# An efficient high dimensional quantum Schur transform

Hari Krovi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Schur transform is a unitary operator that block diagonalizes the action
of the symmetric and unitary groups on an $n$ fold tensor product $V^{\otimes
n}$ of a vector space $V$ of dimension $d$. Bacon, Chuang and Harrow [5] gave
a quantum algorithm for this transform that is polynomial in $n$, $d$ and
$\log\epsilon^{-1}$, where $\epsilon$ is the precision. In a footnote in
Harrow’s thesis [18], a brief description of how to make the algorithm of [5]
polynomial in $\log d$ is given using the unitary group representation theory
(however, this has not been explained in detail anywhere111Specifically,
Harrow sketches a procedure to obtain the same labeling of irreps as the ones
used for Gelfand-Tsetlin basis. Why this procedure leads to the Gelfand-
Tsetlin basis itself (not just the same labeling scheme) could be explained in
more detail.). In this article, we present a quantum algorithm for the Schur
transform that is polynomial in $n$, $\log d$ and $\log\epsilon^{-1}$ using a
different approach. Specifically, we build this transform using the
representation theory of the symmetric group and in this sense our technique
can be considered a “dual” algorithm to [5]. A novel feature of our algorithm
is that we construct the quantum Fourier transform over the so called
_permutation modules_ , which could have other applications.

##  1 Introduction

Schur-Weyl duality is a remarkable correspondence between the irreducible
representations of the symmetric group and those of the unitary group acting
on an $n$ fold tensor product of a vector space $V$. This correspondence
allows one to construct all of the so-called polynomial representations of the
unitary, general linear and special linear groups. Polynomial representations
of matrix groups such as unitary groups are representations whose matrix
entries can be written as polynomials in the entries of the group element
i.e., $\rho(U)$ is a polynomial representation if the entries of $\rho(U)$ are
polynomial in the entries of $U$. Schur-Weyl duality has been generalized to
many other groups and algebras including quantum groups [10, 17].

Schur-Weyl duality has numerous applications in quantum information theory. It
has been used to prove that the tensor product of many copies of a density
operator is close to a projector. In fact, the projector is the one
corresponding to a partition in Schur-Weyl duality that is closest to the
spectrum of $\rho$ [2, 26, 21, 9]. It has also been used to prove de Finetti
theorems [30], which have many applications in security proofs of quantum key
distribution systems. The Schur transform was first constructed for qubits in
the work of Bacon, Chuang and Harrow in [4]. This has been extended to qudits
by the same authors in [5]. A quantum circuit for the Schur transform also has
numerous applications in quantum information theory. It has been applied to
universal distortion-free entanglement concentration [32], universal
compression [21, 22], encoding and decoding into decoherence-free subspaces
[42, 29, 24, 3]. These applications and others are discussed in more detail in
Harrow’s thesis [18]. Recently, the Schur transform has been used as a
primitive in an efficient algorithm for spectrum testing of a density operator
[35] and in algorithms for sample optimal state tomography [34, 16] improving
on previous algorithms [39, 14, 20, 25]. It is also used in a scheme for
optimal compression [41], quantum superreplication [8, 7]. Recently, in [19],
the authors present a classical algorithm to compute the transition amplitudes
of the Schur sampling circuits.

There are two main ways in which Schur sampling is used: weak and strong
sampling. Recall that with any kind of Fourier sampling, we block diagonalize
the group action to obtain the Fourier basis from the computational basis.
Strong (resp. weak) sampling refers to measuring all the registers (resp. only
the label of the irreducible representation) to obtain information from the
Fourier basis. Similarly, a circuit for Schur sampling block diagonalizes the
unitary group (or equivalently the symmetric group) representation on the $n$
fold tensor power of a $d$ dimensional space. In this block diagonalization,
each block essentially contains three pieces of information: (a) the irrep
label, which is common to both the unitary and symmetric groups, (b) the
unitary group irrep register i.e., the register holding the state that
transforms under the unitary group and (c) the symmetric group irrep register
holding the state that transforms under the symmetric group. Strong Schur
sampling refers to measuring all three registers and weak Schur sampling
refers to measuring only the irrep label. One could always measure all three
registers in some basis to obtain more information about the state being
transformed. However, in most of the above applications, one only needs to
measure the irrep label register. In other words, one only needs weak Schur
sampling and the associated probability distribution over the irrep labels to
get enough information for all of the above mentioned applications. Weak Schur
sampling can be performed with a quantum circuit that is polynomial in $\log
d,n$ and $\log(1/\epsilon)$ using the so-called Generalized Phase Estimation
(GPE) procedure. The construction of such a circuit for the weak Schur
transform is described in [18].

In this paper, we present a circuit that can be used for strong Schur sampling
using the representation theory of the symmetric group that runs in time
polynomial in $\log d,n$ and $\log(1/\epsilon)$. In the previous approach by
Bacon, Chuang and Harrow (BCH) [5], the Schur transform is constructed using
the unitary group representation theory. By contrast, our algorithm uses the
symmetric group and, in fact, uses transforms such as Beals’ algorithm for the
Fourier transform over the symmetric group [6] and the GPE algorithm mentioned
above. To give an intuition for how the dual Schur transform works, we should
look at how the original algorithm by BCH for the Schur transform works. BCH
developed a quantum circuit for the Clebsch-Gordan (CG) decomposition problem
for the unitary group, which entails block diagonalizing a tensor product of
two irreps of the unitary group. Then they use this circuit to construct the
Schur transform by applying it iteratively.

This gives us a clue as to how to construct the dual transform if we look at
the so called Littlewood-Richardson (LR) coefficients. The LR coefficients
give the number of times an irrep of the unitary group appears in the CG
decomposition of two unitary group irreps. The same coefficients describe the
decomposition of induced representations of the symmetric group from certain
Young subgroups. This suggests that we need to investigate permutation modules
of the symmetric group since they are exactly such induced representations. We
can now see that block diagonalizing these induced representations gives us
the dual Schur transform. We will not directly work with LR coefficients or
the related Kostka numbers, but embed them in a larger space. This dual
version is polynomial in $\log d,n$ and $\log(1/\epsilon)$, where as the one
in [5] is polynomial in $d,n$ and $\log(1/\epsilon)$. However, as mentioned
above, in Harrow’s thesis [18], a short description is given of how to make
the algorithm from [5] polynomial in $\log d,n$ and $\log(1/\epsilon)$. A
classical algorithm to compute the CG coefficients for SU(d) is given in [1].

This paper is organized as follows. In Section 2, we explain the necessary
background in representation theory, namely the notions of induced
representations, symmetric and unitary group representations and the structure
of permutation modules of the symmetric group. In Section 3, we describe the
construction of quantum Fourier transforms over induced representations. This
construction was also used in [31] for Fourier transforms over quantum doubles
over finite groups. Then, we apply this QFT for permutation modules which are
essentially induced representations. In Section 4, we construct the dual
algorithm for the Schur transform and show that the construction is
O(poly$(n,\log d,\log 1/\epsilon))$ elementary operations. Finally, in Section
5, we present some conclusions and other potential applications of our
subroutines.

##  2 Background in representation theory

###  2.1 Basics of induced representations

In this section, we briefly describe the representation theoretic concepts
such as irreducible representations (irreps, for short), regular
representations and induced representations. Induced representations are
important in this article since the dual Schur transform is essentially a
block diagonalization of induced representations. These concepts for finite
groups are described in several texts such as [37, 12]. Here we follow the
development in [12].

A representation of a finite group on a finite dimensional complex Hilbert
space $V$ is a homomorphism from the group $G$ to the unitary group on the
vector space $\textsf{U}(V)$ i.e., a representation is
$\rho:G\rightarrow\textsf{U}(V)$. For every finite or compact group, any
representation on $V$ can be made unitary i.e., $\rho(g)$ is a unitary matrix
for all $g\in G$. Very often, the space which the representation $\rho$ maps
to, is identified with the representation. Two representations $\rho$ and
$\rho^{\prime}$ of a group $G$ acting on the same vector space $V$ are
considered equivalent if there exists a unitary $U$ such that
$U\rho(g)U^{\dagger}=\rho^{\prime}(g)$ for all $g\in G$. A subspace $W$ of $V$
is called a sub-representation if $\rho(g)$ preserves $W$ for all $g\in G$. In
this case, the orthogonal complement of $W$ in $V$ is also a sub-
representation and $V$ can be viewed as a direct sum of these two sub-
representations. A representation is called an irreducible representation
(irrep) if it does not contain any non-trivial sub-representations. Any
representation $V$ can be broken up into a direct sum of sub-representations
$W$ and its complement as above. Continuing this process further and breaking
up $W$ and its complement into sub-representations, one can arrive at a
decomposition of $V$ into irreps: $V\cong V_{1}\oplus\dots\oplus V_{n}$, where
some of the irreps in the decomposition may be equivalent. A special kind of
irrep is the trivial irrep which acts on a one-dimensional vector space and
takes all the group elements to the identity. Any finite group $G$ has a
finite number of irreducible representations whose number is equal to the
number of conjugacy classes of $G$.

A regular representation of a finite group $G$ acts on the vector space
$\mathbb{C}[G]$, where $g$ acts on any basis vector $h$ by left multiplication
$g:h\rightarrow gh$. The regular representation turns out to have the
following interesting direct sum decomposition into irreps.

$\mathbb{C}[G]\cong\bigoplus_{i}W_{i}^{\oplus d_{i}}\,,$ (2.1)

where $W_{i}$ is an irrep of $G$, $i$ runs over all the different irreps of
$G$ and $d_{i}$ is the vector space dimension of the irrep $W_{i}$. The
quantum Fourier transform (QFT) over a group $G$ usually refers to the
transform that performs the above block diagonalization. This can be defined
as the following basis transformation.

$|g\rangle\rightarrow|\rho,i,j\rangle\,,$

where $\rho$ is the label of the irrep, $i$ is the multiplicity space index
and $j$ is the irrep space index. From the above decomposition, we can see
that the dimension of both the multiplicity space and irrep space are the same
and so $i$ and $j$ run over the same index set labeling the basis vectors.

A type of representation that is of particular importance here is the induced
representation defined as follows. Given a subgroup $H$ of $G$ and a
representation $(\rho,W)$ of $H$, we can construct a representation $V$ of $G$
as follows. As a vector space, it is the tensor product
$\mathbb{C}[G/H]\otimes W$, where $\mathbb{C}[G/H]$ is the space of cosets of
$H$ in $G$. The action of $G$ on this basis can be described using a
transversal $G/H=\\{t_{1},t_{2},\dots,t_{m}\\}$ for $H$ in $G$, where $m$ is
the number of cosets i.e., $m=|G|/|H|$. This means that these elements form an
orthonormal basis of the vector space $\mathbb{C}[G/H]$. Let
$\\{|w_{1}\rangle,\dots,|w_{d}\rangle\\}$ be a basis of $W$. We denote the
induced representation by $\operatorname{\uparrow}_{H}^{G}\rho$ or
$\operatorname{\uparrow}^{G}\rho$ (when $H$ can be inferred). In this basis,
$(\operatorname{\uparrow}_{H}^{G}\rho)(g)$ is the following action on basis
vectors (which can be linearly extended to other vectors).

$(\operatorname{\uparrow}_{H}^{G}\rho)(g):|t\rangle\otimes|w_{i}\rangle\mapsto|t^{\prime}\rangle\otimes\rho(h)|w_{i}\rangle$
(2.2)

where $t^{\prime}\in G/H$ and $h\in H$ are the unique elements for which
$gt=t^{\prime}h$.

###  2.2 Irreducible representations of symmetric and unitary groups

The symmetric group on $n$ letters is denoted $S_{n}$ and consists of all
possible permutations of the $n$ letters. There are $n!$ permutations and
every element can be written as a product of transpositions, where a
transposition is a swap of two letters. Here we denote a transposition between
$a$ and $b$ as $(a,b)$. The representation theory of the symmetric group is
discussed in several books (see for example, [12, 23, 36]). The irreducible
representations of $S_{n}$ are labeled by Young diagrams, which are diagrams
that consist of rows of boxes. A Young diagram corresponds to a partition of
$n$, which is defined as a tuple
$\lambda=(\lambda_{1},\lambda_{2},\dots\lambda_{k})$, where $\lambda_{j}\geq
0$, $\sum_{j}\lambda_{j}=n$ and $\lambda_{k}\geq\lambda_{l}$ for $k<l$. Given
a partition, a Young diagram has $k$ rows and $i_{j}$ boxes in row $j$. In
this case, we say that there are $k$ parts in the partition $\lambda$. For
example, a Young diagram corresponding to the partition $(2,1)$ (not to be
confused with a transposition) is given by . A Young tableau is a Young
diagram with numbers in the boxes. If the numbers are from $1$ to $n$,
increasing from left to right and increasing from top to bottom, then the
Young tableau is called a standard Young tableau (SYT). For example, for the
partition $(2,1)$, an SYT could be
$\vphantom{1}\smash{1}$$\vphantom{1}\smash{3}$$\vphantom{1}\smash{2}$ or
$\vphantom{1}\smash{1}$$\vphantom{1}\smash{2}$$\vphantom{1}\smash{3}$. In
fact, these are the only possible choices. This reflects the fact that the
irrep labeled by this Young diagram is two dimensional. In general, the
dimension of the irrep of $S_{n}$ corresponding to a partition $\lambda$ is
the number of possible standard Young tableau. It is also given by the famous
hook length formula

$d_{\lambda}=\frac{n!}{\Pi_{i,j}\,h_{\lambda}(i,j)}\,,$ (2.3)

where the product in the denominator is over all boxes $(i,j)$ and
$h_{\lambda}(i,j)$ is the hook length of a box, which is defined as the sum of
the number of boxes to the right (including the box $(i,j)$) and the number of
boxes directly below the given box $(i,j)$. We will get back to other aspects
of the symmetric group when we discuss subgroup adapted bases and permutation
modules in the next section.

The unitary group $\textsf{U}(d)$ is the group of $d\times d$ unitary matrices
that is an infinite, though compact, group. The irreducible representations of
this group can be labeled in several ways. Here, we describe the labeling
using both Dynkin labels and Young diagrams. A Dynkin label is the set of
coefficients in the so called basis of fundamental weights. Every irreducible
representation has a basis whose vectors are called weight vectors and have
weights associated with them. In this basis, the highest (and lowest) weight
vectors are unique and, in fact, every irrep of the unitary group can be
associated to a unique highest weight (and there is a one-to-one
correspondence between them). It turns out that the weight vectors lie in a
space (called the root space) spanned by the fundamental weights. This means
that every highest weight (of any irrep) can be written as a linear
combination of the fundamental weights. The weights and weight vectors of a
given irrep form a special basis of the irrep of the unitary group called the
Gelfand-Tsetlin basis. We will discuss this basis in more detail below. It
turns out that one can use Young diagrams to label irreps of the unitary group
as well. One can convert the Dynkin label representation to a Young diagram
representation in the following way: if the Dynkin labels are $(l_{1},\dots
l_{r})$, then the corresponding partition $\lambda$ has components
$\lambda_{i}=l_{i}+\dots+l_{r}$. Conversely, a Young diagram $\lambda$ that
represents an irrep can be converted to Dynkin labels by setting
$l_{i}=\lambda_{i}-\lambda_{i-1}$.

###  2.3 Subgroup adapted bases

A subgroup adapted basis is a canonical basis for an irrep of a group $G$ that
is obtained from a tower of subgroups $G_{0}=1\subset G_{1}\subset\dots
G_{n}=G$ from the identity to $G$. To see how one obtains a canonical basis
from a given subgroup tower, first consider an irrep $\rho$ of $G=G_{n}$.
Suppose we restrict it to the subgroup $G_{n-1}$, then $\rho$ can be
decomposed into irreps of $G_{n-1}$. Suppose that this restriction yields
irreps $\sigma_{i}$ of $G_{n-1}$ each with multiplicity $m_{i}$, then choosing
a basis for the multiplicity spaces and the irrep spaces would give us a basis
for $\rho$. In choosing a basis for the $\sigma_{i}$, we can restrict to
$G_{n-2}$ and so on down the subgroup tower. Finally, we would end up with the
trivial subgroup and since it has only a one-dimensional irrep, this would fix
the entire basis. In the special case that each of the restrictions from
$G_{i}$ to $G_{i-1}$ are multiplicity-free i.e., $m_{i}$ are all zero or one,
we get a canonical basis. In other words, there is no ambiguity in choosing
the basis for the multiplicity space except for the choice of a phase for each
multiplicity space.

In most applications, one makes a projective measurement in this basis, which
makes this phase choice irrelevant. However, if one were to perform other
quantum operations conditioned on basis vectors after the Schur transform,
then the phase choice might be relevant. But in that case, one can incorporate
that phase choice into the conditional operations to be performed. We will see
next that there are special subgroup towers for both the symmetric and unitary
groups that have multiplicity-free branching along the tower and hence lead to
canonical bases.

For the symmetric group, the tower of subgroups $1=S_{1}\subset
S_{2}\subset\dots S_{n}$, where $S_{i}$ permutes the first $i$ letters and
fixes the remaining $n-i$ ones, gives a multiplicity-free branching rule from
one subgroup to the next. This tower is fixed once we number the $n$ qudit
registers in some fashion. The resulting canonical basis is called Young
orthonormal basis (also Young-Yamanouchi basis). This basis can be associated
to Young diagrams with numbers in the boxes with the rule that the numbers are
strictly increasing as one goes from left to right along a row and top to
bottom along a column. As mentioned above, such numbered Young diagrams are
called _standard Young tableaux_ (SYT). An example is given below.

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}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.49998pt}{-9.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\vphantom{1}\smash{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.49998pt}{-9.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\vphantom{1}\smash{5}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.49998pt}{-9.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\vphantom{1}\smash{6}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.49998pt}{-22.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\vphantom{1}\smash{7}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.4)

As is well-known, the symmetric group is generated by adjacent transpositions
$(k,k+1)$. If $k$ and $k+1$ are in different rows and columns in $T$, the
action of any such transposition on an SYT $T$ is given as

$\displaystyle(k,k+1)|T\rangle=a^{T}_{k}|T\rangle+b^{T}_{k}|(k,k+1)T\rangle\,,$
(2.5)
$\displaystyle(k,k+1)|(k,k+1)T\rangle=b^{T}_{k}|T\rangle-a^{T}_{k}|(k,k+1)T\rangle\,,$
(2.6)

where $a^{T}_{k}$ is the inverse of the Manhattan distance in $T$ between the
boxes labeled $k$ and $k+1$ and $b^{T}_{k}=\sqrt{1-(a^{T}_{k})^{2}}$. The
Manhattan distance between two boxes is the number of steps needed to go up
plus number of steps to the right minus number of steps to the left minus
number of steps down. It can be seen easily that this does not depend on the
path taken. We use the notation $|(k,k+1)T\rangle$ to denote the SYT with $k$
and $k+1$ interchanged, which can be seen to be a SYT. If $k$ and $k+1$ are in
the same row, they must be next to each other and the action is given as

$(k,k+1)|T\rangle=|T\rangle\,,$ (2.7)

and if they are in the same column (and necessarily in adjacent rows) the
action is

$(k,k+1)|T\rangle=-|T\rangle\,.$ (2.8)

For the unitary group, a subgroup tower that leads to a canonical basis is
$1=U_{1}\subset U_{2}\subset\dots U_{d}$, where $U_{i}$ is the unitary group
acting on the $i\times i$ minor of the full $d\times d$ matrix. This tower is
determined once we fix a basis for each qudit register. This tower, like the
one for the symmetric group, gives rise to multiplicity-free branching and
hence to a canonical basis. This canonical basis is called the Gelfand-Tsetlin
basis. Given any irrep $\lambda$ of $U_{d}$ in the form of a Young diagram,
one can obtain the diagrams in the restriction to $U_{d-1}$ by removing a box
from end of each column in all possible ways. If two columns have the same
length in $\lambda$ and the choice is to remove a box from the left column,
then one must also remove the box from the right column to ensure that a valid
Young diagram is obtained. An example is given below.

$\lambda=\leavevmode\hbox to39.3pt{\vbox
to39.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.15pt\lower-26.15pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{39.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{38.99pt}{-13.0pt}\pgfsys@lineto{38.99pt}{0.0pt}\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-13.01pt}\pgfsys@lineto{13.0pt}{-13.01pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@moveto{12.99pt}{-26.0pt}\pgfsys@lineto{12.99pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow\left\\{\leavevmode\hbox
to39.3pt{\vbox to39.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-26.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{39.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{38.99pt}{-13.0pt}\pgfsys@lineto{38.99pt}{0.0pt}\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-13.01pt}\pgfsys@lineto{13.0pt}{-13.01pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@moveto{12.99pt}{-26.0pt}\pgfsys@lineto{12.99pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox
to39.3pt{\vbox to26.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-13.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{39.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{38.99pt}{-13.0pt}\pgfsys@lineto{38.99pt}{0.0pt}\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox
to39.3pt{\vbox to39.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-26.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{13.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{12.99pt}{-13.0pt}\pgfsys@lineto{12.99pt}{0.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-13.01pt}\pgfsys@lineto{13.0pt}{-13.01pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@moveto{12.99pt}{-26.0pt}\pgfsys@lineto{12.99pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox
to39.3pt{\vbox to26.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-13.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{13.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{12.99pt}{-13.0pt}\pgfsys@lineto{12.99pt}{0.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right\\}\,.$ (2.9)

Suppose we pick one and proceed with a choice down the tower, we would have
the following possibility.

$\leavevmode\hbox to39.3pt{\vbox to39.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-26.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{39.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{38.99pt}{-13.0pt}\pgfsys@lineto{38.99pt}{0.0pt}\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-13.01pt}\pgfsys@lineto{13.0pt}{-13.01pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@moveto{12.99pt}{-26.0pt}\pgfsys@lineto{12.99pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow\leavevmode\hbox
to39.3pt{\vbox to39.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-26.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{39.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{39.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@moveto{26.0pt}{-13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{38.99pt}{-13.0pt}\pgfsys@lineto{38.99pt}{0.0pt}\pgfsys@moveto{39.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{13.0pt}{-26.0pt}\pgfsys@moveto{0.0pt}{-13.01pt}\pgfsys@lineto{13.0pt}{-13.01pt}\pgfsys@moveto{0.0pt}{-26.0pt}\pgfsys@lineto{0.0pt}{-13.0pt}\pgfsys@moveto{12.99pt}{-26.0pt}\pgfsys@lineto{12.99pt}{-13.0pt}\pgfsys@moveto{13.0pt}{-13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow\leavevmode\hbox
to39.3pt{\vbox to26.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-13.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{13.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{39.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{39.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{26.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@moveto{38.99pt}{0.0pt}\pgfsys@lineto{38.99pt}{13.0pt}\pgfsys@moveto{39.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@closepath\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{13.0pt}{-13.0pt}\pgfsys@moveto{0.0pt}{-0.01pt}\pgfsys@lineto{13.0pt}{-0.01pt}\pgfsys@moveto{0.0pt}{-13.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{12.99pt}{-13.0pt}\pgfsys@lineto{12.99pt}{0.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\rightarrow\leavevmode\hbox
to26.3pt{\vbox to13.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-0.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@lineto{26.0pt}{13.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{26.0pt}{13.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{26.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{12.99pt}\pgfsys@lineto{26.0pt}{12.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{13.0pt}\pgfsys@moveto{13.0pt}{0.0pt}\pgfsys@lineto{13.0pt}{13.0pt}\pgfsys@moveto{25.99pt}{0.0pt}\pgfsys@lineto{25.99pt}{13.0pt}\pgfsys@moveto{26.0pt}{13.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.10)

This sequence gives us a basis vector of the irrep $\lambda$. This can be
encapsulated by putting numbers into the original irrep, which represent the
stage before which the boxes are removed. For the above sequence the following
numbering would hold.

$\leavevmode\hbox to42.67pt{\vbox
to39.3pt{\pgfpicture\makeatletter\hbox{\hskip 1.83302pt\lower-26.15pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} {\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.49998pt}{-22.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$\vphantom{1}\smash{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (2.11)

Notice that the rows are weakly increasing i.e., the numbers either increase
or stay the same as we move right and the columns are strictly increasing.
Such a Young diagram is called a semi-standard Young diagram (SSYT). SSYTs
with numbers taken from the set $[d]$ label basis vectors in the irrep of
$U_{d}$. We will see below that SSYTs also play a role in certain induced
representations of the symmetric group called permutation modules. Efficient
encodings of these bases are constructed in [5] i.e., using poly($\log d$, n,
$\log 1/\epsilon$) bits. For a SSYT, the tuple that contains the number of
boxes labeled by a given integer is called the _content_ of the SSYT. For
instance, in the example above, the content is $(2,2,3)$ corresponding to $2$
boxes numbered one, $2$ boxes numbered two and $3$ boxes numbered three.

SSYTs have an interesting structure that is useful in our algorithms. If we
consider all the boxes containing a specific number, we find that no two of
them appear in the same column. If we isolate these boxes, such a _skew_
diagram is called a _horizontal strip_. An SSYT can be thought of as being
composed of horizontal strips. It turns out that this composition can be made
more precise as we describe in the next subsection. An example of an SSYT and
the associated horizontal strips are given below.

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In terms of horizontal strips, the above decomposition of an irrep $\lambda$
of $U(d)$ into irreps $\mu$ of $U(d-1)$ can be rephrased as the set of all
$\mu$ that can be obtained from $\lambda$ by removing a horizontal strip in
all possible ways.

###  2.4 RSK algorithm and composition of Young tableaux

The discussion in this section is taken from the book by Fulton [11]. This
procedure is used in the algorithm DualSchur (step 4). For a more detailed
explanation of how to obtain SYTs with permuted content, see [11] chapter 4.
The RSK (Robinson-Schensted-Knuth) algorithm establishes a correspondence
between pairs of words and pairs of tableaux. The main part of the RSK is a
procedure called row insertion that lets one insert a letter into a tableau
such that the resulting tableau is semi-standard with one more box. This
correspondence has several applications, but the primary application here is
to produce semi-standard Young tableaux where the content is permuted. In this
subsection, we briefly describe this algorithm. The row insertion procedure
takes as input a tableau $T$ and an integer $x$ and outputs a tableau with one
more box than $T$. The procedure is as follows

  1. 1.

Find the number in the first row of $T$ that is greater than $x$.

  2. 2.

If there is none, then place $x$ in a box at the end of the first row.

  3. 3.

If there are numbers greater than $x$ in the first row, let $y$ be the
smallest among them. Place $x$ in $y$’s position ($x$ ‘bumps’ $y$).

  4. 4.

Repeat the previous steps with $y$ (in the place of $x$) and starting with the
second row.

The RSK algorithm uses the above row bumping procedure. It takes a pair of
words, say $u=u_{1}u_{2}\dots u_{r}$ and $v=v_{1}v_{2}\dots v_{r}$ that has
the following two properties as input. The first is that $u$ is weakly
increasing and second if $u_{k-1}=u_{k}$, then $v_{k-1}\leq v_{k}$. Given such
pairs as input the procedure produces a pair of tableaux ($P,Q$) iteratively
as follows. Start with the base tableaux $\vphantom{1}\smash{x}$ and
$\vphantom{1}\smash{y}$, where $x=v_{1}$ and $y=u_{1}$. Then from any pair
($P_{k-1},Q_{k-1}$), row insert $v_{k}$ into $P_{k-1}$ getting $P_{k}$. Then
add a box to $Q_{k-1}$ in the position where the new box is in $P_{k}$ and put
$u_{k}$ in this box.

This procedure allows us to define a product or composition of tableaux
mentioned in the previous subsection. Suppose $S$ and $T$ are two tableaux,
then $S\cdot T$ is defined as follows. If $T$ consists of only one box, then
the product of $S$ and $T$ is the result of row insertion into $S$. If $T$
contains more than one box, then we row insert them one by one into $S$
starting from the bottom left box and moving left to right along each row and
upwards along the rows.

In this paper, we will need this procedure (in the algorithm DualSchur step 4)
to create an SSYT $V$ from another SSYT $U$ with the content permuted. In
order to describe it for any permutation, we only need to show it for a single
transposition. As noted above, an SSYT consists of horizontal strips each
having the same number. Now suppose that the transposition is $(k,k+1)$ i.e.,
if the original SSYT $U$ contains $n_{k}$ boxes numbered $k$ and $n_{k+1}$
boxes numbered $k+1$, then the new SSYT $V$ should contain $n_{k+1}$ boxes
numbered $k$ and $n_{k}$ boxes numbered $k+1$. This is done by using the
product defined above. It turns out [11] that we can write $U=A\cdot B\cdot
C$, where $A$ is a SSYT that contains only boxes numbered $1$ through $k$, $B$
is an SSYT that contains boxes numbered $k$ and $k+1$ and $C$ is an SSYT that
contains the remaining numbers. Since $B$ contains only two labels, it must be
of the form

$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$st

where $l=k+1$ for brevity in the figure and the overhang consists of $s$ boxes
numbered $k$ and $t$ boxes numbered $l$. When we swap the number of $k$s and
$l$s, we obtain a similar diagram where the overhang contains $t$ $k$s and $s$
$l$s. Denote this diagram by $B^{\prime}$.

$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{k}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$$\vphantom{1}\smash{l}$ts

Now the new SSYT $V$ is obtained by composing $A\cdot B^{\prime}\cdot C$ using
the RSK algorithm.

###  2.5 Gelfand-Tsetlin bases

An alternate way of representing the basis vectors of the unitary group is the
so called Gelfand-Tsetlin (GT) patterns. GT patterns are useful in certain
applications, although one can easily convert an SSYT to a GT pattern and vice
versa. The power of GT patterns comes from the fact that in the GT basis, one
can write the matrix elements of the Lie algebra SU(d) as derived by Gelfand
and Tsetlin [13]. The formulae in this section can be found in the book by
Vilenkin and Klimyk [40].

A GT pattern $M$ is a triangle of numbers such as the one below.

$M=\begin{pmatrix}m_{1,d}&&m_{2,d}&&\dots&&m_{d,d}\\\
&m_{1,d-1}&&\dots&&m_{d-1,d-1}&\\\ &&\ddots&&\reflectbox{$\ddots$}\\\
&&m_{1,2}&m_{2,2}\\\ &&\qquad\qquad\quad m_{1,1}\end{pmatrix}\,.$ (2.13)

These numbers satisfy the _in betweenness_ condition

$m_{k,l}\geq m_{k,l-1}\geq m_{k+1,l}\,,\qquad 1\leq k<l\leq d\,.$ (2.14)

The numbers in the first row of the GT pattern correspond to the number of
boxes in each row of the corresponding SSYT. The number of boxes in the SSYT
with the number $l$ in row $k$ is $m_{k,l}-m_{k,l-1}$. The total number of
boxes with the number $l$ is therefore the difference of the row sums
$\sum_{k}(m_{k,l}-m_{k,l-1})$ (where $m_{k,l}=0$ if $k>l$). More
systematically, in order to convert a GT pattern to an SSYT, we start from
bottom-most row of the GT pattern and create a partial SSYT with one row of
$m_{1,1}$ boxes labelled $1$. Next, we add $m_{1,2}-m_{1,1}$ boxes to this row
labelled $2$ and put $m_{2,2}$ boxes in the second row labelled $2$.
Continuing in this way, we add $m_{1,l}-m_{1,l-1}$ boxes to the first row
labelled $l$, $m_{2,l}-m_{2,l-1}$ boxes in row two labelled $l$ etc. The in-
betweenness conditions guarantee that the skew tableau with boxes labelled $l$
is a _horizontal strip_ i.e., a skew tableau with at most one box in each of
its columns. Now, to convert from an SSYT to a GT pattern, we can use the fact
that the number of boxes labelled $l$ in the $k^{th}$ row is
$m_{k,l}-m_{k,l-1}$ and fill in the $k^{th}$ diagonal.

The generators of the Lie algebra of SU(d) are defined as follows.

$\displaystyle J_{0}^{(l)}=\frac{1}{2}(E^{l,l}-E^{l+1,l+1})\,,$ (2.15)
$\displaystyle J_{+}^{(l)}=E^{l,l+1}\,,$ (2.16) $\displaystyle
J_{-}^{(l)}=E^{l+1,l}\,,$ (2.17)

where $E^{k,l}$ is the matrix with a one in the $(k,l)^{th}$ position and
zeros everywhere else and $1\leq l\leq d-1$. The matrix elements of these
generators can be expressed in the GT basis. The action of these elements on a
basis vector corresponding to a GT pattern $|M\rangle$ is given by

$\displaystyle
J_{0}^{(l)}|M\rangle=\left[\sum_{k=1}^{l}m_{k,l}-\frac{1}{2}(\sum_{k=1}^{l+1}m_{k,l+1}+\sum_{k=1}^{l-1}m_{k,l-1})\right]|M\rangle\,,$
(2.18) $\displaystyle\langle
M+\delta_{k,l}|J_{+}^{(l)}|M\rangle=\left(-\frac{\Pi_{k^{\prime}=1}^{l+1}(m_{k^{\prime},l+1}-m_{k,l}+k-k^{\prime})\Pi_{k^{\prime}=1}^{l-1}(m_{k^{\prime},l-1}-m_{k,l}+k-k^{\prime}-1)}{\Pi_{k^{\prime}=1,k^{\prime}\neq
k}^{l}(m_{k^{\prime},l}-m_{k,l}+k-k^{\prime})(m_{k^{\prime},l}-m_{k,l}+k-k^{\prime}-1)}\right)^{1/2}\,,$
$\displaystyle\langle