A Spin Analogue of Kerov Polynomials

A Spin Analogue of Kerov Polynomials††This paper is a contribution to the
Special Issue on the Representation Theory of the Symmetric Groups and Related
Topics. The full collection is available at
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html

Sho MATSUMOTO

S. Matsumoto

Graduate School of Science and Engineering, Kagoshima University,
Kagoshima 890-0065, Japan<EMAIL_ADDRESS>
Received March 13, 2018, in final form May 29, 2018; Published online June 02,
2018

Kerov polynomials describe normalized irreducible characters of the symmetric
groups in terms of the free cumulants associated with Young diagrams. We
suggest well-suited counterparts of the Kerov polynomials in spin (or
projective) representation settings. We show that spin analogues of
irreducible characters are polynomials in even free cumulants associated with
double diagrams of strict partitions. Moreover, we present a conjecture for
the positivity of their coefficients.

Kerov polynomials; spin symmetric groups; free cumulants; characters

05E10; 20C30; 05E05

##  1 Introduction

###  1.1 Characters of symmetric groups

Irreducible representations of symmetric groups $S_{n}$ are indexed by
partitions $\lambda$ of $n$, or equivalently by Young diagrams of size $n$.
The corresponding character $\chi^{\lambda}$ takes values at conjugacy classes
$C_{\nu}$ in $S_{n}$, which are also indexed by partitions $\nu$ of $n$. The
character values $\chi^{\lambda}_{\nu}=\chi^{\lambda}(C_{\nu})$ have been
studied for a long time by innumerable researchers, see, e.g., [20]. Recently,
for several problems in the asymptotic representation theory, we often deal
with the $\chi^{\lambda}_{\nu}$, fixing $\nu$ and letting $\lambda$ vary. More
precisely, for a partition $\nu$ of $k$ and a partition $\lambda$ of $n$ with
$k\leq n$, we define

$\displaystyle\mathrm{Ch}_{\nu}(\lambda)=n(n-1)\cdots(n-k+1)\frac{\chi^{\lambda}_{\nu\cup(1^{n-k})}}{f^{\lambda}},$
(1.1)

where $f^{\lambda}:=\chi^{\lambda}_{(1^{|\lambda|})}$ denotes the dimension of
the irreducible representation of the symmetric group $S_{|\lambda|}$
associated with $\lambda$. We set $\mathrm{Ch}_{\nu}(\lambda)=0$ whenever
$k>n$. Then the $\mathrm{Ch}_{\nu}$ are functions on all Young diagrams. An
approach from this point of view is sometimes referred to as _dual approach_
or _dual combinatorics_ to characters of symmetric groups [6, 9].

###  1.2 Kerov polynomials

Biane and Kerov discovered that free cumulants play an important role in the
asymptotic representation theory of symmetric groups [1, 2]. Here the free
cumulants $R_{j}(\lambda)$ $(j=1,2,\dots)$ of the transition measure of a
Young diagram $\lambda$ are sequences of real numbers defined in the framework
of the free probability theory. (Note that $R_{1}\equiv 0$.) Then the
normalized character $\mathrm{Ch}_{k}:=\mathrm{Ch}_{(k)}$ for one-row
partition $\nu=(k)$ can be expressed as a _polynomial_ in
$R_{2},\dots,R_{k+1}$ with integer coefficients.

######  Theorem 1.1 (Kerov’s character formula [2]).

For each $k=1,2,3,\dots$, there exists a polynomial $K_{k}$ in $k$ variables
with integer coefficients, such that

$\displaystyle\mathrm{Ch}_{k}(\lambda)=K_{k}(R_{2}(\lambda),R_{3}(\lambda),\dots,R_{k+1}(\lambda))$

for all partitions $\lambda$. Furthermore, $K_{k}$ is of the form

$\displaystyle K_{k}(R_{2},\dots,R_{k+1})=R_{k+1}+(\text{a polynomial in
$R_{2},\dots,R_{k-1}$}).$

######  Example 1.2 ([2]).

$\displaystyle K_{1}=R_{2},$ $\displaystyle K_{2}=R_{3},$ $\displaystyle
K_{3}=R_{4}+R_{2},$ $\displaystyle K_{4}=R_{5}+5R_{3},$ $\displaystyle
K_{5}=R_{6}+15R_{4}+5R_{2}^{2}+8R_{2},$ $\displaystyle
K_{7}=R_{8}+70R_{6}+84R_{4}R_{2}+56R_{3}^{2}+14R_{2}^{3}+469R_{4}+224R_{2}^{2}+180R_{2},$
$\displaystyle
K_{9}=R_{10}+210R_{8}+300R_{6}R_{2}+480R_{5}R_{3}+270R_{4}^{2}+360R_{3}^{2}R_{2}+270R_{4}R_{2}^{2}+30R_{2}^{4}$
$\displaystyle\hphantom{K_{9}=}{}+5985R_{6}+10548R_{4}R_{2}+6714R_{3}^{2}+2400R_{2}^{3}+26060R_{4}+14580R_{2}^{2}+8064R_{2}.$

Here we omit $K_{6}$ and $K_{8}$ because we will compare odd-numbered
polynomials with Example 1.5 below. We can see a complete list of $K_{k}$ up
to $k=11$ in [2].

The polynomials $K_{k}$ in the theorem are called _Kerov polynomials_. We can
observe that all coefficients in the above examples are _nonnegative_
integers. This surprising phenomenon is very nontrivial and had been called
the Kerov (positivity) conjecture. It was finally proved by Féray [7]. The
authors in [5] obtained clearer combinatorial interpretations.

######  Theorem 1.3 (Féray [7]).

All coefficients of $K_{k}$ are nonnegative integers.

Recently, a generalization of Kerov polynomials involving Jack polynomials has
been actively studied in [4, 16, 21]. In this paper, we work on research in
another direction. Our aim here is to present an answer to the following
question: _What is the counterpart of the Kerov polynomials in the spin
representation setting?_

###  1.3 Spin representations

Spin (or projective) representation theory of symmetric groups was introduced
by Schur. Currently, it can be understood via a double covering of the
symmetric group or via the Hecke–Clifford algebra, see, e.g., [22]. In the
spin case, the corresponding character values $X^{\lambda}_{\nu}$ are indexed
by strict partitions $\lambda$ and odd partitions $\nu$. For a fixed odd
partition $\nu$, we can define a spin version $\mathfrak{p}_{\nu}$ of (1.1),
which is a function on the set of all strict partitions. The function
$\mathfrak{p}_{\nu}$ was first introduced by Ivanov [12, 13] (with notation
$p_{\nu}^{\\#}$) and developed in author’s recent work [18].

###  1.4 Results

To define a spin analogue of Kerov polynomials, we need to find a spin
analogue of free cumulants $R_{j}(\lambda)$. Furthermore, it is natural to
expect that the same statement of Theorem 1.3 holds in the spin setting. The
aim in this paper is to establish a spin analogue of Kerov polynomials by
using a spin analogue $\mathfrak{R}_{2j}(\lambda)$ of $R_{j}$, which is
defined as the _half_ of _even_ free cumulants of the _double diagram_ of a
strict partition $\lambda$. Indeed, we obtain the following theorem for
$\mathfrak{p}_{2k-1}$ with one-row odd partition $\nu=(2k-1)$.

######  Theorem 1.4.

For each $k=1,2,3,\dots$, there exists a polynomial $K_{2k-1}^{\mathrm{spin}}$
in $k$ variables with rational coefficients, such that

$\displaystyle\mathfrak{p}_{2k-1}(\lambda)=K_{2k-1}^{\mathrm{spin}}(\mathfrak{R}_{2}(\lambda),\mathfrak{R}_{4}(\lambda),\dots,\mathfrak{R}_{2k}(\lambda))$

for all strict partitions $\lambda$. Furthermore, $K_{2k-1}^{\mathrm{spin}}$
is of the form

$\displaystyle K_{2k-1}^{\mathrm{spin}}=\mathfrak{R}_{2k}+(\text{a polynomial
in $\mathfrak{R}_{2},\mathfrak{R}_{4},\dots,\mathfrak{R}_{2k-2}$ of degree at
most $2k-2$}).$

Here we only show that the coefficients in $K_{2k-1}^{\mathrm{spin}}$ are
rational numbers. The degree of $\mathfrak{R}_{2k}$ is regarded as
$\deg\mathfrak{R}_{2k}=2k-1$. We call the polynomial
$K_{2k-1}^{\mathrm{spin}}$ a _spin Kerov polynomial_.

######  Example 1.5.

$\displaystyle K_{1}^{\mathrm{spin}}=\mathfrak{R}_{2},$ $\displaystyle
K_{3}^{\mathrm{spin}}=\mathfrak{R}_{4}+\mathfrak{R}_{2},$ $\displaystyle
K_{5}^{\mathrm{spin}}=\mathfrak{R}_{6}+15\mathfrak{R}_{4}+10\mathfrak{R}_{2}^{2}+8\mathfrak{R}_{2},$
$\displaystyle
K_{7}^{\mathrm{spin}}=\mathfrak{R}_{8}+70\mathfrak{R}_{6}+168\mathfrak{R}_{4}\mathfrak{R}_{2}+56\mathfrak{R}_{2}^{3}+469\mathfrak{R}_{4}+560\mathfrak{R}_{2}^{2}+180\mathfrak{R}_{2},$
$\displaystyle
K_{9}^{\mathrm{spin}}=\mathfrak{R}_{10}+210\mathfrak{R}_{8}+600\mathfrak{R}_{6}\mathfrak{R}_{2}+540\mathfrak{R}_{4}^{2}+1080\mathfrak{R}_{4}\mathfrak{R}_{2}^{2}+240\mathfrak{R}_{2}^{4}$
$\displaystyle\hphantom{K_{9}^{\mathrm{spin}}=}{}+5985\mathfrak{R}_{6}+23016\mathfrak{R}_{4}\mathfrak{R}_{2}+9120\mathfrak{R}_{2}^{3}+26060\mathfrak{R}_{4}+41628\mathfrak{R}_{2}^{2}+8064\mathfrak{R}_{2}.$

It seems that formulas in Example 1.5 resemble those in Example 1.2. The proof
of Theorem 1.4 is accomplished by observing relations among some collections
of generators in a symmetric function algebra $\Gamma$. We obtain formulas in
Example 1.5 as the by-products. Not only Theorem 1.4 but also Example 1.5 is
an important result in this paper because the latter one indicates a spin
analogue of Theorem 1.3 to us.

######  Conjecture 1.6 (spin Kerov conjecture).

All coefficients in spin Kerov polynomials are nonnegative integers.

###  1.5 Discussion and outline

Comparing Example 1.5 with Example 1.2, we can find some interesting
coincidences for coefficients. For example, the coefficient of linear terms
$\mathfrak{R}_{2j}$ in $K_{2k-1}^{\mathrm{spin}}$ likely coincides with that
of $R_{2j}$ in $K_{2k-1}$. Furthermore, if $2s_{2}+4s_{4}+\cdots=2k-2$, the
coefficient of $\mathfrak{R}_{2}^{s_{2}}\mathfrak{R}_{4}^{s_{4}}\cdots$ in
$K_{2k-1}^{\mathrm{spin}}$ likely coincide with $2^{s_{2}+s_{4}+\cdots-1}$
times the coefficient of $R_{2}^{s_{2}}R_{4}^{s_{4}}\cdots$ in $K_{2k-1}$. We
do not discuss their coincidence or combinatorial interpretations here. We
leave them in future work.

We give the proof of Theorem 1.4 and how to derive Example 1.5 with precise
definitions in Section 3 after the review of preliminary facts for free
cumulants Section 2. As mentioned, the ordinary Kerov conjecture was first
proved by Féray [7]. After that, the authors of [5] gave alternative proof by
using Stanley–Féray polynomials. The Stanley–Féray polynomials were analyzed
by using Young symmetrizers [8]. In order to attack Conjecture 1.6, it is a
natural practice to study projective Young symmetrizers. The projective Young
symmetrizer was studied in [15, 19]. Unfortunately it was quite complicated,
so it seems difficult to apply it to our problem.

In Section 4, we try another choice for free cumulants. The another free
cumulants $\mathtt{R}_{2k}(\lambda)$ comes from a symmetrized double diagram
$\mathtt{D}(\lambda)$ of $\lambda$. The symmetrized diagrams
$\mathtt{D}(\lambda)$ are useful for the study of asymptotics of Plancherel
measures in the spin setting (see [13] and [3, Chapter 4]). Since odd-numbered
quantities $\mathtt{R}_{2k-1}$ vanish by virtue of the symmetry of
$\mathtt{D}(\lambda)$, they are simpler than free cumulants of $D(\lambda)$ in
a sense. Even if we replace $\mathfrak{R}_{2k}$ with $\mathtt{R}_{2k}$, we can
show the existence of “spin Kerov polynomials” easily. However, the
corresponding polynomial for $\mathfrak{p}_{2k-1}$ with $k=2$ has a _non-
integer_ coefficient. In this sense, we view the $\mathtt{R}_{2k}$ as being
unsuitable for spin Kerov polynomials. We thus believe that the
$\mathfrak{R}_{2k}$ are the most appropriate choice in the spin setting.

##  2 Preliminary

In the present section, we review the transition measures of a partition
according to [11, 14].

###  2.1 Partitions

A _partition_ $\lambda=(\lambda_{1},\lambda_{2},\dots)$ is a weakly decreasing
sequence of nonnegative integers satisfying $|\lambda|=\sum\limits_{i\geq
1}\lambda_{i}<\infty$. When $|\lambda|=n$, we say $\lambda$ to be a partition
of $n$ and sometimes write as $\lambda\vdash n$. The number of nonzero
$\lambda_{i}$ is called the length of $\lambda$ and written as
$\ell(\lambda)$.

We usually identity a partition with its Young diagram. We denote by
$\lambda^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots)$ the
conjugate partition of $\lambda$, i.e., the Young diagram of
$\lambda^{\prime}$ is the transpose of that of $\lambda$ with respect to the
diagonal line. Define $d=d(\lambda)$ by the number of boxes on the diagonal
line in the Young diagram of $\lambda$. The _modified Frobenius notation_

$\displaystyle[a_{1},\dots,a_{d}\,|\,b_{1},\dots,b_{d}]$

of $\lambda$ is determined by

$\displaystyle a_{i}=\lambda_{i}-i+\frac{1}{2},\qquad
b_{i}=\lambda_{i}^{\prime}-i+\frac{1}{2},\qquad i=1,2,\dots,d.$

Note that $\sum\limits_{i=1}^{d}(a_{i}+b_{i})=|\lambda|$. For example, the
modified Frobenius notation of $\lambda=(5,4,4,1,1)\vdash 15$ is
$\big{[}4+\frac{1}{2},2+\frac{1}{2},1+\frac{1}{2}\,|\,4+\frac{1}{2},1+\frac{1}{2},0+\frac{1}{2}\big{]}$.

###  2.2 Free cumulants

Let $\mu$ be a probability measure on $\mathbb{R}$ with a compact support.
Define the Cauchy transform of $\mu$ by

$\displaystyle
G_{\mu}(z)=\int_{\mathbb{R}}\frac{1}{z-x}\mu(dx)=\sum_{k=0}^{\infty}\frac{M_{k}[\mu]}{z^{k+1}},\qquad
z\in\mathbb{C}\setminus\mathbb{R},\quad|z|\gg 1,$

where the $M_{k}[\mu]$ are moments of $\mu$:

$\displaystyle M_{k}[\mu]=\int_{\mathbb{R}}x^{k}\mu(dx).$

_Free cumulants_ $R_{k}[\mu]$ $(k=1,2,\dots)$ of $\mu$ are defined via the
famous free cumulant-moment formula, see, e.g., [11, Chapter 1]. We do not
need the explicit definition here. We only use the relation

$\displaystyle
R_{k}=M_{k}+\sum_{\begin{subarray}{c}\nu=(\nu_{1},\nu_{2},\dots)\vdash k\\\
\nu_{1}<k\end{subarray}}c_{\nu}M_{\nu_{1}}M_{\nu_{2}}\cdots$ (2.1)

with some integer coefficients $c_{\nu}$.

###  2.3 Kerov’s transition measures

We draw the Young diagram of a partition $\lambda$ in Russian style, see,
e.g., [11, Fig. 2.1]. Let

$\displaystyle x_{1}<y_{1}<\cdots<x_{r-1}<y_{r-1}<x_{r}$

be the corresponding local minimas and maximas, which are integers by
definition. We call these numbers the _Kerov interlacing coordinates_ of
$\lambda$. It is known that they satisfy the relation [11, Lemma 2.1]

$\displaystyle\sum_{i=1}^{r}x_{r}=\sum_{j=1}^{r-1}y_{j}.$ (2.2)

For each partition $\lambda$, we define a probability measure
$\mathfrak{m}_{\lambda}$ on $\mathbb{R}$ via the Cauchy transform

$\displaystyle
G_{\mathfrak{m}_{\lambda}}(z)=\frac{\prod\limits_{i=1}^{r-1}(z-y_{i})}{\prod\limits_{i=1}^{r}(z-x_{i})}.$

This probability measure, called _Kerov’s transition measure_ of $\lambda$, is
supported by the set $\\{x_{1},\dots,x_{r}\\}$. Denote by $R_{k}(\lambda)$
$(k=1,2,\dots)$ the free cumulant of the measure $\mathfrak{m}_{\lambda}$:
$R_{k}(\lambda)=R_{k}[\mathfrak{m}_{\lambda}]$. It has the expression [2,
Theorem 2]

$\displaystyle
R_{k}(\lambda)=-\frac{1}{k-1}\big{[}z^{-1}\big{]}G_{\mathfrak{m}_{\lambda}}(z)^{-(k-1)}$
(2.3)

for $k\geq 2$.

###  2.4 Super power-sums $\bm{p_{k}^{\mathrm{super}}}$

For each partition $\lambda$ with modified Frobenius expression
$[a_{1},\dots,a_{d}\,|\,b_{1},\dots,b_{d}]$ and for $k=1,2,\dots$, we define

$\displaystyle
p_{k}^{\mathrm{super}}(\lambda)=\sum_{i=1}^{d}(a_{i}^{k}+(-1)^{k-1}b_{i}^{k}).$

These are also determined via

$\displaystyle\phi(z;\lambda):=\prod_{i=1}^{d}\frac{z+b_{i}}{z-a_{i}}=\exp\left(\sum_{k=1}^{\infty}\frac{p_{k}^{\mathrm{super}}(\lambda)}{k}\frac{1}{z^{k}}\right).$
(2.4)

Note that
$p_{1}^{\mathrm{super}}(\lambda)=\sum\limits_{i=1}^{d}(a_{i}+b_{i})=|\lambda|$.

######  Lemma 2.1 ([11, Proposition 2.1]).

For each partition $\lambda$, we have

$\displaystyle
zG_{\mathfrak{m}_{\lambda}}(z)=\frac{\phi\big{(}z-\frac{1}{2};\lambda\big{)}}{\phi\big{(}z+\frac{1}{2};\lambda\big{)}}.$

###  2.5 Rayleigh measures

For each partition $\lambda$ with Kerov’s interlacing coordinate
$x_{1}<y_{1}<\cdots<y_{r-1}<x_{r}$, we introduce an $\mathbb{R}$-valued
measure $\tau_{\lambda}$ on $\mathbb{R}$ by

$\displaystyle\tau_{\lambda}=\sum_{i=1}^{r}\delta_{x_{i}}-\sum_{i=1}^{r-1}\delta_{y_{i}},$

where $\delta_{x}$ is the Dirac measure at a point $x$. This is called the
_Rayleigh measure_ of $\lambda$. The moments are clearly given by

$\displaystyle
M_{k}[\tau_{\lambda}]=\sum_{i=1}^{r}x_{i}^{k}-\sum_{i=1}^{r-1}y_{i}^{k}$

for $k=1,2,\dots$. Note that $M_{1}[\tau_{\lambda}]=0$ by (2.2).

######  Lemma 2.2 ([11, Proposition 2.4]).

For $n\geq 2$,

$\displaystyle
M_{n}[\tau_{\lambda}]=\sum_{j=0}^{\lfloor(n/2)-1\rfloor}\binom{n}{2j+1}2^{-2j}p_{n-2j-1}^{\mathrm{super}}(\lambda)=np_{n-1}^{\mathrm{super}}(\lambda)+\cdots.$

###  2.6 Relations between two measures

Two measures $\mathfrak{m}_{\lambda}$ and $\tau_{\lambda}$ have the following
relation, which is nothing but that the relation between complete symmetric
functions and power-sum symmetric functions.

######  Lemma 2.3 ([11, Proposition 2.2]).

$\displaystyle
1+\sum_{n=1}^{\infty}M_{n}[\mathfrak{m}_{\lambda}]\frac{1}{z^{n}}=\exp\left(\sum_{k=1}^{\infty}\frac{M_{k}[\tau_{\lambda}]}{k}\frac{1}{z^{k}}\right).$

Equivalently,

$\displaystyle
M_{n}[\mathfrak{m}_{\lambda}]=\sum_{\nu=(\nu_{1},\nu_{2},\dots)\vdash
n}z_{\nu}^{-1}M_{\nu_{1}}[\tau_{\lambda}]M_{\nu_{2}}[\tau_{\lambda}]\cdots,$

where $z_{\nu}=\prod\limits_{i\geq 1}i^{m_{i}}m_{i}!$ with
$m_{i}=m_{i}(\nu)=|\\{j\geq 1\,|\,\nu_{j}=i\\}|$.

Combining above formulas, we can express free cumulants $R_{k}(\lambda)$ in
terms of functions $p_{j}^{\mathrm{super}}(\lambda)$ in principle. This fact
will be applied in the next section.

##  3 Spin Kerov polynomials

Our goal of this section is to define the counterpart of the Kerov polynomial
in the spin representation setting.

###  3.1 Strict and odd partitions

A partition $\lambda=(\lambda_{1},\lambda_{2},\dots)$ is said to be _strict_
if its nonzero parts $\lambda_{i}$ are distinct. Let $\mathcal{SP}$ be the set
of all strict partitions. For a strict partition
$\lambda=(\lambda_{1}>\lambda_{2}>\cdots>\lambda_{l}>0)$, we define the
_double_ of $\lambda$ by

$\displaystyle
D(\lambda)=\big{[}\lambda_{1}+\tfrac{1}{2},\lambda_{2}+\tfrac{1}{2},\dots,\lambda_{l}+\tfrac{1}{2}\,|\,\lambda_{1}-\tfrac{1}{2},\lambda_{2}-\tfrac{1}{2},\dots,\lambda_{l}-\tfrac{1}{2}\big{]}$
(3.1)

in the modified Frobenius notation. Note that $|D(\lambda)|=2|\lambda|$.

######  Example 3.1.

For a strict partition $\lambda=(5,4,2,1)$, we have three kinds of diagrams as
below. On the left, the Young diagram (in English style) is a collection of
left-justified rows of boxes where the $i$-th row has $\lambda_{i}$ boxes
reading from top to bottom. On the middle, the shifted diagram is obtained
from the Young diagram by shifting the $i$-th row $(i-1)$ boxes to the right,
for each $i\geq 2$. Moreover, on the right, the Young diagram of $D(\lambda)$,
or the double diagram of $\lambda$, is obtained by combining the shifted
diagram with its reflection (the gray area) on the diagonal.

$\displaystyle\ydiagram{5,4,2,1}\qquad\qquad\ydiagram{1+5,2+4,3+2,4+1}\qquad\qquad\ydiagram{1+5,2+4,3+2,4+1}*[*(gray)]{1,2,3,4,2}$

A (not necessary strict) partition $\rho=(\rho_{1},\rho_{2},\dots)$ is said to
be _odd_ if all nonzero $\rho_{i}$ are odd. It is well known that the number
of strict partitions of $n$ coincides with that of odd partitions of $n$.

###  3.2 Symmetric functions

We review some symmetric functions according to Macdonald’s book [17, Chapter
III.8]. Recall the power-sum symmetric function

$\displaystyle p_{k}(\bm{x})=x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+\cdots$

in infinitely many variables $\bm{x}=(x_{1},x_{2},x_{3},\dots)$. For an odd
partition $\rho=(\rho_{1},\dots,\rho_{l})$, set
$p_{\rho}=p_{\rho_{1}}p_{\rho_{2}}\cdots p_{\rho_{l}}$.

Let $\Gamma$ be the $\mathbb{Q}$-algebra generated by odd power-sum symmetric
functions $p_{k}$ $(k=1,3,5,\dots)$. The degree on $\Gamma$ is naturally
defined by

$\displaystyle\deg p_{k}=k,\qquad k=1,3,5,\dots.$

For each $\lambda\in\mathcal{SP}$, denote by $P_{\lambda}$ Schur’s
$P$-function, see the definition in [17, Chapter III.8]. The family
$\\{P_{\lambda}\,|\,\lambda\in\mathcal{SP}\\}$ and
$\\{p_{\rho}\,|\,\text{$\rho$ are odd partitions}\\}$ form a linear basis of
$\Gamma$, respectively. The quantity $X^{\lambda}_{\rho}$, where $\lambda$ is
a strict partition and $\rho$ is an odd partition with $|\lambda|=|\rho|$, is
defined via the relation

$\displaystyle
p_{\rho}=\sum_{\lambda:|\lambda|=|\rho|}X^{\lambda}_{\rho}P_{\lambda}.$

Put $g^{\lambda}=X^{\lambda}_{(1^{|\lambda|})}$. These quantities
$X^{\lambda}_{\rho}$ are integers and encode character values of irreducible
spin representations of symmetric groups [10]. In particular, a positive
integer $g^{\lambda}$ times a power of $2$ coincides with the dimension of an
irreducible spin representation of symmetric groups.

Each symmetric function $f$ in $\Gamma$ is regarded as a $\mathbb{Q}$-valued
function on $\mathcal{SP}$, by

$\displaystyle
f(\lambda)=f(\lambda_{1},\lambda_{2},\dots,\lambda_{l},0,0,\dots),\qquad\lambda=(\lambda_{1}>\lambda_{2}>\cdots>\lambda_{l}>0).$

In particular, $p_{k}(\lambda)=\sum\limits_{i=1}^{l}\lambda_{i}^{k}$. For two
functions $f,g\in\Gamma$, it holds $f(\lambda)=g(\lambda)$ for all
$\lambda\in\mathcal{SP}$ if and only if $f=g$ [12, Proposition 6.2].

###  3.3 Spin characters

The spin (or projective) analogue of $\mathrm{Ch}_{\nu}$ given in (1.1) is
defined as follows [12, 13, 18]. For an odd partition $\rho$ of $k$ and strict
partition $\lambda$ of $n$ with $k\leq n$, we define

$\displaystyle\mathfrak{p}_{\rho}(\lambda)=n(n-1)\cdots(n-k+1)\frac{X^{\lambda}_{\rho\cup(1^{n-k})}}{g^{\lambda}},$

where
$\rho\cup\big{(}1^{n-k}\big{)}=(\rho_{1},\rho_{2},\dots,\rho_{l},1,1,\dots,1)\vdash
n$. Set $\mathfrak{p}_{\rho}(\lambda)=0$ for $k>n$. In this paper, we focus
the spin characters $\mathfrak{p}_{2k-1}=\mathfrak{p}_{(2k-1)}$ for one-row
odd partitions $\rho=(2k-1)$.

The collection $\\{\mathfrak{p}_{2k-1}\,|\,k=1,2,3,\dots\\}$ forms an
algebraic basis of $\Gamma$ [12, Proposition 6.4]. More specifically, we have

$\displaystyle\mathfrak{p}_{2k-1}=p_{2k-1}+(\text{a polynomial in
$p_{1},p_{3},\dots,p_{2k-3}$ of degree smaller than $2k-1$)}.$ (3.2)

######  Example 3.2.

The functions $\mathfrak{p}_{2k-1}$ for $1\leq k\leq 5$ are expanded in terms
of $p_{j}$ as follows

$\displaystyle\mathfrak{p}_{1}=p_{1},$
$\displaystyle\mathfrak{p}_{3}=p_{3}-3p_{1}^{2}+2p_{1},$
$\displaystyle\mathfrak{p}_{5}=p_{5}-10p_{3}p_{1}+\frac{55}{3}p_{3}+\frac{50}{3}p_{1}^{3}-50p_{1}^{2}+24p_{1},$
$\displaystyle\mathfrak{p}_{7}=p_{7}-14p_{5}p_{1}-7p_{3}^{2}+77p_{5}+98p_{3}p_{1}^{2}-\frac{1862}{3}p_{3}p_{1}-\frac{343}{3}p_{1}^{4}$
$\displaystyle\hphantom{\mathfrak{p}_{7}=}{}+\frac{2128}{3}p_{3}+\frac{2744}{3}p_{1}^{3}-1764p_{1}^{2}+720p_{1},$
$\displaystyle\mathfrak{p}_{9}=p_{9}-18p_{7}p_{1}-18p_{5}p_{3}+222p_{7}+162p_{5}p_{1}^{2}+162p_{3}^{2}p_{1}-2538p_{5}p_{1}-1026p_{3}^{2}-972p_{3}p_{1}^{3}$
$\displaystyle\hphantom{\mathfrak{p}_{9}=}{}+\frac{37401}{5}p_{5}+14094p_{3}p_{1}^{2}+\frac{4374}{5}p_{1}^{5}-52704p_{3}p_{1}-14580p_{1}^{4}$
$\displaystyle\hphantom{\mathfrak{p}_{9}=}{}+47492p_{3}+70632p_{1}^{3}-109584p_{1}^{2}+40320p_{1}.$

These examples are obtained by using the formula given in [13, Proposition
3.3]: For odd $k=1,3,5,\dots$,

$\displaystyle\mathfrak{p}_{k}(\lambda)=\big{[}z^{-1}\big{]}\left(-\frac{1}{4k}\right)(2z-k)\prod_{j=1}^{k-1}(z-j)\cdot\frac{\Phi(z;\lambda)}{\Phi(z-k;\lambda)},$

where $\Phi(z;\lambda)$ is defined by (3.3) or (3.4) below. Here
$\big{[}z^{-1}\big{]}Q(z)$ stands for the coefficient of $z^{-1}$ in the
Laurent series expansion of $Q(z)$ at $z=\infty$.

######  Remark 3.3.

The formulas in Example 3.2 are also obtained in the following way. First, we
expand $\mathfrak{p}_{\rho}$ in terms of factorial Schur $P$-functions
$P^{*}_{\lambda}$. Second, we expand each $P^{*}_{\lambda}$ in terms of
(ordinary) Schur $P$-functions $P_{\nu}$. Finally, we expand each $P_{\nu}$ in
terms of odd power-sum symmetric functions $p_{\sigma}$. See [18, Example 3.3]
for details.

###  3.4 Super symmetric polynomials

Let $\lambda$ be a strict partition. Put

$\displaystyle\Phi(z;\lambda)=\prod_{i=1}^{\ell(\lambda)}\frac{z+\lambda_{i}}{z-\lambda_{i}}.$
(3.3)

It is easy to see that

$\displaystyle\log\Phi(z;\lambda)=2\sum_{k=1}^{\infty}\frac{p_{2k-1}(\lambda)}{2k-1}\frac{1}{z^{2k-1}}.$
(3.4)

Recall functions $p_{n}^{\mathrm{super}}$ introduced in Section 2.4. The
following proposition is a key in the proof of Theorem 1.4.

######  Proposition 3.4.

For each strict partition $\lambda$ and $n=1,2,3,\dots$, we have

$\displaystyle
p_{n}^{\mathrm{super}}(D(\lambda))=\sum_{j=0}^{\lfloor(n-1)/2\rfloor}\binom{n}{2j+1}\frac{1}{2^{n-2j-2}}p_{2j+1}(\lambda).$

In particular, for each $k=1,2,3,\dots$,

$\displaystyle
p_{2k-1}^{\mathrm{super}}(D(\lambda))=2p_{2k-1}(\lambda)+\cdots\qquad\text{and}\qquad
p_{2k}^{\mathrm{super}}(D(\lambda))=2kp_{2k-1}(\lambda)+\cdots,$

where dots are linear combinations of $\\{p_{2j-1}\\}_{j=1,2,\dots,k-1}$ with
$\mathbb{Q}_{\geq 0}$-coefficients. Therefore the family
$\big{\\{}p_{2k-1}^{\mathrm{super}}(D(\cdot))\big{\\}}_{k=1,2,3,\dots}$ is an
algebraic basis of $\Gamma$.

###### Proof.

Using (2.4) and (3.1), we have

$\displaystyle\phi(z;D(\lambda))=\prod_{i=1}^{\ell(\lambda)}\frac{z+\big{(}\lambda_{i}-\frac{1}{2}\big{)}}{z-\big{(}\lambda_{i}+\frac{1}{2}\big{)}}=\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}.$
(3.5)

Taking the logarithm of the right hand side and expanding it at $z=\infty$,

$\displaystyle\log\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}=\sum_{k=0}^{\infty}\frac{2p_{2k+1}(\lambda)}{2k+1}\big{(}z-\tfrac{1}{2}\big{)}^{-(2k+1)}$
$\displaystyle\hphantom{\log\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}}{}=\sum_{k=0}^{\infty}\frac{2p_{2k+1}(\lambda)}{2k+1}\frac{1}{z^{2k+1}}\sum_{j=0}^{\infty}\frac{(2k+1)(2k+2)\cdots(2k+j)}{j!}\frac{1}{(2z)^{j}}.$

Changing variable $j\mapsto n=2k+j+1$, we have

$\displaystyle\log\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}=\sum_{n=1}^{\infty}\frac{1}{z^{n}}\sum_{k=0}^{\lfloor(n-1)/2\rfloor}2p_{2k+1}(\lambda)\frac{(2k+2)(2k+3)\cdots(n-1)}{(n-2k-1)!}\frac{1}{2^{n-2k-1}}$
$\displaystyle\hphantom{\log\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}}{}=\sum_{n=1}^{\infty}\frac{1}{nz^{n}}\sum_{k=0}^{\lfloor(n-1)/2\rfloor}p_{2k+1}(\lambda)\binom{n}{2k+1}\frac{1}{2^{n-2k-2}}.$

On the other hand, from (2.4) we have

$\displaystyle\log\phi(z;D(\lambda))=\sum_{n=1}^{\infty}\frac{1}{nz^{n}}p_{n}^{\mathrm{super}}(D(\lambda)).$

Comparing the coefficient of $z^{-n}$ in the above equations, we obtain the
desired identity. ∎

###  3.5 Moments of Rayleigh measures

Recall the Rayleigh measure $\tau_{\lambda}$ defined in Section 2.5.

######  Proposition 3.5.

For a strict partition $\lambda$ and $k=1,2,3,\dots$, we have

$\displaystyle
M_{2k}[\tau_{D(\lambda)}]=4kp_{2k-1}(\lambda)+\cdots\qquad\text{and}\qquad
M_{2k+1}[\tau_{D(\lambda)}]=2k(2k+1)p_{2k-1}(\lambda)+\cdots,$

where dots are linear combinations of
$\\{p_{2j-1}(\lambda)\\}_{j=1,2,\dots,k-1}$ with $\mathbb{Q}_{\geq
0}$-coefficients. Therefore the family
$\\{M_{2k}[\tau_{D(\cdot)}]\\}_{k=1,2,\dots}$ is an algebraic basis of
$\Gamma$.

###### Proof.

From Lemma 2.2 we have
$M_{n}[\tau_{D(\lambda)}]=np^{\mathrm{super}}_{n-1}(D(\lambda))+\cdots$, where
dots are a linear combination of $p^{\mathrm{super}}_{j}(D(\lambda))$
$(j=1,2,\dots,n-2)$, with $\mathbb{Q}_{\geq 0}$-coefficients. The desired
claim follows from Proposition 3.4 immediately. ∎

###  3.6 Moments and free cumulants of transition measures

Recall the transition measure $\mathfrak{m}_{\lambda}$ defined in Section 2.3.

######  Proposition 3.6.

For strict partition $\lambda$ and $k=1,2,3,\dots$, we have

$\displaystyle
M_{2k}[\mathfrak{m}_{D(\lambda)}]=2p_{2k-1}(\lambda)+\cdots\qquad\text{and}\qquad
M_{2k+1}[\mathfrak{m}_{D(\lambda)}]=2kp_{2k-1}(\lambda)+\cdots,$

where dots are functions in $\Gamma$ of degree at most $2k-2$ with
$\mathbb{Q}_{\geq 0}$-coefficients. Therefore the family
$\\{M_{2k}[\mathfrak{m}_{D(\cdot)}]\\}_{k=1,2,\dots}$ is an algebraic basis of
$\Gamma$.

###### Proof.

Lemma 2.3 implies that

$\displaystyle
M_{n}[\mathfrak{m}_{D(\lambda)}]=\frac{1}{n}M_{n}[\tau_{D(\lambda)}]+\sum_{\begin{subarray}{c}\nu=(\nu_{1},\nu_{2},\dots)\vdash
n\\\
\nu_{1}<n\end{subarray}}z_{\nu}^{-1}M_{\nu_{1}}[\tau_{D(\lambda)}]M_{\nu_{2}}[\tau_{D(\lambda)}]\cdots.$
(3.6)

We here apply Proposition 3.5. Let $n=2k+1$. Then
$\frac{1}{n}M_{n}[\tau_{D(\lambda)}]=2kp_{2k-1}(\lambda)+f_{k}(\lambda)$,
where $f_{k}\\!\in\\!\Gamma$ is of degree at most $2k{-}2$. We will estimate
the degree of each term
$M_{\nu_{1}}[\tau_{D(\lambda)}]M_{\nu_{2}}[\tau_{D(\lambda)}]{\cdots}\\!$ in
(3.6). A partition $\nu\vdash 2k+1$ with $\nu_{1}<2k+1$ has at least one odd
part $\nu_{s}$ with $\deg M_{\nu_{s}}[\tau_{D(\lambda)}]=\nu_{s}-2$ and has
another part $\nu_{t}$ $(t\not=s)$ with $\deg
M_{\nu_{t}}[\tau_{D(\lambda)}]\leq\nu_{t}-1$. Hence the degree of each term
$M_{\nu_{1}}[\tau_{D(\lambda)}]M_{\nu_{2}}[\tau_{D(\lambda)}]\cdots$ is at
most $(2k+1)-3=2k-2$. The case with $n=2k$ is similar (and easier). ∎

Now using the free cumulant-moment formula (2.1), we obtain the following
result for free cumulants
$R_{n}(D(\lambda))=R_{n}[\mathfrak{m}_{D(\lambda)}]$.

######  Corollary 3.7.

For strict partition $\lambda$ and $k=1,2,3,\dots$, we have

$\displaystyle
R_{2k}(D(\lambda))=2p_{2k-1}(\lambda)+\cdots\qquad\text{and}\qquad
R_{2k+1}(D(\lambda))=2kp_{2k-1}(\lambda)+\cdots,$

where dots are functions in $\Gamma$ of degree at most $2k-2$ $($with
$\mathbb{Q}$-coefficients$)$. Therefore the family
$\\{R_{2k}(D(\cdot))\\}_{k=1,2,\dots}$ is an algebraic basis of $\Gamma$.

###  3.7 Proof of main theorem

For each $k=1,2,3,\dots$ and $\lambda\in\mathcal{SP}$, we set

$\displaystyle\mathfrak{R}_{2k}(\lambda)=\frac{1}{2}R_{2k}(D(\lambda)).$

The degree of $\mathfrak{R}_{2k}$ in $\Gamma$ is $2k-1$ as we proved in
Corollary 3.7.

###### Proof of Theorem 1.4.

As we saw in (3.2), the difference $\mathfrak{p}_{2k-1}-p_{2k-1}$ belongs to
$\mathbb{Q}[p_{1},p_{3},$ $\dots,p_{2k-3}]$ and has degree at most $2k-2$. So
does the difference $\mathfrak{R}_{2k}-p_{2k-1}$ by Corollary 3.7. Therefore

$\displaystyle\mathfrak{p}_{2k-1}-\mathfrak{R}_{2k}=(\mathfrak{p}_{2k-1}-p_{2k-1})-(\mathfrak{R}_{2k}-p_{2k-1})$

belongs to $\mathbb{Q}[p_{1},p_{3},\dots,p_{2k-3}]$ and is of degree at most
$2k-2$. Again, using Corollary 3.7, we see that
$\mathbb{Q}[p_{1},p_{3},\dots,p_{2k-3}]=\mathbb{Q}[\mathfrak{R}_{2},\mathfrak{R}_{4},\dots,\mathfrak{R}_{2k-2}]$.
Hence $\mathfrak{p}_{2k-1}-\mathfrak{R}_{2k}$ is a polynomial in
$\mathfrak{R}_{2},\mathfrak{R}_{4},\dots,\mathfrak{R}_{2k-2}$ of degree at
most $2k-2$. ∎

###  3.8 Computations for spin Kerov polynomials

In this subsection, we explain how to obtain formulas in Example 1.5. We use
Okounkov’s idea which is employed in [2] for ordinary Kerov polynomials.
Recall the function $G_{\mathfrak{m}_{\lambda}}(z)$ defined in Section 2.3,
which is the Cauchy transform of the transition measure associated with
$\lambda$.

######  Proposition 3.8.

For each strict partition $\lambda$, we have

$\displaystyle
G_{\mathfrak{m}_{D(\lambda)}}(z)=\frac{\Phi(z-1;\lambda)}{z\Phi(z;\lambda)}.$

###### Proof.

Immediate from Lemma 2.1 and the identity (3.5). ∎

######  Proposition 3.9.

For a strict partition $\lambda$ and each $k=1,2,\dots$, we have

$\displaystyle\mathfrak{R}_{2k}(\lambda)=-\frac{1}{2(2k-1)}\big{[}z^{-2k}\big{]}\left(\frac{\Phi(z;\lambda)}{\Phi(z-1;\lambda)}\right)^{2k-1}.$
(3.7)

###### Proof.

Using (2.3) we have

$\displaystyle\mathfrak{R}_{2k}(\lambda)=\frac{1}{2}R_{2k}(D(\lambda))=-\frac{1}{2(2k-1)}\big{[}z^{-1}\big{]}G_{\mathfrak{m}_{D(\lambda)}}(z)^{-(2k-1)}.$

The desired expression follows immediately from Proposition 3.8. ∎

The expression (3.7) can be rewritten by (3.4) as

$\displaystyle\mathfrak{R}_{2k}(\lambda)=-\frac{1}{2(2k-1)}\big{[}z^{-2k}\big{]}\left\\{\exp\left((2k-1)\sum_{j=1}^{\infty}\frac{2p_{2j-1}(\lambda)}{2j-1}z^{-(2j-1)}\right)\right.$
$\displaystyle\left.\hphantom{\mathfrak{R}_{2k}(\lambda)=}{}\times\exp\left(-(2k-1)\sum_{j=1}^{\infty}\frac{2p_{2j-1}(\lambda)}{2j-1}(z-1)^{-(2j-1)}\right)\right\\}.$

Using this with computer, we can obtain expansions of $\mathfrak{R}_{2k}$ in
terms of power-sums.

######  Example 3.10.

$\displaystyle\mathfrak{R}_{2}=p_{1},$
$\displaystyle\mathfrak{R}_{4}=p_{3}-3p_{1}^{2}+p_{1},$
$\displaystyle\mathfrak{R}_{6}=p_{5}-10p_{3}p_{1}+\frac{10}{3}p_{3}+\frac{50}{3}p_{1}^{3}-15p_{1}^{2}+p_{1},$
$\displaystyle\mathfrak{R}_{8}=p_{7}-14p_{5}p_{1}-7p_{3}^{2}+7p_{5}+98p_{3}p_{1}^{2}-\frac{266}{3}p_{3}p_{1}-\frac{343}{3}p_{1}^{4}+7p_{3}+196p_{1}^{3}-35p_{1}^{2}+p_{1},$
$\displaystyle\mathfrak{R}_{10}=p_{9}-18p_{7}p_{1}-18p_{5}p_{3}+12p_{7}+162p_{5}p_{1}^{2}+162p_{3}^{2}p_{1}-198p_{5}p_{1}-96p_{3}^{2}-972p_{3}p_{1}^{3}$
$\displaystyle\hphantom{\mathfrak{R}_{10}=}{}+\frac{126}{5}p_{5}+1674p_{3}p_{1}^{2}+\frac{4374}{5}p_{1}^{5}-330p_{3}p_{1}-2430p_{1}^{4}+12p_{3}+810p_{1}^{3}-63p_{1}^{2}+p_{1}.$

###### Proof of Example 1.5.

It can be obtained by comparing Examples 3.2 and 3.10. ∎

##  4 Another double diagrams

In this section, we deal with another double diagram of a strict partition
$\lambda$. It is considered in [13] and De Stavola’s thesis [3]. We denote it
by $\mathtt{D}(\lambda)$ and call it the symmetrized double diagram. We do not
give its explicit definition here, see [3, Fig. 11]. The diagram
$\mathtt{D}(\lambda)$ is not a Young diagram associated to any partition but
we can extend notions in Section 2 to such diagrams, see [11, Chapter 2.2].

The diagram $\mathtt{D}(\lambda)$ is drawn in the Russian style and has
Kerov’s interlacing coordinate

$\displaystyle\mathtt{x}_{-m}<\mathtt{y}_{-m}<\mathtt{x}_{-1}<\mathtt{y}_{-1}<\mathtt{x}_{0}<\mathtt{y}_{1}<\mathtt{x}_{1}<\cdots<\mathtt{y}_{m}<\mathtt{x}_{m},$

where $\mathtt{x}_{-i}=-\mathtt{x}_{i}$, $\mathtt{y}_{-i}=-\mathtt{y}_{i}$,
and $\mathtt{x}_{0}=0$. Note that these numbers except $\mathtt{x}_{0}$ belong
to $\mathbb{Z}+\frac{1}{2}$. The corresponding Kerov’s transition measure
$\mathfrak{m}_{\mathtt{D}(\lambda)}$ is characterized by its Cauchy transform

$\displaystyle
G_{\mathfrak{m}_{\mathtt{D}(\lambda)}}(z)=\frac{\prod\limits_{i=1}^{m}(z-\mathtt{y}_{-i})(z-\mathtt{y}_{i})}{(z-\mathtt{x}_{0})\prod\limits_{i=1}^{m}(z-\mathtt{x}_{-i})(z-\mathtt{x}_{i})}=\frac{\prod\limits_{i=1}^{m}\big{(}z^{2}-\mathtt{y}_{i}^{2}\big{)}}{z\prod\limits_{i=1}^{m}\big{(}z^{2}-\mathtt{x}_{i}^{2}\big{)}}.$

######  Lemma 4.1 ([13, Proposition 2.6]).

For a strict partition $\lambda$, we have

$\displaystyle
G_{\mathfrak{m}_{\mathtt{D}(\lambda)}}(z)=\frac{\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}}{z\Phi\big{(}z+\tfrac{1}{2};\lambda\big{)}}.$

Compare this lemma with Proposition 3.8. The remaining discussion is the same
with that in the previous section, so we write only results.

For each $k=1,2,3,\dots$ and $\lambda\in\mathcal{SP}$, we set

$\displaystyle\mathtt{R}_{k}(\lambda)=\frac{1}{2}R_{k}[\mathfrak{m}_{\mathtt{D}(\lambda)}].$

Then it has the expression

$\displaystyle\mathtt{R}_{k}(\lambda)=-\frac{1}{2(k-1)}\big{[}z^{-k}\big{]}\left(\frac{\Phi(z+\tfrac{1}{2};\lambda)}{\Phi\big{(}z-\tfrac{1}{2};\lambda\big{)}}\right)^{k-1}$
$\displaystyle\hphantom{\mathtt{R}_{k}(\lambda)}{}=-\frac{1}{2(k-1)}\big{[}z^{-k}\big{]}\left\\{\exp\left((k-1)\sum_{j=1}^{\infty}\frac{2p_{2j-1}(\lambda)}{2j-1}\big{(}z+\tfrac{1}{2}\big{)}^{-(2j-1)}\right)\right.$
$\displaystyle\left.\hphantom{\mathtt{R}_{k}(\lambda)=}{}\times\exp\left(-(k-1)\sum_{j=1}^{\infty}\frac{2p_{2j-1}(\lambda)}{2j-1}\big{(}z-\tfrac{1}{2}\big{)}^{-(2j-1)}\right)\right\\}.$

It is easy to see that $\mathtt{R}_{k}=0$ for odd $k$. We remark that
$R_{k}(D(\lambda))$ does not vanish even if $k$ is odd.

######  Example 4.2.

$\mathtt{R}_{2}=p_{1}$, $\mathtt{R}_{4}=p_{3}-3p_{1}^{2}+\frac{1}{4}p_{1}$.

######  Proposition 4.3.

For each $k=1,2,3,\dots$, the spin character $\mathfrak{p}_{2k-1}$ is a
polynomial in $\mathtt{R}_{2},\mathtt{R}_{4},$ $\dots,\mathtt{R}_{2k}$ of the
form

$\displaystyle\mathfrak{p}_{2k-1}=\mathtt{R}_{2k}+(\text{a polynomial in
$\mathtt{R}_{2},\mathtt{R}_{4},\dots,\mathtt{R}_{2k-2}$}).$

######  Example 4.4.

$\mathfrak{p}_{1}=\mathtt{R}_{2}$,
$\mathfrak{p}_{3}=\mathtt{R}_{4}+\frac{7}{4}\mathtt{R}_{2}$.

From this example, we find that coefficients of polynomials in Proposition 4.3
are _not_ integers. This indicates that these polynomials are ineligible for
“spin Kerov polynomials”. We thus believe that the $\mathfrak{R}_{2k}$ defined
in Section 3 are the most appropriate choice for spin Kerov polynomials.

### Acknowledgements

The author acknowledges useful discussions with Valentin Féray and Dario De
Stavola. The research was supported by JSPS KAKENHI Grant Number 17K05281.

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##  Appendix A LQR equations

The proposed LQR model can be described as the discrete-time linear-quadratic
optimal control problem with finite horizon $N\in\mathbb{N}$

(39c) $\displaystyle\begin{gathered}\textsl{Minimize}\quad
J_{N}^{\text{(LQR)}}(x,u)=\sum_{n=0}^{N}x_{n}^{\top}Q_{n}x_{n}+\sum_{n=0}^{N-1}u_{n}^{\top}R_{n}u_{n}\\\
\textsl{with respect to
}u=(u_{n})_{n\in\\{0,\dots,N-1\\}}\subset\mathbb{R},\end{gathered}$ where
$x=(x_{n})_{n\in\\{0,\dots,N\\}}\subset\mathbb{R}^{5}$ with
$x_{n}=(p_{n},v_{n},f_{n},g_{n},T)^{\top}$ satisfies (39f)
$\displaystyle\begin{gathered}x_{n+1}=Ax_{n}+Bu_{n},\quad
n\in\\{0,\dots,N-1\\},\\\ x_{0}=\bar{x}_{0},\end{gathered}$ given some
$\bar{x}_{0}\in\mathbb{R}^{5}$, with sampling time $h>0$ and system dynamics
matrices (39g) $\displaystyle A=\begin{pmatrix}1&h&0&0&0\\\ 0&1&h&0&0\\\
0&0&1-\frac{h}{\tau_{2}}&\frac{h}{\tau_{2}}&0\\\
0&0&0&1-\frac{h}{\tau_{1}}&0\\\ 0&0&0&0&1\end{pmatrix},\quad
B=\begin{pmatrix}0\\\ 0\\\ 0\\\ \frac{h}{\tau_{1}}\\\ 0\end{pmatrix}$

corresponding to the combination of a simplified second-order muscle model
with time constants $\tau_{1}$, $\tau_{2}>0$ and a double integrator.
The state cost matrices are defined by

(40) $\displaystyle Q_{n}=\begin{pmatrix}1&0&0&0&-1\\\ 0&\omega_{v}&0&0&0\\\
0&0&\omega_{f}&0&0\\\ 0&0&0&0&0\\\
-1&0&0&0&1\end{pmatrix}\in\mathbb{R}^{5\times 5},\quad n\in\\{0,\dots,N\\},$

which implies

(41) $\displaystyle
x_{n}^{\top}Q_{n}x_{n}=D_{n}^{2}+\omega_{v}v_{n}^{2}+\omega_{f}f_{n}^{2},$

i.e., the distance $D_{n}=|T-p_{n}|$ between mouse and target position as well
as the end-effector velocity $v_{n}$ and force $f_{n}$ are quadratically
penalized at every time step $n\in\\{0,\dots,N\\}$. In our case of one-
dimensional pointing tasks, the control cost matrices are scalar and given by

(42) $\displaystyle
R_{n}=\frac{\omega_{r}}{N-1}\in\mathbb{R},\quad\omega_{r}>0,\quad
n\in\\{0,\dots,N-1\\},$

which yields the quadratic cost terms

(43) $\displaystyle u_{n}^{\top}R_{n}u_{n}=\frac{\omega_{r}}{N-1}u_{n}^{2}.$

It can be shown that the unique solution
$u^{*}=(u_{n}^{*})_{n\in\\{0,\dots,N-1\\}}$ to the optimization problem (39)
is given by

(46) $\displaystyle\begin{gathered}u_{n}^{*}=\pi(x_{n})=-L_{n}x_{n},\quad
n\in\\{0,\dots,N-1\\},\\\
L_{n}=(R_{n}+B^{\top}\mathcal{S}_{n+1}B)^{-1}B^{\top}\mathcal{S}_{n+1}A,\end{gathered}$
(47) $\displaystyle n\in\\{0,\dots,N-1\\},$

where the symmetric matrices $\mathcal{S}_{n}\in\mathbb{R}^{5\times 5}$ can be
determined by solving the Discrete Riccati Equations

(48b)
$\displaystyle\begin{gathered}\mathcal{S}_{n}=Q_{n}+A^{\top}\mathcal{S}_{n+1}A-A^{\top}\mathcal{S}_{n+1}B(R_{n}+B^{\top}\mathcal{S}_{n+1}B)^{-1}B^{\top}\mathcal{S}_{n+1}A\end{gathered}$
for $n\in\\{0,\dots,N-1\\}$ backward in time with initial value (48c)
$\displaystyle\mathcal{S}_{N}=Q_{N}.$

##  Appendix B Supplementary Material

Model | Parameter | Minimum | Maximum | Type
---|---|---|---|---
2OL-Eq | $k$ | 0 | 500 | Continuous
| $d$ | 0 | 500 | Continuous
MinJerk | $N_{MJ}$ | 0 | $N$ | Continuous (Relaxed)
LQR | $\omega_{r}$ | 2e-9 | 20 | Continuous
| $\omega_{v}$ | 0 | 10e-2 | Continuous
| $\omega_{f}$ | 0 | 10e-4 | Continuous
LQG / E-LQG | $\omega_{r}$ | 4e-18 | 7e-3 | Continuous
| $\omega_{v}$ | 0 | 10 | Continuous
| $\omega_{f}$ | 0 | 10 | Continuous
LQG | $\sigma_{u}$ | 10e-10 | 5 | Continuous
| $\sigma_{s}$ | 0 | 5 | Continuous
E-LQG | $\sigma_{u}$ | 10e-10 | 5 | Continuous
| $\sigma_{v}$ | 0 | 10 | Continuous
| $\sigma_{f}$ | 0 | 50 | Continuous
| $\sigma_{e}$ | 0 | 5 | Continuous
| $\gamma$ | 4e-18 | 100 | Continuous
| $n_{s}$ | 0 | $N$ | Continuous (Relaxed)
Table B.1. Boundaries of all model parameters used for parameter fitting.

$\int$$\int$$d$$k$$u(t)$$+$$\ddot{y}(t)$$\dot{y}(t)$$y(t)$$-$$-$

Figure B.1. Control-flow diagram of the second-order lag (2OL).

Fully described in caption and text.

Figure B.2. The relationship between the inverse of the stiffness parameter
$k$ in 2OL-Eq and the surge duration parameter $N_{MJ}$ in MinJerk is captured
well by a linear function.

MinJerk surge duration parameter $N_{MJ}$ plotted against inverse of 2OL-Eq
stiffness parameter $k$ together with its linear approximation, resulting in a
coefficient of determination of $R^{2}=0.8966$.

(a)

(b)

(c)

Figure B.3. Cost weight parameters of the LQG, optimized for the trajectory
sets of all participants, tasks, and directions, grouped by participants. Note
that the optimal values of $\omega_{f}$ and $\omega_{r}$ are plotted on a
logarithmic scale.

Fully described in caption and text.

(a)

(b)

(c)

(d)

(e)

(f)

Figure B.4. Visual observation noise parameters of the E-LQG, optimized for
the mean trajectories of all participants, tasks, and directions, grouped by
participants (left) and by ID (right). For better visibility, both plots for
the position perception noise weight $\gamma$ do not include the 2 largest
outliers with values $19.7$ and $55.4$. Note that the optimal values of
$\sigma_{v}$ are plotted on a logarithmic scale.

Fully described in caption and text.

(a)

(b)

(c)

(d)

Figure B.5. The gaze noise level $\sigma_{e}$ and the saccade time step
$n_{s}$ of the E-LQG, optimized for the mean trajectories of all participants,
tasks, and directions, grouped by participants (left) and by ID (right).

Fully described in caption and text.