
relationships among circuit quantity signs independently of
particular monotone constitutive laws.

\section{Basic}
\label{sec:Basic}

A \textit{subspace pair} $(L_V, L_I)$ is a pair of linear subspaces of
$\Reals^U$, where the elements of finite set $U$ index the coordinates.
The scalar product $v\cdot w = \sum_{e\in U}v_e w_e$ is used to define that
$v, w \in \Reals^U$ are \textit{orthogonal} when $v\cdot w = 0$.  An
\textit{orthogonal subspace pair}  satisfies $v\cdot w = 0$ for
all $v\in L_V$ and $w\in L_I$.  A subspace pair has \textit{full rank} when
$\mathrm{rank}(L_V)+\mathrm{rank}(L_I)= |U|$.  Hence an orthogonal full rank
subspace pair is a linear subspace paired with its orthogonal complement.

The structure of an electrical network is defined
beginning with the \textit{network graph}
$\mathcal{N}$ with \textit{nodes} $N$  and \textit{arcs} $U$.
(The generality obtainable by port, nullator/norator or nullor, and
device characteristic insertions will be treated later.)
Each arc has
a fixed but arbitrary direction to define the sign of its voltage drop and
current flow.  The \textit{incidence matrix} $M_V$, with rows indexed by $N$
and columns indexed by $U$ is defined so $M_V(n,e)=+1$ when the tail of $e$ is
$n$, $-1$ if the head of $e$ is $n$, and $0$ if $n$ and $e$ are not incident.

When $L_V$ is the row space of $M_V$ and $L_I$ is the orthogonal complement
of $L_V$, the members of $L_V$ are voltage drop tuples in $\mathcal{N}$
feasible under Kirchhoff's voltage law and the members of $L_I$ are the
current flow tuples feasible under Kirchhoff's current law.  These facts
restate Kirchhoff's laws and Tellegen's theorem.  Note that we can determine
$(L_V, L_I)$ from one of these subspaces given and Tellegen's theorem: The
role of nodes here is not strictly necessary.

The definition of mechanical network structure begins with the
(undirected) \textit{framework graph} $\mathcal{F}$ with \textit{vertices}
$N$ and \textit{edges} $U$.  A \textit{framework} $\mathcal{F}(\mathbf{p})$
in d $dimentions$ is a framework graph $\mathcal{F}$ and an \textit{embedding}
$\mathbf{p}:N\rightarrow \Reals^d$.
The embedding assigns each vertex to a point in
$d$-dimensional space.  The \textit{rigidity matrix} $M_V$
has $d|N|$ rows, each
indexed by one coordinate in $\Reals^d$ of the point that embeds one vertex.
For edge $e=(i,j)\in U$, column $M_V(e)$ of the rigidity matrix is defined
(when vertices are numbered 0 through $|N|-1$):\\
\begin{minipage}{\linewidth}
\begin{align*}
(0, \ldots,0, &\mathbf{p}(n_i)-\mathbf{p}(n_j),\\
              &\mathrm{positions\ } di\ldots di+d-1
\end{align*}
\end{minipage}
\hfill\begin{minipage}{.85\linewidth}
\begin{align*}
\hfill 0,\ldots,0,&\mathbf{p}(n_j)-\mathbf{p}(n_i), 0, \ldots,0)^T\\
                  &\mathrm{positions\ } dj\ldots dj+d-1
\end{align*}
      \end{minipage}\\
This definition is echoed from the literature \cite{RigidityBook} on rigidity theory, except
we interchange rows and columns.  Just as we deemphasized nodes of electrical
networks, we will use merely the row space of $M_V$ for most of what follows.

\extra{
The rigidity matrix as a function of the embedding $\mathbf{p}$ is denoted
$M_V(\mathbf{p})$.  The row vector $\mathbf{p}$ left multiplied with
$M_V(\mathbf{p})$
is the row tuple denoted $\mathbf{L}=\mathbf{p}M_V(\mathbf{p})$  Then,
$\mathbf{L}_e$ $=$
$(\mathbf{p}M_V(\mathbf{p}))_e$ $=$ $|\mathbf{p}_i-\mathbf{p}_j|^2$ for
each edge $e=(i,j)$.  Now if each $\mathbf{p}_i$ is a differentiable function
of $t$, $d\mathbf{L}/dt$ $=$ $2\mathbf{p}'M_V(\mathbf{p})$.  Framework
$\mathcal{F}(\mathbf{p})$ is \textit{first-order rigid} when
$d\mathbf{L}/dt$ $=$ $0$ for all $\mathbf{p}'$ implies
$|\mathbf{p}_i-\mathbf{p}_j|^2$ is constant for all pairs $i$, $j$, not just
endpoints of edges.
}%extra

Let $\mathbf{p}(i)-\mathbf{p}(j)$ for edge $e=\{i,j\}$ be called the
\textit{vector from } $j$ \textit{to} $i$.
It is known that the row space $L_V$
of $M_V$ consists of tuples in $\Reals^{U}$
such that component $v(e)\in\Reals$ for $e\in U$ is the projection of the
relative velocity of vertex $i$ with respect to vertex $j$ projected onto
the vector from $j$ to $i$, for some combination of vertex velocities
$\mathbf{v}:N\rightarrow\Reals^d$:
$v(e)=(\mathbf{v}(i)-\mathbf{v}(j))\cdot(\mathbf{p}(i)-\mathbf{p}(j))$.

It is also known that the $L_I$, the orthogonal complement of $L_V$,
is comprised of the tuples $\sigma:U\rightarrow\Reals$
of scalars for which the framework is in static
equilibrium when each edge $e$ exerts force
$\sigma(e)(\mathbf{p}(j)-\mathbf{p}(i))$ on vertex $i$.  By this convention,
$\sigma(e)>0$ means $e$ is under tension and $\sigma(e)<0$ means $e$ is under
compression.  Each tuple $\sigma\in L_I$ is called a \textit{self-stress}.

Under this analogy,
(1) KVL corresponds to geometric consistancy of first order
edge length changes under changes in the embedding, (2) KCL corresponds to
Newton's laws of static equilibrium, and (3) Tellegen's theorem corresponds
to a virtual work principle, that static equilibrium is
characterized by
the internal forces of every virtual embedding change
doing zero virtual work.

\subsection{Independent Variables and Bases}
KVL, KCL and analogous mechanical structural or geometric laws
are each formulated above by a constraint of the form
$v\in L$ $=$ $\mbox{row space}(M)$
where $v$ is a tuple of variables indexed by $U$
and $L\subset\Reals^U$ is a linear subspace.  The problem of
reformulating such a law by a system of linear equations is
solved as follows:  A maximal subset $B\subset U$ corresponding to
a linearly independent set of columns of $M$ is found.  Such a $B$ is
called a \textit{basis in the matroid} $\mathcal{M}(L)$
\textit{represented by} the columns of matrix $M$ or the linear subspace
$L$.  Row operations and possibly deletion of zero rows can transform
$M$ to $( I\ ;\ M^{\overline{B}} )$ (after column permutation)
where $I$ is the $r\times r$ identity matrix, where $r$ $=$
$\mbox{rank}(M)$ $=$ $\mbox{dim}(L)$ $=$ $\mbox{rank}(L)$
$=$ $\mbox{rank}(\mathcal{M}(L))$.
It is now clear that $v\in L$ is characterized by
$v_{\overline{B}}$ $=$ $v_{B}M^{\overline{B}}$.
For each independently chosen
$v_{B}\in\Reals^B,$ $v=(v_B;v_{\overline{B}})$ is unique tuple
for which the $B$ coordinates equal $v_B$.

\extra{\section{Elastic Analog of the Nodal Admittance Matrix}

Reduced nodal admittance matrix.  Nodal resistance matrix.
Interaction with a physical framework with it's environment.
A framework is first order rigid iff it ``resolves all applications of
static equilibrium forces''.  However, every physical bar has some
elasticity:  An ideal rigid bar is analogous to an ideal voltage
source.  Hence, given an elastic framework, for every application
of static equilibrium forces on the vertices, the vertex positions
will change as the bars stretch or shrink under the forces they now
carry to resolve the applied force.  These first order vertex position
changes are given by $Z\mathbf{f}$.

The environment might interact by ``forcing'' some vertices to change position
relative to one another.
Intuitively, the framework will ``push back''.  The other vertices are free
to move as adjacent vertices move and incident bars change length in
response to the forces developed in them to resolve the forces required
to hold the framework in its new position.  The position changes of the
free vertices $V$ can be calculated by solving for the unknown position changes
in the system of equations $(Y\mathbf{v}_V)(V)=0$.

For our purposes, we insert port elements in order to make interactions
with the environment explicit.  This enables a coordinate of an
environmental interaction quantity to correspond to an oriented matroid
element, so that its sign can be read off from the corresponding entry
in a covector.

It's yet to be done to handle simutaneous application of force to more
than 2 vertices....

to be written..}

\section{THE SUBSPACE PAIR MODEL}

Ports are introduced so the response of an electrical network to current
and/or voltage sources, and the mechanical analogs, can be formulated.
After modeling device characteristics, questions of existance and
uniquenss of solution for various kinds of sources can be formulated.
Familiar topological conditions on dependencies among source values
pertain to the \textit{matroids} of the subspaces $L_V$ and $L_I$.
Questions about existance and uniqueness of solution
will be answered in terms of supplementary subspace pair models
which are obtained by the familiar operations of opening and shorting ports.
Finally, operations on subspace pairs that model nullor insertion are
defined, so that such ideal elements can be modeled combinatorially or
geometrically.

The supplemental subspace pair derived after nullor
insertion will typically not be orthogonal.  One might also choose to
model linearized CCCSs or VCVSs within one of the subspaces.
Each port insertion generally increases $\rank(L_V)+\rank(L_I)-|U|$;
system behavior for linearized constitutive laws will be shown to be
represented by the intersection of two linear spaces.
Hence we do not assume any rank or orthogonality conditions on subspace
pairs in the definitions below.

Given a subspace pair $(L_V, L_I)$ and element $p\in U$ not already a port,
we define the
operation of \textit{inserting a port at  $p$}
as follows: A new subspace pair $(L'_V, L'_I)$
is defined with $U'=U-\{p\}\cup\{p_V,p_I\}$, $L'_V=L_V\oplus\Reals$ (direct
sum) with $p_V$ replacing $p$ and $p_I$ indexing the coordinate of the added
subspace, and $L'_I=L_I\oplus\Reals$ with $p_I$ replacing $p$ and $p_V$
indexing the coordinate of the added subspace.  Note (going to
$(L'_V, L'_I)$) that the ranks of
$L_V$ and $L_I$ each increase by 1, and $|U'|$ $=$ $|U|+1$.  After $p$ port
insertions, we denote the final $U=E\cup P_V \cup P_I$ with pairwise disjoint
$E$, $P_V$ and $P_I$, $|P_V|$ $=$ $|P_I|$ $=$ $p$, $P_V\cup P_I$ being the
replacement elements.  Let $P$ denote $P_V \cup P_I$.

The \textit{subspace pair model} $\mathbf{M}$ $=$
$(E, \Gamma, P, (L_V, L_I))$ consists of finite set $E$ of
\textit{device elements}, \textit{constitutive law relations}
$\Gamma = \{\Gamma_e\subset\Reals\times\Reals | e \in E\}$, a finite set
$P=P_V \cup P_I$ that result from inserting ports as defined above,
and a subspace pair $(L_V, L_I)$ over $\Reals^U$ with $U=E\cup P$.

The \textit{variables} of $\mathbf{M}$ are
$\{u_{Ve}, u_{Ie} | e \in E\}$  $\cup$ \\
$\{ u_{Vp}, u_{Ip} | p_I, p_V \in P \}$.
(For brevity, subscript ``$Vp$'' means port element
$p_V\in P_V$, etc.)
A \textit{subspace pair model with sources} $S$
is a subspace pair model $(E, \Gamma, P, (L_V, L_I), S)$
together with a subset $S$ of exactly $|P|$ of the $2|P|$ elements
in $P$.  A \textit{$V$-driven port} is a port $p\in P$ for which
$p_V\in S$ and $p_I\not\in S$, then $u_{Vp}$ is called
an \textit{input variable}.   Reverse $V$ and $I$ to define an
\textit{$I$-driven port} and its input variable.

A \textit{solution} of $\mathbf{M}$ with sources
is a real valued extension to \textit{all} variables of $\mathbf{M}$
of a given \textit{input} assignment to the input variables
that satisfies\\
$(u_{VP}, u_{IP}, u_V) \in L_V$,
$(u_{VP}, u_{IP}, u_I) \in L_I$
and
$(u_{Ve}, u_{Ie}) \in \Gamma_e$ for all
$e\in E$.  Note that in this model, the constraint
$(u_{VP}, u_{IP}, u_V) \in L_V$ does not (by itself) imply any constraint
on a ``I'' type port variable $u_{Ip}$, similarly, $u_{Vp}$ is not constrained
by $(u_{VP}, u_{IP}, u_I) \in L_V$.  Port variables are not constrained
by the constitutive laws $\Gamma$ (by themselves) either.

In the language of matroid theory, we can call the element $p_I$ an
\textit{isthmus} of the matroid $\mathcal{M}(L_V)$; similarly, $p_V$ is
an isthmus of $\mathcal{M}(L_I)$.  In general, the matroid represented by
a matrix is characterized by the collection $\mathcal{I}$
of \textit{independent sets}
of matrix columns, where a set of columns is called independent when it is
linearly independent.  (Matroid theory studies what can be deduced by
the following three axioms satisfied by $\mathcal{I}$: (1)
$\mathcal{I}\neq\emptyset$.
(2) If $A\subset B\in\mathcal{I}$ then $A\in\mathcal{I}$.  (3)
If $A$, $B$ $\in\mathcal{I}$ and $|A|<|B|$, then there exists $e\in B-A$
for which $A\cup\{e\}\in\mathcal{I}$.  For example, an isthmus $e$ is
characterized by $A\cup\{e\}\in\mathcal{I}$ for all $A\in\mathcal{I}$.
The \textit{rank} of a subset $C\in U$ is the size of the largest independent
subset of $C$.  A set $D$ is called \textit{co-independent} if $\rank(U-D)$
$=$ $\rank(U)$; in other words, removing $D$ does not diminish the original
matroid's rank.)

We say a subspace pair problem with sources $S$
is \textit{well-posed} when for all input assignments there is a unique
solution.

The condition that there is no cycle of voltage source branches nor a
cutset of current source branches is well-known to be necessary for
an electrical network to have a solution for all choices of source values.
Assume as usual no branch is taken to be both a current and voltage source.
Our generalization of the following:
The set of elements $p_V$
for ports driven by ``V'' sources
is an independent set in the matroid $\mathcal{M}(L_V)$ and
co-independent set
in $\mathcal{M}(L_I)$,
and the elements $p_I$ for ports driven by ``I'' sources comprise an
independent set in $\mathcal{M}(L_I)$ and a co-independent set in
$\mathcal{M}(L_V)$.

When the constitutive laws are linear, the solutions of $\textbf{M}$ are
found from the \textit{intersection} of two linear subspaces:  Let $G$
be the diagonal matrix with ``conductances'' $g_e$ in its positions
indexed by $e \in E$ (so $\Gamma_e$ $=$ $\{ (v, g_e v) | v \in \Reals \}$
and 1 in its other diagonal positions.  The solution set projected onto
the $w_I$ variables is $L_VG\cap L_I$.

\section{DELETION AND CONTRACTION}

Given a subspace $L\subset\Reals^U$ and $e\in U$, the subspace $L-e$
``$L$ \textit{with} $e$ \textit{deleted}'' is the
subspace $L-e\subset\Reals^{U-\{e\}}$ defined by $L-e$ $=$
$\{ l(U-e) | l(U)\in L\}$, where $l(U-e)$ denotes the tuple $l(U)\in\Reals^U$
with component labeled by $e$ dropped.  If $L$ is the row space of matrix $M$,
then $L-e$ is the row space of $M(U-e)$, which is $M$ with column $e$ deleted.

The subspace $L/e$ ``$L$ \textit{with} $e$ \textit{contracted}'' is the
subspace
of $\Reals^{\{U-e\}}$ defined by
$L/e$ $=$
$\{ l(U-e) | l(U)\in L \mathrm{\ and\ } l(e)=0\}$.
In other words, $L/e$ is the
intersection of $L$ with
the (hyperplane) subspace of $\Reals^U$ with $l(e)=0$ followed by the $e$
coordinate removed.

We now define deletion and contraction on subspace pairs:
$(L_V, L_I)-e$ $=$ $(L_V-e, L_I/e)$ and
$(L_V, L_I)/e$ $=$ $(L_V/e, L_I-e)$.  One can recognize that
deleting element $e\in S$ from a subspace pair modeling an electrical network
corresponds to \textit{opening} the corresponding branch.  Dually,
contraction corresponds to \textit{shorting} the branch.  Mechanically,
deletion of an edge corresponds to ``breaking'' the corresponding bar:
ignore any distance change between its ends and transmit no force.
Contraction corresponds to
declaring the bar to be rigid, which rules out
all (first order) distance changes between the endpoints and allows
the bar
to transmit arbitrary force.

\section{SIMPLE EXAMPLES}

\begin{figure}[htb]

\begin{minipage}[c]{.48\linewidth}
  \centering
 \centerline{\input{2res.pstex_t}}
%  \vspace{2.0cm}
\end{minipage}
%
\hfill
\begin{minipage}[c]{.48\linewidth}
\[
\begin{array}{cccc}
\multicolumn{4}{c}{\mbox{($M_V$ matrix)}} \\
0   &  1  &  0  &  0  \\
1   &  0  &  1  &  1  \\ \hline \hline
p_V & p_I & e_1 & e_2 \\ \hline \hline
1   &  0  &  0  &  0  \\
0   &  1  & -1  &  0  \\
0   &  0  &  1  &  -1 \\
\multicolumn{4}{c}{\mbox{($M_I$ matrix)}} \\
\end{array}
\]
\end{minipage}
%
\begin{minipage}[b]{.48\linewidth}
\[
\begin{array}{ccc}
  1  &  0  &  0  \\ \hline
 p_I & e_1 & e_2 \\ \hline
  0  &  0  &  0  \\
  1  & -1  &  0  \\
  0  &  1  &  -1
\end{array}
\]
ZIR analysis for voltage source input:
$p_V$ contracted.  Zero response (unique solution) even if
negative resistances $\neq 0$ are allowed.
\end{minipage}
\hfill
\begin{minipage}[b]{.48\linewidth}
\[
\begin{array}{ccc}
0   &  0  &  0  \\
1   &  1  &  1  \\ \hline
p_V & e_1 & e_2 \\ \hline
1   &  0  &  0  \\
0   &  1  &  -1
\end{array}
\]
ZIR analysis for current source input:
$p_I$ deleted.
Zero response (unique solution) \textit{unless}
$g_1 = -g_2$.
\end{minipage}
\caption{Simple Example.}
\label{Simple}
%
\end{figure}

\section{NO-COMMON-COVECTOR PROPERTY AND $\mathcal{W}_0$ PAIRS}

The following theorem
demonstrates, by means of Sandberg and Willson's theory of
$\mathcal{W}_0$ pairs, that any
subspace pair model (with  its separation of
geometric/topological and constitutive constraints) can be analyzed
for unique solvability from the oriented matroid pair it generates.
Conversely, a matrix pair
$(A,B)\in\mathcal{W}_0$ is characterized by a rank condition and a
no-common-covector property.

\begin{theorem}
The subspace pair model has a unique solution for all source
values when the corresponding oriented matroid pair has a complementary
base pair and no common covector.
\end{theorem}

Proof:  The following matrices must be square and order $|U|$
for solutions of (..) to be unique:
$A$ $=$ $\left(\begin{array}{c}M_V\\ 0\end{array}\right)$ and
$B$ $=$ $\left(\begin{array}{c}0\\ -M_I\end{array}\right)$.
The theorem follows with these matrices used in the equivalance of
2. and 5. in the theorem below.

\begin{theorem}
\label{sandwillompairtheorem}
For a pair
%
% EDITdone
% delete ``order'' --- we understand that when we see
% ``$n\times n$'', and this avoids confusing with order as in
% a linear ordering
%
of $n\times n$
matrices $(A,B)$, the following conditions are
equivalent.
%
% EDITdone
% period, not colon
%
\begin{enumerate}
\item
$(A,B)\in {\mathcal W}_0$ in the sense of
Sandberg and
Willson~\cite{SWExistancePf,W0APPLpaper};
e.g., $|AD+B|\neq 0$ for all positive diagonal $D$, etc.
\item
$\rank{\mathcal M}\hmat{A}{B}=n$ and
${\mathcal L}\hmat{A}{B} \cap {\mathcal L}\hmat{I}{-I}=\{0\}.$
\item
$\rank{\mathcal M}\hmat{A}{B}=n$ and
${\mathcal V}\hmat{A}{B} \cap {\mathcal V}\hmat{I}{-I}=\{0\}.$
\item
\textrm{(}Fundamental theorem of Sandberg and
Willson~\cite{SWExistancePf,W0APPLpaper}\textrm{)}\
%
% EDITdone
% use ~ before \cite so a space is left between the word and the reference
%
For all functions $F:{\bf R}^n\rightarrow {\bf R}^n$ of the form
$F(x)_k=f_k(x_k)$ where each $f_k$ is a strictly
monotone increasing
function from
${\bf R}$ {\em onto} $\bf R$ and for all $c\in {\bf R}^n$, the
equation
\[
AF(x) + Bx = c
\]
has a unique solution $x$.% \cite{W0APPLpaper,W0paper}.
\item
For all functions $G:{\bf R}^n\rightarrow {\bf R}^n$ of the form
$G(w)_k=g_k(w_k)$ where each $g_k$ is a strictly monotone increasing
function from ${\bf R}$ {\em onto} $\bf R$ and for all
$d^{\prime}, d^{\prime\prime}\in {\bf R}^n$, the
equations
\begin{equation}
\label{W0dualproblem}
u^t=z^tA+d^{\prime},\;\; w^t=z^tB+d^{\prime\prime},\;\; u=-G(w)
\end{equation}
have a unique solution $(u,w,z)$.
\end{enumerate}
\end{theorem}

Note:  A direct inductive proof is obtainable by generalizing
the proofs given in \cite{HaslerNeirynck}.  This approach has the advantage
of revealing circuit theoretic concepts that occur.  See also
\cite{Fosseprez}.  One of the steps is to prove that if the no-common-covector
property is true for $(M_V,M_I)$, then it is true for the matrix pair
from the system obtained by replacing one of the non-linear elements by a
source.

% References should be produced using the bibtex program from suitable
% BiBTeX files (here: strings, refs, manuals). The IEEEbib.bst bibliography
% style file from IEEE produces unsorted bibliography list.
% -------------------------------------------------------------------------
\bibliographystyle{IEEEbib}
%\bibliography{strings,refs,manuals}
\bibliography{OMPEMech}

\end{document}


\section{Conclusion}
In this section we want to conclude our results and compare them to what we expected.
\begin{description}
	\item[Solitons in one spatial dimension.]

We analyzed solitonic solutions of the GPE, discretized on a spatial grid. For the case of a black soliton with $\nu=0$, \ we proved, that the distribution does not change over time. At the boundaries of the grid, we could observe some disturbing artefacts, which threaten the stability of our solution after long times.

For a single grey soliton with $\nu = 0.8$ we saw the expected movement in positive $z$-direction. Unfortunately the disturbing effects at the boundaries were a lot stronger in this case. Nevertheless we were able  to reproduce the result one would expect from the analytical solution. The analytical time evolution is also included in the corresponding notebook.

In the case of two grey solitons, moving in opposite directions, we saw the overlapping in the center of the grid and after passing past each other the characteristic single movement, not affected by the collision.

	\item[Perturbed density distributions in two spatial dimensions.]

We tried to perturb the two dimensional grid on two different ways, first by adding some randomized noise and second by adding a periodic, sinusoidal noise structure. Both distributions showed interesting dynamics over time. The randomized noise seems to vanish after long times, i.\,e. the distribution gets smoother. This was not what we expected intuitively at the beginning , but after playing around with the vortex grids in the third part, it seems that the artificially added defects are not stable at all. The second example showed the same long-term behavior.

	\item[Vortices as topological defects.]

In the first constellation, where we observed the evolution of vortex-antivortex pairs of the same charge, we saw, that the vortices unwind and due to the opposite charges, cancel out each other, such that the density distribution becomes homogeneous after some time.

In the second case we had a look at single vortices of higher charges. The absence of antivortices prevents the vortices from unwinding/vanishing and therefore, even after a long time, the density distribution is still covered with artefacts of vortices.

The last situation we observed, was an equidistant positioning of vortices with higher charges. In the density plot, one can still observe the artefacts of the vortices at their initial positions and the phase distribution seems to be more stable than in the first case. In the provided animation we still found the vortices in the density plot after some time, whereas the phase plot became more and more indistinct.
\end{description}