Symmetric polynomials of partitions and isomorphisms of invariant algebras

Changchang Xi∗ and Jinbi Zhang

††∗Corresponding author’s Email<EMAIL_ADDRESS>Fax: 0086 10 68903637.††2020
Mathematics Subject Classification: Primary 16W22, 16S50, 05A17; Secondary
15A30, 11C08, 16K99††Keywords: Elementary symmetric polynomial; Greatest
common divisor; Invariant matrix algebra; Isomorphism; Partition of integer;
Permutation

###### Abstract

A new equivalence relation on partitions is introduced to study the
isomorphism problem of invariant matrix algebras. For an algebraically closed
field $R$ of characteristic $p\geq 0$ and an element $\sigma$ in the symmetric
group $\Sigma_{n}$ of degree $n$, let $S_{n}(\sigma,R)$ be the
$\sigma$-invariant algebra of $n\times n$ matrices over $R$. Suppose
$\sigma\in\Sigma_{n}$ and $\tau\in\Sigma_{m}$ are of cycle types
$\lambda=(\lambda_{1},\cdots,\lambda_{s})$ and $\mu=(\mu_{1},\cdots,\mu_{t})$,
respectively, such that $p\nmid\lambda_{i}$ and $p\nmid\mu_{j}$ for $1\leq
i\leq s$ and $1\leq j\leq t$. We prove that $S_{n}(\sigma,R)$ and
$S_{m}(\tau,R)$ are isomorphic if and only if $m=n$ and both $\lambda$ and
$\tau$ have the same partition polynomial. The two algebras are Morita
equivalent if and only if
$d_{\lambda}\,\epsilon_{\lambda}(1)=d_{\mu}\,\epsilon_{\mu}(1)$, where
$\epsilon_{\lambda}(x)$ is the partition polynomial of $\lambda$ and
$d_{\lambda}$ stands for the greatest common divisor of
$\lambda_{1},\cdots,\lambda_{s}$.

##  1 Introduction

Let $R$ be a unitary ring and $n$ be a natural number, we set
$[n]:=\\{1,2,\cdots,n\\}$ and denote by $\Sigma_{n}$ the symmetric group of
permutations on $[n]$, and by $M_{n}(R)$ the ring of $n\times n$ matrices over
$R$.

For a subgroup $G$ of $\Sigma_{n}$, the $G$-_invariant matrix ring_ over $R$
of degree $n$ is defined by

$S_{n}(G,R):=\\{(a_{ij}\in M_{n}(R)\mid
a_{ij}=a_{(i)\sigma,(j)\sigma},\sigma\in G,1\leq i,j\leq n\\},$

where $(i)\sigma$ stands for the image of $i$ under $\sigma$. If
$G=\langle\sigma\rangle$ for $\sigma\in\Sigma_{n}$, we briefly write
$S_{n}(\sigma,R)$ for $S_{n}(G,R).$

Invariant algebras and rings have been studied in different mathematical
contexts. For instance, on classical invariant theory, the reader is refereed
to the book [9], on matrix theory, to [6, 7], and on groups acting on
associative algebras, to [5]. Also, the algebra $S_{n}(R)$ (see [10] for
notation) of centrosymmetric matrices is just the algebra $S_{n}(\sigma,R)$
for $\sigma$ given by $i\mapsto n+1-i$. This algebra has been studied by many
authors (for example, see [7, 8, 10]) and applied to engineering problems and
quantum physics (see [3]). Recently, invariant matrix algebras are studied as
centralizer rings in [11], where it is shown that $S_{n}(\sigma,R)$ is always
a cellular algebra if $R$ is an algebraically closed field. But even in this
case, $S_{n}(\sigma,R)$ may not be semisimple, as examples show in [11].

In this note, we first show when $S_{n}(\sigma,R)$ is semisimple in case of
$R$ being a field. We then investigate the isomorphism problem:

Let $R$ be an algebraically closed field and $\sigma,\tau\in\Sigma_{n}$. What
are the necessary and sufficient conditions for $S_{n}(\sigma,R)$ and
$S_{n}(\tau,R)$ to be isomorphic?

We will give a complete answer to this question in the semisimple case. Our
conditions are presented in terms of the numerical data of cycle types of
permutations. Here the strategy of handling this situation is, theoretically,
to compare the components of Wedderburn decompositions of the two algebras. In
this course, not only combinatorics of partitions, but also elementary
symmetric polynomials over the canonical quarter-ring of natural numbers play
an intriguing role.

To state our result specifically, we first introduce a few notions and
terminologies. We write $\sigma\in\Sigma_{n}$ as a product of disjoint cycles,
say $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{s}$, where $\sigma_{i}$ is a
$\lambda_{i}$-cycle with $\lambda_{i}\geq 1$. This gives a decomposition of
$n$. By assuming $\lambda_{1}\geq\cdots\geq\lambda_{s}\geq 1$, we get a
partition $\lambda=(\lambda_{1},\cdots,\lambda_{s})$ of $n$ with $s$ parts.
This $\lambda$ is called the _cycle type_ of $\sigma$. We then associate
$\lambda$ with the so-called partition polynomial
$\epsilon_{\lambda}(x)\in\mathbb{Z}[x]$, which is defined by elementary
symmetric polynomials $g_{i}(\lambda)$ in $\lambda_{1},\cdots,\lambda_{s}$,
over the canonical bisemigroup of natural numbers (see Definition 3.4). Here,
the polynomial $g_{i}(\lambda)$ is closely related to a matrix norm of the
triangular divisor matrix of $\lambda$, and the multiplication in the
bisemigroup is given by taking the greatest common divisors of integers (see
Section 3 for details).

Our main result now can be stated as follows.

######  Theorem 1.1.

Let $R$ be a field of characteristic $p\geq 0$, let $\sigma\in\Sigma_{n}$ and
$\tau\in\Sigma_{m}$ be of cycle types
$\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{s})$ and
$\mu=(\mu_{1},\mu_{2},\cdots,\mu_{t})$, respectively, and let $d_{\lambda}$
denote the greatest common divisor of these $\lambda_{i}$’s. Then the
following hold.

$(1)$ $S_{n}(\sigma,R)$ is semisimple if and only if $p\nmid\lambda_{i}$ for
$1\leq i\leq s$.

$(2)$ Assume that $R$ is algebraically closed, $p\nmid\lambda_{i}$ for all
$1\leq i\leq s$ and $p\nmid\mu_{j}$ for all $1\leq j\leq t$. Then

${\rm(i)}$ $S_{n}(\sigma,R)\simeq S_{m}(\tau,R)$ if and only if $m=n$ and
$\epsilon_{\lambda}(x)=\epsilon_{\mu}(x)$, where
$\epsilon_{\lambda}(x)\in\mathbb{Z}[x]$ is the partition polynomial of
$\lambda$.

${\rm(ii)}$ $S_{n}(\sigma,R)$ and $S_{m}(\tau,R)$ are Morita equivalent if and
only if $d_{\lambda}\epsilon_{\lambda}(1)=d_{\mu}\epsilon_{\mu}(1)$.

Thus, the isomorphism problem for semisimple, invariant matrix algebras over
an algebraically closed field can be read off from the numerical values of the
elementary symmetric polynomials evaluated at the corresponding partitions
(see Section 3.1 for more details).

This article is outlined as follows. Section 2 is devoted to some basic facts
on invariant matrix algebras, while Section 3 presents a proof of the main
result. As a preparation for the proof, we introduce a polynomial equivalence
relation and triangular divisor matrices of partitions. We then study
combinatorial aspects of the polynomial equivalence relation based on
elementary symmetric polynomials. All of these contribute to the proof of
Theorem 1.1. At the end of this section, we formulate a few questions
suggested by the main result.

##  2 Preliminaries

In this section, we recall basic notions and establish primary facts on
invariant matrix algebras. Throughout the paper, $R$ denotes a unitary ring.

###  2.1 Invariant matrix algebras

Let $G$ be a subgroup of $\Sigma_{n}$, the symmetric group of all permutations
of the set $[n]:=\\{1,2,\cdots,n\\}$, and $\sigma\in\Sigma_{n}$. Then $G$ acts
on $[n]$ naturally. Clearly, $G$ also acts on $M_{n}(R)$ by

$M_{n}(R)\times G\longrightarrow M_{n}(R),\;(a_{ij})\cdot
g:=(a_{(i)g,(j)g}),\;(a_{ij})\in M_{n}(R),g\in G.$

We define the $G$-_invariant matrix algebra_ , denoted $S_{n}(G,R)$, to be the
$G$-fixed points in $M_{n}(R)$:

$S_{n}(G,R):=(M_{n}(R))^{G}:=\\{a\in M_{n}(R)\mid
a_{ij}=a_{(i)\sigma,(j)\sigma},\;\forall\;\sigma\in
G,i,j\in[n]\\}=\bigcap_{\sigma\in G}S_{n}(\sigma,R),$

where $S_{n}(\sigma,R):=\\{a=(a_{ij})\in M_{n}(R)\mid
a_{ij}=a_{(i)\sigma,(j)\sigma},1\leq i,j\leq n\\}$ is called the
$\sigma$-_invariant matrix ring_ over $R$ of degree $n$.

Let $e_{ij}$ be the matrix units of $M_{n}(R)$ with $i,j\in[n]$, and let
$c_{n}(\sigma):=e_{1,(1)\sigma}+e_{2,(2)\sigma}+\dots+e_{n,(n)\sigma}$ be the
permutation matrix in $M_{n}(R)$, related to $\sigma$. If the subscript $n$ is
clear from the context, we simply write $c_{\sigma}$ for $c_{n}(\sigma).$ In
$M_{n}(R)$, we have $c_{\sigma}c_{\tau}=c_{\sigma\tau}$,
$c^{\prime}_{\sigma}=c_{\sigma^{-1}}=c^{-1}_{\sigma}$ and
$(c_{\sigma})_{ij}=\delta_{\sigma(i),j}$ for $i,j\in[n],$ where $a^{\prime}$
is the transpose of the matrix $a$ and $\delta_{ij}$ is the Kronecker symbol.

######  Lemma 2.1.

$(1)$ Let $a=(a_{ij})\in M_{n}(R)$. Then $a\in S_{n}(G,R)$ if and only if
$c_{\sigma}a=ac_{\sigma}$ for $\sigma\in G$. In particular,
$Z(R)[c_{\sigma}]\subseteq Z(S_{n}(\sigma,R))$, where $Z(R)$ denotes the
center of $R$.

$(2)$ $S_{n}(G,R)$ is a subring of the matrix ring $M_{n}(R)$.

$(3)$ The transpose of matrices is an anti-automorphism of order $2$ of the
ring $S_{n}(G,R)$.

$(4)$ If $G$ is the subgroup of $\Sigma_{n}$ generated by
$\tau_{1},\cdots,\tau_{r}\in\Sigma_{n}$, then
$S_{n}(G,R)=\cap_{i=1}^{r}S_{n}(\tau_{i},R).$

Proof. (1) This is already noticed in [6]. For the convenience of the reader,
we sketch here a proof. By definition, $c_{\sigma}a=ac_{\sigma}$ if and only
if $(c_{\sigma}a)_{ij}=(ac_{\sigma})_{ij}$ for all $i,j\in[n]$. But

$(c_{\sigma}a)_{ij}=\sum_{p=1}^{n}(c_{\sigma})_{ip}a_{pj}=\sum_{p=1}^{n}\delta_{(i)\sigma,p}a_{pj}=a_{(i)\sigma,j},$
$(ac_{\sigma})_{ij}=\sum_{p=1}^{n}a_{ip}(c_{\sigma})_{pj}=\sum_{p=1}^{n}a_{ip}\delta_{(p)\sigma,j}=a_{i,(j)\sigma^{-1}},$
$(c_{\sigma}ac_{\sigma^{-1}})_{ij}=a_{(i)\sigma,(j)\sigma}.$

Thus $c_{\sigma}a=ac_{\sigma}$ if and only if
$a_{(i)\sigma,j}=a_{i,(j)\sigma^{-1}}$ for all $i,j$ if and only if
$a_{ij}=a_{(i)\sigma,(j)\sigma}$ for all $i,j\in[n]$.

The statements remained are clear by definition. $\square$

By Lemma 2.1(1), we may identify $S_{n}(\sigma,R)$ with
$S_{n}(c_{\sigma},R):=\\{a\in M_{n}(R)\mid ac_{\sigma}=c_{\sigma}a\\}$. The
next lemma shows that $S_{n}(\sigma,R)$ depends only upon the cycle type
$\lambda$ of $\sigma$. So we may also write $S_{n}(\lambda,R)$ for
$S_{n}(\sigma,R)$. The proof is similar to the one of [11, Lemma 2.1], we
leave it to the reader.

######  Lemma 2.2.

Let $\tau,\sigma\in\Sigma_{n}$, and let $d\in M_{n}(R)$ be an invertible
matrix. Then

$(1)$ $S_{n}(\sigma^{-1},R)=S_{n}(\sigma,R)$. In particular,
$S_{n}(c_{\sigma},R)=S_{n}(c^{\prime}_{\sigma},R)$.

$(2)$ $S_{n}(dc_{\sigma}d^{-1},R)\simeq S_{n}(\sigma,R).$

$(3)$ $S_{n}(\sigma\tau,R)\simeq S_{n}(\tau\sigma,R).$

Let $\sigma\in\Sigma_{n}$ be of the cyclic type
$(\lambda_{1},\cdots,\lambda_{s})$, $G=\langle\sigma\rangle$, and
$\Lambda:=\\{(i,j)\mid 1\leq i,j\leq n\\}$. Then $G$ acts on $\Lambda$ by
$(i,j)\cdot\sigma=\big{(}(i)\sigma,(j)\sigma\big{)}$. Let $O_{(i,j)}$ denote
the $G$-orbit of $(i,j)$. Then $a\in S_{n}(\sigma,R)$ if and only if $a$ takes
a constant value on each orbit $O_{(i,j)}.$ Here, we regard $a\in M_{n}(R)$ as
a function from $\Lambda$ to $R$. Now we describe the number of $G$-orbits in
$\Lambda.$

We assume $n\geq 2$ and consider the three cases.

(i) $\sigma=(12\cdots n)$. In this case, $G$ has order $n$ and acts on
$\Lambda$ by $(i,j)\sigma=(i+1,j+1)$. Thus the stabilizer subgroup of $G$
fixing $(i,j)\in\Lambda$ is trivial and the $G$-orbit of $(i,j)$ has $|G|$
elements. Hence there are $n$ $G$-orbits of $\Lambda$, they are
$\mathcal{O}_{(1,j)}$, $1\leq j\leq n.$

(ii) $\sigma=(12\cdots p)(p+1,\cdots,p+q)$ with $n=p+q$. In this case,
$|G|=[p,q]$, the least common multiple of $p$ and $q$. Let $X:=\\{(i,j)\mid
1\leq i\leq p,p+1\leq j\leq n\\}$ with the $G$-action given by
$(i,j)\sigma=(i+1,j+1).$ Let $\sigma_{1}=(12\cdots p)\in\Sigma_{p}$ and
$\sigma_{2}=(p+1,\cdots,p+q)\in\Sigma_{q}$, here we consider $\Sigma_{q}$ as
the symmetric group of permutations on $\\{p+1,p+2,\cdots,p+q\\}$. If
$((i)\sigma^{l},(j)\sigma^{l})=(i,j)$ for a natural number $1\leq l\leq[p,q]$,
then $((i)\sigma_{1}^{l},(j)\sigma_{2}^{l})=(i,j)$. This implies that $p|l$
and $q|l$. Thus $[p,q]|l$ and $l=[p,q]$. Hence the stabilizer subgroup of $G$
fixing $(i,j)$ is trivial and the $G$-orbit of $(i,j)$ has $[p,q]$ elements.
Therefore the number of $G$-orbits of $X$ is $pq/[p,q]$.

(iii) $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{s}$, a product of disjoint
cycles, where $\sigma_{i}$ is a cycle of length $\lambda_{i}\geq 1$ with the
context $X_{i}$. Thus $|G|=[\lambda_{1},\cdots,\lambda_{s}]$ and $X_{i}$ is a
$G$-orbit for $i\in[s]$. Therefore $X_{i}\cap X_{j}=\emptyset$ for $1\leq
i\neq j\leq s,$ and $\cup_{j=1}^{s}X_{j}=\\{1,2,\cdots,n\\}.$ The set
$\Lambda$ can be identified with

$\dot{\bigcup}_{1\leq i,j\leq s}X_{i}\times X_{j}.$

For $(x,y)\in X_{i}\times X_{j}$, the $G$-orbit $\mathcal{O}_{(x,y)}$ of
$(x,y)$ depends only upon $\sigma_{i}$ and $\sigma_{j}$. Thus, if $i=j$, then
we are in case (i) and the number of $G$-orbits is $\lambda_{i}$. If $i\neq
j$, then we are in the case (ii) and the number of $G$-orbits of $X_{i}\times
X_{j}$ is $\lambda_{i}\lambda_{j}/[\lambda_{i},\lambda_{j}]$. Thus the number
of $G$-orbits of $\Lambda$ is

$\ell_{n}(\sigma):=\sum_{1\leq i,j\leq
s}\frac{\lambda_{i}\lambda_{j}}{[\lambda_{i},\lambda_{j}]}=\sum_{1\leq i,j\leq
s}\lambda_{i}\centerdot\lambda_{j},$

where $\lambda_{i}\centerdot\lambda_{j}$ stands for
$\gcd(\lambda_{i},\lambda_{j})$, the greatest common divisor of $\lambda_{i}$
and $\lambda_{j}.$

We define $C(\lambda):=(\lambda_{i}\centerdot\lambda_{j})_{s\times s}\in
M_{s}(\mathbb{N}).$ This symmetric matrix is called the _greatest common
divisor matrix_ of $\lambda$ in the literature. In this note, it will be
called the _dimension matrix_ of $S_{n}(\sigma,R)$. Further, let

$f_{i}:=\sum_{j\in X_{i}}e_{jj},\;1\leq i\leq s,\quad
f_{ij}:=\sum_{(p,q)\in\mathcal{O}_{(i,j)}}e_{p,q},\;(i,j)\in\Lambda.$

Then $f_{1},\cdots,f_{s}$ are pairwise orthogonal idempotent elements in
$S_{n}(\sigma,R)$ and $\sum_{j=1}^{s}f_{j}=1$ in $M_{n}(R)$. This yields the
matrix decomposition of $S_{n}(\sigma,R)$

$S_{n}(\sigma,R)=\begin{pmatrix}{f}_{1}S_{n}(\sigma,R)f_{1}&f_{1}S_{n}(\sigma,R)f_{2}&\cdots&f_{1}S_{n}(\sigma,R)f_{s}\\\
f_{2}S_{n}(\sigma,R)f_{1}&f_{2}S_{n}(\sigma,R)f_{2}&\cdots&f_{2}S_{n}(\sigma,R)f_{s}\\\
\vdots&\vdots&\ddots&\vdots\\\
f_{s}S_{n}(\sigma,R)f_{1}&f_{s}S_{n}(\sigma,R)f_{2}&\cdots&f_{s}S_{n}(\sigma,R)f_{s}\end{pmatrix}.$

The transpose of matrices is an $R$-involution on $S_{n}(\sigma,R)$ fixing
$f_{i}$ for all $1\leq i\leq s$.

######  Lemma 2.3.

$(1)$ $S_{n}(\sigma,R)$ is a free $R$-module of rank $\ell_{n}(\sigma)$.

$(2)$ The $R$-rank of $f_{i}S_{n}(\sigma,R)f_{j}$ is
$\lambda_{i}\centerdot\lambda_{j}.$

$(3)$ The dimension matrix $C(\lambda)$ of $S_{n}(\sigma,R)$ is positive
definite if the numbers $\lambda_{i}$, $1\leq i\leq s$, are pairwise distinct.
In this case, the determinant $\det C(\lambda)$ of $C(\lambda)$ satisfies
$\varphi(\lambda_{1})\cdots\varphi(\lambda_{s})\leq\det
C(\lambda)\leq\prod_{i=1}^{s}\lambda_{i}-\frac{s!}{2}$, where
$\varphi(\lambda_{i})$ is the Euler’s totient function of $\lambda_{i}$.

Proof. (1) and (2) are consequences of the definition of $S_{n}(\sigma,R)$ and
the calculation of $G$-orbits of $\Lambda$ (see also [7]). (3) is proved in
[2, Theorem 2] and [4]. $\square$

######  Example 2.4.

$(1)$ $S_{n}((12\cdots n),R)\simeq R[C_{n}]$, where $R[C_{n}]$ is the group
algebra of the cyclic group $C_{n}$ of order $n$ over $R$. Note that
$g=\sum_{j=1}^{n}e_{j,j+1}$ is a generator of $C_{n}$, here we understand
$(n,n+1)=(n,1)$. The transpose of matrices in $S_{n}((12\cdots n),R)$
corresponds to the involution defined by $g\mapsto g^{-1}.$

$(2)$ If $\lambda=(a,\cdots,a)=(a^{s})$ where $a$ appears $s$ times, then
$S_{sa}(\lambda,R)\simeq M_{s}(R[C_{a}])).$ In fact, each element of
$S_{na}(\lambda,R)$ can be partitioned as an $s\times s$ block-matrix such
that each block has entries in $f_{i}S_{sa}(\lambda,R)f_{i}\simeq R[C_{a}]$.
Thus $S_{sa}(\lambda,R)\simeq M_{s}(R[C_{a}]).$

By Lemma 2.2(2), up to isomorphism of rings, we may assume
$\sigma=(12\cdots\lambda_{1})(\lambda_{1}+1,\cdots,\lambda_{1}+\lambda_{2})\cdots(\lambda_{1}+\cdots+\lambda_{s-1}+1,\cdots,\lambda_{1}+\cdots+\lambda_{s})$
as a product of disjoint cycles. Let
$\lambda=(\lambda_{1},\cdots,\lambda_{s})$,
$X_{1}=\\{1,\cdots,\lambda_{1}\\}$,
$X_{i}=\\{j\mid\sum_{\ell=1}^{i-1}\lambda_{\ell}<j\leq\sum_{\ell=1}^{i}\lambda_{i}\\}$
for $2\leq i\leq s$, $d_{ij}:=\lambda_{i}\centerdot\lambda_{j}$. Then

$c_{\sigma}=\begin{pmatrix}P_{\lambda_{1}}&0&\cdots&0\\\
0&P_{\lambda_{2}}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&P_{\lambda_{s}}\end{pmatrix}_{n\times n}\;\mbox{ where
}\,P_{m}:=\begin{pmatrix}0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1\\\ 1&0&\cdots&0\\\ \end{pmatrix}_{m\times m}$

For $r_{0},\cdots,r_{m-1}\in R$, we denote by $C(r_{0},\cdots,r_{m-1})$ (or
$C_{m}(r_{0},\cdots,r_{m-1})$) the circulant matrix

$\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots&r_{m-1}\\\
r_{m-1}&r_{0}&r_{1}&\cdots&r_{m-2}\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
r_{2}&\cdots&\ddots&\ddots&r_{1}\\\ r_{1}&r_{2}&\cdots&r_{m-1}&r_{0}\\\
\end{pmatrix}_{m\times m}=r_{0}I_{m}+r_{1}P_{m}+\cdots+r_{m-1}P_{m}^{m-1}.$

Thus, for a cyclic group $C_{m}$ of order $m$, we have
$R[C_{m}]\simeq\\{C(r_{0},\cdots,r_{m-1})\mid r_{0},\cdots,r_{m-1}\in R\\}.$

Recall that, given $\sigma\in\Sigma_{n}$ and $\tau\in\Sigma_{m}$, an $n\times
m$ matrix $a=(a_{ij})\in M_{n\times m}(R)$ is called a
$(\sigma,\tau)$-_invariant matrix_ if $c_{\sigma}a=ac_{\tau}$.

We see easily that an $s\times s$ block matrix $a=(A_{ij})$ with $A_{ij}\in
M_{\lambda_{i}\times\lambda_{j}}(R)$ lies in $S_{n}(\sigma,R)$ if and only if
$A_{ij}$ is a $(\sigma_{i},\sigma_{j})$-invariant matrix for each pair $1\leq
i,j\leq s$. Thus a matrix $a\in S_{n}(\sigma,R)$ can be written as a block-
matrix form

$a=(A_{ij})=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1s}\\\
A_{21}&A_{22}&\cdots&A_{2s}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{s1}&A_{s2}&\cdots&A_{ss}\end{pmatrix}_{s\times s}$

where the diagonal $A_{ii}$ is a $\lambda_{i}\times\lambda_{i}$ circulant
matrix and the off-diagonal $A_{ij}$ is a $\lambda_{i}\times\lambda_{j}$
matrix over $R$, having the block form:

$A_{ij}=\begin{pmatrix}C(r_{1},\cdots,r_{d_{ij}})&\cdots&C(r_{1},\cdots,r_{d_{ij}})\\\
\vdots&\cdots&\vdots\\\
C(r_{1},\cdots,r_{d_{ij}})&\cdots&C(r_{1},\cdots,r_{d_{ij}})\\\
\end{pmatrix}_{\frac{\lambda_{i}}{d_{ij}}\times\frac{\lambda_{j}}{d_{ij}}}$

for $r_{1},\cdots,r_{d_{ij}}\in R$ (see [7, p.260] for $R=\mathbb{C}$). In
general, we have an isomorphism of $R$-modules

$f_{i}S_{n}(\sigma,R)f_{j}\simeq\\{A_{ij}\in M_{X_{i}\times X_{j}}(R)\mid
r_{1},\cdots,r_{d_{ij}}\in R\\}.$

######  Lemma 2.5.

$(1)$ For $1\leq i\leq s,$ $f_{i}S_{n}(\sigma,R)f_{i}\simeq
R[C_{\lambda_{i}}]$ as rings.

$(2)$ If $\lambda_{1}=\lambda_{2}$, then $S_{n}(\sigma,R)f_{1}\simeq
S_{n}(\sigma,R)f_{2}$ as left $S_{n}(\sigma,R)$-modules.

$(3)$ If $m:=n-\lambda_{i}$ and
$f:=f_{1}+\cdots+f_{i-1}+f_{i+1}+\cdots+f_{s}$, then $fS_{n}(\sigma,R)f\simeq
S_{m}(\sigma_{1}\cdots\sigma_{i-1}\sigma_{i+1}\cdots\sigma_{s},R)$ as rings.

Proof. (1) If $\lambda_{i}=1$, then $f_{i}S_{n}(\sigma,R)f_{i}\simeq R$ and
(1) holds. We assume $\lambda_{i}\geq 2$ and define
$g:=e_{\lambda_{i-1}+1,\lambda_{i-1}+2}+\cdots+e_{\lambda_{i-1}+\lambda_{i}-1,\lambda_{i-1}+\lambda_{i}}+e_{\lambda_{i-1}+\lambda_{i},\lambda_{i-1}+1}$.
Then $g^{\lambda_{i}}=f_{i}$ and $g^{j}\neq f_{i}$ for all $1\leq
j<\lambda_{i}$. One can check that
$\\{f_{i},g,g^{2},\cdots,g^{\lambda_{i}-1}\\}$ is an $R$-basis of
$f_{i}S_{n}(\sigma,R)f_{i}$. Thus $f_{i}S_{n}(\sigma,R)f_{i}\simeq
R[C_{\lambda_{i}}]$.

(2) By definition, $f_{1}=e_{11}+\cdots+e_{\lambda_{1},\lambda_{1}}$ and
$f_{2}=e_{\lambda_{1}+1,\lambda_{1}+1}+e_{\lambda_{1}+2,\lambda_{1}+2}+\cdots+e_{2\lambda_{1},2\lambda_{1}}$.
Let
$a_{0}:=e_{1,\lambda_{1}+1}+e_{2,\lambda_{1}+2}+\cdots+e_{\lambda_{1},2\lambda_{1}}$.
Then $a_{0}\in S_{n}(\sigma,R)$ and $f_{1}a_{0}=a_{0}f_{2}=a_{0}.$ So, we
define a map

$\psi:S_{n}(\sigma,R)f_{1}\longrightarrow S_{n}(\sigma,R)f_{2},\quad
af_{1}\mapsto af_{1}a_{0}\mbox{ for }a\in S_{n}(\sigma,R).$

Note that $\psi$ just moves the first $\lambda_{1}$ columns of $af_{1}$
entirely to the second $\lambda_{1}$ columns of $af_{1}a_{0}$, and sends other
columns of $af_{1}$ to zero. Thus $\psi$ is a homomorphism of
$S_{n}(\sigma,R)$-modules, and it is in fact an isomorphism of
$S_{n}(\sigma,R)$-modules.

(3) We may assume $i=2$ since the argument below works for any $i$. If we
delete the cycle $\sigma_{2}$ from $\sigma$ and consider
$\tau:=\sigma_{1}\sigma_{3}\cdots\sigma_{s}$ as a permutation on
$Y:=[n]\setminus X_{2}=X_{1}\cup(X_{3}\cup\cdots\cup X_{s})$, then
$S_{m}(\sigma_{1}\sigma_{3}\cdots\sigma_{s},R)$ is well defined. Observe that
$\sigma$ and $\tau$ have the same action on $Y$ and that a matrix $a$ in
$fS_{n}(\sigma,R)f$ is of the form

$\begin{pmatrix}A&0&U\\\ 0&0&0\\\ V&0&B\\\ \end{pmatrix}_{n\times n}$

where $A$ is a matrix indexed by $X_{1}\times X_{1}$, $U$ is a matrix indexed
by $X_{1}\times(X_{3}\cup\cdots\cup X_{s})$, $V$ is a matrix indexed by
$(X_{3}\cup\cdots\cup X_{s})\times X_{1}$ and $B$ is a matrix indexed by
$(X_{3}\cup\cdots\cup X_{s})\times(X_{3}\cup\cdots\cup X_{s})$. Since $a\in
fS_{n}(\sigma,R)f$, the block matrix

$b:=\begin{pmatrix}A&U\\\ V&B\\\ \end{pmatrix}$

is in $S_{m}(\tau,R)$. Conversely, given a matrix $b\in S_{m}(\tau,R)$, we
partition $b$ into a block form according to $Y$, say the above form. Then we
get an $n\times n$ matrix by putting $0$-blocks in the middle row and column,
and this matrix is in $fS_{n}(\sigma,R)f$. Note that the identity in
$fS_{n}(\sigma,R)f$ is $f=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&1\\\
\end{pmatrix}.$ In this way, we get a one-to-one correspondence which
preserves identity, addition and multiplication of matrices. This shows
$fS_{n}(\sigma,R)f\simeq S_{m}(\tau,R)$. $\square$

##  3 Isomorphisms of semisimple, invariant matrix algebras

In this section we introduce an equivalence relation on the set of partitions
of $n$ with exactly $s$ parts by employing elementary symmetric polynomials.
We then study combinatorics of this relation by a matrix norm over the
semiring of nonnegative integers. This will later be used for describing
isomorphisms of semisimple, invariant matrix algebras.

###  3.1 Semirings and combinatorics on partitions

Recall that a nonempty set $A$ together with two associative binary operations
$+$ and $\cdot$, named addition and multiplication, respectively, is called a
_bisemigroup_. A bisemigroup $(A,+,\cdot)$ is called a _semiring_ if $(A,+)$
is commutative and the distributive laws hold: $x\cdot(y+z)=x\cdot y+x\cdot z$
and $(x+y)\cdot z=x\cdot z+y\cdot z$ for $x,y,z\in A$. A _commutative_
semiring is a semiring such that the multiplication is commutative.

######  Definition 3.1.

A bisemigroup $(A,+,\centerdot)$ is called a quarter-ring if

$(1)$ $(A,+)$ and $(A,\centerdot)$ are commutative,

$(2)$ $a\centerdot a=a$ for $a\in A$, and

$(3)$ $(a+b)\centerdot a=b\centerdot a$ for $a,b\in A$.

In a bisemgroup $(A,+,\cdot)$, for any positive integer $m$ and $a_{i},b,d\in
A$, we write $\sum_{i=1}^{m}a_{i}$ for $a_{1}+\cdots+a_{m}$, particularly,
$ma$ for $\sum_{i=1}^{m}a$, and $a^{\cdot m}$ for $\underbrace{a\cdot
a\cdot\cdots\cdot a}_{m}$, the product of $m$ copies of $a$.

In a quarter-ring $(A,+,\centerdot)$, the following hold.

$\bullet\quad$ If $a\centerdot b=d$, then $a\centerdot d=d.$ In particular, if
$a_{i}\centerdot b=d$ for $1\leq i\leq n$, then
$a_{1}\centerdot\cdots\centerdot a_{n}\centerdot b=d.$

$\bullet\quad$ $(ma+b)\centerdot a=a\centerdot b$ and $(ma)\centerdot a=a$ for
any positive integer $m$.

$\bullet\quad$ If $(A,+)$ is additionally a monoid with zero element $0$, then
$0\centerdot a=a$ for $a\in A$.

$\bullet\quad$ $a_{1}\centerdot a_{2}\centerdot\cdots\centerdot
a_{n}=(a_{1}\centerdot a_{2})\centerdot(a_{2}\centerdot
a_{3})\centerdot\cdots\centerdot(a_{n-1}\centerdot a_{n}).$

$\bullet\quad$ $(a_{1}\centerdot a_{2}\centerdot\cdots\centerdot
a_{n})\centerdot b=(a_{1}\centerdot
b)\centerdot(a_{2}\centerdot\cdots\centerdot a_{n})=(a_{1}\centerdot
b)\centerdot(a_{2}\centerdot b)\centerdot(a_{3}\centerdot\cdots\centerdot
a_{n})=\cdots=(a_{1}\centerdot b)\centerdot(a_{2}\centerdot
b)\centerdot\cdots\centerdot(a_{n}\centerdot b).$

######  Example 3.2.

(1) Let $\mathbb{N}$ be the set of natural numbers including $0$. Then
$\mathbb{N}$ with the usual addition and multiplication forms a semiring. The
polynomial semiring $\mathbb{N}[x_{1},\cdots,x_{s}]$ over $\mathbb{N}$ in
variables $x_{1},\cdots,x_{s}$ is the set of all polynomials over $\mathbb{N}$
with the usual addition and multiplication of polynomials. Similarly, we
define the matrix semiring $M_{n}(R)$ over a semiring $R$.

(2) Let $a\centerdot b:=\gcd(a,b)$ for $a,b\in\mathbb{N}$. Then
$(\mathbb{N},+,\centerdot)$ is a quarter-ring, but not a semiring. It is
called the _canonical quarter-ring_ of natural numbers.

Recall that a _multiset_ is a collection of elements possibly containing
duplicate. For a multiset $\lambda=\\{\lambda_{1},\cdots,\lambda_{s}\\}$ of
$s$ elements in $\mathbb{N}$, we define a map $\varphi_{\lambda}$ from the
polynomial semiring $\mathbb{N}[x_{1},\cdots,x_{s}]$ to
$(\mathbb{N},+,\centerdot)$ as follows:

$\varphi_{\lambda}:\mathbb{N}[x_{1},\cdots,x_{s}]\longrightarrow\mathbb{N},\qquad\sum_{i_{1},\cdots,i_{s}}a_{i_{1},\cdots,i_{s}}x_{1}^{i_{1}}\cdots
x_{s}^{i_{s}}\mapsto\sum_{i_{1},\cdots,i_{s}}a_{i_{1},\cdots,i_{s}}\lambda_{1}^{i_{1}}\centerdot\cdots\centerdot\lambda_{s}^{i_{s}}.$

Note that $\varphi_{\lambda}$ is additive and can be extended to the matrix
semiring over $\mathbb{N}[x_{1},\cdots,x_{s}]$:

$\|-\|_{{}_{\lambda}}:M_{n}(\mathbb{N}[x_{1},\cdots,x_{s}])\longrightarrow\mathbb{N},\qquad(a_{ij})_{n\times
n}\mapsto\|(a_{ij})\|_{{}_{\lambda}}:=\sum_{1\leq i,j\leq
n}(a_{ij})\varphi_{\lambda},$

where $(a_{ij})\varphi_{\lambda}$ denotes the image of $a_{ij}$ under
$\varphi_{\lambda}.$ The map $\|-\|_{{}_{\lambda}}$ is called the _norm map_.

Now we recall the following elementary fact in combinatorics.

######  Lemma 3.3.

Let $V_{i}$ be a finite set for $1\leq i\leq s$. We define

$V:=\bigcup_{1\leq j\leq s}V_{j},\;\;g_{i}:=\sum_{1\leq
k_{1}<k_{2}<\cdots<k_{i}\leq s}\big{|}V_{k_{1}}\cap\cdots\cap
V_{k_{i}}\big{|},\;1\leq i\leq s,$

where $|S|$ indicates the number of elements of a set $S$. Then

$(1)$ For $1\leq i\leq s$, the number $h_{i}$ of the elements of $V$ which are
members of exactly $i$ sets of $V_{1},V_{2},\cdots,V_{s}$ is given by
$h_{i}=\sum_{k=0}^{s-i}(-1)^{k}C_{i+k}^{i}g_{i+k}$, where
$C_{i+k}^{i}:=\binom{i+k}{i}=\frac{(i+k)!}{i!k!}$ is the number of $i$-subsets
of an $(i+k)$-element set.

$(2)$ Principle of Inclusion-Exclusion:
$\big{|}V\big{|}=\sum_{i=1}^{s}h_{i}=\sum_{i=1}^{s}(-1)^{i+1}g_{i}.$ Moreover,
$\sum_{i=1}^{s}|V_{i}|=\sum_{i=1}^{s}ih_{i}.$

Let $R$ be an algebraically closed field of characteristic $p\geq 0$ and
$\sigma\in\Sigma_{n}$ be of cycle type
$(\lambda_{1},\lambda_{2},\cdots,\lambda_{s})$. Suppose $p\nmid\lambda_{i}$
for $1\leq i\leq s$. For $1\leq i\leq s$, we denote by $V_{i}$ the set of all
$\lambda_{i}$-th roots of unity in $R$ and define $V:=\bigcup_{1\leq j\leq
s}V_{j}.$ It follows from $p\nmid\lambda_{i}$ that $V_{i}$ is a cyclic group
of order $\lambda_{i}$. Moreover, $V_{k_{1}}\cap\cdots\cap
V_{k_{i}}=\\{\omega\in R\mid\omega^{d}=1\\}$ for $1\leq
k_{1}<k_{2}<\cdots<k_{i}\leq s$, where
$d:=\gcd(\lambda_{k_{1}},\lambda_{k_{2}},\cdots,\lambda_{k_{i}})$ is the
greatest common divisor of $\lambda_{k_{j}},1\leq j\leq i$. Thus
$\big{|}V_{k_{1}}\cap\cdots\cap
V_{k_{i}}\big{|}=\gcd(\lambda_{k_{1}},\lambda_{k_{2}},\cdots,\lambda_{k_{i}})$
for $1\leq k_{1}<k_{2}<\cdots<k_{i}\leq s.$ We further define

$g_{i}(\sigma):=\sum_{1\leq k_{1}<k_{2}<\cdots<k_{i}\leq
s}\gcd(\lambda_{k_{1}},\lambda_{k_{2}},\cdots,\lambda_{k_{i}}),$

and $H_{i}(\sigma)$ to be the subset of $V$ consisting of all elements which
belong to exactly $i$ sets of the given $V_{1},V_{2},\cdots,V_{s}$, namely

$H_{i}(\sigma)=\\{v\in V\mid\exists\,1\leq k_{1}<k_{2}<\cdots<k_{i}\leq
s,v\in\bigcap_{1\leq j\leq i}V_{k_{j}},\,v\not\in\bigcup_{r\neq k_{j},1\leq
j\leq i}V_{r}\\}.$

Clearly, $H_{i}(\sigma)\cap H_{j}(\sigma)=\emptyset$ if $1\leq i\neq j\leq s$.
By Lemma 3.3(1),
$h_{i}(\sigma):=|H_{i}(\sigma)|=\sum_{k=0}^{s-i}(-1)^{k}C_{i+k}^{i}g_{i+k}(\sigma).$
Obviously, $g_{i}(\sigma)$ and $H_{i}(\sigma)$ depend merely on the type of
$\sigma$. So $g_{i}(\lambda)$ and $h_{i}(\lambda)$ are well-defined functions
from the set of all partitions of positive integers to $\mathbb{N}$ if we
define $g_{i}=0$ and $h_{i}=0$ for $i>s$.

As in Example 3.2, we define $a\centerdot b=\gcd(a,b)$ for $a,b\in\mathbb{Z}$.
Then $g_{i}(x_{1},\cdots,x_{s})$ is just the $i$-th elementary symmetric
polynomial over the canonical quarter-ring of natural numbers:

$g_{i}(x_{1},\cdots,x_{s})=\sum_{1\leq j_{1}<j_{2}<\cdots<j_{i}\leq
s}x_{j_{1}}\centerdot x_{j_{2}}\centerdot\cdots\centerdot x_{j_{i}}$

for $1\leq i\leq s$. We set
$g_{0}(x_{1},\cdots,x_{s})=g_{i}(x_{,}\cdots,x_{s})=0$ for $i>s$. Clearly,
$g_{i}(x_{1},\cdots,x_{s})$ does not depend on any order of $x_{1}\geq
x_{2}\geq\cdots\geq x_{s}$, and $g_{s}(x_{1},\cdots,x_{s})\mid
g_{i}(x_{1},\cdots,x_{s})$ for $1\leq i\leq s$ and all $x_{i}\in\mathbb{N}$.

Recall that a _partition_ $\lambda$ of a positive integer $n$ is an $s$-tuple
$(\lambda_{1},\cdots,\lambda_{s})$ of integers satisfying
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{s}\geq 1$ and
$\sum_{i=1}^{s}\lambda_{i}=n$. We write
$\lambda=(\lambda_{1},\cdots,\lambda_{s})\vdash n$ for simplicity. Each
$\lambda_{i}$ is called a _part_ of $\lambda$. Sometimes it is also convenient
to think of partitions as multisets and to write $\lambda$ in the form
$(\lambda_{1}^{a_{1}},\lambda_{2}^{a_{2}},\cdots,\lambda_{t}^{a_{t}})$ with
$\lambda_{1}>\lambda_{2}>\cdots>\lambda_{t}\geq 1$, where $\lambda_{i}$
appears $a_{i}\geq 1$ times in the multiset of $\lambda$.

Let $P(n)$ be the set of all partitions of $n$, $P(s,n)$ the set of partitions
of $n$ with exactly $s$ parts, and $P_{s}^{*}(n)$ the set of partitions of $n$
with the largest part $s$. Then $P(s,n)$ and $P_{s}^{*}(n)$ have the same
cardinality. This can be seen from Ferrers graphs (or Young diagrams) of
partitions.

Convention: Given partitions $\lambda\in P(n)$ and $\mu\in P(m)$, we write
$(\lambda,\mu)$ for the partition $\gamma\in P(n+m)$ such that the parts of
$\gamma$ is the disjoint union of the parts of $\lambda$ with the parts of
$\mu$. For example, if $\lambda=(4,3,2,1)$ and $\mu=(5,4,4,2,2,1,1)$, then
$(\lambda,\mu)=(5,4,4,4,3,2,2,2,1,1,1)$. For $1\leq d\in\mathbb{N}$, we define
$d\lambda=(d\lambda_{1},\cdots,d\lambda_{s})\in P(s,dn)$ and
$\lambda_{-}:=(\lambda_{1},\cdots,\lambda_{s-1})=(\lambda_{1},\cdots,\lambda_{s-1},\hat{\lambda}_{s})\in
P(s-1,n-\lambda_{s})$. Inductively,
$\lambda_{-j}=(\lambda_{1},\cdots,\lambda_{s-j})$. If $j\geq s$, we define
$\lambda_{s-j}=(0,\cdots,0)$. Then $g_{i}(d\lambda)=d\,g_{i}(\lambda)$, and

$g_{i}(\lambda)=g_{i}(\lambda_{-})\;+\sum_{1\leq j_{1}<\cdots<j_{i-1}\leq
s-1}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{i-1}}\centerdot\lambda_{s}\leq
g_{i}(\lambda_{-})\;+\sum_{1\leq j_{1}<\cdots<j_{i-1}\leq
s-1}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{i-1}},$

and therefore, if $\lambda_{s}=1$, then
$g_{i}(\lambda)=g_{i}(\lambda_{-})+C_{s-1}^{i-1}.$ In general, we have

$g_{i}(\lambda_{-})+C^{i-1}_{s-1}\leq g_{i}(\lambda)\leq
g_{i}(\lambda_{-})+\min\\{g_{i-1}(\lambda_{-}),\lambda_{s}C_{s-1}^{i-1}\\}.$

Now, we introduce a new equivalence relation $\sim$ on $P(s,n)$ and define a
polynomial $\epsilon_{\lambda}(x)$ for each $\lambda\vdash n$.

######  Definition 3.4.

$(1)$ The partition polynomial $\epsilon_{\lambda}(x)$ of $\lambda\in P(s,n)$
is

$\begin{array}[]{rl}\epsilon_{\lambda}(x):=&\sum_{i=0}^{s-1}(-1)^{s-1-i}\frac{g_{i+1}(\lambda)}{g_{s}(\lambda)}x^{i}\\\
\\\
=&x^{s-1}-\frac{g_{s-1}(\lambda)}{g_{s}(\lambda)}x^{s-2}+\cdots+(-1)^{s-2}\frac{g_{2}(\lambda)}{g_{s}(\lambda)}x+(-1)^{s-1}\frac{g_{1}(\lambda)}{g_{s}(\lambda)}\in\mathbb{N}[x]\subseteq\mathbb{Z}[x].\end{array}$

$(2)$ The equivalence relation $\sim$ on $P(s,n)$ (or $P(n))$ is given by

$\lambda\sim\mu\;\mbox{ if }\;\epsilon_{\lambda}(x)=\epsilon_{\mu}(x).$

Equivalently, $\lambda\sim\mu$ if and only if $g_{i}(\lambda)=g_{i}(\mu)\mbox{
for }1\leq i\leq s$. This equivalence relation is called the _polynomial
equivalence_ of partitions in $P(n)$.

######  Example 3.5.

(1) $\epsilon_{(1^{n})}(x)=\sum_{i=0}^{n-1}(-1)^{n-1-i}C^{i+1}_{n}x^{i},$
while
$\epsilon_{(m,1^{n-m})}(x)=\sum_{i=1}^{n-m}(-1)^{n-m-i}C_{n-m+1}^{i+1}x^{i}+(-1)^{n-m}n$
for $1\leq m<n$ and $\epsilon_{(n)}(x)=1$.

(2) In $P(2,p)$ with $p$ a prime, any two-part partitions are polynomial
equivalent because the partition polynomial of each such partition is of the
form $x-p$. In $P(3,11)$, there hold
$(8,2,1)\sim(7,2,2)\sim(6,4,1)\sim(5,4,2)$, but $(8,2,1)\nsim(6,3,2)$.

Remark that $(-1)^{s-1}g_{s}(\lambda)\epsilon_{\lambda}(1)$ counts the number
of distinct eigenvalues of $c_{\sigma}$ in $\mathbb{C}$, where
$\sigma\in\Sigma_{n}$ has the cycle type $\lambda$ (see [7, Lemma 6.3]).

Next, we consider when two partitions are polynomial equivalent. We first give
a matrix interpretation of the polynomial equivalence.

######  Definition 3.6.

Given $\lambda=(\lambda_{1},\cdots,\lambda_{s})\in P(s,n)$, we associate it
with a matrix $\Lambda_{\lambda}\in M_{s}(\mathbb{N})$ by setting

$\Lambda_{\lambda}:=\begin{pmatrix}0&d_{12}&d_{13}&\dots&d_{1s}\\\
0&0&d_{23}&\dots&d_{2s}\\\ \vdots&\vdots&\ddots&\ddots&\vdots\\\
0&0&\cdots&\ddots&d_{s-1,s}\\\ 0&0&0&\dots&0\\\ \end{pmatrix}$

where $d_{ij}=\lambda_{i}\centerdot\lambda_{j}$ is the product in the
canonical quarter-ring of natural numbers. This matrix is called the
_triangular divisor matrix_ of $\lambda$.

Now we define a norm $\|\Lambda_{\lambda}\|$ of $\Lambda_{\lambda}$. First, we
consider $\lambda_{j},1\leq j\leq s,$ as variables and then regard
$\Lambda_{\lambda}$ as a matrix with the entries in the polynomial semiring
$\mathbb{N}[\lambda_{1},\cdots,\lambda_{s}]$, thinking of
$\lambda_{i}\centerdot\lambda_{j}$ as the usual monomial
$\lambda_{i}\lambda_{j}$ in $\mathbb{N}[\lambda_{1},\cdots,\lambda_{s}]$. By
applying $\|-\|_{{}_{\lambda}}$ to this matrix, we get the norm
$\|\Lambda_{\lambda}\|$ of the triangular divisor matrix $\Lambda_{\lambda}$.
That is, $\|\Lambda_{\lambda}\|=\sum_{1\leq i<j\leq
s}\lambda_{i}\centerdot\lambda_{j}.$ Similarly, we consider the $i$-th power
$\Lambda_{\lambda}^{i}$ of $\Lambda_{\lambda}$ in the $s\times s$ matrix
semiring $M_{s}(\mathbb{N}[\lambda_{1},\cdots,\lambda_{s}])$ over
$\mathbb{N}[\lambda_{1},\cdots,\lambda_{s}]$ and then define
$\|\Lambda_{\lambda}^{i}\|:=\|\Lambda_{\lambda}^{i}\|_{{}_{\lambda}}=(\Lambda_{\lambda}^{i})\varphi_{\lambda}$.
Note that $\dim_{R}S_{n}(\lambda,R)=g_{1}(\lambda)+2\|\Lambda_{\lambda}\|$
(see Lemma 2.3(1)).

The triangular divisor matrix $\Lambda_{\lambda}$ of $\lambda$ depends upon
the ordering of $\lambda_{1},\cdots,\lambda_{s}$. Nevertheless, we will show
that the norm $\|-\|$ of triangular divisor matrices is independent of the
choices of orderings.

######  Proposition 3.7.

Let $\lambda,\mu\in P(s,n)$. Then

$(1)$ $\|\Lambda^{i}_{\lambda}\|=g_{i+1}(\lambda)$ for $1\leq i\leq s-1$.

$(2)$ $\lambda\sim\mu$ if and only if
$\|\Lambda^{i}_{\lambda}\|=\|\Lambda^{i}_{\mu}\|$ for all $1\leq i\leq s-1$,
that is, the norms of $\Lambda^{i}_{\lambda}$ and $\Lambda^{i}_{\mu}$ are
equal for all $1\leq i\leq s-1$.

Proof. (1) We check $\|\Lambda^{i}_{\lambda}\|=g_{i+1}(\lambda)$ for $1\leq
i\leq s-1$. Since the summands of
$\|\Lambda^{i}_{\lambda}\|=\|\Lambda^{i-1}_{\lambda}\Lambda_{\lambda}\|$ are
of the form
$\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{i}}\centerdot\lambda_{j_{i+1}}$
with $j_{1}<\cdots<j_{i}<j_{i+1}$, these are precisely the summands of
$g_{i+1}(\lambda)$. Hence $\|\Lambda^{i}_{\lambda}\|=g_{i+1}(\lambda)$.

(2) Clearly, $g_{1}(\lambda)=g_{1}(\mu)=n$. Hence $g_{j}(\lambda)=g_{j}(\mu)$
for all $1\leq j\leq s$ if and only if
$\|\Lambda^{i}_{\lambda}\|=\|\Lambda^{i}_{\mu}\|$ for all $1\leq i\leq s-1$.
$\square$.

An immediate consequence of Proposition 3.7 is that if
$\\{\mu_{1},\cdots,\mu_{s}\\}$ is a permutation of
$\\{\lambda_{1},\cdots,\lambda_{s}\\}$ then
$\|\Lambda^{i}_{\lambda}\|=\|\Lambda^{i}_{\mu}\|$ for all $1\leq i\leq s-1$
because the $i$-th elementary symmetric polynomial $g_{i}(\lambda)$ does not
depend on the ordering of its variables $\lambda_{i}$.

Another consequence is the following result which is quite useful for deciding
whether two partitions are polynomial equivalent.

######  Corollary 3.8.

Let $\lambda,\mu\in P(s,n)$ with the corresponding triangular divisor matrices
$\Lambda_{\lambda}=(d_{ij})$ and $\Lambda_{\mu}=(c_{ij})$, respectively. If
the multisets $\\{d_{ij}\mid 1\leq i<j\leq s\\}$ and $\\{c_{ij}\mid 1\leq
i<j\leq s\\}$ are equal, then $\lambda\sim\mu$.

Proof. That the multisets $\\{d_{ij}\mid 1\leq i<j\leq s\\}$ and
$\\{c_{ij}\mid 1\leq i<j\leq s\\}$ are equal means
$g_{2}(\lambda)=g_{2}(\mu)$. Since any summand
$\lambda_{j_{1}}\centerdot\lambda_{j_{2}}\centerdot\cdots\centerdot\lambda_{j_{i}}$
of $g_{i}(\lambda)$ for $i\geq 3$ is a product $d_{j_{1},j_{2}}\centerdot
d_{j_{2},j_{3}}\centerdot\cdots\centerdot d_{j_{i-1},j_{i}}$ (in the canonical
quarter-ring $(\mathbb{N},+,\centerdot)$ of $i-1$ elements of the mutiset
$\\{d_{ij}\mid 1\leq i<j\leq s\\}$ with the increasing indices, the summation
of all such products of elements in the multiset of $\lambda$ equals the one
in the multiset of $\mu$. Hence $g_{i}(\lambda)=g_{i}(\mu)$ for $2\leq i\leq
s$. Clearly, $g_{1}(\lambda)=g_{1}(\mu)$. Thus $\lambda\sim\mu.$ $\square$

Note that the converse of Lemma 3.8 fails. For example,
$\lambda=(12,4,3,1)\sim\mu=(10,5,3,2)$, but the corresponding multisets for
$\lambda$ and $\mu$ are $\\{4,3,1,1,1,1\\}$ and $\\{5,1,2,1,1,1\\}$,
respectively, they are clearly different. This example also shows that if the
common part $3$ is removed from $\lambda$ and $\mu$, then the resulting
partitions $(12,4,1)$ and $(10,5,2)$ are no longer polynomial equivalent.

######  Corollary 3.9.

Let $\lambda=(\lambda_{1},\cdots,\lambda_{s}),\mu=(\mu_{1},\cdots,\mu_{s})\in
P(s,n)$. If $\lambda_{i}\centerdot\lambda_{j}=d=\mu_{p}\centerdot\mu_{q}$ for
all $1\leq i\neq j\leq s$ and $1\leq p\neq q\leq s$, then $\lambda\sim\mu.$ In
particular, if $\lambda_{1},\cdots,\lambda_{s}$ are pairwise coprime and if
$\mu_{1},\cdots,\mu_{s}$ are pairwise coprime, then $\lambda\sim\mu$.

Proof. Under the assumption on $\lambda$, the triangular divisor matrix of
$\lambda$ is

$\Lambda_{\lambda}=\begin{pmatrix}0&d&\cdots&d\\\ &0&\ddots&\vdots\\\
&&\ddots&d\\\ &&&0\\\ \end{pmatrix}.$

Thus $\Lambda_{\lambda}=\Lambda_{\mu}$. Hence Corollary 3.9 follows
immediately from Corollary 3.8. $\square$

By the definition of partition polynomials $\epsilon_{\lambda}(x)$, we have
the following useful fact.

######  Lemma 3.10.

Let $\lambda,\mu\in P(s,n)$. Then $\lambda\sim\mu$ in $P(s,n)$ if and only if
$d\lambda\sim d\mu$ in $P(s,dn)$ for some integer $d\geq 1$ if and only if
$d\lambda\sim d\mu$ in $P(s,dn)$ for all integers $d\geq 1.$

Proof. The coefficients of the monic partition polynomial
$\epsilon_{d\lambda}(x)$ is given by $g_{i}(d\lambda)/g_{s}(d\lambda)$. As we
know, $g_{i}(d\lambda)=dg_{i}(\lambda)$ for all $i$. Thus
$\epsilon_{d\lambda}(x)=\epsilon_{\lambda}(x)$. $\square$

Thus, when considering $\lambda\sim\mu$, we can always assume that
$\lambda_{1},\cdots,\lambda_{s}$ are coprime and that $\mu_{1},\cdots,\mu_{s}$
are coprime.

######  Lemma 3.11.

Let $\lambda,\mu\in P(s,n)$ and $1\leq m\in\mathbb{N}$ such that
$\\{\lambda_{j}\centerdot m\mid 1\leq i\leq s\\}=\\{\mu_{j}\centerdot m\mid
1\leq j\leq s\\}$ as multisets. Then $\lambda\sim\mu$ if and only if
$(\lambda,m)\sim(\mu,m)$ in $P(s+1,n+m)$. In particular, suppose
$\lambda_{j}\centerdot m=d=\mu_{j}\centerdot m$ for $1\leq j\leq s$. Then
$\lambda\sim\mu$ if and only if
$(\frac{\lambda}{d},\frac{m}{d})\sim(\frac{\mu}{d},\frac{m}{d})$ in
$P(s+1,\frac{n+m}{d})$.

Proof. The last conclusion in Lemma 3.11 follows from Lemma 3.10. We prove the
other one. Let $\bar{\lambda}=(\lambda,m)\vdash n+m\in P(s+1,n+m)$. Since the
multisets $\\{\lambda_{j}\centerdot m\mid 1\leq i\leq s\\}$ and
$\\{\mu_{j}\centerdot m\mid 1\leq j\leq s\\}$ are equal, the summations
$\sum_{j_{1}<j_{2}<\cdots<j_{i-1}}(\lambda_{j_{1}}\centerdot
m)\centerdot\cdots\centerdot(\lambda_{j_{i-1}}\centerdot m)$ equals the
summation $\sum_{j_{1}<j_{2}<\cdots<j_{i-1}}(\mu_{j_{1}}\centerdot
m)\centerdot\cdots\centerdot(\mu_{j_{i-1}}\centerdot m).$ In particular,
$g_{s+1}(\bar{\lambda})=g_{s+1}(\bar{\mu})$. Clearly, for $1\leq i\leq s$, we
have $g_{i}(\bar{\lambda})=g_{i}(\lambda)+\sum_{1\leq j_{i}<\cdots<j_{i-1}\leq
s}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{i-1}}\centerdot
m=g_{i}(\lambda)+\sum_{1\leq j_{i}<\cdots<j_{i-1}\leq
s}(\lambda_{j_{1}}\centerdot m)\cdots\centerdot(\lambda_{j_{i-1}}\centerdot
m)$. Thus, for $1\leq i\leq s$, $g_{i}(\bar{\lambda})=g_{i}(\bar{\mu})$ if and
only if $g_{i}(\lambda)=g_{i}(\mu)$. This shows Lemma 3.11. $\square$.

A special case of Lemma 3.11 is that $\lambda\sim\mu$ if and only if
$(\lambda,1)\sim(\mu,1)$. A slightly generalized form of the last statement of
Lemma 3.11 is as follows.

######  Lemma 3.12.

Assume $\lambda\sim\mu$ in $P(s,n)$ and $\gamma\sim\delta$ in $P(t,m)$. If
$\lambda_{i}\centerdot\gamma_{j}=\mu_{p}\centerdot\delta_{q}$ for all $1\leq
i,p\leq s,1\leq j,q\leq t$, then $(\lambda,\gamma)\sim(\mu,\delta)$ in
$P(s+t,n+m).$

Proof. By assumption, we suppose
$d:=\lambda_{i}\centerdot\gamma_{j}=\mu_{p}\centerdot\delta_{q}$. Now, we
calculate $g_{i}(\lambda,\gamma)$ and $g_{i}(\mu,\delta)$ for $1\leq i\leq
s+t.$ Clearly, $g_{1}(\lambda,\gamma)=g_{1}(\mu,\delta)$ and
$g_{s+t}(\lambda,\gamma)=g_{s+t}(\mu,\delta)$. We may assume $s\leq t$ and
consider the following cases.

(1) $1<i\leq s\leq t$.

$\begin{array}[]{rl}g_{i}(\bar{\lambda})&=\sum_{1\leq j_{1}<\cdots<j_{i}\leq
s}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{i}}+\sum_{1\leq
k_{1}<\cdots<k_{i}\leq
t}\gamma_{k_{1}}\centerdot\cdots\centerdot\gamma_{k_{i}}\\\
&+\sum_{(p,q):p+q=i,1\leq p,q\leq i}\sum_{1\leq j_{1}<\cdots<j_{p}\leq s,1\leq
k_{1}<\cdots<k_{q}\leq
t}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{p}}\centerdot\gamma_{k_{1}}\centerdot\cdots\centerdot\gamma_{k_{q}}\\\
&=g_{i}(\lambda)+g_{i}(\gamma)+\sum_{(p,q):p+q=i,1\leq p\leq s,1\leq q\leq
t}dC^{p}_{s}C^{q}_{t}\\\ &=g_{i}(\mu)+g_{i}(\delta)+\sum_{(p,q):p+q=i,1\leq
p\leq s,1\leq q\leq t}dC^{p}_{s}C^{q}_{t}\\\
&=g_{i}(\mu)+g_{i}(\delta)+\sum_{(p,q):p+q=i,1\leq p,q\leq i}\sum_{1\leq
j_{1}<\cdots<j_{p}\leq s,1\leq k_{1}<\cdots<k_{q}\leq
t}\mu_{j_{1}}\centerdot\cdots\centerdot\mu_{j_{p}}\centerdot\delta_{k_{1}}\centerdot\cdots\centerdot\delta_{k_{q}}\\\
&=g_{i}(\bar{\mu}).\end{array}$

(2) $s+1\leq i<s+t$.

In this case, we first assume $i\leq t$.

$\begin{array}[]{rl}g_{i}(\bar{\lambda})&=\sum_{1\leq k_{1}<\cdots<k_{i}\leq
t}\gamma_{k_{1}}\centerdot\cdots\centerdot\gamma_{k_{i}}\\\
&+\sum_{(p,q):p+q=i,1\leq p,q\leq i}\sum_{1\leq j_{1}<\cdots<j_{p}\leq s,1\leq
k_{1}<\cdots<k_{q}\leq
t}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{p}}\centerdot\gamma_{k_{1}}\centerdot\cdots\centerdot\gamma_{k_{q}}\\\
&=g_{i}(\gamma)+\sum_{(p,q):p+q=i,1\leq p\leq s,1\leq q\leq
t}dC^{p}_{s}C^{q}_{t}\\\ &=g_{i}(\delta)+\sum_{(p,q):p+q=i,1\leq p\leq s,1\leq
q\leq t}dC^{p}_{s}C^{q}_{t}\\\ &=g_{i}(\delta)+\sum_{(p,q):p+q=i,1\leq p,q\leq
i}\sum_{1\leq j_{1}<\cdots<j_{p}\leq s,1\leq k_{1}<\cdots<k_{q}\leq
t}\mu_{j_{1}}\centerdot\cdots\centerdot\mu_{j_{p}}\centerdot\delta_{k_{1}}\centerdot\cdots\centerdot\delta_{k_{q}}\\\
&=g_{i}(\bar{\mu}).\end{array}$

Next, we assume $t<i$. Then

$\begin{array}[]{rl}g_{i}(\bar{\lambda})&=\sum_{(p,q):p+q=i,1\leq p,q\leq
i}\sum_{1\leq j_{1}<\cdots<j_{p}\leq s,1\leq k_{1}<\cdots<k_{q}\leq
t}\lambda_{j_{1}}\centerdot\cdots\centerdot\lambda_{j_{p}}\centerdot\gamma_{k_{1}}\centerdot\cdots\centerdot\gamma_{k_{q}}\\\
&=\sum_{(p,q):p+q=i,1\leq p\leq s,1\leq q\leq t}dC^{p}_{s}C^{q}_{t}\\\
&=\sum_{(p,q):p+q=i,1\leq p,q\leq i}\sum_{1\leq j_{1}<\cdots<j_{p}\leq s,1\leq
k_{1}<\cdots<k_{q}\leq
t}\mu_{j_{1}}\centerdot\cdots\centerdot\mu_{j_{p}}\centerdot\delta_{k_{1}}\centerdot\cdots\centerdot\delta_{k_{q}}\\\
&=g_{i}(\bar{\mu}).\end{array}$

Thus $g_{i}(\bar{\lambda})=g_{i}(\bar{\mu})$ for all $1\leq i\leq s+t$.
$\square$

###  3.2 Isomorphisms of semisimple invariant matrix algebras

This section is devoted to a proof of Theorem 1.1. We first show an auxiliary
lemma.

######  Lemma 3.13.

Let $R$ be a division ring of characteristic $p\geq 0$ and
$\sigma\in\Sigma_{n}$ be of cycle type
$(\lambda_{1},\lambda_{2},\cdots,\lambda_{s})$ with $\lambda_{i}\geq 1$ for
all $1\leq i\leq s$. Then

$(1)$ $S_{n}(\sigma,R)$ is semisimple if and only if $p\nmid\lambda_{i}$ for
$1\leq i\leq s$.

$(2)$ If $R$ is an algebraically closed field and $p\nmid\lambda_{i}$ for all
$i$, then $S_{n}(\sigma,R)\simeq R^{\oplus h_{1}(\sigma)}\times
M_{2}(R)^{\oplus h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus
h_{s}(\sigma)}$ as algebras, where $R^{\oplus h_{1}(\sigma)}$ denotes the
direct product of $h_{1}(\sigma)$ copies of the ring $R$.

Proof. (1) Clearly, $M_{n}(R)$ is a simple artinian ring if $R$ is a division
ring. Let ${\rm rad}(A)$ denote the Jacobson radical of a ring $A$. If
$S_{n}(\sigma,R)$ is semisimple, that is ${\rm
rad}\big{(}S_{n}(\sigma,R)\big{)}=0$, then ${\rm
rad}\big{(}f_{i}S_{n}(\sigma,R)f_{i}\big{)}$ = $f_{i}{\rm
rad}\big{(}S_{n}(\sigma,R)\big{)}f_{i}=0$ for all $1\leq i\leq s$. This means
that all $f_{i}S_{n}(\sigma,R)f_{i}$ are semisimple. By Lemma 2.5(1), the
group algebra $R[C_{\lambda_{i}}]$ is semisimple. Hence $p\nmid\lambda_{i}$
for all $i$. Conversely, if $p\nmid\lambda_{i}$ for all $i$, then $p$ does not
divide the least common multiple of these $\lambda_{i}$, that is, the order of
$G=\langle\sigma\rangle$ is invertible in $R$. In this case, $S_{n}(\sigma,R)$
is semisimple by [5, Theorem 1.15, p.15], which says that if a finite group
acts on a semisimple artinian ring with its order invertible in the ring, then
the fixed ring is semisimple.

(2) Let $c_{\sigma}:=e_{1,(1)\sigma}+e_{2,(2)\sigma}+\cdots+e_{n,(n)\sigma}$
be the permutation matrix in $M_{n}(R)$ corresponding to $\sigma$. Then
$c_{\sigma}={\rm
diag}\\{c_{\sigma_{1}},c_{\sigma_{2}},\cdots,c_{\sigma_{s}}\\}$, where
$c_{\sigma_{i}}\in M_{\lambda_{i}}(R)$ is defined similarly. For $1\leq i\leq
s$, the eigenvalues of $c_{\sigma_{i}}$ are distinct and denoted by $V_{i}$.
Hence the Jordan canonical form of $c_{\sigma_{i}}$ has only Jordan blocks of
order $1$, namely the diagonal matrix ${\rm
diag}(1,\omega_{i},\cdots,\omega_{i}^{\lambda_{i}-1})$, where $\omega_{i}$ is
a primitive root of the unity of order $\lambda_{i}$ in $R$. This implies that
the Jordan canonical form of $c_{\sigma}$ is the diagonal matrix

$c:={\rm
diag}(1,\omega_{1},\cdots,\omega_{1}^{\lambda_{1}-1},\cdots,1,\omega_{i},\cdots,\omega_{i}^{\lambda_{i}-1},\cdots,1,\omega_{s},\cdots,\omega_{s}^{\lambda_{s}-1}).$

For $1\leq i\leq s$, we write
$H_{i}(\sigma)=\\{r_{i1},r_{i2},\cdots,r_{ih_{i}(\sigma)}\\}$ (see the
notation above Definition 3.4). Then the algebraic and geometric
multiplicities of the eigenvalue $r_{ij}$ of $c_{\sigma}$ are just $i$ for
$1\leq j\leq h_{i}(\sigma)$. Thus $c$ is similar to

$d:={\rm
diag}(r_{11}I_{1},r_{12}I_{1},\cdots,r_{1h_{1}(\sigma)}I_{1},\cdots,r_{i1}I_{i},r_{i2}I_{i},\cdots,r_{ih_{i}(\sigma)}I_{i},\cdots,r_{s1}I_{s},r_{s2}I_{s},\cdots,r_{sh_{s}(\sigma)}I_{s})$

with $\sum_{i=1}^{s}ih_{i}(\sigma)=n.$ Recall that $I_{i}$ denotes the
identity matrix in $M_{i}(R)$. Hence the matrix $c_{\sigma}$ is similar to the
matrix $d$. By Lemma 2.2 (see also [11, Lemma 2.1]),
$S_{n}(\sigma,R)=S_{n}(c_{\sigma},R):=\\{x\in M_{n}(R)\mid
xc_{\sigma}=c_{\sigma}x\\}\simeq S_{n}(d,R):=\\{x\in M_{n}(R)\mid xd=dx\\}$ as
algebras. Note that $r_{ij}\neq r_{kl}$ for $(i,j)\neq(k,l)$ with $1\leq
i,k\leq s,1\leq j\leq h_{i}(\sigma),1\leq l\leq h_{k}(\sigma)$. It follows
from [11, Lemma 2.2] that the centralizer algebra $S_{n}(d,R)$ is isomorphic
to $R^{\oplus h_{1}(\sigma)}\times M_{2}(R)^{\oplus
h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus h_{s}(\sigma)}$. Thus
$S_{n}(\sigma,R)\simeq R^{\oplus h_{1}(\sigma)}\times M_{2}(R)^{\oplus
h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus h_{s}(\sigma)}$ as algebras.
$\square$

Lemma 3.13(2) shows that the multiplicity of the matrix algebra $M_{i}(R)$ in
a Wedderburn decomposition of the semisimple algebra $S_{n}(\lambda,R)$ is
given by $h_{i}(\lambda)$.

Proof of Theorem 1.1. (1) follows from Lemma 3.13(1).

(2) By Lemma 3.13(1), $S_{n}(\sigma,R)\simeq R^{\oplus h_{1}(\sigma)}\times
M_{2}(R)^{\oplus h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus
h_{s}(\sigma)}$ and $S_{m}(\tau,R)\simeq R^{\oplus h_{1}(\tau)}\times
M_{2}(R)^{\oplus h_{2}(\tau)}\times\cdots\times M_{t}(R)^{\oplus
h_{t}(\tau)}$.

(i) It follows from $h_{i}(\sigma)\geq 0,h_{i}(\tau)\geq 0$ for $1\leq i\leq
s$, $h_{s}(\sigma)=\lambda_{1}\centerdot\cdots\centerdot\lambda_{s}\neq 0$ and
$h_{t}(\tau)\neq 0$ that

$R^{\oplus h_{1}(\sigma)}\times M_{2}(R)^{\oplus
h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus h_{s}(\sigma)}\simeq
R^{\oplus h_{1}(\tau)}\times M_{2}(R)^{\oplus h_{2}(\tau)}\times\cdots\times
M_{t}(R)^{\oplus h_{t}(\tau)}$

as algebras if and only if $s=t$ and $h_{i}(\sigma)=h_{i}(\tau)$ for all
$1\leq i\leq s.$

Suppose $S_{n}(\sigma,R)\simeq S_{m}(\tau,R)$, that is, we assume that $s=t$
and $h_{i}(\sigma)=h_{i}(\tau)$ for all $i$. We show that $m=n$ and
$\epsilon_{\lambda}(x)=\epsilon_{\mu}(x)$. In fact, since
$\sum_{i=1}^{s}ih_{i}(\sigma)=n$ and $\sum_{i=1}^{t}ih_{i}(\tau)=m,$ we have
$m=n$. By the definitions of $h_{s}(\sigma)$ and $h_{s}(\tau)$, we get
$h_{s}(\sigma)=g_{s}(\sigma)$ and $h_{s}(\tau)=g_{s}(\tau)$. It follows from
$h_{s}(\sigma)=h_{s}(\tau)$ that $g_{s}(\sigma)=g_{s}(\tau)$. Now, by the
definitions $h_{s-1}(\sigma)$ and $h_{s-1}(\tau)$ and Lemma 3.3(1), we have
$h_{s-1}(\sigma)=g_{s-1}(\sigma)-C_{s}^{s-1}g_{s}(\sigma)$ and
$h_{s-1}(\tau)=g_{s-1}(\tau)-C_{s}^{s-1}g_{s}(\tau)$. It then follows from
$h_{s-1}(\sigma)=h_{s-1}(\tau)$ and $g_{s}(\sigma)=g_{s}(\tau)$ that
$g_{s-1}(\sigma)=g_{s-1}(\tau)$. Continuing this procedure, we get
$g_{i}(\sigma)=g_{i}(\tau)$ for $1\leq i\leq s-2$. Thus, if
$h_{i}(\sigma)=h_{i}(\tau)$ for $1\leq i\leq s$, then
$g_{i}(\sigma)=g_{i}(\tau)$ for $1\leq i\leq s$ and
$\epsilon_{\lambda}(x)=\epsilon_{\mu}(x)$.

Conversely, if $m=n$ and $\epsilon_{\lambda}(x)=\epsilon_{\mu}(x)$, then $s=t$
and $g_{s}(\tau)g_{i}(\sigma)=g_{s}(\sigma)g_{i}(\tau)$ for $1\leq i\leq s-1$.
As $g_{1}(\lambda)=n=m=g_{1}(\mu)$, we get $g_{s}(\sigma)=g_{s}(\tau)$. Thus
it follows from $\epsilon_{\lambda}(x)=\epsilon_{\mu}(x)$ that
$g_{i}(\lambda)=g_{i}(\tau)$ for all $1\leq i\leq s$. By Lemma 3.3(2), we
obtain $h_{i}(\sigma)=h_{i}(\tau)$ for all $i$. Hence $S_{n}(\sigma,R)\simeq
S_{m}(\tau,R)$.

(ii) Since $S_{n}(\sigma,R)\simeq R^{\oplus h_{1}(\sigma)}\times
M_{2}(R)^{\oplus h_{2}(\sigma)}\times\cdots\times M_{s}(R)^{\oplus
h_{s}(\sigma)}$ by Lemma 3.13(2), the basic algebra $B(\sigma)$ of
$S_{n}(\sigma,R)$ is isomorphic to $R^{\oplus h_{1}(\sigma)}\times R^{\oplus
h_{2}(\sigma)}\times\cdots\times R^{\oplus h_{s}(\sigma)}$. As is known,
$S_{n}(\sigma,R)$ and $B(\sigma)$ are always Morita equivalent, while
$S_{n}(\sigma,R)$ and $S_{m}(\tau,R)$ are Morita equivalent if and only if
their basic algebras $B(\sigma)$ and $B(\tau)$ are isomorphic. Clearly,
$B(\sigma)\simeq B(\tau)$ if and only if
$\sum_{i=1}^{s}h_{i}(\sigma)=\sum_{i=1}^{t}h_{i}(\tau)$. By Lemma 3.3(2),
$\sum_{i=1}^{s}h_{i}(\sigma)=\sum_{i=1}^{t}h_{i}(\tau)$ if and only if
$\sum_{i=1}^{s}(-1)^{i+1}g_{i}(\sigma)=\sum_{i=1}^{t}(-1)^{i+1}g_{i}(\tau)$.
Therefore $S_{n}(\sigma,R)$ and $S_{m}(\tau,R)$ are Morita equivalent if and
only if
$\sum_{i=1}^{s}(-1)^{i+1}g_{i}(\sigma)=\sum_{i=1}^{t}(-1)^{i+1}g_{i}(\tau)$ if
and only if $g_{s}(\lambda)\epsilon_{\lambda}(1)=g_{t}(\mu)\epsilon_{\mu}(1)$.
Note that $g_{s}(\lambda)$ is the greatest common divisor of
$\lambda_{1},\cdots,\lambda_{s}$. $\square$

Theorem 1.1 tells us that the foregoing consideration on the polynomial
equivalence $\lambda\sim\mu$ in $P(s,n)$ provides us with many isomorphisms of
invariant matrix algebras. For example,
$S_{38}\big{(}(17,11,8,2),\mathbb{C}\big{)}\simeq
S_{38}\big{(}(17,11,6,4),\mathbb{C}\big{)}$ by repeatedly applying Corollary
3.11 to the polynomial equivalence $(8,2)\sim(6,4)$ in $P(2,10)$. Also, for
any prime number $p\geq 5$ and two-part partitions $\lambda$ and $\mu$ of $p$,
it follows from Theorem 1.1 that $S_{p}(\lambda,\mathbb{C})\simeq
S_{p}(\mu,\mathbb{C})$. For instance, $S_{5}((4,1),\mathbb{C})\simeq
S_{5}((3,2),\mathbb{C})\simeq\mathbb{C}\times\mathbb{C}\times\mathbb{C}\times
M_{2}(\mathbb{C})$.

Remark that the condition $m=n$ in Theorem 1.1(2)(i) is needed. For example,
both $\lambda=(4,2,2)$ and $\mu=(2,1,1)$ have the same partition polynomial
$x^{2}-3x+4$, but $S_{8}(\lambda,R)\not\simeq S_{4}(\mu,R)$ for any ring $R$.

As a consequence of Theorem 1.1 and Corollary 3.9. we have immediately the
corollary.

######  Corollary 3.14.

Let $R$ be an algebraically closed field of characteristic $p\geq 0$ and let
$\sigma\in\Sigma_{n}$ and $\tau\in\Sigma_{m}$ be of cycle types
$(\lambda_{1},\lambda_{2},\cdots,\lambda_{s})$ and
$(\mu_{1},\mu_{2},\cdots,\mu_{t})$, respectively. Assume that
$p\nmid\lambda_{i}$ and $p\nmid\mu_{j}$ for $1\leq i\leq s$ and $1\leq i\leq
t$. If $\\{\lambda_{1},\cdots,\lambda_{s}\\}$ are pairwise coprime and
$\\{\mu_{1},\cdots,\mu_{t}\\}$ are pairwise coprime, then

$(1)$ $S_{n}(\sigma,R)\simeq S_{m}(\tau,R)$ if and only if $n=m$ and $s=t$.

$(2)$ $S_{n}(\sigma,R)$ and $S_{m}(\tau,R)$ are Morita equivalent if and only
if $n-s=m-t$.

Finally, we mention that there are a few issues worthy of further
consideration, which have directly been suggested in this section but not
answered. We formalize some of them as the following questions.

Questions. (1) What should be Theorem 1.1 for arbitrary field $R$? Or more
generally, when are $S_{n}(\lambda,R)$ and $S_{n}(\mu,R)$ isomorphic (or
derived equivalent) for $\lambda,\mu\in P(s,n)$ and $R$ a unitary ring?

(2) Given a polynomial
$f(x)=x^{s-1}-a_{s-2}x^{s-2}+\cdots+(-1)^{s-i-1}a_{i}x^{i}+\cdots+(-1)^{s-2}a_{1}x+(-1)^{s-1}a_{0}\in\mathbb{Z}[x]$
with $s\geq 3$ and $a_{i}\geq C_{s}^{i+1}$ for $1\leq i\leq s-2$, is there a
partition $\lambda\in P(s,n)$ for some $n$ such that
$\epsilon_{\lambda}(x)=f(x)$? In other words, what are the necessary and
sufficient conditions for a monic polynomial $f(x)$ to be a partition
polynomial?

(3) Are $\epsilon_{\lambda}(x)\epsilon_{\mu}(x)$ and
$\epsilon_{\lambda}(\epsilon_{\mu}(x))$ again partition polynomials? Or more
generally, how do the operations of partitions correspond to the ones of
partition polynomials?

(4) Let $I(s,n)$ be the set of equivalence classes of $P(s,n)$ with respect to
the polynomial equivalence relation. Then $I(s,n)$ is the union of distinct
equivalence classes. Let $E(s,n,j)$ be the set of those equivalence classes
that contain $j$ elements of $P(s,n)$. Define $p(s,n)$, $i(s,n)$ and
$e(s,n,j)$ to be the numbers of elements in $P(s,n)$, $I(s,n)$ and $E(s,n,j)$,
respectively. What are the generating functions of $i(s,n)$ and $e(s,n,j)$?

Note that the generating function for $p(s,n)$ is
$\sum_{n=0}^{\infty}p(s,n)q^{n}=\frac{q^{s}}{(1-q)(1-q^{2})\cdots(1-q^{s})}$
(see [1, Chapter 6]).

(5) Describe partitions $\lambda\in P(s,n)$ that are self-equivalent, that is,
if $\mu\in P(s,n)$ such that $\mu\sim\lambda$, then $\mu=\lambda$. For
instance, $\lambda=(4,4,1)$ is self-equivalent with
$\epsilon_{(4,4,1)}(x)=(x-3)^{2}$, but $\mu=(5,2,2)$ is not self-equivalent.

(6) For which partitions $\lambda$ are $\epsilon_{\lambda}(x)$ irreducible
polynomials over $\mathbb{Q}$?

(7) What are the necessary and sufficient conditions for $\lambda\sim\mu$ in
$P(s,n)$ in terms of the conjugate partitions of $\lambda$ and $\mu$?

Acknowledgement. The research work of both authors was partially supported by
the National Natural Science Foundation of China (12031014) and Beijing
Natural Science Foundation (1192004). The corresponding author CCX thanks
Professor Hourong Qin from Nanjing University for discussion on greatest
common divisor matrices.

## References

  * [1] G. E. Andrews and K. Eriksson, _Integer partitions_ , Cambridge University Press, Cambridge, 2004.
  * [2] S. Beslin and S. Ligh, Greatest common divisor matrices. _Linear Algebra Appl._ 118 (1989) 69-76.
  * [3] L. Datta and S. D. Morgera, On the reducibility of centrosymmetric matrices-Applications in engineering problems, _Circuits Systems Sig. Proc._ 8 (1) (1989) 71-96.
  * [4] Z. S. Li, The determinants of GCD matrices, _Linear Algebra Appl._ 134 (1990) 137-143.
  * [5] S. Montgomery, _Fixed rings of finite automorphism groups of associative rings._ Lecture Notes in Mathematics 818. Springer, Berlin, 1980.
  * [6] J. L. Stuart, Inflation matrices and ZME-matrices that commute with a permutation matrix. _SIAM J. Matrix Anal. Appl._ 9 (3) (1988) 408-418.
  * [7] J. L. Stuart and J. R. Weaver, Matrices that commute with a permutation matrix. _Linear Algebra Appl._ 150 (1991) 255-265.
  * [8] J. R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. _Amer. Math. Monthly_ 92 (10) (1985) 711-717.
  * [9] H. Weyl, _The classical groups. Their invariants and representations_ , Princeton University Press, Princeton, N.J., 1939.
  * [10] C. C. Xi and S. J. Yin, Cellularity of centrosymmetric matrix algebras and Frobenius extensions, _Linear Algebra Appl._ 590 (2020) 317-329.
  * [11] C. C. Xi and J. B. Zhang, Structures of centralizer matrix algebras. Preprint is available at: arXiv: 2012.11089.

Changchang Xi, School of Mathematical Sciences, Capital Normal University,
100048 Beijing, P.R. China; and School of Mathematics and Statistics, Central
China Normal University, 430079 Wuhan, P.R. China

Email<EMAIL_ADDRESS>
Jinbi Zhang, School of Mathematical Sciences, Capital Normal University,
100048 Beijing, P.R. China

Email<EMAIL_ADDRESS>


# Anisotropic swelling due to hydration in fibrous biomaterials

Xander A. Gouws  Laurent Kreplak  Andrew D. Rutenberg<EMAIL_ADDRESS>Department
of Physics & Atmospheric Science, Dalhousie University, Halifax, Nova Scotia,
B3H 4R2, Canada

###### Abstract

Naturally occurring protein fibres often undergo anisotropic swelling when
hydrated. Within a tendon, a hydrated collagen fibril’s radius expands by 40%
but its length only increases by 5%. The same effect, with similar magnitude,
is observed for keratin microfibrils within hair. Nevertheless, current
explanations for swelling anisotropy are based on molecular details that are
unique to each material. Here, we describe a coarse-grained liquid-crystalline
elastomer model for biomaterial hydration that allows for anisotropic
swelling. We show that our model is consistent with previously observed
behavior for both hair fibers and collagen fibrils. Despite a non-linear
relationship between water saturation and relative humidity, we find that the
squared deformation in the axial and radial directions is approximately linear
with water saturation. We further find that anisotropic swelling is linear
with respect to volume and shape changes. Hair and collagen exhibit remarkably
similar behavior under hydration. We suggest that our model may also be useful
for other biomaterials that exhibit anisotropic swelling.

##  I Introduction

Many biomaterials absorb water in response to changes in the ambient humidity.
Leonardo da Vinci used this effect to make the first hygrometer, or humidity
sensor, in 1481 using the changing mass of cotton with humidity. In 1783, hair
hygrometers were developed that used the changing length of hair with humidity
[1], and were widely used until the last century. More generally, any
biomaterial or biotextile will have changing water content with humidity (see,
e.g., [2]) — and will also have changing structure at microscopic, mesoscopic,
and bulk scales. The increasing development and use of biomaterials and
biotextiles makes the systematic effects of hydration important to
characterize.

For relatively well-studied biological materials, such as hair fibers and
collagen fibrils, the effects of hydration are dramatic. Mechanical properties
of hair depend strongly on water content [3], as do modes of mechanical
failure [4]. The Young’s modulus of collagen fibrils varies by orders of
magnitude between dry and aqueous conditions [5]; similar scale changes are
seen in bending [6] or indentation [7, 5] and in keratin [8]. Furthermore,
mechanical effects of hydrated filaments are strongly affected by solution
conditions [9, 10, 11].

Both hair and collagen are strongly anisotropic materials, with keratin
intermediate filaments or collagen molecules predominantly aligned with the
cylindrical axis of individual hair fibers or collagen fibrils, respectively.
Substantial anisotropy is also observed with hydration effects. When hydrated,
a collagen fibril’s radius may expand by 40% while its length only increases
by 5% [11]. Similarly, when hair is hydrated, its radius expands by 14%, but
its length expands by only 2% [12, 13].

While both hair and collagen have anisotropic structures and exhibit
anisotropic swelling with similar relative magnitudes, microstructural
explanations of their anisotropic response to hydration are quite different.
For collagen fibrils, length changes are thought to be minimized by the
sliding of collagen molecules past each other [11]. In contrast, intermediate
filaments in hair do not themselves restructure, but rather guide the swelling
of their surrounding amorphous matrix [14, 15]. We are agnostic with respect
to microstructural models, rather we want to develop coarse grained
explanations of the hydration of anisotropic fibers such as hair and collagen.

Coarse-grained liquid-crystal elastomers (LCE) have recently proven useful for
modelling mechanical effects of anisotropic collagen fibrils [16, 17]. The
essential structural components of these LCE models is anisotropic alignment
of collagen molecules together with mechanical cross-linking that orients with
respect to the molecules. Hair exhibits similar anisotropic alignment of
cross-linked keratin microfibrils [18] and so could be addressed by similar
models. Characterizing hydration effects in such coarse-grained models would
then enable the systematic theoretical characterization of the mechanical
effects of hydration in these systems.

Our goal in this paper is to simplify and summarize the observed phenomenology
of anisotropic swelling due to hydration in both hair and collagen fibrils.
Since the water content of protein fibers can vary by osmolarity even at a
fixed relative humidity [9, 10, 11], we investigate how shape depends on water
content. We obtain a simple phenomenological model for this relationship, and
show how it can be naturally described within an LCE model. For simplicity,
here we neglect molecular tilts observed in collagen fibrils [19] and the hair
cortex [18], and so we consider an LCE fiber in the nematic limit — with
molecules completely aligned with the cylindrical axis.

##  II Methods and Results

Figure 1: Previously reported hydration data. a) radial deformation of hair
[13] and collagen [10], b) axial deformation of hair [13] and collagen [10],
c) water saturation (g water/g material) for hair [20] and collagen [21]. The
hydration curve for hair is shown. All are vs. relative humidity ($\%$). Hair
is given by orange triangles, collagen by blue circles. Standard errors are
indicated. Unfilled points indicate the reference state.

Relative dimensional changes induced by variations in relative humidity have
been experimentally characterized by several groups for both collagen [10] and
hair [13]. We have extracted digitized collagen data [10], and used raw hair
measurements [13]. Fig. 1a shows the radial deformation

$\lambda_{r}=R/R_{0},$ (1)

for both collagen and hair vs. relative humidity, where $R$ is the cylindrical
radius and $R_{0}$ is the reference radius. Error bars are standard errors of
the means. Fig. 1b shows the corresponding axial deformations

$\lambda_{z}=L/L_{0},$ (2)

where $L$ is the axial length and $L_{0}$ is the reference length at the
relative humidity indicated by the unfilled point. We observe that
deformations are nonlinear functions of relative humidity.

Figure 2: Squared deformations vs water saturation. a) Squared radial
deformation, $\lambda_{r}^{2}$ vs. saturation $\theta$ (grams of water
absorbed per gram of dry mass). b) Squared axial deformation $\lambda_{z}^{2}$
vs. $\theta$. Unfilled points indicate the reference states. Hair or collagen
fibrils as indicated by legend. Linear best fit lines are given for hair
(orange dashed) and collagen (solid blue).

We are interested in how the water content of hair and collagen affects their
shapes and structures. Independent studies have examined the relationship
between water content and humidity both in hair [20] and in collagen [21], as
shown in Fig. 1c. These experiments report saturation $\theta$ as the grams of
water absorbed per gram of protein dry mass. Errors were not reported for
these saturation studies. How deformations change with saturation $\theta$ has
not been previously reported, but can be determined by also using the
deformation studies exploring ambient humidity [13, 10].

Figure 3: Geometric deformations vs water saturation. a) Deformation
anisotropy $\alpha=\lambda_{r}/\lambda_{z}$ vs. saturation $\theta$. b)
Swelling ratio $\beta=\lambda_{r}^{2}\lambda_{z}$ vs. $\theta$. For both hair
and collagen fibrils as indicated by legend. Lines are model curves for
$\alpha$ and $\beta$ derived from Eqns. 3 and 4, using the fitted slope
parameters.

In Fig. 2, we re-plot the deformation data as a function of saturation. This
data is from Fig. 1. Note that the hair study used a dehydrated baseline (at
$\theta_{0}\approx 0$) and explored the effects of increased hydration while
the collagen study used a hydrated baseline ($\theta_{0}\approx 0.5$) and then
dehydrated the samples. We see that the squared deformations vary
approximately linearly with saturation.

Accordingly, our phenomenological model is that $\lambda_{r}^{2}$ and
$\lambda_{z}^{2}$ are linear functions of the saturation $\theta$;

$\lambda_{r}^{2}=B_{\perp}(\theta-\theta_{0})+1$ (3)

and

$\lambda_{z}^{2}=B_{\parallel}(\theta-\theta_{0})+1.$ (4)

The initial saturation $\theta_{0}$ is the saturation at which the reference
lengths $R_{0}$ and $L_{0}$ have been measured, where $\lambda=1$.

For hair and collagen, the best-fit slope parameters are

$\displaystyle B_{\perp}^{\text{collagen}}$ $\displaystyle=0.62\phantom{0}\pm
0.01,$ (5) $\displaystyle B_{\parallel}^{\text{collagen}}$
$\displaystyle=0.110\pm 0.005,$ (6) $\displaystyle B_{\perp}^{\text{hair}}$
$\displaystyle=1.05\phantom{0}\pm 0.03,\text{ and}$ (7) $\displaystyle
B_{\parallel}^{\text{hair}}$ $\displaystyle=0.20\phantom{0}\pm 0.03.$ (8)

These parameters have been determined using Deming regression with the SciPy
version 1.10.1 orthogonal distance regression subpackage. This performs
ordinary least squares fitting whilst accounting for uncertainty in the data.
Our final uncertainties are underestimates, because the saturation and
deformation measurements were done by different groups and because saturation
errors were not reported.

We are interested in shape changes with water content, and so we introduce the
deformation anisotropy

$\alpha={\lambda_{r}}/{\lambda_{z}},$ (9)

which is a measure of how anisotropic the deformations are. We also introduce
a geometric measure of water content, the swelling ratio

$\beta=\lambda_{r}^{2}\lambda_{z}=V/V_{0},$ (10)

which is how much the volume has changed with respect to the baseline. In Fig.
3, these quantities have been plotted as functions of the saturation. Lines
from our model Eqns. 3 and 4 have been provided, using the fitted slope
parameters in Eqns. 5–8. While the model lines are nonlinear functions, they
appear quite linear. This is because the squared deformations in Fig. 2 do not
deviate too much from $1$, so the square-root is approximately linear, i.e.,
$\lambda=\sqrt{\lambda^{2}}$, and $\sqrt{1+x}\approx 1+x/2$, where $x$
represents saturation effects.

Figure 4: Deformation anisotropy $\alpha=\lambda_{r}/\lambda_{z}$ vs. swelling
ratio $\beta=\lambda_{r}^{2}\lambda_{z}=V/V_{0}$, for both hair and collagen
as indicated. Unfilled points indicate the reference state at
$\alpha_{0}=\beta_{0}=1$. The lines indicate model lines derived from Eqns. 3
and 4 using fitted slope parameters.

Finally, $\alpha(\theta)$ and $\beta(\theta)$ can be plotted against each
other as a parametric function of the saturation $\theta$. This is shown in
Fig. 4, and reflects the geometric response to water hydration: how shape
changes anisotropically as the volume changes. We observe that both collagen
and hair lie on approximately the same gently-curving line. While the
reference point at $\alpha_{0}=\beta_{0}=1$ must be shared, the slopes are
approximately equal despite being unconstrained. Expanding about $\beta=1$,
the slope $d\alpha/d\beta$ is

$\left.\frac{d\alpha}{d\beta}\right|_{\theta=\theta_{0}}=\frac{d\alpha/d\theta|_{\theta=\theta_{0}}}{d\beta/d\theta|_{\theta=\theta_{0}}}=\frac{B_{\perp}/B_{\parallel}-1}{2B_{\perp}/B_{\parallel}+1}.$
(11)

Using the fitted slope parameters in Eqns. 5–8, we find that

$\displaystyle(B_{\perp}/B_{\parallel})_{\text{collagen}}$
$\displaystyle=5.6\pm 0.3\text{ and}$ (12)
$\displaystyle(B_{\perp}/B_{\parallel})_{\text{hair}}$ $\displaystyle=5.3\pm
0.9.$ (13)

These are equal within uncertainties, consistent with the observation that
hair and collagen lie on approximately the same model curve in Fig. 4.

###  II.1 Liquid Crystal Elastomer Theory

Our analysis of swelling data for both hair and collagen fibrils shows a
striking linearity of the squared deformations with respect to saturation.
Considering that both hair [18] and collagen [22] are comprised of long
helical molecules which are cross-linked together, this phenomenological
observation can be motivated within the context of liquid crystal elastomer
(LCE) theory [23]. In this theory, cross-links are anisotropic Gaussian random
walks with step-lengths $\ell^{\perp}$ and $\ell^{\parallel}$ in directions
perpendicular and parallel to the local director field
${\bm{\mathbf{\hat{n}}}}$, respectively. Here, we take the director field to
be in the nematic limit, with
${\bm{\mathbf{\hat{n}}}}={\bm{\mathbf{\hat{e}}}}_{z}$.

Water is proposed to be a good solvent for protein chains [24]. Assuming this
applies to hair and collagen, then as water saturation increases from dry to
wet, both step-lengths are expected to increase due to repulsive interactions
within and between chains. Under the assumption that added water do not
interact, we expect a linear dependence of both step-lengths as a function of
saturation:

$\ell^{\perp}=\left(1-\frac{\theta}{\theta_{\text{max}}}\right)\ell^{\perp}_{\text{dry}}+\left(\frac{\theta}{\theta_{\text{max}}}\right)\ell^{\perp}_{\text{wet}}$
(14)

and

$\ell^{\parallel}=\left(1-\frac{\theta}{\theta_{\text{max}}}\right)\ell^{\parallel}_{\text{dry}}+\left(\frac{\theta}{\theta_{\text{max}}}\right)\ell^{\parallel}_{\text{wet}},$
(15)

where $\theta_{\text{max}}$ is the largest accessible saturation. With these
expressions, we can easily obtain the LCE anisotropy parameter
$\zeta=\ell^{\parallel}/\ell^{\perp}$. Using the fits in Fig. 1c,
$\theta_{\text{max}}$ can be estimated to be
$0.61\text{\,}\mathrm{g}\mathrm{/}\mathrm{g}$ for collagen and
$0.30\text{\,}\mathrm{g}\mathrm{/}\mathrm{g}$ for hair.

A three-dimensional random walk with a total length $L_{\text{chain}}$ has
root-mean-squared end-to-end distances of
$\sqrt{\ell^{\perp}L_{\text{chain}}/3}$ and
$\sqrt{\ell^{\parallel}L_{\text{chain}}/3}$ in the two radial and the axial
directions, respectively [23]. This gives

$\lambda_{r}=\sqrt{\ell^{\perp}/\ell^{\perp}_{0}}$ (16)

and

$\lambda_{z}=\sqrt{\ell^{\parallel}/\ell^{\parallel}_{0}}.$ (17)

Looking at Eqns. 16 and 17 in the context of Eqns. 14 and 15, one directly
recovers our phenomenological observation that the squared deformations
increase linearly with saturation. Furthermore, the ratio of wet and dry step-
lengths are related to the $B$-parameters by

$\displaystyle\frac{\ell^{\perp}_{\text{wet}}}{\ell^{\perp}_{\text{dry}}}$
$\displaystyle=\frac{B_{\perp}\theta_{\text{max}}}{1-B_{\perp}\theta_{0}}+1\text{
and}$ (18)
$\displaystyle\frac{\ell^{\parallel}_{\text{wet}}}{\ell^{\parallel}_{\text{dry}}}$
$\displaystyle=\frac{B_{\parallel}\theta_{\text{max}}}{1-B_{\parallel}\theta_{0}}+1,$
(19)

and using our $B$ values we estimate

$\displaystyle(\ell^{\perp}_{\text{wet}}/\ell^{\perp}_{\text{dry}})_{\text{collagen}}$
$\displaystyle=1.55\phantom{0}\pm 0.02,$ (20)
$\displaystyle(\ell^{\parallel}_{\text{wet}}/\ell^{\parallel}_{\text{dry}})_{\text{collagen}}$
$\displaystyle=1.071\pm 0.004,$ (21)
$\displaystyle(\ell^{\perp}_{\text{wet}}/\ell^{\perp}_{\text{dry}})_{\text{hair}}$
$\displaystyle=1.320\pm 0.008,\text{ and}$ (22)
$\displaystyle(\ell^{\parallel}_{\text{wet}}/\ell^{\parallel}_{\text{dry}})_{\text{hair}}$
$\displaystyle=1.06\phantom{0}\pm 0.01.$ (23)

##  III Discussion

We have modeled the relationship between water content and anisotropic
swelling in two protein fibres — hair and collagen. We have shown that
experimental data exhibits a strong linear dependence between the squared
axial and radial deformations and the water saturation (Fig. 2). These trends
are directly predicted by an LCE framework assuming that cross-link steps
swell linearly as water saturation increases. Allowing anisotropic step-
lengths to vary linearly with water saturation is enough to capture trends in
current experimental data.

Considering that the experimental data used in this work are for hair fibres
with a typical diameter of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and for collagen fibrils with
diameters around $100\text{\,}\mathrm{nm}$, it is remarkable that both systems
show such similar behaviours. When the water content is implicit, such as in
the relationship between the deformation anisotropy
$\alpha=\lambda_{r}/\lambda_{z}$ and the swelling factor
$\beta=\lambda_{r}^{2}\lambda_{z}$, the similarity is even more striking (Fig.
4). A similar plot of the strain anisotropy $\epsilon_{r}/\epsilon_{z}$ (where
$\epsilon=\lambda-1$) highlights this similarity as well as the remarkable
${\sim}5$-fold strain anisotropies that are observed (Fig. 5).

Figure 5: Strain anisotropy vs. swelling ratio. Using the same data as Fig. 4,
we plot the strain anisotropy $\varepsilon_{r}/\varepsilon_{z}$ vs the
swelling ratio $\beta=\lambda_{r}^{2}\lambda_{z}=V/V_{0}$ — for both hair and
collagen as indicated. Note that $\epsilon=\lambda-1$. The lines indicate
model lines derived from Eqns. 3 and 4 using fitted slope parameters.

From our LCE framework and the similarity of hair and collagen fibril swelling
behavior, we would expect some structural similarities between the two
materials at coarse-grained length-scales. Indeed, each hair strand contains
keratin intermediate filaments and sulfur-rich associated proteins that
assemble into macrofibril bundles with a diameter in the order of 100 to 200
nm — like collagen fibrils [25]. Furthermore, the keratin filaments are
embedded in a protein matrix crosslinked via disulfide bonds [26]. The matrix
provides the elastomeric network required by our LCE framework, whereas in
collagen fibrils this is obtained through peptide bonds linking disordered end
domains of each collagen molecule. Furthermore, both the keratin filaments and
the triple-helical collagen molecules have the large aspect ratio required to
support nematic order. Interestingly, this nematic order can be suppressed
using salts like LiBr that denature keratin filaments within hair [27] or
temperature that denature collagen molecules within fibrils in tendons [28].
Fibres treated in this way show classical rubber elasticity in the presence of
water [29, 30], as would be expected for an isotropic LCE.

The connection between our phenomenological deformation model (Eqns. 3 and 4)
and microscopic LCE step-lengths (Eqns. 14 and 15) allows for future extension
of our results by generalizing from a simple nematic anisotropy (with
molecules aligned along a cylindrical axis) to more general orientational
structures. Recent theoretical studies of collagen fibrils have shown that
nearly-axial double-twist structures, in which molecules are tilted by as
little as $5\text{\,}\mathrm{\SIUnitSymbolDegree}$ with respect to the axis,
lead to rich couplings between structure and mechanical deformations that
depend on the LCE anisotropy parameter $\zeta$ [16, 17]. We have shown here
that water content modifies both $\ell_{\perp}$ and $\ell_{\parallel}$ — and
so it will modify $\zeta\equiv\ell_{\parallel}/\ell_{\perp}$ and therefore, we
expect, mechanical properties that depend on the double-twist. Considering
that hair macrofibrils also have a double-twist structure with an even higher
tilt [18] than collagen fibrils, one can expect interesting coupling between
water content and mechanical deformations in hair as well.

We expect that our approach would also apply to anisotropic synthetic
materials. For example, Mredha et al. [31] created a millimeter diameter bio-
mimetic “hair” by crosslinking a water soluble polymer solution containing
aligned collagen fibrils. Depending on the formulation, they achieved
hydrogels with axial deformations after hydration that ranged from 1.04 to
1.29, and radial deformations from 1.18 to 1.7. Despite using very different
chemistries from hair, all formulations showed significant strain anisotropy
between 2 and 5, i.e., within the range of what is observed for hair and
collagen (Fig. 5). It will be interesting to see whether squared-deformations
or geometries in such synthetic biomaterials also have the same linear
dependence on water content as we have described.

In summary, we have described linear behavior of anisotropic swelling due to
hydration in both collagen fibrils and hair. How much our results generalize
to other biomaterials and bio-mimetic materials remains to be determined.
Whether water content controlled by solution conditions (see e.g. [9, 10, 11])
or cross-linking density (see e.g. [32, 33, 34]) has similar effects also
remains an open question, since our phenomenological results are based on
studies that only varied relative humidity.

_Acknowledgments_.—We thank the Natural Sciences and Engineering Research
Council of Canada (NSERC) for operating Grants RGPIN-2018-03781 (LK) and
RGPIN-2019-05888 (ADR).

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