# Invariants of partitions

Bao Shou  Center of Mathematical Sciences, Zhejiang University, Hangzhou
310027, China<EMAIL_ADDRESS>
###### Abstract.

The symbol invariant is used to describe the Springer correspondence for the
classical groups by Lusztig. And the fingerprint invariant can be used to
describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple
operators labeled by pairs of partitions
$(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})$. It is conjectured that
the symbol invariant is equivalent to the fingerprint invariant. We prove that
the former imply the latter one. We make a classification of the maps
preserving symbol, and then prove these maps preserve fingerprint. We also
discuss the classification of the maps preserving fingerprint, which is found
related to the conditions of the definition of the fingerprint. The
constructions of the symbol and the fingerprint invariants in previous works
play significant roles in the proof.

###### Key words and phrases:

partition, symbol invariant, fingerprint invariant, rigid seimisimple
operator, Springer correspondence, Kazhdan Lusztig map

###### 2010 Mathematics Subject Classification:

05E10

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
    1. 2.1 Rigid Partitions in the $B_{n},C_{n}$, and $D_{n}$ theories
    2. 2.2 Symbol invariant of partitions
    3. 2.3 Fingerprint invariant of partitions
  3. 3 Examples of maps preserving symbol
    1. 3.1 Maps $See1$ and $See2$
    2. 3.2 Map $Seo2$
  4. 4 Classification of maps preserving symbol
    1. 4.1 Maps $TE$ and $TO$
    2. 4.2 Stopping phenomenon and Splitting theorem
    3. 4.3 $S$-type maps
    4. 4.4 $D$-type maps
    5. 4.5 Special blocks
    6. 4.6 $I$-type maps
  5. 5 Classification of $\lambda$ with the same fingerprint
    1. 5.1 Classification of $\lambda$
  6. 6 $B_{n}$ and $D_{n}$ theories
  7. A Special case of the splitting theorem
  8. B Special blocks
  9. C $I$-type maps
  10. D Condition $(ii)$ of the definition of fingerprint

##  1\. Introduction

A partition $\lambda$ of the positive integer $n$ is a decomposition
$\sum_{i=1}^{l}\lambda_{i}=n$
($\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{l}$; $l$ is the length).
Partitions are in a one to one correspondence with Young tableaux. For
instance the partition $3^{3}2^{3}1$ corresponds to

Young tableaux occur in a number of branches of mathematics and physics. They
are also tools for the construction of the eigenstates of Hamiltonian System
[28] [29] [30] and label surface operators[4]..

Surface operators are two-dimensional defects supported on a two-dimensional
submanifold of spacetime, which are natural generalisations of the ’t Hooft
operators. In [2], Gukov and Witten initiated a study of surface operators in
$\mathcal{N}=4$ super Yang-Mills theories in the ramified case of the
Geometric Langlands Program. $S$-duality for certain subclass of surface
operators is discussed in [3][5]. The $S$-duality [10] assert that
$S:\;(G,\tau)\rightarrow(G^{L},-1/n_{\mathfrak{g}}\tau)$ (where
$n_{\mathfrak{g}}$ is 2 for $F_{4}$, 3 for $G_{2}$, and 1 for other semisimple
classical groups [2]; $\tau=\theta/2\pi+4\pi i/g^{2}$ is usual gauge coupling
constant ). This transformation exchanges gauge group $G$ with the Langlands
dual group $G^{L}$. For example, the Langlands dual groups of
$\mathrm{Spin}(2n{+}1)$ are $\mathrm{Sp}(2n)/\mathbb{Z}_{2}$. And the
langlands dual groups of $\mathrm{SO}(2n)$ are themselves.

In [4], Gukov and Witten extended their earlier analysis [2] of surface
operators of which based on the invariants. They identified a subclass of
surface operators called ’rigid’ surface operators expected to be closed under
$S$-duality. There are two types rigid surface operators: unipotent and
semisimple. The rigid semisimple surface operators are labelled by pairs of
partitions $(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})$. Unipotent
rigid surface operators arise when one of the partitions is empty. Invariants
of the surface operators $(\lambda^{\prime};\lambda^{\prime\prime})$ do not
change under the $S$-duality map [4][5]. The dimension $d$ [4][1] is the most
basic invariant of a rigid surface operator. The invariant symbol based on the
Springer correspondence[1], which can be extended to rigid semisimple
conjugacy classes. There is another invariant fingerprint  related to the
Kazhdan-Lusztig map for the classical groups [26][27]. The Kazhdan-Lusztig map
is a map from the unipotent conjugacy classes to the set of conjugacy classes
of the Weyl group. It can be extended to the case of rigid semisimple
conjugacy classes[25]. The rigid semisimple conjugacy classes and the
conjugacy classes of the Weyl group are described by pairs of partitions
$(\lambda^{{}^{\prime}};\lambda^{{}^{\prime\prime}})$ and pairs of partitions
$[\alpha;\beta]$, respectively. The fingerprint invariant is a map between
these two classes of objects. In [4], it was pointed that two discrete quantum
numbers ’center’ and ’topology’ are interchanged under $S$-duality.

Compared to the fingerprint, the symbol is much easier to be calculated and
more convenient to find the $S$-duality maps of surface operators. By using
symbol invariant, Wyllard made some explicit proposals for how the $S$-duality
maps should act on rigid surface operators in [5]. In [23], we find a new
subclass of rigid surface operators related by S-duality. In [17], we found a
construction of symbol for the rigid partitions in the $B_{n},C_{n}$, and
$D_{n}$ theories. Another construction of symbol is given in [18]. The
calculation of fingerprint is a complicated and tedious work. Its basic
properties and constructions is discussed in [19].

A problematic mismatch in the total number of rigid surface operators between
the $B_{n}$ and the $C_{n}$ theories was pointed out in [4] [5]. Fortunately,
the construction of symbol [6] and the classification of symbol preserving
maps are helpful to address this problem. In [24], we clear up cause of this
discrepancy and construct all the $B_{n}/C_{n}$ rigid surface operators which
can not have a dual. A classification of the problematic surface operators is
made.

The fingerprint invariant is assumed to be equivalent to the symbol invariant
[5]. It is checked that the symbol of a rigid surface operator contains the
same amount of information as the fingerprint [5]. In this paper, we attempt
to prove the symbol invariant of partitions implies the fingerprint invariant
of partitions. The constructions of the symbol and the fingerprint in previous
works play significant roles in the proof of the conjecture. With no
noncentral rigid conjugacy classes in the $A_{n}$ theory, we do not discuss
surface operators in this case. We also omit the discussion of the exceptional
groups, which are more complicated. We will focus on theories with gauge
groups $\mathrm{SO}(2n)$ and the gauge groups $\mathrm{Sp}(2n)$ whose
Langlands dual group is $\mathrm{SO}(2n+1)$.

The following is an outline of this article. In Section 2, we introduce some
basic results related to the rigid partition and the definition of symbol in
[5][17]. We also introduce the construction of symbol in [17] which is the
basics of this study, as well as that of the fingerprint invariant in [19]. In
Section 3, we introduce examples of maps preserving symbol and prove they
preserve fingerprint as preparation for introducing the general results in the
following sections. In Section 4, we prove two lemmas which are useful for the
classifications of maps preserving symbol. Then we make a classification of
the maps preserving symbol and prove these maps preserve fingerprint, which
are the main results in this paper. In Section 5, we discuss the
classification of the maps preserving fingerprint, which is found related to
the conditions of the definition of the fingerprint.

In the appendix, we prove several revelent facts which are the supplement to
the proofs of the main results.

##  2\. Preliminary

In this section, we introduce the rigid partitions in the $B_{n}$, $C_{n}$,
and $D_{n}$ theories and the construction of the symbol of them [17]. Then we
introduce the definition of fingerprint [5] and its construction [19].

###  2.1. Rigid Partitions in the $B_{n},C_{n}$, and $D_{n}$ theories

A $B_{n}$($D_{n}$)theories partition is a partition of $2n{+}1$($2n$) where
all even integers appear an even number of times. A $C_{n}$ partition is a
partition of $2n$ where all odd integers appear an even number of times. If it
has no gaps (i.e. $\lambda_{i}-\lambda_{i+1}\leq 1$ for all $i$) and no odd
(even) integer appears exactly twice, a partition in the $B_{n}$ or $D_{n}$
($C_{n}$) theories is called rigid. We will focus on rigid partition
corresponding to rigid operators. The operators will refer to rigid operators
in the rest of paper. The addition rule of two partitions $\lambda$ and $\mu$
is defined by the additions of each part $\lambda_{i}+\mu_{i}$.

The following facts play important roles in this study.

######  Proposition 2.1.

The longest row in a rigid $B_{n}$ partition always contains an odd number of
boxes. The following two rows of the first row are either both of odd length
or both of even length. This pairwise pattern then continues. If the Young
tableau has an even number of rows the row of shortest length has to be even.

Remark: If the last row of the partition is odd, the number of rows is odd.

######  Proposition 2.2.

For a rigid $D_{n}$ partition, the longest row always contains an even number
of boxes. And the following two rows are either both of even length or both of
odd length. This pairwise pattern then continue. If the Young tableau has an
even number of rows the row of the shortest length has to be even.

######  Proposition 2.3.

For a rigid $C_{n}$ partition, the longest two rows both contain either an
even or an odd number number of boxes. This pairwise pattern then continues.
If the Young tableau has an odd number of rows the row of shortest length has
contain an even number of boxes.

Examples of partitions in the $B_{n}$, $D_{n}$, and $C_{n}$ theories as shown
in Fig.(1):

Figure 1.  Partitions in the $B_{n}$, $D_{n}$, and $C_{n}$ theories.

###  2.2. Symbol invariant of partitions

In [17], we proposed equivalent definitions of symbol for the $C_{n}$ and
$D_{n}$ theories which are consistent with that for the $B_{n}$ theory as much
as possible.

######  Definition 1.

The symbol of a partition in the $B_{n}$, $C_{n}$, and $D_{n}$ theories.

  * •

$B_{n}$ theory: first add $l-k$ to the $k$th part of the partition. Next
arrange the odd parts of the sequence $l-k+\lambda_{k}$ and the even parts in
an increasing sequence $2f_{i}+1$ and in an increasing sequence $2g_{i}$,
respectively. Then calculate the terms

$\alpha_{i}=f_{i}-i+1\quad\quad\quad\beta_{i}=g_{i}-i+1.$

Finally write the symbol as follows

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\alpha_{3}&&\cdots\\\
&\beta_{1}&&\beta_{2}&&\cdots&\end{array}\right).$

  * •

$C_{n}$ theory:

1:

If the length of partition is even, compute the symbol as in the $B_{n}$ case,
and then append an extra ${}^{\prime}0^{\prime}$ on the left of the top row of
the symbol.

2:

If the length of the partition is odd, first append an extra
${}^{\prime}0^{\prime}$ as the last part of the partition. Then compute the
symbol as in the $B_{n}$ case. Finally, we delete a ${}^{\prime}0^{\prime}$ in
the first entry of the bottom row of the symbol.

  * •

$D_{n}$ theory: first append an extra ${}^{\prime}0^{\prime}$ as the last part
of the partition, and then compute the symbol as in the $B_{n}$ case. We
delete two 0’s in the first two entries of the bottom row of the symbol.

Parity of row | Parity of $i+t+1$ | Contribution to symbol | $L$
---|---|---|---
odd | even | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots\overbrace{1\;\;1\cdots 1}^{L}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}+1)$
even | odd | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots\overbrace{1\;\;1\cdots 1}^{L}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k})$
even | even | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k})$
odd | odd | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$
Table 1.  Contribution to symbol of the $i$ th row($B_{n}(t=-1)$,
$C_{n}(t=0)$, and $D_{n}(t=1)$).

In [17], the construction of symbol is given by Table 1. We determine the
contribution to symbol for each row of a partition in the different theories
uniformly. For the rigid semisimple operators
$(\lambda^{\prime},\lambda^{\prime\prime})$, the symbol is constructed by
adding the entries that are ‘in the same place’ of symbols of
$\lambda^{\prime}$ and $\lambda^{\prime\prime}$. An example illustrates the
addition rule:

(2.1)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&0&&0&&0&&0&&1&&1\\\
&1&&1&&1&&1&&1&&2&\end{array}\right)+\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&0&&0&&1&&1&&1\\\
&1&&1&&1&&1&&1&\end{array}\right)=\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&0&&0&&0&&1&&2&&2\\\
&1&&2&&2&&2&&2&&3&\end{array}\right).$

###  2.3. Fingerprint invariant of partitions

The fingerprint invariant of rigid semisimple surface operator
$(\lambda^{\prime},\lambda^{\prime\prime})$ is constructed as follows [5].
First, add the two partitions $\lambda=\lambda^{{}^{\prime}}+\lambda^{"}$, and
then construct the partition $\mu$

(2.2) $\displaystyle\mu_{i}=Sp(\lambda)_{i}=\left\\{\begin{aligned}
&\lambda_{i}+p_{\lambda}(i)\quad\quad\textrm{if}\quad\lambda_{i}\textrm{ is
odd and}\quad\lambda_{i}\neq\lambda_{i-p_{\lambda}(i)},\\\
&\lambda_{i}\quad\quad\quad\quad\quad\textrm{ otherwise.}\end{aligned}\right.$

Next define the function $\tau$ from an even positive integer $m$ to $\pm$ as
follows. For $B_{n}$($D_{n}$) partitions, $\tau(m)$ is $-1$ if at least one
$\mu_{i}$ such that $\mu_{i}=m$ and either of the following three conditions
is satisfied.

$\displaystyle(i)\quad\quad\quad\quad\quad\,\,\mu_{i}\neq\lambda_{i}$
$\displaystyle(ii)\quad\quad\quad\sum^{i}_{k=1}\mu_{k}\neq\sum^{i}_{k=1}\lambda_{k}$
$\displaystyle(iii)_{SO}\quad\quad\quad\lambda^{{}^{\prime}}_{i}\quad\textrm{is
odd}.$

Otherwise $\tau$ is 1. For $C_{n}$ partitions, the definition is exactly the
same except the condition $(iii)_{SO}$ is replaced by

$(iii)_{Sp}\quad\quad\lambda^{{}^{\prime}}_{i}\quad\textrm{is even}.$

Finally construct a pair of partitions $[\alpha;\beta]$. For each pair of
parts of $\mu$ both equal to $a$, satisfying $\tau(a)=1$, retain one part $a$
as a part of the partition $\alpha$. For each part of $\mu$ of size $2b$,
satisfying $\tau(2b)=-1$, retain $b$ as a part of the partition $\beta$.

According to the formula (2.2), $\mu_{i}\neq\lambda_{i}$ only happen at the
end of a row. We have the following important lemma.

######  Lemma 2.1.

Under the map $\mu$, the change of rows depend on the parity of the
$\lambda_{i}$th row and the sign of $p_{\lambda}(i)$.

Parity of row | Sign | Change
---|---|---
odd | $-$ | $\mu_{i}=\lambda_{i}-1$
even | $-$ | $\mu_{i}=\lambda_{i}+1$
even | $+$ | $\mu_{i}=\lambda_{i}$
odd | $+$ | $\mu_{i}=\lambda_{i}$

Figure 2.  The difference of the height of the $(j-1)$th row and the $(j-3)$th
row is two, violating the rigid condition $\lambda_{i}-\lambda_{i+1}\leq 1$.

It is easy to prove the following fact.

######  Proposition 2.4.

[19] For the partition $\lambda$ with $\lambda_{i}-\lambda_{i+1}\leq 1$, the
condition (2.3) $(i)$ imply $(ii)$.

We would give an example where the condition $(ii)$ works. As shown in
Fig.(2), the heights of the $(j-1)$th row, the $(j-3)$th row, and the
$(j-5)$th are even. And the difference of the height of the $(j-1)$th row and
the $(j-3)$th row violate the rigid condition $\lambda_{i}-\lambda_{i+1}\leq
1$ as well as that of the heights of the $(j-3)$th row and the $(j-5)$th row.
The part $(j-3)$ satisfy the condition (2.3)$(ii)$ but do not satisfy the
condition $(i)$. In the rest of the paper, we would focus on the partition
$\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$ with
$\lambda_{i}-\lambda_{i+1}\leq 1$. In Appendix D, we would like to give the
reasons that the neglect of the condition (2.3)$(ii)$ would not change the
conclusions in this paper.

Now we review the algorithm for calculating the fingerprints of rigid
semisimple operators $(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})$ in
the $B_{n}$, $C_{n}$, and $D_{n}$ theories [19].

Operators $\mu_{e11}$, $\mu_{e12}$, $\mu_{e21}$, and $\mu_{e22}$

The partition $\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$ is
constructed by inserting rows of $\lambda^{{}^{\prime}}$ into
$\lambda^{{}^{\prime\prime}}$ one by one. Without confusion, the image of the
map $\mu$ is also denoted as $\mu$. The partition
$\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$ is decomposed into
several blocks such as blocks $I,II,III$ as shown in Fig.(3). ( We will refine
this model in Section 4.5.)

Figure 3.  Blocks in the partition $\lambda$.

The $III$ type block consist of pairwise rows of $\lambda^{{}^{\prime}}$ and
$\lambda^{{}^{\prime\prime}}$ as shown in Fig.(3). The upper boundary and the
lower boundary of the block consist of a row of $\lambda^{{}^{\prime}}$ and
one pairwise rows of $\lambda^{{}^{\prime\prime}}$. When the pairwise rows of
$\lambda^{{}^{\prime}}$ are even, we define four operators $\mu_{e11}$,
$\mu_{e12}$, $\mu_{e21}$, and $\mu_{e22}$ according to the positions of
insertions of the rows $b$, $c$ of $\lambda^{{}^{\prime}}$, as shown in
Fig.(4). The first and last row of the block do not change under the map $\mu$
as well as the even pairwise rows of $\lambda^{{}^{\prime\prime}}$. We append
a box to the first row of odd pairwise rows of $\lambda^{{}^{\prime\prime}}$
and delete a box at the end of the second row. If $b$ is not the first row of
the block, it will behave as the first row of pairwise rows of
$\lambda^{{}^{\prime\prime}}$ as shown in Fig.(3). If $c$ is not the last row
of the block, it will behave as the second row of pairwise rows of
$\lambda^{{}^{\prime\prime}}$. When the pairwise rows of
$\lambda^{{}^{\prime}}$ inserted into $\lambda^{{}^{\prime\prime}}$ are odd,
we can define four operators $\mu_{o11}$, $\mu_{o12}$, $\mu_{o21}$, and
$\mu_{o22}$ similarly. However the odd rows and the even rows interchange
roles in this case. We can also consider $III$ type block consisting of one
pairwise rows $\lambda^{{}^{\prime\prime}}$ and pairwise rows of
$\lambda^{{}^{\prime}}$.

Figure 4. The images of $III$ type blocks under the map $\mu$.

The $II$ type block only consist of pairwise rows of $\lambda^{{}^{\prime}}$
or $\lambda^{{}^{\prime\prime}}$ as shown in Fig.(5) which do not change under
the map $\mu$.

Figure 5.  $II$ type block and its image under the map $\mu$.

The $I$ type block consist of pairwise rows of $\lambda^{{}^{\prime\prime}}$,
besides of the first row $a$ of $\lambda^{{}^{\prime}}$ and the first row of
$\lambda^{{}^{\prime\prime}}$. We define two operators $\mu_{e1}$and
$\mu_{e2}$ as shown in Fig.(6). The row $a$ is above the pairwise rows in
Fig.(6)(a) and is between the pairwise rows in Fig.(6)(b). The last row of the
block and the even pairwise rows of $\lambda^{{}^{\prime\prime}}$ do not
change under the map $\mu$. We append a box to the first row of odd pairwise
rows of $\lambda^{{}^{\prime\prime}}$ and delete a box at the end of the
second row of odd pairwise rows. If $a$ is not the last row of the block, it
will behave as the second row of pairwise rows of
$\lambda^{{}^{\prime\prime}}$ as shown in Fig.(6)(b). We can also consider $I$
type block consist of pairwise rows of $\lambda^{{}^{\prime}}$ besides of the
first rows of $\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$. And
we define two operators $\mu_{o1}$ and $\mu_{o2}$ similarly. However the odd
rows and the even rows interchange roles in this case. There is no $I$ type
block in the $C_{n}$ theory. So the fingerprint of the $C_{n}$ rigid
semisimple operators is a simplified version of that in the $B_{n}$ and
$D_{n}$ theories.

Figure 6.  $I$ type block and its image under the map $\mu$.

##  3\. Examples of maps preserving symbol

In this section, we introduce several maps preserving symbol as examples and
prove they preserve the fingerprint. The parities of the length of the rows
will not change under the first class of maps and the parities do change under
the second class of the maps.

###  3.1. Maps $See1$ and $See2$

First, we introduce a map swapping one row of $\lambda^{{}^{\prime}}$ with one
row of $\lambda^{{}^{\prime\prime}}$. To preserve the symbol invariant, the
second row of even pairwise rows of $\lambda^{{}^{\prime}}$ swap with the
second row of even pairwise rows of $\lambda^{{}^{\prime\prime}}$ as shown in
in Fig.(7).

Figure 7.  The map
$See1:(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})\rightarrow(\lambda^{T^{\prime}},\lambda^{T^{\prime\prime}})$.

Or the first row of even pairwise rows in $\lambda^{{}^{\prime}}$ swap with
the first row of even pairwise rows in $\lambda^{{}^{\prime\prime}}$ as shown
in in Fig.(8).

Figure 8.  The map
$See2:(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})\rightarrow(\lambda^{T^{\prime}},\lambda^{T^{\prime\prime}})$.

For the maps $See1,See2$, we have the following proposition.

######  Proposition 3.1.

The maps $See1,See2$ preserve the fingerprint invariant

###### Proof.

First, we prove the proposition by computing the fingerprints of the two sides
of the maps. The partition
$\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$ is shown in
Fig.(9)

Figure 9.  Partition
$\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$.

We can calculate the fingerprint as follows

| $\lambda_{2n_{2}+1}$ | $\cdots$ | $\lambda_{2m_{2}}$
---|---|---|---
$\lambda_{i}$ | $2(l+k)+5$ | $2(l+k)+5$ | $2(l+k)+5$
$\mu_{i}$ | $2(l+k)+5$ | $2(l+k)+5$ | $2(l+k)+5$
$(i)$ |  |  |
$(ii)$ |  |  |
$(iii)$ |  |  |
$\tau(m)$ | 1 | 1 | 1

The fingerprint of the parts $\lambda_{2n_{2}+1}\cdots\lambda_{2m_{2}}$ is

(3.4) $[{(2(l+k)+5)}^{(m_{2}-n_{2})},\emptyset].$

The partition $\lambda^{T}=\lambda^{T^{\prime}}+\lambda^{T^{\prime\prime}}$ of
the right of the maps is the same with the partition $\lambda$ as shown in
Fig.(9). We can calculate the fingerprint as follows

| $\lambda^{T}_{2n_{2}+1}$ | $\cdots$ | $\lambda^{T}_{2m_{2}}$
---|---|---|---
$\lambda^{T}_{i}$ | $2(l+k)+5$ | $2(l+k)+5$ | $2(l+k)+5$
$\mu_{i}$ | $2(l+k)+5$ | $2(l+k)+5$ | $2(l+k)+5$
$(i)$ |  |  |
$(ii)$ |  |  |
$(iii)$ |  |  |
$\tau(m)$ | 1 | 1 | 1

And the fingerprint of the parts $\lambda_{2n_{2}+1}\cdots\lambda_{2m_{2}}$ is

$[{(2(l+k)+5)}^{(m_{2}-n_{2})},\emptyset]$

which is equal to (3.4). ∎

Figure 10. The maps $Se2$ and $Ie2$ would not change the parities of the rows.

The maps in Fig.(10) also preserve symbol and fingerprint invariants. In fact,
we have the following general result.

######  Proposition 3.2.

The maps preserve fingerprint, which preserve symboland do not change parities
of rows.

###### Proof.

The partition $\lambda$ will not change under the map in the position, through
the partitions $\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$
change. The condition $(i)$ and $(ii)$ in the definition of the fingerprint do
not change under the map. So we only need to consider the condition $(iii)$.
If $\lambda_{i}=\lambda^{{}^{\prime}}_{i}+\lambda^{{}^{\prime\prime}}_{i}$ is
even and both $\lambda^{{}^{\prime}}_{i}$ and
$\lambda^{{}^{\prime\prime}}_{i}$ are odd, we have
$\lambda_{i}=\lambda^{{}^{\prime}}_{i}+\lambda^{{}^{\prime\prime}}_{i}$ on the
left hand side of the maps. Then we have
$\lambda_{i}=(\lambda^{{}^{\prime}}_{i}-2)+(\lambda^{{}^{\prime\prime}}_{i}+2)$
on the right hand side of the maps. So the condition $(iii)$ do not change on
the two sides of the maps. ∎

###  3.2. Map $Seo2$

Next, we consider a map swapping a row of $\lambda^{{}^{\prime}}$ with a row
of different parity of $\lambda^{{}^{\prime\prime}}$. To preserve the symbol
invariant, the first row of even pairwise rows of $\lambda^{{}^{\prime}}$
increasing one box swap with the second row of odd pairwise rows in
$\lambda^{{}^{\prime\prime}}$ decreasing one box as shown in Fig.(11).

Figure 11.  Map $Seo2$.

Or the second row of even pairwise rows in $\lambda^{{}^{\prime}}$ increasing
one box swap with the first row of odd pairwise rows of
$\lambda^{{}^{\prime}}$ decreasing one box.

######  Proposition 3.3.

The map $Seo2$ preserve the fingerprint invariant

###### Proof.

We prove the proposition by computing the fingerprints of the two sides of the
map. Without lose of generality, we assume the map $Seo2$ acting on a pair of
partitions as shown in Fig.(12).

Figure 12.  The pair of partition:
$(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})$.

And then $\lambda=\lambda^{{}^{\prime}}+\lambda^{{}^{\prime\prime}}$ is

We can calculate the fingerprint as follows.

$\lambda_{i}$ | $\lambda_{2n_{2}+1}$ | $\cdots$ | $\lambda_{2m_{1}}$ | $\lambda_{2m_{1}+1}$
---|---|---|---|---
| 5 | 4 | 4 | 3
$\mu_{i}$ | 4 | 4 | 4 | 4
$(i)$ | y |  |  | y
$(ii)$ | y | y | y |
$(iii)$ |  |  |  |
$\tau(m)$ | -1 | -1 | -1 | -1

The fingerprint of the parts $\lambda_{2n_{2}}\cdots\lambda_{2m_{2}+1}$ is

(3.5) $[\emptyset,2^{2(m_{1}-n_{2})+1}].$

Figure 13.  The image of the map $I$:
$(\lambda^{I^{\prime}},\lambda^{I^{\prime\prime}})$.

Upon map $Seo2$, $(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})$ become
the partition pair $(\lambda^{I^{\prime}},\lambda^{I^{\prime\prime}})$ as
shown in Fig.(13). And then
$\lambda^{I}=\lambda^{I^{\prime}}+\lambda^{I^{\prime\prime}}$ is

We can calculate the fingerprint as follows

$\lambda_{i}$ | $\lambda_{2n_{2}+1}$ | $\cdots$ | $\lambda_{2m_{1}}$ | $\lambda_{2m_{1}+1}$
---|---|---|---|---
| 4 | 4 | 4 | 4
$\mu_{i}$ | 4 | 4 | 4 | 4
$(i)$ |  |  |  |
$(ii)$ |  |  |  |
$(iii)$ | y | y | y | y
$\tau(m)$ | -1 | -1 | -1 | -1

and the fingerprint of the parts $\lambda_{2n_{2}}\cdots\lambda_{2m_{2}+1}$ is

$[\emptyset,2^{2(m_{1}-n_{2})+1}]$

which is equal to (3.5). ∎

Remarks:We have the similar maps for the odd pairwise rows.

##  4\. Classification of maps preserving symbol

In this section, we give a classification of the maps preserving symbol and
prove they preserve the fingerprint. We find two kinds of fundamental maps:
the $S$ type and $D$ type. In the first subsection, we prove two theorems
which are the bases of the classification of the maps.

###  4.1. Maps $TE$ and $TO$

With two special partitions $\lambda^{{}^{\prime}}$ and
$\lambda^{{}^{\prime\prime}}$, we introduce two special symbol preserving maps
$TE$ and $TO$ as shown in Fig.(14) and Fig.(15).

The row $a$, the first row of even pairwise rows, is inserted into partition
the $\lambda^{{}^{\prime\prime}}$ as the first row of even pairwise rows in
Fig.(14). To preserve symbol, the parities of the lengths and heights of the
rows above $a$ of the partition $\lambda^{{}^{\prime}}$ would change under the
map $TE$, as well as that of the rows above $a$ of the partition
$\lambda^{{}^{\prime\prime}}$.

Figure 14. The row $a$ is inserted into $\lambda^{{}^{\prime\prime}}$.

The row $a$, the first row of even pairwise rows, is inserted into partition
the $\lambda^{{}^{\prime\prime}}$ as the second row of odd pairwise in
Fig.(15). To preserve symbol, parities of the lengths and heights of the rows
above $a$ of $\lambda^{{}^{\prime}}$ would change under the map $TO$. The rows
above $a$ of the partition $\lambda^{{}^{\prime\prime}}$ become even.

Figure 15. The row $a$ is inserted into $\lambda^{{}^{\prime\prime}}$.

The partitions $\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$
suffer severe constrain to preserve the rigid conditions of the partitions
under the maps $TE$ and $TO$. The number of rows of $\lambda^{{}^{\prime}}$ is
even. The row $a$ and the rows above it are all even. The number of rows of
$\lambda^{{}^{\prime\prime}}$ is odd. The last parts of rows are even as shown
in Fig.(15). Similarly, we can insert an odd row of $\lambda^{{}^{\prime}}$
into $\lambda^{{}^{\prime\prime}}$, leading to another two cases. We will omit
these two kinds maps in the following sections.

The rows changed and the partition changed are also denoted as $a$, $b$ and
$\lambda^{{}^{\prime}}$, $\lambda^{{}^{\prime\prime}}$ in the maps $TE$ and
$TO$ to save notations.

###  4.2. Stopping phenomenon and Splitting theorem

The following Lemma help us to search fundamental building blocks of the
symbol preserving maps.

######  Lemma 4.1 (Stopping phenomenon).

To preserve symbol, insertion of one row of $\lambda^{{}^{\prime}}$ into
$\lambda^{{}^{\prime\prime}}$ leads to the changes of the parities of lengths
and heights of all the rows above the insertion position. Insertion of another
shorter row would stop the changes of the parities of lengths and heights of
all rows above the new insertion position.

Figure 16. Insertion of the row $b$ into $\lambda^{{}^{\prime\prime}}$.

###### Proof.

First, we consider the case that the rows $a$ and $b$ are of pairwise rows. We
insert the row $b$ of $\lambda^{{}^{\prime}}$ into
$\lambda^{{}^{\prime\prime}}$ as shown in Fig.(16)(a). To preserve symbol, the
parities of lengths and heights of the rows in the region $I$ change similar
to the maps $TE$ and $TO$(We omit the condition of the orthogonal partition
temporary), as well as the rows above the row $a$ of $\lambda^{{}^{\prime}}$.
Next, we insert the row $a$ into $\lambda^{{}^{\prime\prime}}$ as shown in
Fig.(17). The parities of lengths and heights of the rows in the region $III$
change again, recovering the original parities of lengths and heights as in
Fig.(16)(a), as well as that of the rows above $a$ of $\lambda^{{}^{\prime}}$.

For the case that rows $a$ and $b$ are not of pairwise rows, we can prove the
proposition with minor modification.

Figure 17. Insertion of the row $a$ into $\lambda^{{}^{\prime\prime}}$.

∎

We can insert several rows of $\lambda^{{}^{\prime}}$ into
$\lambda^{{}^{\prime\prime}}$ at the same time. However, we can prove that
this map can be decomposed into more fundamental maps.

######  Proposition 4.1.

The insertion of three rows of $\lambda^{{}^{\prime}}$ into
$\lambda^{{}^{\prime\prime}}$ can be decomposed into two independent fundament
maps.

###### Proof.

Figure 18. Insertion of the rows $a$, $b$, and $c$ into
$\lambda^{{}^{\prime\prime}}$.

For a rigid semisimple operator as shown in Fig.(18)(a), we insert the three
rows $a$, $b$, and $c$ into $\lambda^{{}^{\prime\prime}}$. According to Lemma
(4.1), the parities of the lengths of all the rows in the region $I$ would not
change under the map $O$. To preserve symbol, the parities of lengths and
heights of all the rows in the region $I$ would change according to the maps
$TE$ and $TO$ as shown in Fig.(14) and Fig.(15). Then the map $O$ can be
decomposed into two maps $O1$ and $O2$ as shown in Fig.(19) and Fig.(19).

Figure 19. Insertion of the rows $a$ and $b$ into
$\lambda^{{}^{\prime\prime}}$.  Figure 20. Insertion of the rows $c$ into
$\lambda^{{}^{\prime\prime}}$.

∎

######  Remark 4.1.

If even number of rows are inserted into the partition
$\lambda^{{}^{\prime\prime}}$, the operation can be decomposed into several
operations $O1$. If odd number of rows are inserted into the partition
$\lambda^{{}^{\prime\prime}}$, the operation can be decomposed into several
operations $O1$ and one operation $O2$.

Next we consider the case that two rows of $\lambda^{{}^{\prime}}$ are
inserted into $\lambda^{{}^{\prime\prime}}$ and two rows in
$\lambda^{{}^{\prime\prime}}$ are inserted into $\lambda^{{}^{\prime}}$ at the
same time. The following theorem imply that this map can be decomposed into
two independent fundamental ones.

Figure 21. The partitions $\lambda^{{}^{\prime}}$ and
$\lambda^{{}^{\prime\prime}}$ with lengths $L(b)>L(d)>L(a)>L(c)$.

######  Proposition 4.2 (Splitting theorem).

$a$ and $b$ are rows of $\lambda^{{}^{\prime}}$ and $c$ and $d$ are rows of
$\lambda^{{}^{\prime\prime}}$, with lengths $L(b)>L(d)>L(a)>L(c)$ as shown in
Fig.(21). The insertion of the rows $a$ and $b$ into
$\lambda^{{}^{\prime\prime}}$ and the rows $c$ and $d$ into
$\lambda^{{}^{\prime}}$ at the same time can be decomposed into two maps: one
is the insertion of the rows $a$ into $\lambda^{{}^{\prime\prime}}$ and the
rows $c$ into $\lambda^{{}^{\prime}}$ at same time, the another one is the
insertion of the row $b$ into $\lambda^{{}^{\prime\prime}}$ and the row $d$
into $\lambda^{{}^{\prime}}$ at same time.

Figure 22. Insertions of the rows $a$ and $b$ into
$\lambda^{{}^{\prime\prime}}$ and insertion of the rows $c$ and $d$ into
$\lambda^{{}^{\prime}}$ at the same time.

###### Proof.

The insertions of the rows $a$ and $b$ into $\lambda^{{}^{\prime\prime}}$ and
the rows $c$ and $d$ into $\lambda^{{}^{\prime}}$ at the same time are shown
in Fig.(22). The parities of lengths and heights of the rows in the region $I$
and $II$ do not change under this map according to Lemma 4.1. So we can insert
the row $a$ into $\lambda^{{}^{\prime\prime}}$ and insert the row $c$ into
$\lambda^{{}^{\prime}}$ at the same time firstly as shown in Fig.(23), and
then insert the row $b$ into $\lambda^{{}^{\prime\prime}}$ and insert the row
$d$ into $\lambda^{{}^{\prime}}$ at the same time. Or we can insert the row
$b$ into $\lambda^{{}^{\prime\prime}}$ and insert the row $d$ into
$\lambda^{{}^{\prime}}$ at the same time firstly as shown in Fig.(24), and
then insert the row $a$ into $\lambda^{{}^{\prime\prime}}$ and insert the row
$c$ into $\lambda^{{}^{\prime}}$ at the same time.

Figure 23. Insertion of the row $a$ into $\lambda^{{}^{\prime\prime}}$ and
insertion of the row $c$ into $\lambda^{{}^{\prime}}$ at the same time. Figure
24. Insertion of the row $b$ into $\lambda^{{}^{\prime\prime}}$ and insertion
of the row $d$ into $\lambda^{{}^{\prime}}$ at the same time.

∎

######  Remark 4.2.

The rows $a$ and $b$ need not to be pairwise rows, as well as the rows $c$ and
$d$.

When the blocks $I$ and $II$ do not exist, we prove Proposition 4.1 in
Appendix B. Combining Lemma 4.1 and Proposition 4.2, we can prove the
following theorem, which is the base of the classification of the maps
preserving symbol.

######  Theorem 4.1.

The maps preserving symbol consist of movements of two rows at once111We omit
the extreme maps $TE$ and $TO$.

According to Lemma 4.1 and Proposition 4.2, there are two types of maps
preserving symbol consisting of movement of two rows at a time. The fist one
denoted as the $D$ type are the maps that the two moving rows are in one
partition. The second one denoted as the $S$ type are the maps that the two
moving rows are in different partitions. We will introduce these two type of
maps in the following sections.

###  4.3. $S$-type maps

Figure 25. Taking one row from $\lambda^{{}^{\prime}}$ to
$\lambda^{{}^{\prime\prime}}$ and taking one row from
$\lambda^{{}^{\prime\prime}}$ to $\lambda^{{}^{\prime}}$.

In this subsection, we introduce the $S$ type symbol preserving maps which
take one row from $\lambda^{{}^{\prime}}$ to $\lambda^{{}^{\prime\prime}}$ and
take one row from $\lambda^{{}^{\prime\prime}}$ to $\lambda^{{}^{\prime}}$.
The most general model is shown in Fig.(25). We can put the first or the
second row of the pairwise rows in the region $l1$ to the positions $1$ or $2$
in the region $l2$. We can put the first or the second row of the pairwise
rows in the region $l3$ to the positions $1$ or $2$ in the region $l4$.

We can prove that the rows in the regions $I$ and $II$ must have the opposite
parities. We illustrate it through an example as shown in Fig.(26). Assuming
the rows in the regions $I$ and $II$ are all odd, we put the first row of
pairwise rows in the region $l1$ under the pairwise rows in the region $l2$
and put one row in the region $l3$ to the positions ’1’ or ’2’ in the region
$l4$. The parity of the first row of pairwise rows in the region $l1$ do not
change under the operation. However, this operation changes the parities of
the rows above the first row of the pairwise rows in the region $l1$. To
preserve symbol, the pairwise rows in the region $l1$ must be odd. This
operation also changes the parities of the rows above the region $l2$. To
preserve symbol, the pairwise rows of the region $l2$ must be odd. And the
first row of the pairwise rows in the region $l1$ is even after the operation,
which is a contradiction. For the other situations, it is can be proved
similarly.

Figure 26.  A counter example.

Since the parities of the rows in the regions $I$ and $II$ are $(odd,even)$ or
$(even,odd,)$, the number of the $S$ type maps is $2^{5}$. One example of the
$S$ type maps $So1121$ is shown in Fig.(27). We put the first row $b1$ of the
pairwise rows in the region $l1$ under the pairwise rows of the region $l2$ as
shown in Fig.(27). And we put the second row $c6$ of the pairwise rows in the
region $l3$ between the pairwise rows of the region $l4$. The superscript $o$
means that we choice the parities $(odd,even)$ for the rows in the regions $I$
and $II$. And the number following ${}^{\prime}o^{\prime}$ means the positions
in regions $l1$, $l2$, $l3$, and $l4$ that the row chosen and inserted.

Figure 27. The map $So1121$.

Next, we prove that this type map preserve fingerprint invariant.

######  Theorem 4.2.

The $S$-type maps preserve the fingerprint invariant.

###### Proof.

We prove the proposition for the map $So1121$ as shown in Fig.(27) above as an
example. The rigid semisimple operator
$(\lambda^{{}^{\prime},},\lambda^{{}^{\prime\prime}})_{T}$ is the image of the
rigid semisimple operator
$(\lambda^{{}^{\prime},},\lambda^{{}^{\prime\prime}})$ under the map $So1121$.
The rows of the partition $\lambda^{{}^{\prime}}_{T}$ are all odd while the
rows of the partition $\lambda^{{}^{\prime\prime}}$ are all even. To preserve
symbol after insertion, the parities of the rows of
$\lambda^{{}^{\prime\prime}}_{T}$ above the $b1$ are odd and the parities of
the rows of $\lambda^{{}^{\prime}}$ above the $b4$ are even. While the parity
of the row $b4$ do not change.

The partitions $\lambda$, $\lambda_{T}$ and their images are shown in
Fig.(28). This is one of the possibilities for the additions of the two
partition. The black boxes are deleted and the gray boxed are appended in the
partition $\lambda_{T}$. The images of the partitions $\lambda$ and
$\lambda_{T}$ are denoted as $\mu$ and $\mu_{T}$ under the map $\mu$.

Figure 28. The images of the partitions $\lambda$ and $\lambda_{T}$ under the
map $\mu$.

Now we prove the fingerprint invariant on the two sides of the map $So1121$
are the same. We decompose the images $\mu$ and $\mu_{T}$ into several blocks
using the same strategy to compute fingerprint in Section 2.3. The first three
ones are shown in Figs.(29), (30), and (31). For the block $I$ of $\lambda$ as
shown in Figs.(29), the rows $c6$, $c5$ belong to the partition
$\lambda^{{}^{\prime\prime}}$ are even. And the rows belong to
$\lambda^{{}^{\prime}}$ are odd. The partition $\mu$ of this block is given by
the operator $\mu_{e11}$ in the Fig.(4). According to the definition of
$\mu_{e11}$, all the rows belonging to $\lambda^{{}^{\prime}}$ become even,
which means the even parts of $\lambda$ satisfy the condition
$(i)\quad\,\,\mu_{i}\neq\lambda_{i}$. For the even parts
$\lambda^{{}^{\prime}}$, the parts $\lambda^{{}^{\prime}}_{i}$ are even. For
the block $I$ of $\lambda_{T}$ as shown in Figs.(29), all the rows including
the row $c6$ are of the partition $\lambda^{{}^{\prime}}$. The partition $\mu$
of this block is given by the operator $\mu_{e}$. The block $I$ of $\lambda$
has two rows more than the block $I$ of $\lambda_{T}$. All the rows excepting
the row $b4$ change their parities under the map $So1121$. In fact the two
blocks have the same $\mu$ partition for the same row as shown in Figs.(29).
However, the even parts of the partition $\lambda_{T}$ satisfy the condition
$(iii)$ in the definition of the fingerprint but do not satisfy the condition
$(i)$. Up to this point, block $I$ of $\lambda$ and $\lambda_{T}$ lead to the
same fingerprint for the same row in the blocks.

Figure 29. The blocks $I$ and their images under the map $\mu$.

Next we prove that the fingerprint of block $II$ on the two sides of the map
$So1121$ are the same. The block $II$ of $\lambda_{T}$ has four rows more than
the block $I$ of $\lambda$. The row $c5$ and the row above it have not been
discussed in the block $I$ of $\lambda_{T}$. The rows $c4$, $c5$ belonging to
the partition $\lambda^{{}^{\prime\prime}}$ are odd. And the rows belonging to
$\lambda^{{}^{\prime}}$ become even under the map $So1121$. The partition
$\mu$ of this block is given by the operator $\mu_{o11}$. According to the
definition of $\mu_{o11}$, all the rows belonging to
$\lambda^{{}^{\prime\prime}}$ become odd, which means the even parts satisfy
the condition $(i)\quad\,\,\mu_{i}\neq\lambda_{i}$. For the even parts of
$\lambda$, the parts $\lambda^{{}^{\prime}}_{i}$ are even. For the block $II$
of $\lambda$ as shown in Figs.(29), all the rows are of the partition
$\lambda^{{}^{\prime}}$ and are odd. The partition $\mu$ of this block is
given by the operator $\mu_{e}$. In fact the two blocks have the same $\mu$
partition for the same row as shown in Figs.(30). However, the even parts of
the partition $\lambda$ satisfy the condition $(iii)$ in the definition of the
fingerprint but do not satisfy the condition $(i)$. Up to this point, block
$II$ of $\lambda$ and $\lambda_{T}$ lead to the same fingerprint for the same
row in the blocks.

Figure 30. The blocks $II$ and their images under the map $\mu$.

Now we prove that the fingerprint of block $III$ on the two sides of the map
$So1121$ are the same. For the block $III$ of $\lambda$ as shown in Figs.(31),
the rows $c3$, $c4$ belonging to the partition $\lambda^{{}^{\prime\prime}}$
are even and the rows belonging to $\lambda^{{}^{\prime}}$ are odd. The
partition $\mu$ of this block is given by the operator $\mu_{e21}$ in the
Fig.(4). According to the definition of $\mu_{e21}$, all the rows belonging to
$\lambda^{{}^{\prime}}$ become even, which means the even parts satisfy the
condition $(i)\quad\,\,\mu_{i}\neq\lambda_{i}$. For the even parts $\lambda$,
the parts $\lambda^{{}^{\prime}}_{i}$ are even, which means they do not
satisfy the condition $(iii)$. For the block $III$ of $\lambda_{T}$ as shown
in Figs.(31), all the rows are of the partition $\lambda^{{}^{\prime}}$ and
the partition $\mu$ of this block is given by the operator $\mu_{e}$. The
block $III$ of $\lambda$ has four rows more than the block $III$ of
$\lambda_{T}$. All the rows change their parities under the map $So1121$. In
fact the two blocks have the same $\mu$ partition for the same row as shown in
Figs.(29). However, the even parts of the partition $\lambda_{T}$ satisfy the
condition $(iii)$ in the definition of the fingerprint but do not satisfy the
condition $(i)$. Up to this point, block $III$ of $\lambda$ and $\lambda_{T}$
lead to the same fingerprint for the same row in the blocks.

Figure 31. The blocks $III$ and their images under the map $\mu$.

For the blocks following, we can also prove that the fingerprint on the two
sides of the map $So1121$ are the same. We can prove the other $S$-type maps
preserve the fingerprint invariant similarly. ∎

###  4.4. $D$-type maps

Figure 32.  Example of the $D$ type map.

In this subsection, we introduce the $D$ type symbol preserving maps which
take two rows from $\lambda^{{}^{\prime}}$ to $\lambda^{{}^{\prime\prime}}$.
The simplest example of the $D$ type map is shown in Fig.(32). A pair of even
rows can be obtained from five different blocks $\lambda^{{}^{\prime}}$
indexed with number under the partitions as shown in Fig.(33). This pair of
rows can be inserted into block of $\lambda^{{}^{\prime\prime}}$, leading to
five different blocks as shown in Fig.(34). Considering the odd version of
these processes, the number of the $D$ type maps is 5*5*2.

Figure 33. Two rows of the block comprise even pairwise rows.  Figure 34. Even
pairwise rows are inserted into a block.

The above symbol preserving maps preserve fingerprint.

######  Theorem 4.3.

The $D$-type maps preserve the fingerprint invariant.

###### Proof.

Figure 35. Example of the $D$ type map.

We prove the proposition for the map $De24$ as an example as shown in
Fig.(35). The row $b$ is odd and is the second row of pairwise rows of
$\lambda^{{}^{\prime}}$. We insert it into $\lambda^{{}^{\prime\prime}}$ above
the row $A$ as the second row of pairwise rows. The row $c$ is odd and is the
first row of pairwise rows of $\lambda^{{}^{\prime}}$. We insert it into
$\lambda^{{}^{\prime\prime}}$ above the row $D$ as the second row of pairwise
rows. The rows between $b$ and $c$ are all even and become odd under the map
$De24$. The rows of the partition $\lambda^{{}^{\prime\prime}}$ are all odd
and the rows above $A$ become even under the map $De24$.

The rigid semisimple operator
$(\lambda^{{}^{\prime},},\lambda^{{}^{\prime\prime}})_{T}$ is the image of the
rigid semisimple operator
$(\lambda^{{}^{\prime},},\lambda^{{}^{\prime\prime}})$ under the map $De24$.
The partitions $\lambda$, $\lambda_{T}$ and their images under the map $\mu$
are shown in Fig.(36). This is one of the possibilities for the additions of
the two partition. The black boxes are deleted and the gray boxes are appended
in the partition $\lambda_{T}$. Under the map $\mu$, we denote the images of
the partitions $\lambda$ and $\lambda_{T}$ as $\mu$ and $\mu_{T}$.

Figure 36. The images of the partitions $\lambda$ and $\lambda_{T}$ under the
map $\mu$.

Next, we prove that the partitions $\mu$ and $\mu_{T}$ lead to the same
fingerprint. We decompose the images $\mu$ and $\mu_{T}$ into several blocks
using the same strategy to compute fingerprint in Section 2.3. The first three
ones are shown in Figs.(37), (38), and (39 ). For the block $I$ of $\lambda$
as shown in Fig.(37), all the rows including the row $c$ are of the partition
$\lambda^{{}^{\prime}}$. The partition $\mu$ of this block is given by the
operator $\mu_{e}$. The even parts of the partition $\lambda$ satisfy the
condition $(iii)$ in the definition of the fingerprint but do not satisfy the
condition $(i)$. The block $I$ of $\lambda_{T}$ has two rows more than the
block $I$ of $\lambda$. All the rows change their parities under the map
$De24$. In fact the two blocks have the same $\mu$ partition for the same row
as shown in Figs.(36). However, the even parts of the partition $\lambda_{T}$
satisfy the condition $(i)$ in the definition of the fingerprint but do not
satisfy the condition $(iii)$. Up to this point, block $I$ of $\lambda$ and
$\lambda_{T}$ lead to the same fingerprint for the same row in the blocks.

Figure 37. The blocks $I$ and their images under the map $\mu$.

For the block $II$ of $\lambda$ as shown in Figs.(30), the rows $D$, $C$
belong to the partition $\lambda^{{}^{\prime\prime}}$ are odd. And the rows
belonging to $\lambda^{{}^{\prime}}$ are even. The partition $\mu$ of this
block is given by the operator $\mu_{o22}$. According to the definition of
$\mu_{o22}$, all the rows belonging to $\lambda^{{}^{\prime}}$ become even,
which means the even parts satisfy the condition
$(i)\quad\,\,\mu_{i}\neq\lambda_{i}$. For the even parts $\lambda$, the parts
$\lambda^{{}^{\prime}}_{i}$ are even. For the block $II$ of $\lambda_{T}$ as
shown in Figs.(38), all the rows are of the partition
$\lambda^{{}^{\prime\prime}}$ and the partition $\mu$ of this block is given
by the operator $\mu_{e}$. The block $II$ of $\lambda$ has four rows more than
the block $II$ of $\lambda_{T}$. In fact the two blocks have the same $\mu$
partition for the same row as shown in Figs.(36). However, the even parts of
the partition $\lambda_{T}$ satisfy the condition $(iii)$ in the definition of
the fingerprint but do not satisfy the condition $(i)$. Up to this point,
block $I$ of $\lambda$ and $\lambda_{T}$ lead to the same fingerprint for the
same row in the blocks.

Figure 38. The blocks $II$ and their images under the map $\mu$.

For the block $III$ of $\lambda$ as shown in Figs.(39), all the rows are of
the partition $\lambda^{{}^{\prime}}$. The partition $\mu$ of this block is
given by the operator $\mu_{e}$. The even parts of the partition $\lambda$
satisfy the condition $(iii)$ in the definition of the fingerprint but do not
satisfy the condition $(i)$. The block $III$ of $\lambda_{T}$ has four rows
more than the block $III$ of $\lambda$. All the rows change their parities
under the map $De24$. In fact the two blocks have the same $\mu$ partition for
the same row as shown in Figs.(36). However, the even parts of the partition
$\lambda_{T}$ satisfy the condition $(i)$ in the definition of the fingerprint
but do not satisfy the condition $(iii)$. Up to this point, block $III$ of
$\lambda$ and $\lambda_{T}$ lead to the same fingerprint for the same row in
the blocks.

Figure 39. The blocks $III$ and their images under the map $\mu$.

For the blocks following, we can prove that the fingerprint of blocks on the
two sides of the map $So1121$ are the same. We can prove the other $S$-type
maps preserve the fingerprint invariant similarly. ∎

###  4.5. Special blocks

Figure 40.  Blocks of $\lambda$.

We ignore one situation in the above discussions. If the first row of a block
belong to the partition $\lambda^{{}^{\prime}}$ and the second one belong to
the partition $\lambda^{{}^{\prime\prime}}$, then the row under the block
would belong to another block as shown in Fig.(40). However, if the first two
rows of a block belong to the same the partition $\lambda^{{}^{\prime}}$, the
situation become complicated as shown in Fig.(41). The fingerprint of the
$III$ type block can be still calculated by previous formulas except the first
row. We construct a $S$ type block including one row of
$\lambda^{{}^{\prime}}$ and even number of rows of
$\lambda^{{}^{\prime\prime}}$ as little as possible. In Append B, we will
prove that the special blocks on the two sides of the symbol preserving maps
lead to the same fingerprint through an example.

Figure 41.  $S$ type block.

###  4.6. $I$-type maps

Figure 42. The map $I$ interchange block $\lambda^{{}^{\prime}}$ with block
$\lambda^{{}^{\prime\prime}}$.

In this subsection, we introduce $I$-type symbol preserving maps which
interchange a block of $\lambda^{{}^{\prime}}$ with a block of
$\lambda^{{}^{\prime\prime}}$. We omit other parts and also denote these two
blocks as $\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$ as shown
in Fig.(42). The simplest example is shown in Fig.(43).

Figure 43.  The simplest example of the $I$ type map.

This type of maps are not a fundamental symbol preserving maps and can be
decomposed into $S$ and $D$ types maps. For clear presentation, we put the
proof in the appendix.

##  5\. Classification of $\lambda$ with the same fingerprint

In this section, we discuss the possible strategies to prove the fingerprint
imply the symbol invariant, which is the other half of the conjecture.

###  5.1. Classification of $\lambda$

First, we have the following lemma.

######  Lemma 5.1.

Given the fingerprint and the partition $\lambda$, we would recover the rigid
semisimple operator.

To prove the fingerprint invariant imply the symbol invariant, according to
Lemma 5.1, we must classify the partition $\lambda$ with the same fingerprint
and prove that they have the same symbol invariant. The $\lambda$ with the
same fingerprint are different since they satisfy different conditions in the
definition of the fingerprint. They can be connected each other by the maps
adjusting these conditions. So we classify $\lambda$ by classify the maps
adjusting conditions in the definition of fingerprint. To prove the symbol
invariant is equal to the fingerprint invariant, we have to prove the
following conjecture.

######  Conjecture 5.1.

With the same fingerprint invariant and satisfying different conditions in the
definition of the fingerprint, the rigid semisimple operators have the same
symbol invariant.

##  6\. $B_{n}$ and $D_{n}$ theories

According to the preceding discussions, we have the following theorem.

######  Theorem 6.1.

The symbol invariant for rigid semisimple operator imply the fingerprint
invariant.

###### Proof.

We draw the conclusion using Theorems 4.1, 4.2, and 4.3. ∎

For the $C_{n}$ theory, we could prove the similar results using the same
strategy with minor modification.

##  Appendix A Special case of the splitting theorem

We discuss one special situation of the splitting theorem which is omitted in
the proof of Proposition 4.1.

######  Proposition A.1.

$a$ and $b$ are rows of $\lambda^{{}^{\prime}}$ and $c$ and $d$ are rows of
$\lambda^{{}^{\prime\prime}}$, with lengths $L(b)>L(d)>L(a)>L(c)$ as shown in
Fig.(21). The insertion of the rows $a$ and $b$ into
$\lambda^{{}^{\prime\prime}}$ and the rows $c$ and $d$ into
$\lambda^{{}^{\prime}}$ at the same time without the regions $I$ and $II$ as
shown in Fig.(22) can be decomposed into $D$ and $S$ type fundamental maps.

###### Proof.

We prove all the situations of the map can be decomposed into two type
fundamental maps we found. The pairwise rows containing $a$ of
$\lambda^{{}^{\prime}}$ are $a$ and $b$. And the pairwise rows containing $d$
of $\lambda^{{}^{\prime}}$ are denoted as $A$ and $B$. We only need to
consider the changes of these pairwise rows under the map. Without lose
generality, we assume that they are even. The map of the first case as shown
in Fig.(44) can be decomposed into two $S$ type maps.

Figure 44. $a$ and $b$ are pairwise rows of $\lambda^{{}^{\prime}}$. $A$ and
$B$ are pairwise rows of $\lambda^{{}^{\prime\prime}}$. This map can be
decomposed into two $S$ type maps.

The maps shown in Fig.(45) can be decomposed into two $D$ type maps.

Figure 45. $a$ and $b$ are pairwise rows of $\lambda^{{}^{\prime}}$. $A$ and
$B$ are pairwise rows of $\lambda^{{}^{\prime\prime}}$. This map can be
decomposed into two $D$ type maps.

The map shown in Fig.(46) can be decomposed into two $S$ type maps which is
the same as the maps in Fig.(44).

Figure 46. $a$ and $b$ are pairwise rows of $\lambda^{{}^{\prime}}$. $A$ and
$B$ are pairwise rows of $\lambda^{{}^{\prime\prime}}$. This map can be
decomposed into two $S$ type maps.

Next, we consider the situation that the two pairwise rows are of different
parities as shown in Fig.(47). Without lose generality, we assume the two
pairwise rows are even and odd, respectively. The map shown in Fig.(47) can be
decomposed into two $S$-type maps.

Figure 47. $a$ and $b$ are pairwise rows of $\lambda^{{}^{\prime}}$. $A$ and
$B$ are pairwise rows of $\lambda^{{}^{\prime\prime}}$. This map can be
decomposed into two $S$ type maps.

∎

##  Appendix B Special blocks

Figure 48. Special block

In this section, we prove that the special blocks lead to the same fingerprint
on the two sides of the symbol preserving maps.

######  Lemma B.1.

The special blocks under the symbol preserving map preserve the fingerprint
invariant.

###### Proof.

We will prove the proposition for the special block in Fig.(4)(a) under the
map $De24$ as example. Using the construction of the fingerprint in Section
2.3, the partitions $\lambda$ and $\lambda_{T}$ of the
$(\lambda^{{}^{\prime\prime}},\lambda^{{}^{\prime\prime}})$ and
$(\lambda^{{}^{\prime}},\lambda^{{}^{\prime\prime}})_{T}$ correspond to the
two sides of the map $D_{e24}$ are shown in Fig.(48). Next, we will prove that
the images of the partitions $\lambda$ and $\lambda_{T}$ under the map $\mu$
lead to the same fingerprint.

We have proved the rows of the block $I$ above the row $l2$ lead to the same
fingerprint on the two sides of the symbol preserving maps in Proposition 4.3.
Then we would like to prove the rows $l2$, $l3$, $l4$, $l5$, and $l6$ of the
block $S$ lead to the same fingerprint on the two sides of the map $De24$.

The red rows are in the partition $\lambda_{T}$ and odd. The while rows are in
the partition $\lambda$ and odd. First we determine the partition $\mu$ of the
partition $\lambda$ under the map $\mu$. If $l2$ is row of
$\lambda^{{}^{\prime}}$ which is even, all the rows above the row $l3$ are
pairwise rows or even. Thus the number of boxes above the row $l3$ is even. If
$l2$ is row of $\lambda^{{}^{\prime\prime}}$ which is odd, then the number of
boxes above the row $l3$ is odd which means the sign of the last box of the
row is ${}^{\prime}-^{\prime}$. The height of the row $l3$ is odd, thus we
should delete a box according to table in Lemma 2.1. Since red rows are even,
the sign of the last boxes of these rows are ${}^{\prime}-^{\prime}$. The sign
of the last box of the row $l6$ is also ${}^{\prime}-^{\prime}$ since the
height of this row is even. Next, we determine the partition $\mu_{T}$ of the
partition $\lambda_{T}$ under the map $\mu$. The while rows become odd and the
red rows become even under the map $De24$. The block $I$ and block $S$ of
$\lambda$ decompose into blocks $I$, $II$ and two rows $l5$, $l6$ in another
block. The block $II$ are all rows of $\lambda^{{}^{\prime\prime}}$ and do not
change under the map $\mu$. Since the row $l5$ is the last row of a block it
will not change under the map $\mu$ according to Fig.(4). Under the map
$De24$, the last box of the row $l6$ is deleted since the row $l6$ is odd and
its height is odd. The partitions $\lambda$, $\lambda_{T}$ and their images
are shown in Fig.(48). We denote the images of the partitions $\lambda$ and
$\lambda_{T}$ as $\mu$ and $\mu_{T}$. The black boxes are deleted and the gray
boxed are appended in the partition $\lambda_{T}$.

We can prove the proposition for the special block in Fig.(4)(b) under the map
$De24$ similarly. ∎

##  Appendix C $I$-type maps

The map $I$ interchange a block of $\lambda^{{}^{\prime}}$ with a block of
$\lambda^{{}^{\prime\prime}}$, preserving symbol. Omitting other parts, we
denote these two blocks as $\lambda^{{}^{\prime}}$ and
$\lambda^{{}^{\prime\prime}}$ again, as shown in Fig.(42). In this section, we
prove the following proposition.

######  Proposition C.1.

The $I$-type map can be decomposed into the $S$-type and $D$-type maps.

###### Proof.

Figure 49. The lengths of rows $A$ and $B$ is longer than the lengths of rows
$a$ and $b$.

$L(a)$ denote the length of a row $a$. If the lengths of pairwise rows $A$ and
$B$ of $\lambda^{{}^{\prime\prime}}$ are longer than the lengths of pairwise
rows $a$ and $b$ of $\lambda^{{}^{\prime}}$, we can insert the rows $A$ and
$B$ under the row $a$ as shown in Fig.(49). Conversely, we can insert the two
rows $a$ and $b$ under the row $A$. Similarly, we put all the pairwise rows of
$\lambda^{{}^{\prime\prime}}$ with rows longer than $a$ under the row $a$.
Then, we place the rows $a$ and $b$ under the block
$\lambda^{{}^{\prime\prime}}$ if length of $b$ is longer than the length of
$\lambda^{{}^{\prime\prime}}$ left. In this way, part of the block
$\lambda^{{}^{\prime}}$ and part of the block $\lambda^{{}^{\prime\prime}}$
have swaped places which are omitted in the proof following to keep the
presentation simple.

Figure 50. $L(b)<L(B)<L(A)<L(a)$.

Then the length of $b$ is shorter than the length of
$\lambda^{{}^{\prime\prime}}$. And there are several situations we should pay
attentions to. For the first case, the two pairwise rows have opposite
parities as shown in Fig.(50). If $L(b)<L(B)<L(A)<L(a)$, we can pace the rows
$A$ and $B$ between $a$ and $b$ by using the map of $D$ type. Then place the
rows $a$ and $b$ under $\lambda^{{}^{\prime\prime}}$ by using the map of $D$
type again. Finally, we have swap the two pairwise rows $a,b$ and $A,B$.
Similarly, for $L(B)<L(b)<L(a)<L(A)$, we can swap the two pairwise rows $a,b$
and $A,B$.

Figure 51. $L(b)<L(B)<L(a)<L(A)$.

For the second case, the two pairwise have opposite parities as shown in
Fig.(51). If $L(b)<L(B)<L(a)<L(A)$, we can interchange the row $a$ with $B$,
reducing to the map $It$.

Figure 52. $L(a)<L(d)<L(b)<L(c)$.

For the last situation, the rows $A$, $B$, $a$, and $b$ are of the same
parities as shown in Fig.(52), with $L(a)<L(d)<L(b)<L(c)$. However, we would
like to prove a more general result as follows.

######  Lemma C.1.

For the blocks as shown in Fig.(52) with $L(a)<L(e)$ and $L(b)<L(d)$, the
interchange of the block $\lambda^{{}^{\prime}}$ and the block
$\lambda^{{}^{\prime\prime}}$ can be decomposed into the $S$-type and $D$-type
maps.

###### Proof.

We propose the following algorithm to prove this lemma. Without lose
generality, we assume the rows $a$ and $b$ in Fig.(52) are even. We prove the
lemma for even rows $f$, $e$, $c$, and $d$ in Fig.(53), with $L(a)<L(f)$ and
$L(b)>L(c)$. We reduce other cases to this one.

First, we place all the odd pairwise rows of $\lambda^{{}^{\prime\prime}}$ to
$\lambda^{{}^{\prime}}$ by using the $D$-type maps as shown in Fig.(53). Then
the rows of $\lambda^{{}^{\prime}}$ and $\lambda^{{}^{\prime\prime}}$ are all
even. Next we place rows $a$ and $b$ to $\lambda^{{}^{\prime\prime}}$ as shown
in Fig.(54). Finally, we insert all the odd pairwise rows of
$\lambda^{{}^{\prime\prime}}$ into $\lambda^{{}^{\prime}}$ by using the
$D$-type maps as shown in Fig.(55). Thus the block $\lambda^{{}^{\prime}}$
swap with the block $\lambda^{{}^{\prime\prime}}$.

Figure 53. Insertions of all the odd pairwise rows of
$\lambda^{{}^{\prime\prime}}$ to $\lambda^{{}^{\prime}}$ by using the $D$-type
maps.  Figure 54. $a$ and $b$ are placed at the upper boundary and the lower
boundary of the block $\lambda^{{}^{\prime\prime}}$.  Figure 55. Insertions of
all the odd pairwise rows of $\lambda^{{}^{\prime\prime}}$ into
$\lambda^{{}^{\prime}}$ by using the $D$-type maps.

For the case $L(b)<L(c)$ or $L(a)>L(f)$, we can interchange the row $b$ with
$c$ or interchange the row $a$ with $f$ firstly, reducing to the cases as
shown in Fig.(56). We interchange the row $b$ with $c$ or interchange the row
$a$ with $f$ in the last step as shown in Fig.(57).

Figure 56. Interchange of the row $b$ and $c$.  Figure 57. Interchange of the
row $b$ and $c$.

For the case that the rows $d$ and $c$ are odd, the rows $f$ and $e$ are even,
and $L(b)>L(c)$, we can place the rows $c$ and $d$ between the row $a$ and
$b$, reducing to the case as shown in Fig.(53). For the case that the rows $d$
and $c$ are odd and $L(b)<L(c)$, we interchange the row $b$ with $c$ firstly
as shown in Fig.(58). Then we place the rows $c$ and $b$ under the row $d$ as
shown in Fig.(59). We can interchange the rows $a$ and $d$ as pairwise rows
with the other parts of the block $\lambda^{{}^{\prime\prime}}$. We
interchange the row $b$ with $c$ or interchange the row $b$ with $d$ in the
last step as shown in Fig.(60).

Figure 58. Interchange of the row $b$ and $c$.  Figure 59. The rows $c$ and
$b$ are placed under the row $d$ .  Figure 60. Interchange of the row $b$ and
$d$.

∎

We can deal with the case that the rows $f$ and $e$ are odd similarly. ∎

##  Appendix D Condition $(ii)$ of the definition of fingerprint

In Section 2.3, we point out that the conditions following $(i)$ imply $(ii)$
for the partition with $\lambda_{i}-\lambda_{i+1}\leq 1$.

$\displaystyle(i)\quad\quad\quad\quad\quad\,\,\mu_{i}\neq\lambda_{i}$ (D.6)
$\displaystyle(ii)\quad\quad\quad\sum^{i}_{k=1}\mu_{k}\neq\sum^{i}_{k=1}\lambda_{k}$
$\displaystyle(iii)_{SO}\quad\quad\quad\lambda^{{}^{\prime}}_{i}\quad\textrm{is
odd}.$

We would like to prove that it is not necessary to consider the condition
$(ii)$ above. For the $S$ type map, $\lambda_{i}-\lambda_{i+1}\geq 2$ will not
appear for the partition $\lambda=\lambda_{I}+\lambda_{II}$ since the parities
of rows in the region $I$ and $II$ are different. Assuming $a$ and $b$ belong
to different partitions under the $D$ type maps, we would check case by case
and find that $\lambda_{i}-\lambda_{i+1}\geq 2$ will not appear in the region
including the rows $a$, $b$ and the rows between them.

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