# Symbol Invariant of Partition and the Construction

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

The symbol is used to describe the Springer correspondence for the classical
groups. We propose equivalent definitions of symbols for rigid partitions in
the $B_{n}$, $C_{n}$, and $D_{n}$ theories uniformly. Analysing the new
definition of symbol in detail, we give rules to construct symbol of a
partition, which are easy to remember and to operate on. We introduce formal
operations of a partition, which reduce the difficulties in the proof of the
construction rules. According these rules, we give a closed formula of symbols
for different theories uniformly. As applications, previous results can be
illustrated more clearly by the construction rules of symbol.

Keywords. Partition, symbol, construction, surface operator, Young diagram

MSC (2010).05E10

We would like to thank Ming Huang and Xiaobo Zhuang for many helpful
discussions. This work was supported by a grant from the Postdoctoral
Foundation of Zhejiang Province.

###### Contents

  1. 1 Introduction
  2. 2 Symbol of rigid partitions in the $B_{n}$, $C_{n}$, and $D_{n}$ theories
    1. 2.1 Old definition of symbol
    2. 2.2 New definition of symbol
    3. 2.3 Formal operations of a partition
  3. 3 Construction of symbol
    1. 3.1 Three Lemmas
    2. 3.2 Symbol of the rigid partition in the $B_{n}$ theory
      1. 3.2.1 Closed formula of symbol
    3. 3.3 Symbol of rigid partitions in the $C_{n}$ theory
    4. 3.4 Symbol of rigid partitions in the $D_{n}$ theory
    5. 3.5 Closed formula of symbols in the $B_{n}$, $C_{n}$, and $D_{n}$ theories
  4. 4 Applications
    1. 4.1 Maps: $X_{S}$ and $Y_{S}$
    2. 4.2 $S$-duality maps for rigid surface operators

##  1\. Introduction

The Springer correspondence [1] is an injective map from the unipotent
conjugacy classes of a simple group to the set of unitary representations of
the Weyl group. For the classical groups this map can be described explicitly
in terms of certain symbols introduced in [1]. For the Weyl group in the
$B_{n}$, $C_{n}$, and $D_{n}$ theories, irreducible unitary representations
are in one-to one correspondence with ordered pairs of partitions
$[\alpha;\beta]$ where $\alpha$ is a partition of $n_{\alpha}$ and $\beta$ is
a partition of $n_{\beta}$ satisfying $n_{\alpha}+n_{\beta}=n$.

Gukov and Witten initiated a study of the $S$ duality of surface operators in
$\mathcal{N}=4$ super Yang-Mills theories in [2][3][4][14]. The $S$-duality
conjecture suggest that surface operators in the theory with gauge group $G$
should have a counterpart in the dual theory with gauge group $G^{L}$(the
Langlands dual group of $G$)[4][10]. The surface operator is defined by
prescribing a certain singularity structure of fields near the surface
supported by the operator.

A subclass of surface operators called ’rigid’ surface operators corresponding
to rigid partition are expected to be closed under $S$-duality. There are two
types rigid surface operators: unipotent and semisimple. They correspond to
certain ( unipotent and semisimple) conjugacy classes of the (complexified)
gauge group. The unipotent surface operators are classified by partitions. The
semisimple surface operators in the theories are labelled by pair of
partitions. Unipotent surface operators arise when one of these two partitions
is empty. The symbol is an invariant of partitions under the $S$ duality map.
Using the symbol invariant, Wyllard made some explicit proposals for how the
$S$-duality map should act on rigid surface operators in [5]. Other aspects of
the surface operators have been studied in [12][13][14][15].

In [6], relying on the characteristic of symbol, we found that a new subclass
of rigid surface operators in $B_{n}$ and $C_{n}$ theories are related by
$S$-duality map. To find more $S$-duality maps for rigid surface operators, we
need characterize the structure of symbol further. However, litter work has
been done on it. The calculation of symbol is a complicated and tedious work.
In [6], we simplify the construction of symbol by mapping a partition into two
partitions with only even rows. A simple and even explicit formula of symbol
need to be found.

In this paper, we attempt to extend the analysis in [5][6] to study the
construction of symbol further. Because of no noncentral rigid conjugacy
classes in $A_{n}$ series, we do not discuss surface operators in this case.
We also omit exceptional groups, which are more complicated. We will
concentrate on the $B_{n},C_{n}$, and $D_{n}$ series. Our main result is the
following table of computational rules of symbols. We determine the
contributions to symbol for each row of a partition in the $B_{n}$, $C_{n}$,
and $D_{n}$ theories uniformly.

Contribution to symbol of the $i$ th row
---
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 | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$

These rules are easy to remember and operator on, which is equivalent to the
following closed formula of symbol.

######  Proposition 1.

For a partition $\lambda=m^{n_{m}}{(m-1)}^{n_{m-1}}\cdots{1}^{n_{1}}$, we
introduce two notations

$\Delta^{T}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i+1}}{2}),\quad\quad
P^{T}_{i}=\frac{1+\pi_{i}}{2}$

where the superscript $T$ indicates it is related to the top row of the symbol
and

$\pi_{i}=(-1)^{\sum^{m}_{k=i}n_{k}}\cdot(-1)^{i+1+t},$

for $B_{n}(t=-1)$, $C_{n}(t=0)$, and $D_{n}(t=1)$ theories. Other parallel
notations

$\Delta^{B}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i}}{2}),\quad\quad
P^{B}_{i}=\frac{1-\pi_{i}}{2}$

where the superscript $B$ indicates it is related to the bottom row of the
symbol and $P^{B}_{i}$ is a projection operator similar to $P^{T}_{i}$. Then
the symbol $\sigma(\lambda)$ is

(1.1)
$\sigma(\lambda)=\sum^{m}_{i=1}\big{\\{}\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{P^{T}_{i}\Delta^{T}_{i}}\\\
\;\;\;\underbrace{0\cdots 0\;\;0\cdots
0}_{l+t}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{0\;\;0\cdots
0\;\;0\cdots 0}^{l}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1}_{P^{B}_{i}\Delta^{B}_{i}}\end{array}\Bigg{)}\big{\\}}$

with $l=(m+(1-(-1)^{m})/2)/2$.

The new Definitions 3 of symbols we proposed paly are critical role. The proof
of the computational rules is based on three Lemmas 3.1, 3.3, and 3.4. The
three lemmas are clearly proved by introducing formal operators of partition
Lemmas 2.10 and 2.12.

The following is an outline of this article. In Section 2, we propose an
equivalent definition of symbols for the $B_{n},C_{n}$, and $D_{n}$ theories
uniformly. Then we introduce formal operations of a partition. These formal
operations are helpful for understanding the new definitions of symbol. In
Section 3, we discuss the contribution to symbol of each row in a partition
formally. For an even row, it is natural to calculate the contribution. For an
odd row, the calculation is artificial. However, it is reasonable to compute
the contributions to symbol of a pair of odd rows together. The new definition
of symbol and the formal operations of a partition are crucial for computing
the contribution of odd rows. We also give rules to construct symbol which are
easy to memory and to operate on. Finally, according these construction rules
of symbol, we give a closed formula of symbols for the $B_{n},C_{n}$, and
$D_{n}$ theories uniformly. In Section 4, as applications, we prove several
propositions which are consistent with the results in [6]. The construction
rules of symbol are helpful in searching $S$-duality map of surface operators
in $\mathcal{N}=4$ super-Yang-Mills theories.

In fact, we give another completely new construction in [16], discussions
made. There is another invariant of partition called fingerprint related to
the Kazhdan-Lusztig map[7][8][9]. It is assumed that the fingerprint invariant
is equivalent to the symbol invariant. Hopefully our constructions will be
helpful in the proof of the equivalence. The mismatch of the total number of
rigid surface operators was found in the $B_{n}/C_{n}$ theories through the
study of the generating functions [5][17][18][19]. Clearly more work is
required.

##  2\. Symbol of rigid partitions in the $B_{n}$, $C_{n}$, and $D_{n}$
theories

In this section, firstly, we recall briefly the definitions of symbols in the
$B_{n}$, $C_{n}$, and $D_{n}$ theories described in [1]. The definitions of
symbols in different theories are not identical, which is an obstruction to
study the constructions of symbols of different theories uniformly. We find
equivalent definitions of symbols which are convenient to compute symbols for
different theories uniformly. We also introduce several formal operations
acting on the partition which are not only helpful to understand the new
definition of symbol but also helpful for the proofs of lemmas in the next
section.

###  2.1. Old definition of symbol

Unipotent conjugacy classes in the $B_{n}$, $C_{n}$, and $D_{n}$ theories are
classified by partitions. A partition $\lambda$ of the positive integer $n$ is
a decomposition $\sum_{i=1}^{l}\lambda_{i}=n$
($\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{l}$). The integer $l$ is
called the length of the partition. There is a one-to-one correspondence
between partition and Young tableaux. For instance the partition $3^{2}2^{3}1$
corresponds to

Young diagrams turn out to be extremely useful in the study of symmetric
functions and group representation theory. They are also useful tools for the
construction of the eigenstates of Hamiltonian System [20]. For two partitions
$\lambda$ and $\kappa$, $\lambda+\kappa$ is the partition with parts
$\lambda_{i}+\kappa_{i}$.

Unipotent conjugacy classes in the $B_{n}$($D_{n}$)theories are in one-to-one
correspondence with partitions of $2n{+}1$($2n$) where all even integers
appear an even number of times. Unipotent conjugacy classes in the $C_{n}$
theories are in one-to-one correspondence with partitions of $2n$ for which
all the odd integers appear an even number of times. A partition in the
$B_{n}$ and $D_{n}$($C_{n}$) theories is called rigid 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. We focus on rigid partitions in this paper.

A partition $\lambda$ is called special if its transpose partitions satisfy
following condition

$\displaystyle B_{n}:$ $\displaystyle\quad\lambda^{t}$
$\displaystyle\;\mbox{is orthogonal}\,,$ (2.1) $\displaystyle C_{n}:$
$\displaystyle\quad\lambda^{t}$ $\displaystyle\;\mbox{is symplectic}\,,$
$\displaystyle D_{n}:$ $\displaystyle\quad\lambda^{t}$
$\displaystyle\;\mbox{is symplectic}\,.$

It is easy to find that all rows of a rigid special partition are odd in the
$B_{n}$ theory. On the contrary, all rows in the Young tableau corresponding
to a rigid special partition are even in the $C_{n}$ and $D_{n}$ cases.

We introduce the construction of symbol introduced in [1] [5] as follows

######  Definition 1.

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

(2.2) $\alpha_{i}=f_{i}-i+1\quad,\quad\quad\beta_{i}=g_{i}-i+1.$

Finally the symbol is written as

(2.3)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\alpha_{3}&&\cdots\\\
&\beta_{1}&&\beta_{2}&&\cdots&\end{array}\right).$

Example: $B_{10}$, $\lambda=3^{3}\,2^{4}\,1^{4}$, the symbol is

(2.4)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&0&&1&&1&&1&&1\\\
&1&&1&&1&&1&&2&\end{array}\right).$

We can write the two rows of this symbol as two partitions $[1^{4},21^{4}]$
corresponding to a unitary representation of the Weyl group.

For a partition in the $C_{n}$ theory:  we append an extra $0$ as the last
part of the partition if the length of the partition is even. Next we
calculate terms $f_{i}$ and $g_{i}$ as in the $B_{n}$ case. Then we calculate
terms $\alpha_{i}=g_{i}-i+1$ and terms $\beta_{i}=f_{i}-i+1$.

Example: $C_{10}$, $\lambda=3^{2}\,2^{6}\,1^{2}$, the length of the partition
is even and the symbol is

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

Example: $C_{12}$, $\lambda=3^{2}\,2\,1^{4}$, the length of the partition is
odd and the symbol is

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&1&&1&&1&&1\\\
&&&&&0&&0&&2&\end{array}\right)$

For a partition in the $D_{n}$ theory: we calculate $f_{i}$ and $g_{i}$
exactly as in the $B_{n}$ case, and then calculate terms
$\alpha_{i}=g_{i}-i+1$ and $\beta_{i}=f_{i}-i+1$. A rigid partitions in the
$D_{n}$ theory always have at least one part equal to 1, so $\beta_{1}=0$. We
omit this entry and relabel $\beta_{2}\rightarrow\beta_{1}$ etc.

Example: $D_{10}$, $\lambda=4^{2}\,3\,2^{2}\,1^{5}$, the length of the
partition is even and the symbol is

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&1&&1&&2&&2&&2\\\
&&&0&&0&&0&&2&\end{array}\right)$

######  Remark 2.1.

  1. (1)

The form of the symbol (2.3) is a convenient notation. The location between
the entries $\alpha_{*}$ and $\beta_{*}$ the symbol are not essential.

  2. (2)

$l$ is the length of the partition including the extra 0 appended if not
special specified. The following table illustrates the first step of the
computation of symbol

$\lambda_{k}$ | $\lambda_{1}$ | $\lambda_{2}$ | $\cdots$ | $\lambda_{l}$
---|---|---|---|---
$l-k$ | $l-1$ | $l-2$ | $\cdots$ | $0$
$l-k+\lambda_{k}$ | $l-1+\lambda_{1}$ | $l-2+\lambda_{2}$ | $\cdots$ | $\lambda_{l}$

Note that $l-k$ is an increasing sequence from right to left, whose first term
is zero. This fact will be used frequently. The terms $f_{i}$ and $g_{i}$ are
calculated as follows

$2f_{i}+1$ | $\cdots$ | $2f_{2}+1$ | $2f_{1}+1$
---|---|---|---
$f_{i}$ | $\cdots$ | $f_{2}$ | $f_{1}$
$2g_{i}$ | $\cdots$ | $2g_{2}$ | $2g_{1}$
---|---|---|---
$g_{i}$ | $\cdots$ | $g_{2}$ | $g_{1}$

The sequences $f_{i}$ and $g_{i}$ increase from right to left.

  3. (3)

The terms in the sequence $\lambda_{i}$ are independent to each other as well
as the terms in the sequence $l-k+\lambda_{k}$.

In [5], it is pointed out that $\alpha_{*}$ is one more than $\beta_{*}$ in
the symbols in the $B_{n},C_{n}$, and $D_{n}$ theories. Before showing it, we
give some basic facts. For the part $\lambda_{i}$ appearing even ($2m$) times
in the partition, according to the first step of the computation of symbol, we
get the following sequence

$\displaystyle(\lambda_{i}+l-i,\lambda_{i+1}+l-(i+1),\cdots,\lambda_{i+2m-1}+l-(i+2m-1))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad=(\lambda_{i}+l-i,\lambda_{i}+l-(i+1),\cdots,\lambda_{i}+l-(i+2m-1)),$

where $l$ is the length of the partition. With equal number of old terms and
even terms in (2.1), this sequence leads to equal number of $\alpha_{*}$ and
$\beta_{*}$ in the symbol. For the terms left in the sequence $l-k$, the
number of the difference between the even terms and old terms is

(2.6) $\frac{1-(-1)^{l}}{2}$

which is 1 if $l$ is odd and 0 if $l$ is even.

######  Lemma 2.2.

The number of $\alpha_{i}$’s is always one more than the number of
$\beta_{i}$’s in the symbol of a partition $\lambda$ in the $B_{n}$ theory.
The symbol has the following form

(2.7)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m}\\\
&\beta_{1}&&\cdots&&\beta_{m-1}&&&\end{array}\right).$

###### Proof.

According to the definition of orthogonal partition, all the even integers
appear an even number of times . By using the formula (2.1), all the even
parts lead to equal number of $\alpha_{*}$ and $\beta_{*}$.

Since the length $l$ of the partition is odd111 It will be proved in
Proposition 4. and the number of the total boxes in $\lambda$ is old, the
number of odd parts is odd. According to the formula (2.6), for these odd
parts $\lambda_{i}$, the number of odd terms $l-i+\lambda_{i}$ corresponding
to $f_{i}$ which lead to $\alpha_{*}$ is one more than the number of even
terms $l-i+\lambda_{i}$ corresponding to $g_{i}$ which lead to $\beta_{*}$.

Combing the results of the above paragraphs, we draw the conclusion. ∎

######  Remark 2.3.

The total number of $\alpha_{*}$ and $\beta_{*}$ is $l$.

######  Lemma 2.4.

The number of $\alpha_{i}$’s is always one more than the number of
$\beta_{i}$’s in the symbol of a partition $\lambda$ in the $C_{n}$ theory.

###### Proof.

When the length of the partition is even, we should append an extra 0 as the
last part of the partition. So the length of the partition is odd.

According to the definition of symplectic partition, all the odd integers
appear an even number of times. According to the formula (2.1), all these old
parts lead to equal number of $\alpha_{*}$ and $\beta_{*}$.

Since $l$ is odd and the number of the total boxes in $\lambda$ is even, the
number of the even parts in the partition is odd. According to the formula
(2.6), for these odd parts $\lambda_{i}$, the number of even terms
$l-i+\lambda_{i}$ corresponding to $g_{i}$ which lead to $\alpha_{*}$ is one
more than the number of old terms $l-i+\lambda_{i}$ corresponding to $f_{i}$
which lead to $\beta_{*}$.

Combing the results of the above paragraphs, we draw the conclusion. ∎

######  Remark 2.5.

If the length of the partition $l$ is odd, the total number of $\alpha_{*}$
and $\beta_{*}$ is $l$. If $l$ is even, the total number of $\alpha_{*}$ and
$\beta_{*}$ is $l+1$ because of the extra 0 appended.

######  Lemma 2.6.

The number of $\alpha_{i}$’s is always one more than the number of
$\beta_{i}$’s in the symbol of a partition $\lambda$ in the $D_{n}$ theory.

###### Proof.

For a partition in the $D_{n}$ theory, the length of the partition $l$ is
even222It will be proved in Proposition 7.and all the odd integers appear an
even number times. According to the formula (2.1), these odd parts lead to
equal number of $\alpha_{*}$ and $\beta_{*}$.

Since $l$ is even and the number of the total boxes in $\lambda$ is even, the
number of the even parts in the partition is even. According to the formula
(2.6), for these even parts $\lambda_{i}$, the number of even terms
$l-i+\lambda_{i}$ corresponding to $g_{i}$ which lead to $\alpha_{*}$ is equal
to the number of old terms $l-i+\lambda_{i}$ corresponding to $f_{i}$ which
lead to $\beta_{*}$.

For rigid partition, $\lambda_{l}=1$, so $l-l+\lambda_{l}=l-l+1$ corresponds
to $2f_{1}+1$. Thus $f_{1}=0$ which means $\beta_{1}=f_{1}-1+1=0$. After
$\beta_{1}$ is omitted, the number of $\alpha_{i}$’s in the top row of the
symbol is always one more than the number of $\beta_{i}$’s on the bottom row.
∎

######  Remark 2.7.

Note that we delete one $\beta_{1}$ artifically and relabel
$\beta_{2}\rightarrow\beta_{1}$ etc. The total number of $\alpha_{*}$ and
$\beta_{*}$ is $l-1$.

###  2.2. New definition of symbol

In this subsection, we propose 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 2.

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

  * •

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

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

Finally we write the symbol as

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

  * •

For the $C_{n}$ theory:

1:

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

2:

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

  * •

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

######  Remark 2.8.

Compared to the odd definitions of symbol, $\alpha_{*}$ are all related to
$f_{*}$ and $\beta_{*}$ are all related to $g_{*}$ for the new definitions of
symbol in the $B_{n}$, $C_{n}$, and $D_{n}$ theories.

Before proving the equality between the new definition of symbol and odd one,
we give several universal results. For the first step of the computation of
the symbol, we have the following sequence

(2.8) $S=(l-1+\lambda_{1},l-2+\lambda_{2},\cdots,l-l+\lambda_{l}).$

Arrange the odd parts in a decreasing sequence

(2.9)
$(l-j_{1}+\lambda_{j_{1}},l-j_{2}+\lambda_{j_{2}},\cdots,l-j_{m}+\lambda_{j_{m}})=(2f_{{m}}+1,2f_{{m-1}}+1,\cdots,2f_{1}+1).$

And arrange the even parts in a decreasing sequence

(2.10)
$(l-i_{1}+\lambda_{i_{1}},l-i_{2}+\lambda_{i_{2}},\cdots,l-{i_{n}}+\lambda_{i_{n}})=(2g_{{n}},2g_{{n-1}},\cdots,2g_{1}).$

Introduce the following notations

(2.11) $A_{a}=f_{a}-a+1,\quad\quad B_{b}=g_{b}-b+1,$

where $A_{a}$ corresponding to $f_{a}$ and $B_{b}$ corresponding to $g_{b}$.
We write $A_{*}$ and $B_{*}$ as follows

(2.12)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}A_{1}&&A_{2}&&\cdots&&A_{m}\\\
&B_{1}&&\cdots&&B_{n}&&&\end{array}\right).$

If we append an extra 0 as the last part of the partition, the sequence
corresponding to (2.8) is

(2.13)
$S^{+}=((l+1)-1+\lambda_{1},(l+1)-2+\lambda_{2},\cdots,(l+1)-l+\lambda_{l},0).$

Let $f^{{}^{\prime}}_{a}$ and $g^{{}^{\prime}}_{a}$ be the notations $f_{a}$
and $g_{a}$ for this sequence. Arrange the odd parts in a decreasing sequence

(2.14)
$(l+1-i_{1}+\lambda_{i_{1}},l+1-i_{2}+\lambda_{i_{2}},\cdots,l+1-{i_{n}}+\lambda_{i_{n}})=(2f{{}^{\prime}}_{{n}}+1,2f{{}^{\prime}}_{{n-1}}+1,\cdots,2f{{}^{\prime}}_{1}+1).$

Comparing the left hand side of formula (2.14) with that of formula (2.10), we
have

$(2f{{}^{\prime}}_{{n}}+1,2f{{}^{\prime}}_{{n-1}}+1,\cdots,2f{{}^{\prime}}_{1}+1)=(2g_{{n}}+1,2g_{{n-1}}+1,\cdots,2g_{1}+1).$

Arrange the even parts of sequence (2.13) in a decreasing sequence

(2.15)
$(l+1-j_{1}+\lambda_{j_{1}},l+1-j_{2}+\lambda_{j_{2}},\cdots,l+1-j_{m}+\lambda_{j_{m}},0)=(2g{{}^{\prime}}_{{m+1}},2g{{}^{\prime}}_{{m}},\cdots,2g{{}^{\prime}}_{2},0).$

Comparing the left hand side of formula (2.15) with that of formula (2.9), we
have

$(2g{{}^{\prime}}_{{m+1}},2g{{}^{\prime}}_{{m}},\cdots,2g{{}^{\prime}}_{2},0)=(2(f_{{m}}+1),2(f_{{m-1}}+1),\cdots,2(f_{1}+1),0).$

Let $A^{{}^{\prime}}_{a}$ and $B^{{}^{\prime}}_{a}$ be the notations $A_{a}$
and $B_{a}$ for the sequence (2.13)

(2.16) $A^{{}^{\prime}}_{b}=f^{{}^{\prime}}_{b}-b+1=g_{b}-b+1=B_{b},\quad\quad
B^{{}^{\prime}}_{a}=g^{{}^{\prime}}_{a}-a+1=(f_{{a-1}}+1)-a+1=A_{a-1}.$

Since the last term in sequence (2.16) is 0, it corresponds to
$B^{{}^{\prime}}_{1}=A_{0}=0$ which means

(2.17)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&A^{{}^{\prime}}_{1}&&\cdots&&A^{{}^{\prime}}_{n}&\\\
&&&B^{{}^{\prime}}_{1}&&\cdots&&B^{{}^{\prime}}_{m}&&B^{{}^{\prime}}_{m+1}\end{array}\right)=\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&0&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

######  Proposition 2.

For a partition $\lambda$ in the $C_{n}$ theory, the new definition of symbol
is equivalent to the odd one.

###### Proof.

We need to consider two cases:

  * •

The length of $\lambda$ is even. First, we compute the symbol by the new
definition. we have

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

Using the formula (2.12), we get

(2.18)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}A_{1}&&A_{2}&&\cdots&&A_{m}\\\
&B_{1}&&\cdots&&B_{n}&&&\end{array}\right).$

After appending an extra 0 on the left of the bottom row of the formula
(2.18), the symbol of $\lambda$ is

(2.19)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&A_{1}&&A_{2}&&\cdots&&A_{m}\\\
&&&B_{1}&&\cdots&&B_{n}&&&\end{array}\right).$

Next, we compute the symbol by the odd definition. After appending an extra 0
as the last part of the partition $\lambda$, using the formula (2.17), we have

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&0&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

Let $f^{{}^{\prime}}_{a}$ and $g^{{}^{\prime}}_{a}$ be the notations $f_{a}$
and $g_{a}$ for the sequence (2.13) corresponding to the partition
$\lambda\oplus 0$. According to the odd definition, we have

$\alpha^{{}^{\prime}}_{i}=g^{{}^{\prime}}_{i}-i+1=B^{{}^{\prime}}_{i}=A_{i-1},\quad\quad\beta^{{}^{\prime}}_{i}=f^{{}^{\prime}}_{i}-i+1=A^{{}^{\prime}}_{i}=B_{i}.$

Thus the symbol of $\lambda$ is

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&A_{1}&&A_{2}&&\cdots&&A_{m}\\\
&&&B_{1}&&\cdots&&B_{n}&&&\end{array}\right)$

which is equal to the formula (2.19).

  * •

The length of $\lambda$ is odd. First, we compute the symbol by the old
definition, we have

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

which mean the symbol of $\lambda$ is

(2.20)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

Next, we compute symbol by the new definition. After appending an extra 0 as
the last part of partition $\lambda$, using the formula (2.17), we have

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&0&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

After deleting the 0 in the first entry of the bottom row of the symbol, we
have

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right)$

which is equal to the formula (2.20).

∎

######  Proposition 3.

For a partition $\lambda$ in the $D_{n}$ theory, the new definition of symbol
is equivalent to the odd one.

###### Proof.

According to the old construction of the symbol, we have

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

leading to

(2.21)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

After omitting the term $\beta_{1}=A_{1}=0$, we have the symbol as follows

(2.22)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&&&A_{2}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

Next, we compute the symbol by the new definition. After appending an extra 0
as the last part of partition $\lambda$, using the formula (2.17), we have

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&0&&A_{1}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right).$

After deleting two 0 in the first two terms of the bottom row of the symbol,
we have

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&B_{1}&&\cdots&&B_{n}&\\\
&&&A_{2}&&\cdots&&A_{m-1}&&A_{m}\end{array}\right)$

which is equal to the formula (2.22). ∎

According to proofs of the above propositions, operations of deleting a 0 or
appending a 0 in the last step of the calculation of symbol are not essential.
So we can regard the following symbols are equivalent.

Example: For the $C_{10}$ partition $3^{2}\,2^{6}\,1^{2}$, the length of the
partition is even

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&0&&1&&1&&1&&1\\\
&1&&1&&1&&1&&2&\end{array}\right)\sim\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&0&&1&&1&&1&&1\\\
&1&&1&&1&&1&&2&\end{array}\right)$

where $\sim$ means that two symbols are equivalent.

Example: For the $C_{12}$ partition $3^{2}\,2\,1^{4}$, the length of the
partition is odd. We have

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

Example: For the $D_{10}$ partition $4^{2}\,3\,2^{2}\,1^{5}$, we have

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

The above equivalences imply the following definition of symbol.

######  Definition 3.

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

  * •

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

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

Finally we write the symbol as

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

  * •

For the $C_{n}$ theory: if the length of partition is even, compute the symbol
as the $B_{n}$ case. If the length of the partition is odd, first append an
extra 0 as the last part of the partition. Then compute the symbol as in the
$B_{n}$ case. The symbol has the following form

(2.23)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\beta_{1}&&\beta_{2}&&\cdots&&\beta_{m-1}&&\beta_{m}&\end{array}\right)$

  * •

For the $D_{n}$ theory: first append an extra 0 as the last part of the
partition, then compute the symbol as in the $B_{n}$ case. The symbol has the
following form

(2.24)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\beta_{1}&&\beta_{2}&&\beta_{3}&&\cdots&&\beta_{m}&&\beta_{m+1}&\end{array}\right)$

######  Remark 2.9.

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

(2.25)
$\boxed{\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m}\\\
&\beta_{1}&&\cdots&&\beta_{m+t}&&&\end{array}\right)}.$

where $m=(l+(1-(-1)^{l})/2)/2$, $l$ is the length of the partition. $t=-1$ for
the $B_{n}$ theory, $t=0$ for the $C_{n}$ theory, and $t=1$ for the $D_{n}$
theory. The above definition of symbol will be the starting point of the
proofs in the following sections.

###  2.3. Formal operations of a partition

In this subsection, we discuss formal operations of a partition which will be
used in the construction of symbol in the next section.

The first operation is to append an extra 0 as the last part of the partition
$\lambda$. Then we compute the symbol of the new partition $\lambda\oplus 0$.
In fact, this operation leads to the formula (2.1). Note that for a partition
in the $D_{n}$ theory and for a partition with the first row is odd in the
$C_{n}$ theory we need append an extra 0 as the last part of the partition.
The partition appended an extra 0 is notated as $\lambda^{0}$. And $l^{0}$ is
the length of the partition $\lambda^{0}$.

######  Lemma 2.10.

For a partition in the $B_{n}$, $C_{n}$, and $D_{n}$ theories, the symbol is

(2.26)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m}\\\
&\beta_{1}&&\cdots&&\beta_{m+t}&&&\end{array}\right)$

where $t=-1$ for the $B_{n}$ theory, $t=0$ for the $C_{n}$ theory, and $t=1$
for the $D_{n}$ theory. $\lambda^{0}$ denote the partition appended an extra 0
if needed for the computation of symbol. Then the symbol of
$\lambda^{{}^{\prime}}=\lambda^{0}\oplus 0^{1}$ computed as a partition in the
$B_{n}$ theory is

(2.27)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&&&\beta_{1}&&\cdots&&\beta_{m+t}&\\\
&0&&\alpha_{1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\end{array}\right).$

And the symbol of $\lambda^{0}\oplus 0^{2}$ computed as a partition in the
$B_{n}$ theory is

(2.28)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}0&&\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m}\\\
&0&&\beta_{1}&&\cdots&&\beta_{m+t}&&&\end{array}\right).$

###### Proof.

The formula (2.27) is the result of the formula (2.17). Applying the formula
(2.27) twice, we get the formula (2.28). ∎

######  Remark 2.11.

  1. (1)

After appending an extra 0, we have $\lambda^{0}_{l+1}=0$ which means

$l^{0}+1-(l^{0}+1)+\lambda_{l^{0}+1}=0.$

The superscript ’0’ means it is an index in the partition $\lambda^{0}$. So we
get $g^{{}^{\prime}}_{1}=0$ which means
$\beta^{{}^{\prime}}_{1}=g^{{}^{\prime}}_{1}-1+1=0$.
$\beta^{{}^{\prime}}_{1}=0$ is in the first entry on the left side of the
bottom row in the symbol (2.27).

  2. (2)

After appending an extra 0, one increase in the length of the partition. The
even terms of the sequence $l^{0}-k$ become the odd terms of the the sequence
$l^{0}+1-k$. And the odd terms of the sequence $l^{0}-k$ become the even terms
in the sequence $l^{0}+1-k$. The bottom row and top row in symbol reverse
roles by comparing the formula (2.28) with (2.27).

Next, we introduce another two formal operations: the first one is to add a
row $0^{l^{0}+2a}$ to a partition $\lambda$. And the second one is to add a
column $\lambda_{0}(=\lambda_{1})$ to a partition $\lambda^{0}$. Then we
discuss what happen to the symbol.

######  Lemma 2.12.

For a partition $\lambda$ in the $B_{n}$, $C_{n}$, and $D_{n}$ theories, the
symbol is

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

where $t=-1$ for the $B_{n}$ theory, $t=0$ for the $C_{n}$ theory, and $t=1$
for the $D_{n}$ theory. $\lambda^{0}$ denote the partition appended an extra 0
if needed for the computation of symbol. Then the symbol of the partition
$\lambda^{{}^{\prime}}=\lambda^{0}+0^{l^{0}+2a}$ computed as a partition in
the $B_{n}$ theory is

(2.29)
$\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{0\;\cdots\;0}^{a}\;\;\alpha_{1}\;\;\alpha_{2}\cdots\alpha_{m}\\\
\underbrace{0\;\cdots\;0}_{a}\;\;\beta_{1}\cdots\beta_{m+t}\end{array}\Bigg{)}.$

If we append an extra $\lambda_{0}(=\lambda_{1})$ as the first part of the
partition $\lambda$, then the symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda_{0}\oplus\lambda^{0}$ computed as a
partition in the $B_{n}$ theory has the following form

(2.30)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\alpha_{1}&&\alpha_{2}&&\cdots&&\alpha_{m}&&\square^{t}\\\
&\beta_{1}&&\cdots&&\beta_{m+t}&&\square_{b}&&&\end{array}\right)$

where $(\square^{t},\square^{b})=(\alpha_{m+1},\emptyset)$ or
$(\square^{t},\square^{b})=(\emptyset,\beta_{m+t+1})$.

###### Proof.

After appending a row $0^{l^{0}+2a}$ under the first row of $\lambda^{0}$,
nothing changed for the first $l^{0}$ terms in the sequence
$l^{0}-k+\lambda_{k}$. Using the formula (2.28) $m$ times, we get the formula
(2.29).

Because of the extra part $\lambda_{0}$, there will be an extra term in the
symbol of the partition $\lambda^{{}^{\prime\prime}}$ compared with that of
$\lambda^{0}$. It appear at the end of the row of the symbol, since
$(l^{0}+1)-0+\lambda_{0}$ is the biggest one of the sequence
$(l^{0}+1)-k+\lambda_{k}$. ∎

##  3\. Construction of symbol

Our main strategy to construct symbol is to sum the contribution of each row
in the partition. We find that it is natural to consider the contribution of
each even row independently. The addition of symbol is defined by writing the
symbols right adjusted and simply add the entries that are ’in the same
place’. Unfortunately, it is not natural to consider the contribution of each
odd row independently. However, it is natural to consider the contribution of
two odd rows together. In the first subsection, we introduce three lemmas
which are crucial for the construction of symbol. Using these lemmas, we give
simple rules to calculate symbol. In the final subsection, we give a uniform
closed formula of symbol in the $B_{n}$, $C_{n}$, and $D_{n}$ theories.

###  3.1. Three Lemmas

First, we analyse the contribution to symbol of even rows in the partition.
Added an even row to a partition, the result partition is still in the same
theory. Excepting appending the extra 0, the processes of the computation of
symbol are the same in the $B_{n}$, $C_{n}$, and $D_{n}$ theories.

######  Lemma 3.1.

For the partition $\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots),(a>2b)$, the
symbol has the following form

(3.1)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+1+t}&&\cdots&&\beta_{m+t}&\end{array}\right).$

$l$ is the length of the partition including the extra 0 appended if needed
for the computation of the symbol. If $l+\lambda_{1}$ is odd, the symbol of
$\lambda^{{}^{\prime}}=\lambda+1^{2b}$ has the following form

(3.2)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\beta_{m-b+1+t}&&\cdots&&\beta_{m+t}&\end{array}\right).$

If $l+\lambda_{1}$ is even, the symbol of
$\lambda^{{}^{\prime}}=\lambda+1^{2b}$ has the following form

(3.3)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+1+t}+1&&\cdots&&\beta_{m+t}+1&\end{array}\right).$

###### Proof.

The formula (3.1) is the result of Definition 3. For the first step of the
computation of symbol, we have the following sequence

(3.4)
$(l-1+\lambda_{1},\cdots,l-2b+\lambda_{2b},l-(2b+1)+\lambda_{2b+1},\cdots,0+\lambda_{l}).$

Since $\lambda_{1}=\cdots=\lambda_{2b}$, the first $2b$ terms of the sequence
(3.4) are successive

(3.5) $(l-1+\lambda_{1},\cdots,l-2b+\lambda_{1}).$

According to Definition 3, all even terms in the sequence (3.5) can be written
in a decreasing sequence as follows

$(2g_{m+t},2g_{m+t-1},\cdots,2g_{m+t-b+1})$

which are one to one correspondence with terms

$(\beta_{m+t},\beta_{m+t-1},\cdots,\beta_{m+t-b+1}).$

The odd terms in the sequence (3.5) can be written in a decreasing sequence as
follows

$(2f_{m}+1,2f_{m-2}+1,\cdots,2f_{m-b+1}+1)$

which are one to one correspondence with terms

$(\alpha_{m},\alpha_{m-1},\cdots,\alpha_{m-b+1}).$

After adding an even row $1^{2b}$ to $\lambda$, the sequence corresponding to
(3.5) for the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b}$ is

(3.6) $(l-1+\lambda_{1}+1,\cdots,l-2b+\lambda_{1}+1).$

Let $g^{{}^{\prime}}_{i}$ and $f^{{}^{\prime}}_{i}$ be the notations $g_{i}$
and $f_{i}$ for the sequence (3.6).

  * •

If $l+\lambda_{1}$ is odd, the even terms in the sequence (3.5) can be
rewritten as follows

(3.7)
$(l-1+\lambda_{1},l-3+\lambda_{1},\cdots,l-(2b-1)+\lambda_{1})=(2g_{m+t},2g_{m+t-1},\cdots,2g_{m+t-b}).$

While the even terms in the sequence (3.6) are

(3.8)
$(l-1+\lambda_{1},l-3+\lambda_{1},\cdots,l-(2b-1)+\lambda_{1})=(2g^{{}^{\prime}}_{m+t},2g^{{}^{\prime}}_{m+t-1},\cdots,2g^{{}^{\prime}}_{m+t-b}).$

Because of no differences between the sequence (3.8) and (3.7), we have

(3.9)
$(\beta^{{}^{\prime}}_{m+t},\beta^{{}^{\prime}}_{m+t-1},\cdots,\beta^{{}^{\prime}}_{m+t-b})=(\beta_{m+t},\beta_{m+t-1},\cdots,\beta_{m+t-b}).$

The odd terms in the sequence (3.5) can be rewritten as follows

(3.10)
$(l-2+\lambda_{1},l-4+\lambda_{1},\cdots,l-(2b)+\lambda_{1})=(2f_{m}+1,2f_{m-2}+1,\cdots,2f_{m-b}+1)$

While the odd terms in the sequence (3.6) are

(3.11)
$(l-1+\lambda_{1}+1,l-3+\lambda_{1}+1,\cdots,l-(2b-1)+\lambda_{1}+1)=(2f^{{}^{\prime}}_{m}+1,2f^{{}^{\prime}}_{m-2}+1,\cdots,2f^{{}^{\prime}}_{m-b}+1)$

Comparing the sequence (3.11) with (3.10), using
$\alpha^{{}^{\prime}}_{i}=f^{{}^{\prime}}_{i}-i+1=(f_{i}+1)-i+1$, we have

(3.12)
$(\alpha^{{}^{\prime}}_{m},\alpha^{{}^{\prime}}_{m-1},\cdots,\alpha^{{}^{\prime}}_{m-b})=(\alpha_{m}+1,\alpha_{m-1}+1,\cdots,\alpha_{m-b}+1).$

Combing the formula (3.9) and (3.12), we have the formula (3.2).

  * •

If $l+\lambda_{1}$ is even, the even terms in the sequence (3.5) of the
partition $\lambda$ can be rewritten as follows

(3.13)
$(l-2+\lambda_{1},l-4+\lambda_{1},\cdots,l-(2b)+\lambda_{1})=(2g_{m+t},2g_{m+t-1},\cdots,2g_{m+t-b}).$

While the even terms in the sequence (3.6) of $\lambda^{{}^{\prime}}$ are

(3.14)
$(l-1+\lambda_{1}+1,l-3+\lambda_{1}+1,\cdots,l-(2b-1)+\lambda_{1}+1)=(2g^{{}^{\prime}}_{m+t},2g^{{}^{\prime}}_{m+t-1},\cdots,2g^{{}^{\prime}}_{m+t-b}).$

Comparing the sequence (3.13) with (3.14), using
$\beta^{{}^{\prime}}_{i}=g^{{}^{\prime}}_{i}-i+1=(g_{i}+1)-i+1$, we have

(3.15)
$(\beta^{{}^{\prime}}_{m+t},\beta^{{}^{\prime}}_{m+t-1},\cdots,\beta^{{}^{\prime}}_{m+t-b})=(\beta_{m+t}+1,\beta_{m+t-1}+1,\cdots,\beta_{m+t-b}+1).$

These odd terms in the sequence (3.5) of $\lambda$ are

(3.16)
$(l-1+\lambda_{1},l-3+\lambda_{1},\cdots,l-(2b-1)+\lambda_{1})=(2f_{m}+1,2f_{m-2}+1,\cdots,2f_{m-b}+1)$

While odd terms in the sequence (3.6) of $\lambda^{{}^{\prime}}$ are

(3.17)
$(l-1+\lambda_{1},l-3+\lambda_{1},\cdots,l-(2b-1)+\lambda_{1})=(2f^{{}^{\prime}}_{m}+1,2f^{{}^{\prime}}_{m-2}+1,\cdots,2f^{{}^{\prime}}_{m-b}+1)$

Since nothings changed in the sequence (3.17) compared with (3.16), we have

$(\alpha^{{}^{\prime}}_{m},\alpha^{{}^{\prime}}_{m-1},\cdots,\alpha^{{}^{\prime}}_{m-b})=(\alpha_{m},\alpha_{m-1},\cdots,\alpha_{m-b}).$

Combing the formula (3.15) and (3.17), we have the formula (3.3).

∎

######  Remark 3.2.

One increase for the entries in the same row of the symbol from right to left.
The number of the changed entries is half length of the even row added to the
partition.

######  Lemma 3.3.

For a partition $\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots)$, the last
$b,(a>2b),$ terms in the top and bottom rows of the symbol has the following
form

(3.18)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m+t}&&\cdots&&\beta_{m+t}&\end{array}\right),$

where $t=-1$ for the $B_{n}$ theory , $t=0$ for the $C_{n}$ theory, and $t=1$
for the $D_{n}$ theory. $l$ is the length of the partition including the extra
0 appended if needed for the computation of the symbol. If $l+\lambda_{1}$ is
odd, $\beta_{m+t}=\alpha_{m}-t+1.$ If $l+\lambda_{1}$ is even,
$\beta_{m+t}=\alpha_{m}-t.$

###### Proof.

For the first step of the computation of symbol, we have the following
sequence

(3.19)
$(l-1+\lambda_{1},\cdots,l-2b+\lambda_{2b},l-(2b+1)+\lambda_{2b+1},\cdots,0+\lambda_{l}).$

Since $\lambda_{1}=\cdots=\lambda_{2b}$, the first $2b$ terms of the sequence
(3.19) are successive

(3.20) $(l-1+\lambda_{1},\cdots,l-2b+\lambda_{1}).$

According to Definition 3, the even terms in the sequence (3.20) can be
rewritten as

$(2g_{m+t},2g_{m+t-1},\cdots,2g_{m+t-b+1})$

which are one to one correspondence with terms

$(\beta_{m+t},\beta_{m+t-1},\cdots,\beta_{m+t-b+1}).$

Since $g_{i}=g_{i-1}+1$ for $i\geq m+t-b$, we have

$\beta_{i}=\beta_{i-1},\quad i\geq m+t-b.$

The odd terms in the sequence (3.20) can be written as

$(2f_{m}+1,2f_{m-2}+1,\cdots,2f_{m-b+1}+1)$

which are one to one correspondence with terms

$(\alpha_{m},\alpha_{m-1},\cdots,\alpha_{m-b+1}).$

Since $f_{i}=f_{i-1}$ for $i\geq m-b$, we have

$\alpha_{i}=\alpha_{i-1},\quad i\geq m-b.$

  * •

If $l+\lambda_{1}$ is odd, so the largest number $l-1+\lambda_{1}$ in the
sequence (3.19) is even, thus corresponding to $\beta_{m+t}$. We have
$g_{m+t}=f_{m}+1$ which mean

$\beta_{m+t}=g_{m+t}-(m+t)+1=f_{m}+1-(m+t)+1=f_{m}-m+1-t+1=\alpha_{m}-t+1.$

  * •

If $l+\lambda_{1}$ is even, so the largest number $l-1+\lambda_{1}$ in the
sequence (3.19) is odd, thus corresponding to $\alpha_{m}$. We have
$g_{m+t}=f_{m}$ which mean

$\beta_{m+t}=g_{m+t}-(m+t)+1=f_{m}-(m+t)+1=f_{m}-m+1-t=\alpha_{m}-t.$

∎

Next, we discuss what happen to the symbol of a partition when added an old
row. We can reduce the proof to the case of Lemma 3.1 by introducing an
virtual column before the first part of the partition. We find that it is more
natural to consider the contributions of two odd rows together.

######  Lemma 3.4.

For a partition $\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots)$, $(a>2b+1)$,
the symbol has the following form

(3.21)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&\end{array}\right).$

$l$ is the length of the partition including the extra 0 appended if
necessary. If $l+\lambda_{1}$ is odd, the symbol of the partition
$\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.22)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\alpha_{m}\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}+1&&\cdots&&\beta_{m+t-1}+1&&&&&\end{array}\right).$

If $l+\lambda_{1}$ is even, the symbol of the partition
$\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.23)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&&&\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&&\beta_{m+t}&\end{array}\right).$

###### Proof.

To use Lemma 3.1, we append a virtual row $\lambda_{0}(=\lambda_{1})$ before
the first part $\lambda_{1}$, thus forming a new partition
$\lambda^{V}(\lambda_{0}=\lambda_{1}=\cdots=\lambda_{a}>\cdots)$ with length
of $l+1$. For the first step of computation of symbol, we have the following
sequence

(3.24)
$(\underline{l+1-1+\lambda_{0}},l-1+\lambda_{1},\cdots,l-2b+\lambda_{2b},l-(2b+1)+\lambda_{2b+1},\cdots,0+\lambda_{l}).$

The underline indicate that it is an auxiliary term. After adding a row
$1^{2b+2}$ to the partition, we have the following sequence

(3.25)
$(\underline{l+1-1+\lambda_{0}+1},l-1+\lambda_{1}+1,\cdots,l-2b+\lambda_{2b}+1+1,l-(2b+1)+\lambda_{2b+1}+1,\cdots,0+\lambda_{l}).$

  * •

If $l+\lambda_{1}$ is odd, $l+1-1+\lambda_{0}$ in the sequence (3.24)
corresponds to $\alpha_{m+1}$. Then the symbol of the partition $\lambda^{V}$
is

(3.26)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\alpha_{m+1}\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&&&\end{array}\right).$

According to Lemma 3.3, we have $\alpha_{m}=\alpha_{m+1}$. For the partition
$\lambda^{V}+1^{2b+2}$, $l+1+\lambda_{0}$ is even. Using Lemma 3.1, we have

(3.27)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\alpha_{m}\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}+1&&\cdots&&\beta_{m+t-1}+1&&\beta_{m+t}+1&&&\end{array}\right).$

Since the auxiliary term $\underline{l-0+\lambda_{0}+1}$ in the sequence
(3.25) is even, it corresponds to the last one on the bottom row of the symbol
which is $\beta_{m+t}+1$. After omitting this term, the symbol of
$\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.28)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\alpha_{m}\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}+1&&\cdots&&\beta_{m+t-1}+1&&&&&\end{array}\right)$

which is the formula (3.22).

  * •

$l+\lambda_{1}$ is even, $l+1-1+\lambda_{0}$ in the sequence (3.24)
corresponds to $\beta_{m+t+1}$. Then the symbol of the partition $\lambda^{V}$
is

$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&&\beta_{m+t+1}&\end{array}\right).$

According to Lemma 3.3, we have $\beta_{m+t}=\beta_{m+t+1}$. For the partition
$\lambda^{V}+1^{2b+2}$, $l+1+\lambda_{0}$ is old. Using Lemma 3.1, we have

(3.29)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&\alpha_{m}+1&&\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&&\beta_{m+t}&\end{array}\right).$

Since the auxiliary term $\underline{l-0+\lambda_{0}+1}$ in the sequence
(3.25) is odd, it corresponds to $\alpha_{m}$ which should be omitted in the
end. So the symbol of $\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.30)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&&&\\\
&\cdots&&\beta_{m+t-b}&&\beta_{m+t-b+1}&&\cdots&&\beta_{m+t-1}&&\beta_{m+t}&&\beta_{m+t}&\end{array}\right)$

which is the formula (3.23).

∎

######  Remark 3.5.

Note that we omit the entries in the symbols (3.27), (3.29), which correspond
to the auxiliary term in the sequence.

###  3.2. Symbol of the rigid partition in the $B_{n}$ theory

The following fact [5] is useful to study the structure of symbol, so we give
the proof in detail.

######  Proposition 4.

The longest row in a rigid $B_{n}$ partition always contains an odd number of
boxes. And 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.

###### Proof.

For a partition in the $B_{n}$ theory, even integers appear an even number of
times. So the sum of all odd integers is odd which implies that the number of
odd integers is odd. It imply that the length of the partition which is the
sum of the number of odd integers and even integers is odd. So the longest row
contains an odd number of boxes.

If the following two rows of the first row are of different parities, then the
difference of the number of boxes between these two rows is odd. It imply that
part ’2’ appears odd number of times in the partition, which is a
contradiction. So the following two rows are either both of odd length or both
of even length. In the same way, we can prove the next two rows are of the
same parities. This pairwise pattern continues.

The number of total boxes of a pairwise pattern is even. If the Young tableau
has an even number of rows, then the number of the total boxes of the first
row and shortest row is old. Since the longest row contains an odd number of
boxes, the shortest row is even. ∎

######  Remark 3.6.

If the last row is odd, there is an odd number of rows in the partition.

Using above characteristics of the partition in the $B_{n}$ theory, we can
refine Lemma 3.3

######  Lemma 3.7.

Let $\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots>\lambda_{l})$ is a
partition in the $B_{n}$ theory and the last two rows have same parity. If
$a>2b$, the symbol of $\lambda$ has the following form

(3.31)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b}&&\cdots&&\beta_{m-1}&\end{array}\right)=\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m}&&\cdots&&\alpha_{m}\\\
&\cdots&&\alpha_{m}+1&&\cdots&&\alpha_{m}+1&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b}$ is

(3.32)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b}+1&&\cdots&&\beta_{m-1}+1&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda^{{}^{\prime}}+1^{2c}$, $b>c$, is

(3.33)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\alpha_{m-b}+1&&\cdots&&\alpha_{m-b+1}+1&&\cdots&&\beta_{m-1}+1&\end{array}\right).$

###### Proof.

According to Lemma 3.3, we get the formula (3.31). According to Proposition 4,
$\lambda_{1}$ is odd which means $l+\lambda_{1}$ is even, thus formula (3.32)
is the result of formula (3.3). For partition $\lambda^{{}^{\prime}}$,
$l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd, thus formula (3.33) is
the result of formula (3.2). ∎

######  Remark 3.8.

  1. (1)

Compared the formula (3.32) with (3.31), the contribution to symbol of row
$1^{2b}$ is

(3.34) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{0\cdots 0}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1\;1\cdots 1}_{b}\end{array}\Bigg{)}$

  2. (2)

Compared the formula (3.33) with (3.32), the contribution to symbol of row
$1^{2c}$ is

(3.35) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{0\cdots
0\;0\cdots 0}_{b}\end{array}\Bigg{)}$

  3. (3)

This pattern of the contribution to symbol for the even rows in a pairwise
patten will continue.

If a partition in the $B_{n}$ theory is added a even row, the new partition is
in the same theory by Proposition 4. However, if it is added an odd row, the
result partition is not in the same theory. If it is added two odd rows
together, the new partition is also in the same theory. We can regard adding
an odd row to a partition as a formal operation. And Lemma 3.4 can be
rewritten as follows

######  Lemma 3.9.

The partition $\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots\geq\lambda_{l})$
is in the $B_{n}$ theory and the last two rows have the same parity. If
$a>2b+1$, the symbol of $\lambda$ has the following form

(3.36)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&\alpha_{m}+1&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.37)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&&&\\\
&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&\alpha_{m}+1&&\alpha_{m}+1&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda+1^{2b+1}+1^{2c+1},\,(c<b)$ is

(3.38)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\alpha_{m-b}+1&&\cdots&&\alpha_{m-c+1}+2&&\cdots&&\alpha_{m}+2&\end{array}\right).$

###### Proof.

The formula (3.36) is the result of Lemma 3.3. Since the last two rows have
the same parity, $\lambda_{1}$ is odd which means $l+\lambda_{1}$ is even and
thus the formula (3.37) is the result of the formula (3.23). For the partition
$\lambda^{{}^{\prime}}$, $l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd,
and thus the formula (3.38) is the result of the formula (3.22). ∎

######  Remark 3.10.

Compared the formula (3.38) with (3.31), the contribution to symbol of the two
rows $1^{2b+1}+1^{2c+1}$ is

(3.39) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}=\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{0\cdots
0}_{c}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{0\cdots 0\;\;0\cdots 0}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots 1}_{c}\end{array}\Bigg{)}.$

Formally, the first term on the right side can be regarded as the contribution
of the row $1^{2b+1}$ by comparing the formula (3.37) with (3.31). And the
second term can be regarded as the contribution of the row $1^{2c+1}$ by
comparing the formula (3.38) with (3.32).

####  3.2.1. Closed formula of symbol

We summary the remarks of Lemmas 3.7 and 3.9 as follows333Here, we discuss the
contribution to symbol of odd rows formally. We will find that this method is
reasonable after introducing the map $X_{S}$ (4.7)in Section 5.

Contribution to symbol of the $i$ th row
---
Parity of row | Parity of $i$ | 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 | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$

To compute symbol, the contribution to symbol of the first row of a partition
should be calculated as an initial condition besides these rules in the above
table. Although, the contribution of the first row can be calculated directly,
there is another method which is particularly revealing. According to the
formula (2.29), the symbol of the first row $1^{l}$ is equal to the symbol of
the partition $1^{l}+0^{l}$. By using the formula (3.27), its contribution is

(3.40) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{0\;\;0\cdots\cdots
0}^{(l+1)/2}\\\ \;\;\;\underbrace{1\cdots\cdots
1}_{(l-1)/2}\end{array}\Bigg{)}$

which is consistent with the remark of Lemma 3.9. We draw the conclusion that
the contribution to symbol of the first row can be seen as $i=1$ case in the
above table.

We can describe the above table concisely as follows

Rules ($B_{n}$): Formally, a row in a partition contribute ’1’ in succession
from right to left in the same row of the symbol. The contribution of
adjoining rows with the different parity occupy the same row of symbol,
otherwise occupy the other row. The number of ’1’ contributed by even row is
one half length of the row. The number of ’1’ contributed by the first odd row
of a pairwise patten is one half of the number which is the length of the row
plus one. The number of ’1’ contributed by the second odd row of a pairwise
patten is one half of the number which is the length of the row minus one.

Compared to original definition of symbol, theses rules are easy to remember
and convenient to operator on. According to the rules, we give a closed
formula of the symbol of a rigid partition in the $B_{n}$ theory

######  Proposition 5.

For a partition $\lambda=m^{n_{m}}{(m-1)}^{n_{m-1}}\cdots{1}^{n_{1}}$ in the
$B_{n}$ theory, we introduce the following notations

$\Delta^{T}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i+1}}{2}),\quad
P^{T}_{i}=\frac{1+\pi_{i}}{2}$

where the superscript $T$ indicate it is related to the top row of the symbol
and

$\pi_{i}=(-1)^{\sum^{m}_{k=i}n_{k}}\cdot(-1)^{i}.$

Other parallel notations

$\Delta^{B}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i}}{2}),\quad
P^{B}_{i}=\frac{1-\pi_{i}}{2}$

where the superscript $B$ indicates it is related to the bottom row of the
symbol and $P^{b}_{i}$ is a projection operator similar to $P^{t}_{i}$.
Notating the symbol of $\lambda$ as $\sigma(\lambda)$, then we have

(3.41)
$\sigma(\lambda)=\sum^{m}_{i=1}\big{\\{}\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{P^{T}_{i}\Delta^{T}_{i}}\\\
\;\;\;\underbrace{0\cdots 0\;\;0\cdots
0}_{\frac{l-1}{2}}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{0\;\;0\cdots
0\;\;0\cdots 0}^{\frac{l+1}{2}}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1}_{P^{B}_{i}\Delta^{B}_{i}}\end{array}\Bigg{)}\big{\\}}.$

###  3.3. Symbol of rigid partitions in the $C_{n}$ theory

The contents of this subsection and next subsection are parallel to the
previous subsection. First, we prove the following proposition

######  Proposition 6.

The longest two rows in a rigid $C_{n}$ partition 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.

###### Proof.

For a partition in the $C_{n}$ theory, odd integers appear an even number of
times. If the first two rows are of different parities, then the number of
boxes of the difference between them will be odd. It imply that part ’1’
appear odd number of times which is a contradiction. So we have proved that
the length of the first two rows are either both odd or even. In the same way,
we can prove that the next two rows will be of the same parity. This pairwise
pattern then continues. ∎

Using the above proposition, we can refine Lemmas 3.1 and 3.3 as follows

######  Lemma 3.11.

For a partition
$\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots\geq\lambda_{l})$ in the $C_{n}$
theory, the last two rows have the same parity. $l$ is the length of the
partition including the extra 0 if necessary. If $a>b$, the symbol of
$\lambda$ has the following form

(3.42)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+1}&&\cdots&&\beta_{m}&\end{array}\right)=\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m}&&\cdots&&\alpha_{m}\\\
&\cdots&&\alpha_{m}&&\cdots&&\alpha_{m}&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b}$ is

(3.43)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+1}+1&&\cdots&&\beta_{m}+1&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda^{{}^{\prime}}+1^{2c}$ is

(3.44)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\beta_{m}+1&\end{array}\right).$

###### Proof.

According to remark of Definition 3, $l$ is even. According to Lemma 3.3, we
get the formula (3.42). According to Proposition 6, $\lambda_{1}$ is even
which means $l+\lambda_{1}$ is even, and thus the formula (3.44) is the result
of the formula (3.3). Similarly, for partition $\lambda^{{}^{\prime}}$,
$l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd, thus the formula (3.32)
is the result of the formula (3.2). ∎

######  Remark 3.12.

  1. (1)

Compared the formula (3.43) with (3.42), the contribution to symbol of row
$1^{2b}$ is

(3.45) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{0\cdots 0}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1\;1\cdots 1}_{b}\end{array}\Bigg{)}$

  2. (2)

Compared the formula (3.44) with (3.43), the contribution to symbol of row
$1^{2c}$ is

(3.46) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{0\cdots
0\;0\cdots 0}_{b}\end{array}\Bigg{)}$

Next, we discuss the contributions to symbol of odd rows formally.

######  Lemma 3.13.

For a partition
$\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots\geq\lambda_{l})$ in the $C_{n}$
theory, the last two rows have the same parity. $l$ is the length of the
partition including the extra 0 in the last part of partition if necessary. If
$a>2b+1$, the symbol of $\lambda$ has the following form

(3.47)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.48)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&&&\\\
&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}&&\alpha_{m}&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda+1^{2b+1}+1^{2c+1},\,(c<b),$ is

(3.49)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\alpha_{m-b}&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1&\end{array}\right).$

###### Proof.

The formula (3.47) is the result of Lemma 3.3. Since $\lambda_{1}$ is even
which means $l+\lambda_{1}$ is even, thus the formula (3.48) is the result of
the formula (3.23). For partition $\lambda^{{}^{\prime}}$,
$l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd, thus the formula (3.49)
is the result of the formula (3.22). ∎

######  Remark 3.14.

Compared the formula (3.49) with (3.42), the contribution to symbol of
$1^{2b+1}+1^{2c+1}$ is

(3.50) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}=\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{0\cdots
0}_{c}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{0\cdots 0\;\;0\cdots 0}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots 1}_{c}\end{array}\Bigg{)}.$

Formally, the first term on the right side can be regarded as the contribution
of row $1^{2b+1}$ by comparing formula (3.48) with (3.42). And the second term
can be regarded as the contribution of row $1^{2c+1}$ by comparing formula
(3.49) with (3.43).

We can summary the above results as the following stable

Contribution to symbol of the $i$ th row
---
Parity of row | Parity of $i+1$ | 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 | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$

It is easy to check that the contributions to symbol of the first two rows are
consistent with this table. Note that this table is the same with the table in
the $B_{n}$ case except check of the parity of the index $i+1$.

###  3.4. Symbol of rigid partitions in the $D_{n}$ theory

We can prove the following proposition similar to the proof of Proposition 4.

######  Proposition 7.

The longest row in a rigid $D_{n}$ partition 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.

Using the above proposition, we can refine Lemma 3.1 and 3.3 as follows

######  Lemma 3.15.

For a partition
$\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots\geq\lambda_{l})$ in the $D_{n}$
theory, the last two rows have same parity. $l$ is the length of the partition
including the extra 0 as the last part of partition if necessary. If $a>b$,
the symbol of $\lambda$ has the following form

(3.51)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+2}&&\cdots&&\beta_{m+1}&\end{array}\right)=\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m}&&\cdots&&\alpha_{m}\\\
&\cdots&&\alpha_{m}-1&&\cdots&&\alpha_{m}-1&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b}$ is

(3.52)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m}\\\
&\cdots&&\beta_{m-b+2}+1&&\cdots&&\beta_{m+1}+1&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda^{{}^{\prime}}+1^{2c}$ $(c<b)$ is

(3.53)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\beta_{m-b+2}+1&&\cdots&&\beta_{m-c+2}+1&&\cdots&&\beta_{m+1}+1&\end{array}\right).$

###### Proof.

According to the remark of Definition 3, $l$ is odd. Since $t=1$ in the
$D_{n}$ theory, combining Lemma 3.3, we get the formula (3.51). According to
Proposition 7, $\lambda_{1}$ is odd which means $l+\lambda_{1}$ is even, thus
the formula (3.53) is the result of the formula (3.3). Similarly, for
partition $\lambda^{{}^{\prime}}$,
$l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd, thus the formula (3.52)
is the result of the formula (3.2). ∎

######  Remark 3.16.

  1. (1)

Compared the formula (3.52) with (3.51), the contribution to symbol of the row
$1^{2b}$ is

(3.54) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{0\cdots 0}_{c}\end{array}\Bigg{)}$

  2. (2)

Compared the formula (3.53) with (3.52), the contribution to symbol of the row
$1^{2c}$ is

(3.55) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{0\cdots 0\;\;0\cdots 0}^{b}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots 1}_{c}\end{array}\Bigg{)}$

Next, we discuss the contribution to symbol of the odd rows formally.

######  Lemma 3.17.

For a partition
$\lambda(\lambda_{1}=\cdots=\lambda_{a}>\cdots\geq\lambda_{l})$ in the $D_{n}$
theory, the last two rows have the same parity. $l$ is the length of the
partition including the extra 0 if necessary for the computation of symbol. If
$a>2b+1$, the symbol of $\lambda$ has the following form

(3.56)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}&&\alpha_{m-b+1}&&\cdots&&\alpha_{m-1}&&\alpha_{m}\\\
&\cdots&&\alpha_{m-b}-1&&\alpha_{m-b+1}-1&&\cdots&&\alpha_{m-1}-1&&\alpha_{m}-1&\end{array}\right).$

The symbol of the partition $\lambda^{{}^{\prime}}=\lambda+1^{2b+1}$ is

(3.57)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\alpha_{m-b+1}+1&&\cdots&&\alpha_{m-1}+1&&&&\\\
&\cdots&&\alpha_{m-b}-1&&\alpha_{m-b+1}-1&&\cdots&&\alpha_{m-1}-1&&\alpha_{m}-1&&\alpha_{m}-1&\end{array}\right).$

The symbol of the partition
$\lambda^{{}^{\prime\prime}}=\lambda+1^{2b+1}+1^{2c+1},\,(c<b)$ is

(3.58)
$\left(\begin{array}[]{@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}}\cdots&&\cdots&&\alpha_{m-b}+1&&\cdots&&\alpha_{m-c+1}+1&&\cdots&&\alpha_{m}+1\\\
&\cdots&&\alpha_{m-b}&&\cdots&&\alpha_{m-c+1}&&\cdots&&\alpha_{m}&\end{array}\right).$

###### Proof.

The formula (3.56) is the result of Lemma 3.3. Since $\lambda_{1}$ is odd ,
$l+\lambda_{1}$ is even, thus the formula (3.57) is the result of the formula
(3.23). For partition $\lambda^{{}^{\prime}}$,
$l+\lambda^{{}^{\prime}}_{1}=l+\lambda_{1}+1$ is odd, thus the formula (3.58)
is the result of the formula (3.22). ∎

######  Remark 3.18.

Compared the formula (3.58) with (3.56), the contributions to symbol of rows
$1^{2b+1}+1^{2c+1}$ is

(3.59) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}=\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{0\cdots
0}_{c}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{0\cdots 0\;\;0\cdots 0}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots
0\;\underbrace{1\cdots 1}_{c}\end{array}\Bigg{)}.$

Formally, the first term on the right side of the above formula can be
regarded as the contribution to symbol of the $1^{2b+1}$ by comparing the
formula (3.57) with (3.51). And the second term can be regarded as the
contribution of $1^{2c+1}$ by comparing the formula (3.58) with (3.57).

We summary these results as the follows table

Contribution to symbol of the $i$ th row
---
Parity of row | Parity of $i+2$ | 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 | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$

It is easy to check that the contribution to symbol of the first two rows as
initial condition are consistent with this table. Note that this table is
exact with the table in the $B_{n}$ case.

###  3.5. Closed formula of symbols in the $B_{n}$, $C_{n}$, and $D_{n}$
theories

With the experiences learned in the previous subsections, we find that there
are some common characteristics of the computational rules of symbol in
$B_{n}$, $C_{n}$, and $D_{n}$ theories. From Lemmas 3.1 and 3.4, the
difference of the construction of symbol in different theories lie in the
definition of the length of partition. From Definition 3, we should append a 0
as the last part of partition $\lambda$ in the $D_{n}$ theory and in the
$C_{n}$ theory with odd length of the partition. For a partition
$\lambda=m^{n_{m}}{(m-1)}^{n_{m-1}}\cdots{1}^{n_{1}}$, we denote

$n_{0}=\frac{1-(-1)^{\sum^{m}_{k=1}n_{k}}}{2}+t$

where $t=-1$ for $B_{n}$ theory, $t=0$ for $C_{n}$ theory, and $t=1$ for
$D_{n}$ theory. Then the length $l$ of the partition in Lemmas 3.1 and 3.4 is
equal to the length of the following partition

$\lambda=m^{n_{m}}{(m-1)}^{n_{m-1}}\cdots{1}^{n_{1}}0^{n_{0}}.$

By using Lemmas 3.1 and 3.4, the contributions to symbol of a pairwise patten
fit into the same forms as shown in the formulas
(3.34),(3.35),(3.39),(3.45),(3.46),(3.50),(3.54),(3.55), and (3.59).

Note that the tables of computational rules of symbols in the $B_{n}$,
$C_{n}$, and $D_{n}$ theories are exact in the same pattern. We can describe
these tables uniformly as follows444Here, we discuss the contribution to
symbol of the odd rows formally. We will find that this formal method is
reasonable after we introduce the maps $X_{S}(\ref{XS}),Y_{S}(\ref{YS})$ in
Section 5.

Contribution to symbol of the $i$ th row
---
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 | $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\ \;\;\;0\cdots\underbrace{1\;\;1\cdots 1}_{L}\end{array}\Bigg{)}$ | $\frac{1}{2}(\sum^{m}_{k=i}n_{k}-1)$

We can describe the above table concisely as the following rules.

Rules : Formally, a row in a partition contribute ’1’ in succession from right
to left in the same row of the symbol. The contribution of adjoining rows with
the different parity occupy the same row of symbol, otherwise occupy the other
row. The number of ’1’ contributed by even row is one half length of the row.
The number of ’1’ contributed by the first odd row of a pairwise patten is one
half of the number which is the length of the row plus one. The number of ’1’
contributed by the second odd row of a pairwise patten is one half of the
number which is the length of the row minus one.

According to the above table, it is easy to get a closed formula of symbol for
the rigid partition in the $B_{n}$, $C_{n}$, and $D_{n}$ theories

######  Proposition 8.

For a partition $\lambda=m^{n_{m}}{(m-1)}^{n_{m-1}}\cdots{1}^{n_{1}}$, we
introduce two notations

$\Delta^{T}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i+1}}{2}),\quad\quad
P^{T}_{i}=\frac{1+\pi_{i}}{2}$

where the superscript $T$ indicates it is related to the top row of the symbol
and

$\pi_{i}=(-1)^{\sum^{m}_{k=i}n_{k}}\cdot(-1)^{i+1+t},$

for $B_{n}(t=-1)$, $C_{n}(t=0)$, and $D_{n}(t=1)$ theories. Other parallel
notations

$\Delta^{B}_{i}=\frac{1}{2}(\sum^{m}_{k=i}n_{k}+\frac{1+(-1)^{i}}{2}),\quad\quad
P^{B}_{i}=\frac{1-\pi_{i}}{2}$

where the superscript $B$ indicates it is related to the bottom row of the
symbol and $P^{B}_{i}$ is a projection operator similar to $P^{T}_{i}$. Then
the symbol $\sigma(\lambda)$ is

(3.60)
$\sigma(\lambda)=\sum^{m}_{i=1}\big{\\{}\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{P^{T}_{i}\Delta^{T}_{i}}\\\
\;\;\;\underbrace{0\cdots 0\;\;0\cdots
0}_{l+t}\end{array}\Bigg{)}+\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{0\;\;0\cdots
0\;\;0\cdots 0}^{l}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1}_{P^{B}_{i}\Delta^{B}_{i}}\end{array}\Bigg{)}\big{\\}}$

with $l=(m+(1-(-1)^{m})/2)/2$.

##  4\. Applications

As an invariant of the partition, symbol can be used to study the $S$ duality
pair of rigid unipotent partition [4] [5]. In [6], we propose a simple rule to
compute symbols of partitions with only even rows. Symbol of a general
partition in $B_{n},C_{n}$, and $D_{n}$ theories can be calculated by reducing
a partition into two partitions with only even rows through two maps $X_{S}$
and $Y_{S}$. These two maps can be explained by the table in the previous
section.

###  4.1. Maps: $X_{S}$ and $Y_{S}$

There are partitions with only even rows in the $C_{n}$ and $D_{n}$ theories.
For partitions with only even rows, the rules in the previous section recover
the following Rule proposed in [6]

Rule: For a partition with only even rows in the $C_{n}$ and $D_{n}$ theories,
the row with $2m$ boxes contribute $m$ ’1’ in sequence from right to left in
the same row of symbol, while other entries of the symbol are ’0’. The
contribution to symbol of the adjoining even rows are formed in the same way,
excepting the ’1’s occupy another row of symbol.

A general partition $\lambda$ in the $B_{n},C_{n}$, and $D_{n}$ theories can
be reduced to two partitions with only even rows by the maps $X_{S}$ and
$Y_{S}$ [5]. The symbol of the partition $\lambda$ is the sum of the symbols
of these two partitions. Before introducing these two maps, we prove the
following proposition.

######  Proposition 9.

For a partition in the $B_{n}$ theory and a partition with only even rows in
the $C_{n}$ theory, we have $\alpha_{i}\leq\beta_{i+t}$. For a partition in
the $D_{n}$ theory and a partition with only odd rows in the $C_{n}$ theory,
we have $\alpha_{i}\geq\beta_{i+t}$.

###### Proof.

For a partition in the $B_{n}$ theory, according to the formula (3.40), the
symbol of the first row with $2m+1$ boxes is

$\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\
\;\;\;\underbrace{1\cdots 1\;\;1\cdots 1}_{m}\end{array}\Bigg{)}.$

So, we draw the conclusion for the first row.

According to Proposition 4, the following two rows of the first row are either
both odd or even length. By using the formula (3.32), the contributions to
symbol of two even rows $1^{2b}+1^{2c},(b>c),$ are

(4.1) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1\;1\cdots 1}_{b}\end{array}\Bigg{)}.$

By using the formula (3.37), the contributions to symbol of two odd rows
$1^{2b+1}+1^{2c+1},(b>c),$ are

(4.2) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;\overbrace{1\cdots
1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}.$

Note that for these three rows, we have $\alpha_{i}=\beta_{i-1}+1$, $i\geq
m-c+1$. So, we draw the conclusion for the first three rows of the partition.

This pairwise patten then continue. If the partition has an odd number of
rows, we draw the conclusion. If the partition has an even number of rows,
then the last row is even. By using the formula (3.32), the contribution to
symbol of the last row is

$\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\
\;\;\;0\cdots 0\;\;1\cdots 1\end{array}\Bigg{)},$

which has no influence on the conclusion.

For a partition with only odd rows in the $C_{n}$ theory, by using the formula
(3.48), the contribution to symbol of the first two odd rows
$1^{2b+1}+1^{2c+1},(b>c),$ is

(4.3) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;\overbrace{1\cdots
1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}.$

This pairwise patten then continue, which imply the conclusion.

For a partition with only even rows in the $C_{n}$ theory, by using the
formula (3.43), they contribute to symbol of the first two even row
$1^{2b}+1^{2c},(b>c),$

(4.4) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1\;1\cdots 1}_{b}\end{array}\Bigg{)}.$

This pairwise patten then continue. If the partition has an even number of
row, then we draw the conclusion. If the partition has an odd number of row,
the last row is even according to Proposition 6. By using the formula (3.43),
the contribution to symbol of the last row is

$\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\
\;\;\;0\cdots 0\;\;1\cdots 1\end{array}\Bigg{)}$

which have no influence on the conclusion.

For a partition in the $D_{n}$ theory, the contribution to symbol of the first
row with $2m$ boxes is

$\Bigg{(}\\!\\!\\!\begin{array}[]{c}\overbrace{1\;\;1\cdots 1\;\;1\cdots
1}^{m}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\end{array}\Bigg{)}.$

So, we draw the conclusion for the first row.

According to Proposition 7, the following two rows of the first row are either
both of odd length or both of even. By using the formula (3.53), the
contributions to symbol of two even rows $1^{2b}+1^{2c},(b>c),$ are

(4.5) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots
0\;\;\overbrace{1\cdots 1}^{c}\\\ \;\;\;0\cdots 0\;\;\underbrace{1\cdots
1\;1\cdots 1}_{b}\end{array}\Bigg{)}.$

Note that for these three rows, we have $\alpha_{i}=\beta_{i+1}-1$ for $i\geq
m-c+1$. By using the formula (3.58), the contributions to symbol of two odd
rows $1^{2b+1}+1^{2c+1},(b>c),$ is

(4.6) $\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;\overbrace{1\cdots
1\;\;1\cdots 1}^{b+1}\\\ \;\;\;0\cdots 0\;\;0\cdots 0\;\underbrace{1\cdots
1}_{c}\end{array}\Bigg{)}.$

So, we draw the conclusion for the first three rows in the partition.

This pairwise patten then continue. If the partition has an odd number of row,
we draw the conclusion. If the partition has an even number of row, the last
row is even. By using the formula (3.53), the contribution to symbol of the
last row has the following form

$\Bigg{(}\\!\\!\\!\begin{array}[]{c}0\;\;0\cdots 0\;\;0\cdots 0\\\
\;\;\;0\cdots 0\;\;1\cdots 1\end{array}\Bigg{)}$

which have no influence on the conclusion. ∎

######  Remark 4.1.

For these partitions in the proposition, the parity of the first row determine
the structure of symbol.

Symbol can be seen as an invariant of the partition. This proposition imply
two kinds of maps preserving symbol possibly. The first one is the map between
rigid partitions in the $B_{n}$ theory and rigid partitions with only even
rows in the $C_{n}$ theory. The second one is the map between rigid partitions
in the $D_{n}$ theory and rigid partitions with only odd rows in the $C_{n}$
theory. We propose one map for each case.

First, we propose the following map which take a special rigid partition with
only odd rows in the $B_{n}$ theory to a special rigid partition (2.1) with
only even rows in the $C_{n}$ theory

(4.7) $\displaystyle X_{S}:$ $\displaystyle
m^{2n_{m}+1}\,(m-1)^{2n_{m-1}}\,(m-2)^{2n_{m-2}}\cdots 2^{n_{2}}\,1^{2n_{1}}$
$\displaystyle\mapsto$ $\displaystyle
m^{2n_{m}}\,(m-1)^{2n_{m-1}+2}\,(m-2)^{2n_{m-2}-2}\cdots
2^{n_{2}+2}\,1^{2n_{1}-2}\,,$

where $m$ has to be odd in order for the first object to be a partition in the
$B_{n}$ theory. It is clear that the map is a bijection so that $X_{S}^{-1}$
is well defined. This symbol preserving map is exact the map $X_{S}$ in [5].
The map (4.7) is essentially the ’$p_{C}$ collapse’ described in [1]. The
inver map $X_{S}^{-1}$ is essentially the ’$p^{B}$ expansion’ described in
[1].

Second, we propose the following map which take a special rigid partition with
only odd rows in the $C_{n}$ theory to a special rigid partition with only
even rows in the $D_{n}$ theory

(4.8) $\displaystyle Y_{S}:$ $\displaystyle
m^{2n_{m}+1}\,(m-1)^{2n_{m-1}}\,(m-2)^{2n_{m-2}}\cdots 2^{n_{2}}\,1^{2n_{1}}$
$\displaystyle\mapsto$ $\displaystyle
m^{2n_{m}}\,(m-1)^{2n_{m-1}+2}\,(m-2)^{2n_{m-2}-2}\cdots
2^{n_{2}-2}\,1^{2n_{1}+2}\,.$

where $m$ has to be even in order for the first element to be a $C_{k}$
partition. This symbol preserving map is exact the map $Y_{S}$ in [5]. This is
a bijection. The map (4.7) is essentially the ’$p_{D}$ collapse’ described in
[1]. The inver map $Y_{S}^{-1}$ is essentially the ’$p^{C}$ expansion’
described in [1].

###  4.2. $S$-duality maps for rigid surface operators

For the $B_{n}$ , $C_{n}$, and $D_{n}$ theories, the rigid semisimple
conjugacy classes $S$ correspond to diagonal matrices with elements $+1$ and
$-1$ along the diagonal [Gw08]. The centraliser of the diagonal matrices $S$
as follows (at the Lie algebra level)

$\displaystyle\mathrm{so}(2n{+}1)$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{so}(2k{+}1)\oplus\mathrm{so}(2n-2k)\,,$ (4.9)
$\displaystyle\mathrm{sp}(2n)$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{sp}(2k)\oplus\mathrm{sp}(2n-2k)\,,$
$\displaystyle\mathrm{so}(2n)$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{so}(2k)\oplus\mathrm{so}(2n-2k)\,.$

The rigid semisimple surface operators correspond to pairs of partitions
$(\lambda^{\prime};\lambda^{\prime\prime})$ [Gw08]. In the $B_{n}$ case,
$\lambda^{\prime}$ is a rigid $B_{k}$ partition and $\lambda^{\prime\prime}$
is a rigid $D_{n-k}$ partition. For the $C_{n}$ theories, $\lambda^{\prime}$
is a rigid $C_{k}$ partition and $\lambda^{\prime\prime}$ is a rigid $C_{n-k}$
partition. For the $D_{n}$ theories, $\lambda^{\prime}$ is a rigid $D_{k}$
partition and $\lambda^{\prime\prime}$ is a rigid $D_{n-k}$ partition. In the
theories under consideration, the rigid unipotent surface operators can be
seen as a special case $\lambda^{\prime\prime}=0$.

The Langlands dual group of $B_{n}(SO(2n+1))$ is $C_{n}(Sp(2n))$. Thus it is
expected

$BC:(\lambda_{B};\rho_{D})_{B}\rightarrow(\lambda^{\prime}_{C};\rho^{{}^{\prime\prime}}_{C})_{C}$

where the subscripts $B$ imply it is a partition in the $B_{n}$ theory. This
map $BC$ preserve the symbol in the sense that the addition of symbols on the
two sides are equal

$\sigma^{B}(\lambda)+\sigma^{D}(\rho)=\sigma^{C}(\lambda^{\prime})+\sigma^{C}(\rho^{{}^{\prime\prime}}).$

The group $D_{n}(SO(n))$ is self-duality, which implies

$DD:(\lambda_{D};\rho_{D})_{D}\rightarrow(\lambda^{\prime}_{D};\rho^{{}^{\prime\prime}}_{D})_{D}.$

Using maps $X_{S}$ and $Y_{S}$, we can propose dual maps of rigid surface
operators between different theories. The map $X_{S}$ imply that the special
rigid unipotent surface operators in the $B_{n}$ and $C_{n}$ theories are
related by $S$-duality. For rigid unipotent surface operators in $B_{n}$
theory, the following map proposed in [5]

(4.10)
$WB:\,\,\,(\lambda_{B},\emptyset)_{B}\rightarrow(\lambda_{old}+\lambda_{even},\emptyset)\rightarrow(X_{S}\lambda_{old},\lambda_{even})_{C}.$

First, we split the Young tableau into one Young tableau constructed from the
odd rows $\lambda_{old}$ and one Young tableau constructed from the even rows
$\lambda_{even}$. The first tableaux $\lambda_{old}$ is always a special rigid
partition in the $B_{k}$ theory and the map $X_{S}$ (4.7) turns this into a
special rigid partition in the $C_{k}$ theory. While the second partition
$\lambda_{even}$ already corresponds to a special rigid partition in the
$C_{n-k}$ theory and is left untouched.

For rigid unipotent surface operators in $C_{n}$ theory, the following map is
proposed

(4.11)
$WC:\,\,\,(\lambda_{C},\emptyset)_{C}\rightarrow(\lambda_{old}+\lambda_{even},\emptyset)\rightarrow(X_{S}^{-1}\lambda_{even},Y_{S}\lambda_{old})_{B},$

For rigid unipotent surface operators in $C_{n}$ theory, the following map is
proposed

(4.12)
$WD:\,\,\,(\lambda_{D},\emptyset)_{D}\rightarrow(\lambda_{\mathrm{old}}+\lambda_{\mathrm{even}},\emptyset)\rightarrow(\lambda_{\mathrm{even}},Y_{S}\lambda_{\mathrm{old}})_{D}.$

These are more maps proposed in [5][6], which we refer the reader for more
details.

Comparing the table in Section 3.2.1 with that in Section 3.3, we find that
contributions to symbol of the even rows in the $B_{n}$ theory satisfy the
same rules with even rows in the $C_{n}$ theory. So we have

(4.13)
$\sigma^{B}(\lambda)=\sigma^{B}(\lambda_{old})+\sigma^{C}(\lambda_{even}).$

Comparing the table in Section 3.3 with that in Section 3.4, we find that the
odd rows in the $C_{n}$ theory satisfy same rules with old rows in the $D_{n}$
theory. So we have

(4.14)
$\sigma^{D}(\lambda)=\sigma^{D}(\lambda_{even})+\sigma^{C}(\lambda_{old}).$

Using the maps (4.13), (4.14) and another two maps $X_{S}$, $Y_{S}$ preserving
symbol, we can prove the following identities proposed in [6]

  * •

For a rigid semisimple surface operator $(\lambda,\rho)$ in the $B_{n}$
theory, the symbol is

(4.15)
$\sigma^{B}_{(\lambda,\rho)}=\sigma^{B}_{(\lambda)}+\sigma^{D}_{(\rho)}=\sigma^{C}_{(\lambda_{even})}+\sigma^{C}_{(X_{S}\lambda_{old})}+\sigma^{D}_{(\rho_{even})}+\sigma^{D}_{(Y_{S}\rho_{old})}.$

  * •

For a rigid semisimple surface operator $(\lambda,\rho)$ in the $C_{n}$
theory, the symbol is

(4.16)
$\sigma^{C}_{(\lambda,\rho)}=\sigma^{C}_{(\lambda)}+\sigma^{C}_{(\rho)}=\sigma^{C}_{(\lambda_{even})}+\sigma^{D}_{(Y_{s}\lambda_{old})}+\sigma^{C}_{(\rho_{even})}+\sigma^{D}_{(Y_{S}\rho_{old})}.$

  * •

For a rigid semisimple surface operator $(\lambda,\rho)$ in the $D_{n}$
theory, the symbol is

(4.17)
$\sigma^{D}_{(\lambda,\rho)}=\sigma^{D}_{(\lambda)}+\sigma^{D}_{(\rho)}=\sigma^{D}_{(\lambda_{even})}+\sigma^{D}_{(Y_{s}\lambda_{old})}+\sigma^{D}_{(\rho_{even})}+\sigma^{D}_{(Y_{S}\rho_{old})}.$

Using the identities (4.15), (4.16), and (4.17), we can prove the maps $WB$
(4.10), $WC$(4.12), and $WC$(4.12) preserve symbol.

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