# Universal characters twisted by roots of unity

Seamus P. Albion  Fakultät für Mathematik, Universität Wien, Oskar-
Morgenstern-Platz 1, A-1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

A classical result of Littlewood gives a factorisation for the Schur function
at a set of variables “twisted” by a primitive $t$-th root of unity,
characterised by the core and quotient of the indexing partition. While
somewhat neglected, it has proved to be an important tool in the character
theory of the symmetric group, the cyclic sieving phenomenon, plethysms of
symmetric functions and more. Recently, similar factorisations for the
characters of the groups $\mathrm{O}(2n,\mathbb{C})$,
$\mathrm{Sp}(2n,\mathbb{C})$ and $\mathrm{SO}(2n+1,\mathbb{C})$ were obtained
by Ayyer and Kumari. We lift these results to the level of universal
characters, which has the benefit of making the proofs simpler and the
structure of the factorisations more transparent. Our approach also allows for
universal character extensions of some factorisations of a different nature
originally discovered by Ciucu and Krattenthaler, and generalised by Ayyer and
Behrend.

Keywords: Schur functions, symplectic characters, orthogonal characters,
universal characters, $t$-core, $t$-quotient.

###### 2020 Mathematics Subject Classification:

15A15, 20C15, 05E05, 05E10

##  1\. Introduction

In his $1940$ book _The Theory of Group Characters and Matrix Representations
of Groups_ , D. E. Littlewood devotes a section to the evaluation of the Schur
function $s_{\lambda}$ at a set of variables “twisted” (not his term) by a
primitive $t$-th root of unity $\zeta$ [27, §7.3]. In modern terminology,
Littlewood’s theorem asserts that $s_{\lambda}$ evaluated at the variables
$\zeta^{j}x_{i}$ for $1\leqslant i\leqslant n$ and $0\leqslant j\leqslant t-1$
is zero unless the $t$-core of $\lambda$ is empty. Moreover, when it is
nonzero, it factors as a product of Schur functions indexed by the elements of
the $t$-quotient of $\lambda$, each with the variables
$x_{1}^{t},\dots,x_{n}^{t}$.

The Schur functions are characters of the irreducible polynomial
representations of the general linear group $\mathrm{GL}(n,\mathbb{C})$. Ayyer
and Kumari [5] have recently generalised Littlewood’s theorem to the
characters of the other classical groups $\mathrm{O}(2n,\mathbb{C})$,
$\mathrm{Sp}(2n,\mathbb{C})$ and $\mathrm{SO}(2n+1,\mathbb{C})$ indexed by
partitions. While their factorisations are still indexed by the $t$-quotient
of the corresponding partition, the vanishing is governed by the $t$-core
having a particular form. More precisely, $t\textup{-core}(\lambda)$ is of the
form $(a\mid a+z)$ in Frobenius notation, where $z=-1,1,0$, for
$\mathrm{O}(2n,\mathbb{C})$, $\mathrm{Sp}(2n,\mathbb{C})$ and
$\mathrm{SO}(2n+1,\mathbb{C})$ respectively. Note that these are the same
partitions occurring in Littlewood’s Schur expansion of the Weyl denominators
for types $\mathrm{B}_{n}$, $\mathrm{C}_{n}$ and $\mathrm{D}_{n}$ [27, p. 238]
(see also [31, p. 79]).

Littlewood’s proof, and the proofs of Ayyer and Kumari, use the Weyl-type
expressions for the characters as ratios of alternants. In the Schur case,
Chen, Garsia and Remmel [7] and independently Lascoux [24, Theorem 5.8.2] have
given an alternate proof based on the Jacobi–Trudi formula (2.3). This
approach was already known to Farahat, who used it to extend Littlewood’s
theorem to skew Schur functions $s_{\lambda/\mu}$ where $\mu$ is the $t$-core
of $\lambda$ [12, Theorem 2]. The full skew Schur case was then given by
Macdonald [31, p. 91], again proved using the Jacobi–Trudi formula; see
Theorem 3.1 below.

In this article we lift the results of Ayyer and Kumari to the much more
general universal characters of the groups $\mathrm{O}(2n,\mathbb{C})$,
$\mathrm{Sp}(2n,\mathbb{C})$ and $\mathrm{SO}(2n+1,\mathbb{C})$ as defined by
Koike and Terada [20]. These are symmetric functions indexed by partitions
which, under appropriate specialisation of the variables, become actual
characters of their respective groups. In fact, these generalise the
Jacobi–Trudi-type formulas for the characters of these groups, which were
first written down by Weyl [43, Theorems 7.8.E & 7.9.A]. For the universal
characters we generalise the notion of “twisting” a set of variables by
introducing operators $\varphi_{t}:\Lambda\longrightarrow\Lambda$ for each
integer $t\geqslant 2$ which act on the complete homogeneous symmetric
functions as

(1.1) $\varphi_{t}h_{r}=\begin{cases}h_{r/t}&\text{if $t$ divides $r$},\\\
0&\text{otherwise}.\end{cases}$

It is not at all hard to show that the image of $\varphi_{t}$ acting on a
symmetric function at the variables $x_{1}^{t},\dots,x_{n}^{t}$ agrees with
the result of twisting the variables $x_{1},\dots,x_{n}$ by $\zeta$. The
advantages of this framework for such factorisations are that the proofs are
much simpler, and the structure of the factorisations is made transparent.
Moreover, we are able to discuss dualities between these objects which are
only present at the universal level. A particularly important tool for our
purposes is Koike’s universal character $\mathrm{rs}_{\lambda,\mu}$ (2.10)
associated with a rational representation of $\mathrm{GL}(n,\mathbb{C})$. This
object, which is used later in Subsection 6.3 to prove other character
factorisations, appears to be the correct universal character analogue of the
Schur function with variables $(x_{1},1/x_{1},\dots,x_{n},1/x_{n})$.

The remainder of the paper reads as follows. In the next section we outline
the preliminaries on partitions and symmetric functions needed to state our
main results, which follow in Section 3. In the following Section 4 we prepare
for the proofs of these results by giving a series of lemmas regarding cores
and quotients and their associated signs. The factorisations are then proved
in Section 5, including a detailed proof of the Schur case, following
Macdonald. The final Section 6 concerns other factorisation results relating
to Schur functions and other characters. This includes universal extensions of
factorisations very different from those already discussed originally due to
Ciucu and Krattenthaler, later generalised by Ayyer and Behrend.

##  2\. Preliminaries

###  2.1. Partitions

A _partition_ $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\dots)$ is a
weakly decreasing sequence of nonnegative integers such that only finitely
many of the $\lambda_{i}$ are nonzero. The nonzero $\lambda_{i}$ are called
_parts_ and the number of parts the _length_ , written $l(\lambda)$. We say
$\lambda$ is _a partition of $n$_ if
$\lvert\lambda\rvert:=\lambda_{1}+\lambda_{2}+\lambda_{3}+\cdots=n$. Two
partitions are regarded as the same if they agree up to trailing zeroes, and
the set of all partitions is written $\mathscr{P}$. A partition is identified
with its _Young diagram_ , which is the left-justified array of squares
consisting of $\lambda_{i}$ squares in row $i$ with $i$ increasing downward.
For example

is the Young diagram of $(6,4,3,2)$. We define the _conjugate_ partition
$\lambda^{\prime}$ by reflecting the diagram of $\lambda$ in the main diagonal
$x=y$, so that the conjugate of $(6,4,3,2)$ above is $(4,4,3,2,1,1)$. If
$\lambda=\lambda^{\prime}$ then $\lambda$ is called _self-conjugate_. For a
square at coordinate $(i,j)$ where $1\leqslant i\leqslant l(\lambda)$ and
$1\leqslant j\leqslant\lambda_{i}$ the _hook length_ is
$h(i,j)=\lambda_{i}+\lambda_{j}^{\prime}-i-j+1$. For example the square
$(1,2)$ in

has hook length $8$, with its hook shaded. A partition $\lambda$ _is a
$t$-core_ if it contains no squares of hook length $t$, the set of which is
denoted $\mathscr{C}_{t}$. For a pair of partitions $\lambda,\mu$ we write
$\mu\subseteq\lambda$ if the diagram of $\mu$ can be drawn inside the diagram
of $\lambda$, i.e., if $\mu_{i}\leqslant\lambda_{i}$ for all $i\geqslant 1$.
In this case we can form the _skew shape_ $\lambda/\mu$ by removing the digram
of $\mu$ from that of $\lambda$. For example $(3,2,1,1)\subseteq(6,4,3,2)$ and
the diagram of $(6,4,3,2)/(3,2,1,1)$ is given by the non-shaded squares of

A skew shape is called a _ribbon_ (or _border strip_ , _rim hook_ , _skew
hook_) if its diagram is connected and contains no $2\times 2$ square. A
$t$-ribbon is a ribbon with $t$ boxes. The _height_ of a $t$-ribbon $R$,
written $\mathrm{ht}(R)$, is one less than the number of rows it occupies. In
our example above $R=(6,4,3,2)/(3,2,1,1)$ is an $8$-ribbon with height
$\mathrm{ht}(R)=3$. We say a skew shape is _tileable by $t$-ribbons_ or _$t$
-tileable_ if there exists a sequence of partitions

(2.1)
$\mu=\nu^{(0)}\subseteq\nu^{(1)}\subseteq\cdots\subseteq\nu^{(k-1)}\subseteq\nu^{(k)}=\lambda$

such that $\nu^{(i)}/\nu^{(i-1)}$ is a $t$-ribbon for $1\leqslant i\leqslant
k$. A sequence $D=(\nu^{(0)},\dots,\nu^{(k)})$ (not to be confused with the
$t$-quotient of $\nu$ below, for which we use the same notation) satisfying
(2.1) is called a _ribbon decomposition_ (or _border strip decomposition_) of
$\lambda/\mu$. We define the height of a ribbon decomposition to be the sum of
the heights of the individual ribbons:
$\mathrm{ht}(D):=\sum_{i=1}^{k}\mathrm{ht}(\nu^{(i)}/\nu^{(i-1)})$. As shown
by van Leeuwen [26, Proposition 3.3.1] and Pak [35, Lemma 4.1] (also in [2,
§6]), the quantity $(-1)^{\mathrm{ht}(D)}$ is the same for every ribbon
decomposition of $\lambda/\mu$. We therefore define the _sign_ of a
$t$-tileable skew shape $\lambda/\mu$ as

(2.2) $\operatorname{sgn}_{t}(\lambda/\mu):=(-1)^{\mathrm{ht}(D)}.$

Let $\mathrm{rk}(\lambda)$ be the greatest integer such that
$\mathrm{rk}(\lambda)\geqslant\lambda_{\mathrm{rk}(\lambda)}$, usually called
the _Frobenius rank_ of $\lambda$. Equivalently, $\mathrm{rk}(\lambda)$ is the
side length of the largest square which fits inside the diagram of $\lambda$
(the _Durfee square_). A partition can alternatively be written in _Frobenius
notation_ as

$\lambda=\big{(}\lambda_{1}-1,\dots,\lambda_{\mathrm{rk}(\lambda)}-\mathrm{rk}(\lambda)\mid\lambda^{\prime}_{1}-1,\dots,\lambda^{\prime}_{\mathrm{rk}(\lambda)}-\mathrm{rk}(\lambda)\big{)}.$

Any pair of integer sequences $a_{1}>\cdots>a_{k}\geqslant 0$ and
$b_{1}>\cdots>b_{k}\geqslant 0$ thus determines a partition $\lambda=(a\mid
b)$ with $\mathrm{rk}(\lambda)=k$. For $z\in\mathbb{Z}$ and an integer
sequence of predetermined length $a=(a_{1},\dots,a_{k})$ we write
$a+z:=(a_{1}+z,\dots,a_{k}+z)$. Following Ayyer and Kumari [5, Definition
2.9], $\lambda$ is called _$z$ -asymmetric_ if it is of the form
$\lambda=(a\mid a+z)$ for some integer sequence $a$ and integer $z$. Clearly a
$0$-asymmetric partition is self-conjugate. Partitions which are $-1$\- and
$1$-asymmetric are called _orthogonal_ and _symplectic_ respectively.

###  2.2. Cores and quotients

We now describe the $t$-core and $t$-quotient of $\lambda$ arithmetically
following [31, p. 12]. There are many equivalent descriptions, see for
instance [13, 15, 16, 42]. We begin with the _beta set_ of a partition, which
is simply the set of $n$ integers

$\beta(\lambda;n):=\\{\lambda_{1}+n-1,\lambda_{2}+n-2,\dots,\lambda_{n-1}+1,\lambda_{n}\\},$

where $n\geqslant l(\lambda)$ is fixed. The number of elements in this set
congruent to $r$ modulo $t$ is denoted by $m_{r}(\lambda;n)=m_{r}$. Each
element which falls into residue class $r$ for $0\leqslant r\leqslant t-1$ can
be written as $\xi^{(r)}_{k}t+r$ for some integers
$\xi^{(r)}_{1}>\cdots>\xi^{(r)}_{m_{r}}\geqslant 0$. These integers are used
to define a partition with parts
$\lambda^{(r)}_{k}=\xi^{(r)}_{k}-m_{r}(\lambda;n)+k$ where $1\leqslant
k\leqslant m_{r}(\lambda;n)$, and the ordered sequence
$(\lambda^{(0)},\dots,\lambda^{(t-1)})$ of these partitions is called the _$t$
-quotient_. The precise order of the constituents of the $t$-quotient depends
on the residue class of $n$ modulo $t$. However, the orders only differ by
cyclic permutations, and Macdonald comments that it is best to think of the
quotient as a sort-of “necklace” of partitions. To simplify things somewhat,
we adopt the convention that the $t$-quotient is always computed with $n$ a
multiple of $t$, so that the order of its constituents is fixed. To define the
$t$-core, one writes down the $n$ distinct integers $kt+r$ where $0\leqslant
k\leqslant m_{r}(\lambda;n)-1$ and $0\leqslant r\leqslant t-1$ in descending
order, say as $\tilde{\xi}_{1}>\dots>\tilde{\xi}_{n}$. Then
$t\textup{-core}(\lambda)_{i}:=\tilde{\xi}_{i}-n+i$. If
$t\textup{-core}(\lambda)$ is empty then we say $\lambda$ _has empty
$t$-core_.

It will prove useful later on to work with the _bead configurations_ (or _bead
diagrams_ , _abacus model_) of James and Kerber [16, §2.7], which give a
different model for $t$-cores and $t$-quotients. The “board” for a bead
configuration is the set of nonnegative integers arranged in $t$ downward-
increasing columns, called _runners_ , according to their residues modulo $t$.
A bead is then placed at the space corresponding to each element of
$\beta(\lambda;n)$. For an example, let $\lambda=(4,4,3,2,1)$ so that
$\beta(\lambda;6)=\\{9,8,6,4,2,0\\}$. Then the bead configuration for
$\lambda$ with $t=3$ and $n=6$ and the beads labelled by their position is

$\scriptstyle 0$$\scriptstyle 2$$\scriptstyle 4$$\scriptstyle 6$$\scriptstyle
8$$\scriptstyle 9$

Moving a bead up one space is equivalent to reducing one of the elements of
$\beta(\lambda;n)$ by $t$. This is, in turn, equivalent to removing a
$t$-ribbon from $\lambda$ such that what remains is still a Young diagram (see
for instance [31, p. 12]). Pushing all beads to the top will give the bead
configuration of $t\textup{-core}(\lambda)$, and this is clearly independent
of the order in which the beads are pushed. It follows that
$t\textup{-core}(\lambda)$ is the unique partition obtained by removing
$t$-ribbons (in a valid way) from the diagram of $\lambda$ until it is no
longer possible to do so. We note that if removing a ribbon $R$ corresponds to
moving a bead from position $b$ to $b-t$, then $\mathrm{ht}(R)$ is equal to
the number of beads lying at the positions strictly between $b-t$ and $b$. The
$t$-quotient can be obtained from the bead configuration by reading the $r$-th
runner, bottom-to-top, as a bead configuration with $m_{r}(\lambda;n)$ beads.
For our example, this means that
$\beta(t\textup{-core}(\lambda);6)=\\{6,5,3,2,1,0\\}$, so that
$t\textup{-core}(\lambda)=(1,1)$, with the quotient $((1,1),(1),(1))$ computed
similarly.

The above procedure of computing the $t$-core and $t$-quotient actually
encodes a bijection

$\displaystyle\phi_{t}:\mathscr{P}$
$\displaystyle\longrightarrow\mathscr{C}_{t}\times\mathscr{P}^{t}$
$\displaystyle\lambda$
$\displaystyle\longmapsto\big{(}t\textup{-core}(\lambda),(\lambda^{(0)},\dots,\lambda^{(t-1)})\big{)},$

such that $\lvert\lambda\rvert=\lvert
t\textup{-core}(\lambda)\rvert+t(\lvert\lambda^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}\rvert)$.
The arithmetic description of this correspondence was first written down by
Littlewood [29]. The idea of removing ribbons from a partition until a unique
core is obtained goes back to Nakayama [33]. The $t$-quotient of a partition
has its origin in the _star diagrams_ of Nakayama, Osima, Robinson and Staal
[34, 39, 40], which were shown to be equivalent to Littlewood’s $t$-quotient
by Farahat [10].

Let $w_{t}(\lambda;n)$ be the permutation of $\beta(\lambda;n)$ which sorts
the elements of the beta set so that their residues modulo $t$ are increasing,
and the elements within each residue class decrease. The sign of
$w_{t}(\lambda;n)$ will be denoted $\operatorname{sgn}(w_{t}(\lambda;n))$. The
permutation $w_{t}(\lambda;n)$ can also be read off the bead configuration by
first labelling the beads “backwards”: label the bead with largest place $1$,
second-largest $2$, and so on. Reading the labels column-wise from bottom-to-
top gives $w_{t}(\lambda;n)$ in one-line notation. An inversion in this
permutation corresponds to a pair of beads $b_{1},b_{2}$ such that $b_{2}$
lies weakly below and strictly to the right of $b_{1}$. With the same example
as before $w_{3}((4,4,3,2,1);6)=136425$ and the bead at position $0$ generates
three inversions, as it “sees” the beads $2,4$ and $8$.

As follows from the above, a partition $\lambda$ has empty $t$-core if and
only if it is $t$-tileable. In our results below we will need the following
characterisation of when a skew shape is $t$-tileable, generalising the notion
of “empty $t$-core” to this case. We briefly recall our convention that
$t$-quotients are always computed with the number of beads in the bead
configuration a multiple of $t$.

######  Lemma 2.1.

A skew shape $\lambda/\mu$ is tileable by $t$-ribbons if and only if
$t\textup{-core}(\lambda)=t\textup{-core}(\mu)$ and
$\mu^{(r)}\subseteq\lambda^{(r)}$ for each $0\leqslant r\leqslant t-1$.

###### Proof.

The skew shape being $t$-tileable is equivalent to the diagram of $\mu$ being
obtainable from the diagram of $\lambda$ by removing $t$-ribbons. In other
words, we can obtain the bead configuration of $\mu$ from that of $\lambda$,
where both have $nt$ beads, by moving beads upwards. Assume that this is the
case. Then $m_{r}(\lambda;nt)=m_{r}(\mu;nt)$ for each $0\leqslant r\leqslant
t-1$, so that the $r$-th runner has the same number of beads in each diagram.
This implies that $t\textup{-core}(\lambda)=t\textup{-core}(\mu)$. It also
follows that the $i$-th bead in each runner of $\lambda$’s bead configuration
must lie weakly below the $i$-th bead in the same runner of $\mu$’s bead
configuration. Equivalently, $\mu^{(r)}_{i}\leqslant\lambda^{(r)}_{i}$ for all
$0\leqslant r\leqslant t-1$ and $1\leqslant i\leqslant m_{r}(\lambda;nt)$,
which in turn is equivalent to $\mu^{(r)}\subseteq\lambda^{(r)}$. The reverse
direction is now clear. ∎

Note that the lemma is also true when the $t$-quotients of $\lambda$ and $\mu$
are computed using the same integer $n$ of any residue class modulo $t$. If
$\lambda/\mu$ is $t$-tileable, then we think of
$\lambda^{(0)}/\mu^{(0)},\dots,\lambda^{(t-1)}/\mu^{(t-1)}$ as its
$t$-quotient. When $\lambda/\mu$ is not $t$-tileable, it is not so clear how
to define the $t$-quotient.

###  2.3. Symmetric functions and universal characters

Here we discuss some basics of the theory of symmetric functions, following
[31]. Let $\Lambda$ denote the _ring of symmetric functions_ in an arbitrary
countable set of variables $X=(x_{1},x_{2},x_{3},\dots)$, called an
_alphabet_. Where possible, we write elements of $\Lambda$ without reference
to an alphabet if the expression is independent of the chosen alphabet. If for
a positive integer $n$ one sets $x_{i}=0$ for all $i>n$ then the elements of
$\Lambda$ reduce to symmetric polynomials in the variables
$(x_{1},\dots,x_{n})$. Another common specialisation sets $x_{n+i}=x_{i}^{-1}$
for $1\leqslant i\leqslant n$ and $x_{i}=0$ for $i>2n$. This gives Laurent
polynomials in the $x_{i}$ invariant under permutation and inversion of the
variables (i.e., $\mathrm{BC}_{n}$-symmetric functions). We will later write
$(x_{1}^{\pm},\dots,x_{n}^{\pm})$ for this alphabet.

Two fundamental algebraic bases for $\Lambda$ are the _complete homogeneous
symmetric functions_ and the _elementary symmetric functions_ , defined for
any positive integer $r$ by

$h_{r}(X):=\sum_{1\leqslant i_{1}\leqslant\cdots\leqslant
i_{r}}x_{i_{1}}\cdots x_{i_{r}}\quad\text{and}\quad e_{r}(X):=\sum_{1\leqslant
i_{1}<\cdots<i_{r}}x_{i_{1}}\cdots x_{i_{r}},$

respectively. We further set $h_{0}=e_{0}:=1$ and $h_{-r}=e_{-r}=0$ for
positive $r$. These admit the generating functions

$\displaystyle H_{z}(X)$ $\displaystyle:=\sum_{r\geqslant
0}z^{r}h_{r}(X)=\prod_{i\geqslant 1}\frac{1}{1-zx_{i}}$ $\displaystyle
E_{z}(X)$ $\displaystyle:=\sum_{r\geqslant 0}z^{r}e_{r}(X)=\prod_{i\geqslant
1}(1+zx_{i}).$

The $h_{r}$ and $e_{r}$ for $r\geqslant 1$ are algebraically independent over
$\mathbb{Z}$ and generate $\Lambda$. In view of this, we can define a
homomorphism $\omega:\Lambda\longrightarrow\Lambda$ by $\omega h_{r}=e_{r}$.
It then follows from the relation $H_{z}(X)E_{-z}(X)=1$ that $\omega
e_{r}=h_{r}$, so that $\omega$ is an involution. We also define the power sums
by

$p_{r}(X):=\sum_{i\geqslant 1}x_{i}^{r},$

for $r\geqslant 1$ and $p_{0}:=1$. These satisfy $\omega
p_{r}=(-1)^{r-1}p_{r}$.

The most important family of symmetric functions are the _Schur functions_.
These have several definitions, but for our purposes it is best to define
them, already for skew shapes, by the _Jacobi–Trudi formula_. If $\lambda/\mu$
is a skew shape and $n$ an integer such that $n\geqslant l(\lambda)$ we define

(2.3) $s_{\lambda/\mu}:=\det_{1\leqslant i,j\leqslant
n}(h_{\lambda_{i}-\mu_{j}-i+j}).$

This is independent of $n$ as long as $n\geqslant l(\lambda)$. If
$\mu\not\subseteq\lambda$ then we set $s_{\lambda/\mu}:=0$. There is also an
equivalent formula in terms of the $e_{r}$, called the _dual Jacobi–Trudi
formula_ (rarely also the _Nägelsbach–Kostka identity_)

$s_{\lambda/\mu}=\det_{1\leqslant i,j\leqslant
m}(e_{\lambda_{i}^{\prime}-\mu_{j}^{\prime}-i+j}).$

Restricting to the $\mu$ empty case, we have $s_{(r)}=h_{r}$ and
$s_{(1^{r})}=e_{r}$. Moreover, it is clear that $\omega
s_{\lambda/\mu}=s_{\lambda^{\prime}/\mu^{\prime}}$.

If the set of variables $(x_{1},\dots,x_{n})$ is finite then the Schur
function for $\mu=0$ admits another definition as a ratio of alternants

(2.4) $s_{\lambda}(x_{1},\dots,x_{n})=\frac{\det_{1\leqslant i,j\leqslant
n}(x_{i}^{\lambda_{j}+n-j})}{\det_{1\leqslant i,j\leqslant n}(x_{i}^{n-j})}.$

The denominator is the Vandermonde determinant and has the product
representation $\det_{1\leqslant i,j\leqslant
n}(x_{i}^{n-j})=\prod_{1\leqslant i<j\leqslant n}(x_{i}-x_{j})$. In this case
we also define $s_{\lambda}(x_{1},\dots,x_{n})=0$ if $l(\lambda)>n$. If
$\lambda$ is a partition of length at most $n$, then

(2.5) $s_{(\lambda_{1}+1,\dots,\lambda_{n}+1)}(x_{1},\dots,x_{n})=(x_{1}\cdots
x_{n})s_{(\lambda_{1},\dots,\lambda_{n})}(x_{1},\dots,x_{n}).$

This allows for Schur functions with a finite set of $n$ variables to be
extended to weakly decreasing sequences of integers of length exactly $n$.

Following Koike and Terada we define the _universal characters_ for
$\mathrm{O}(2n,\mathbb{C})$ and $\mathrm{Sp}(2n,\mathbb{C})$ as the symmetric
functions [20, Definition 2.1.1]

(2.6) $\displaystyle\mathrm{o}_{\lambda}$ $\displaystyle:=\det_{1\leqslant
i,j\leqslant n}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j}\big{)}$ (2.7)
$\displaystyle\mathrm{sp}_{\lambda}$
$\displaystyle:=\frac{1}{2}\det_{1\leqslant i,j\leqslant
n}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+2}\big{)},$

where $n\geqslant l(\lambda)$. Like the Schur functions, these determinants
also have dual versions

$\displaystyle\mathrm{o}_{\lambda}$ $\displaystyle=\frac{1}{2}\det_{1\leqslant
i,j\leqslant
m}\big{(}e_{\lambda_{i}^{\prime}-i+j}+e_{\lambda_{i}^{\prime}-i-j+2}\big{)}$
(2.8) $\displaystyle\mathrm{sp}_{\lambda}$ $\displaystyle=\det_{1\leqslant
i,j\leqslant
m}\big{(}e_{\lambda_{i}^{\prime}-i+j}-e_{\lambda_{i}^{\prime}-i-j}\big{)},$

where $m\geqslant\lambda_{1}$. From this it is clear that
$\omega\mathrm{o}_{\lambda}=\mathrm{sp}_{\lambda^{\prime}}$. Koike alone added
a third universal character for the group $\mathrm{SO}(2n+1,\mathbb{C})$ [19,
Definition 6.4] (see also [25, Equation (3.8)])

$\mathrm{so}_{\lambda}:=\det_{1\leqslant i,j\leqslant
n}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+1}\big{)}=\det_{1\leqslant
i,j\leqslant
m}\big{(}e_{\lambda_{i}^{\prime}-i+j}+e_{\lambda_{i}^{\prime}-i-j+1}\big{)}.$

This universal character is self-dual under $\omega$, so
$\omega\mathrm{so}_{\lambda}=\mathrm{so}_{\lambda^{\prime}}$. For later
convenience we also define a variant of the above as

$\mathrm{so}_{\lambda}^{-}:=\det_{1\leqslant i,j\leqslant
n}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j+1}\big{)}=\det_{1\leqslant
i,j\leqslant
m}\big{(}e_{\lambda_{i}^{\prime}-i+j}-e_{\lambda_{i}^{\prime}-i-j+1}\big{)}.$

If $X=(x_{1},x_{2},x_{3},\dots)$ is a set of variables (which may be finite or
countable) and $-X:=(-x_{1},-x_{2},-x_{3},\dots)$, then

$\mathrm{so}_{\lambda}^{-}(X)=(-1)^{\lvert\lambda\rvert}\mathrm{so}_{\lambda}(-X),$

since the $h_{r}$ and $e_{r}$ are homogeneous of degree $r$.

For $l(\lambda)\leqslant n$, each of the three above universal characters
become actual characters of irreducible representations of their associated
groups when specialised to $(x_{1}^{\pm},\dots,x_{n}^{\pm})$ (hence the name
universal characters).

The irreducible polynomial representations of $\mathrm{GL}(n,\mathbb{C})$ are
indexed by partitions of length at most $n$. On the other hand, the
irreducible rational representations are indexed by weakly decreasing
sequences of integers of length $n$, which are called _staircases_ by
Stembridge [41]. Such sequences are equivalent to pairs of partitions
$\lambda,\mu$ such that $l(\lambda)+l(\mu)\leqslant n$. Given such a pair, one
defines the associated staircase $[\lambda,\mu]$ by
$[\lambda,\mu]_{i}:=\lambda_{i}-\mu_{n-i+1}$ for $1\leqslant i\leqslant n$.
The characters of the rational representations of $\mathrm{GL}(n,\mathbb{C})$
are then given by $s_{[\lambda,\mu]}(x_{1},\dots,x_{n})$ for all staircases
with $n$ entries. Note that (2.5) implies that this object is just a Schur
function up to a power of $x_{1}\cdots x_{n}$. In [28], Littlewood gave the
expansion

(2.9)
$s_{[\lambda,\mu]}(x_{1},\dots,x_{n})=\sum_{\nu}(-1)^{\lvert\nu\rvert}s_{\lambda/\nu}(x_{1},\dots,x_{n})s_{\mu/\nu^{\prime}}(1/x_{1},\dots,1/x_{n}).$

For a pair of partitions $\lambda,\mu$ and sets of indeterminates $X$, $Y$,
this may be used to define the universal character associated to a rational
representation of $\mathrm{GL}(n,\mathbb{C})$ as

(2.10)
$\mathrm{rs}_{\lambda,\mu}(X;Y):=\sum_{\nu}(-1)^{\lvert\nu\rvert}s_{\lambda/\nu}(X)s_{\mu/\nu^{\prime}}(Y).$

Note that the only terms which contribute are those with $\nu\subseteq\lambda$
and $\nu^{\prime}\subseteq\mu$. If we let $\omega_{X}$ and $\omega_{Y}$ denote
the involution $\omega$ acting on the set of variables in its subscript, then

$\displaystyle\omega_{X}\omega_{Y}\mathrm{rs}_{\lambda,\mu}(X;Y)$
$\displaystyle=\sum_{\nu}(-1)^{\lvert\nu\rvert}s_{\lambda^{\prime}/\nu^{\prime}}(X)s_{\mu^{\prime}/\nu}(Y)$
$\displaystyle=\sum_{\nu^{\prime}}(-1)^{\lvert\nu\rvert}s_{\lambda^{\prime}/\nu}(X)s_{\mu^{\prime}/\nu^{\prime}}(Y)$
$\displaystyle=\mathrm{rs}_{\lambda^{\prime},\mu^{\prime}}(X;Y).$

As shown by Koike [18], this object has a Jacobi–Trudi-type expression as a
block matrix

(2.11) $\mathrm{rs}_{\lambda,\mu}(X;Y)=\det_{1\leqslant i,j\leqslant
n+m}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j}(X)\big{)}_{1\leqslant
i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+1}(X)\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n\\\ 1\leqslant j\leqslant m\end{subarray}}\\\
\big{(}h_{\mu_{i}-i-j+1}(Y)\big{)}_{\begin{subarray}{c}1\leqslant i\leqslant
m\\\ 1\leqslant j\leqslant
n\end{subarray}}&\big{(}h_{\mu_{i}-i+j}(Y)\big{)}_{1\leqslant i,j\leqslant
m}\end{pmatrix},$

where $n\geqslant l(\lambda)$ and $m\geqslant l(\mu)$. As for the other
determinants, this is independent of $n$ and $m$ as long as $n\geqslant
l(\lambda)$ and $m\geqslant l(\mu)$. The relation under $\omega_{X}\omega_{Y}$
implies we have the dual form [18, Definition 2.1]

(2.12) $\mathrm{rs}_{\lambda,\mu}(X;Y)=\det_{1\leqslant i,j\leqslant
n+m}\begin{pmatrix}\big{(}e_{\lambda_{i}^{\prime}-i+j}(X)\big{)}_{1\leqslant
i,j\leqslant
n}&\big{(}e_{\lambda_{i}^{\prime}-i-j+1}(X)\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n\\\ 1\leqslant j\leqslant m\end{subarray}}\\\
\big{(}e_{\mu_{i}^{\prime}-i-j+1}(Y)\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant m\\\ 1\leqslant j\leqslant
n\end{subarray}}&\big{(}e_{\mu_{i}^{\prime}-i+j}(Y)\big{)}_{1\leqslant
i,j\leqslant m}\end{pmatrix},$

where $n\geqslant\lambda_{1}$ and $m\geqslant\mu_{1}$. The definition (2.10)
and the determinants (2.11) and (2.12) are related by taking the Laplace
expansion of each determinant according to its presented block structure; see
[18, Equation (2.1)]. Also, by the definition (2.10) and (2.9) it is immediate
that, for $l(\lambda)+l(\mu)\leqslant n$,

$\mathrm{rs}_{\lambda,\mu}(x_{1},\dots,x_{n};1/x_{1},\dots,1/x_{n})=s_{[\lambda,\mu]}(x_{1},\dots,x_{n}).$

We will always take $X=Y$ in $\mathrm{rs}_{\lambda,\mu}(X;Y)$, which we write
as $\mathrm{rs}_{\lambda,\mu}$ in the rest of the paper. In particular we note
that $\mathrm{rs}_{\lambda,\mu}=\mathrm{rs}_{\mu,\lambda}$.

As mentioned in the introduction, the notion of twisting the set of variables
$x_{1},\dots,x_{n}$ by a primitive $t$-th root of unity $\zeta$ is replaced by
the operator $\varphi_{t}$ (1.1), which has been considered by Macdonald [31,
p. 91] and, for $t=2$, by Baik and Rains [6, p. 25]. Let
$X^{t}:=(x_{1}^{t},x_{2}^{t},x_{3}^{t}\dots)$ and denote by $\psi_{t}$ the
homomorphism

$\displaystyle\psi_{t}:\Lambda$ $\displaystyle\longrightarrow\Lambda_{X^{t}}$
$\displaystyle f$ $\displaystyle\longmapsto f(X,\zeta X,\dots,\zeta^{t-1}X).$

Since $\psi_{t}H_{z}(X)=H_{z^{t}}(X^{t})$, both $\varphi_{t}$ and $\psi_{t}$
act on the $h_{r}$ in the same way, i.e., the diagram

(2.13)
${\Lambda}$${\Lambda}$${\Lambda_{X^{t}}}$$\scriptstyle{\varphi_{t}}$$\scriptstyle{\psi_{t}}$$\scriptstyle{X^{t}}$

commutes, where the arrow labelled $X^{t}$ is the substitution map. This
implies the claim of the introduction that the action of $\varphi_{t}$ is
equivalent to twisting the alphabet $X$ by a primitive $t$-th root of unity
$\zeta$. If one wishes to think about this as a map
$\Lambda_{X}\longrightarrow\Lambda_{X}$ where $X$ is some concrete alphabet,
then substitute each $x\in X$ by its set of $t$-th roots $x^{1/t},\zeta
x^{1/t},\dots,\zeta^{t-1}x^{1/t}$ and evaluate this expression. By the action
of this map on the $h_{r}$, such a map gives a symmetric function again in the
variables $X$. Using the generating function $E_{z}(X)$ one may also show that
[24, §5.8]

$\varphi_{t}e_{r}=\begin{cases}(-1)^{r(t-1)/t}e_{r/t}&\text{if $t$ divides
$r$},\\\ 0&\text{otherwise}.\end{cases}$

Therefore $\omega$ and $\varphi_{t}$ commute if $t$ is odd, but not in
general. Proposition 3.5 shows that, in some cases, the maps commute up to a
computable sign. Different, but closely related operators are discussed at the
end of this paper.

##  3\. Summary of results

With the preliminary material of the previous section under our belts, we are
now ready to state our main results regarding factorisations of universal
characters under $\varphi_{t}$. The first of these is the action of the map on
the skew Schur functions.

######  Theorem 3.1.

We have that $\varphi_{t}s_{\lambda/\mu}=0$ unless $\lambda/\mu$ is tileable
by $t$-ribbons, in which case

$\varphi_{t}s_{\lambda/\mu}=\operatorname{sgn}_{t}(\lambda/\mu)\prod_{r=0}^{t-1}s_{\lambda^{(r)}/\mu^{(r)}}.$

For $\mu$ empty, this result is due to Littlewood [27, p. 131], who proves it
by direct manipulation of the ratio of alternants (2.4). By (2.13) with
$X=(x_{1},\dots,x_{n})$, one can recover Littlewood’s result by simply
evaluating the right-hand side of the equation at
$(x_{1}^{t},\dots,x_{n}^{t})$. He also states the $\mu$ empty case of the
theorem in the language of symmetric group characters: see both [27, p. 144]
and [29, p. 340]. The generalisation to skew characters was discovered by
Farahat [11] (see also [9, Theorem 3.3]). The form we state here is precisely
that of Macdonald [31, p. 91]. Curiously, Prasad recently rediscovered the
$\mu$ empty case independently, with a proof identical to Littlewood’s, but in
a more representation-theoretic context [37]. A version of the result for
Schur’s $P$-and $Q$-functions has been given by Mizukawa [32, Theorem 5.1].

Theorem 3.1 has been rediscovered many times for both skew and straight
shapes, and often only in special cases. We make no attempt to give a complete
history, but it appears to us that the theorem deserves to be better known.
The interested reader can consult [44] for some exposition on the character
theory side of this story. On the symmetric functions side, such an exposition
is lacking in the literature.

We now state, in sequence, the three factorisations lifting [5, Theorems 2.11,
2.15 & 2.17] to the level of universal characters, beginning with the
universal orthogonal character.

######  Theorem 3.2.

Let $\lambda$ be a partition of length at most $nt$. Then
$\varphi_{t}\mathrm{o}_{\lambda}=0$ unless $t\textup{-core}(\lambda)$ is
orthogonal, in which case

$\varphi_{t}\mathrm{o}_{\lambda}=(-1)^{\varepsilon^{\mathrm{o}}_{\lambda;nt}}\operatorname{sgn}(w_{t}(\lambda;nt))\mathrm{o}_{\lambda^{(0)}}\prod_{r=1}^{\lfloor(t-1)/2\rfloor}\mathrm{rs}_{\lambda^{(r)},\lambda^{(t-r)}}\times\begin{cases}\mathrm{so}_{\lambda^{(t/2)}}^{-}&\text{$t$
even},\\\ 1&\text{$t$ odd},\end{cases}$

where

$\varepsilon^{\mathrm{o}}_{\lambda;nt}=\sum_{r=\lfloor(t+2)/2\rfloor}^{t-1}\binom{m_{r}(\lambda;nt)+1}{2}+\mathrm{rk}(t\textup{-core}(\lambda))+\begin{cases}\binom{n+1}{2}+n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$
even},\\\ 0&\text{$t$ odd}.\end{cases}$

Our next result is the same factorisation for the symplectic character.

######  Theorem 3.3.

Let $\lambda$ be a partition of length at most $nt$. Then
$\varphi_{t}\mathrm{sp}_{\lambda}=0$ unless $t\textup{-core}(\lambda)$ is
symplectic, in which case

$\varphi_{t}\mathrm{sp}_{\lambda}=(-1)^{\varepsilon^{\mathrm{sp}}_{\lambda;nt}}\operatorname{sgn}(w_{t}(\lambda;nt))\mathrm{sp}_{\lambda^{(t-1)}}\prod_{r=0}^{\lfloor(t-3)/2\rfloor}\mathrm{rs}_{\lambda^{(r)},\lambda^{(t-r-2)}}\times\begin{cases}\mathrm{so}_{\lambda^{((t-2)/2)}}&\text{$t$
even},\\\ 1&\text{$t$ odd},\end{cases}$

where

$\varepsilon^{\mathrm{sp}}_{\lambda;nt}=\sum_{r=\lfloor
t/2\rfloor}^{t-2}\binom{m_{r}(\lambda;nt)+1}{2}+\begin{cases}\binom{n+1}{2}+n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$
even},\\\ 0&\text{$t$ odd}.\end{cases}$

Finally, we can claim a similar factorisation for $\mathrm{so}_{\lambda}$.

######  Theorem 3.4.

Let $\lambda$ be a partition of length at most $nt$. Then
$\varphi_{t}\mathrm{so}_{\lambda}=0$ unless $t\textup{-core}(\lambda)$ is
self-conjugate, in which case

$\varphi_{t}\mathrm{so}_{\lambda}=(-1)^{\varepsilon^{\mathrm{so}}_{\lambda;nt}}\operatorname{sgn}(w_{t}(\lambda;nt))\prod_{r=0}^{\lfloor(t-2)/2\rfloor}\mathrm{rs}_{\lambda^{(r)},\lambda^{(t-r-1)}}\times\begin{cases}1&\text{$t$
even},\\\ \mathrm{so}_{\lambda^{((t-1)/2)}}&\text{$t$ odd},\end{cases}$

where111We have corrected the lower bound in the sum defining
$\varepsilon^{\mathrm{so}}_{\lambda;nt}$ from $\lfloor t/2\rfloor$ in [5,
Theorem 2.17] (there denoted $\epsilon$) to $\lfloor(t+1)/2\rfloor$.

$\varepsilon^{\mathrm{so}}_{\lambda;nt}=\sum_{r=\lfloor(t+1)/2\rfloor}^{t-1}\binom{m_{r}(\lambda;nt)+1}{2}+\begin{cases}0&\text{$t$
even},\\\ n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$ odd}.\end{cases}$

Some remarks are in order. Firstly, the three signs
$\operatorname{sgn}(w_{t}(\lambda;nt))(-1)^{\varepsilon_{\lambda;nt}^{\bullet}}$
are actually independent of $n$ as long as $nt\geqslant l(\lambda)$, a fact
which we prove in Lemma 4.8 below. As remarked by Ayyer and Kumari [5, Remark
2.19], the order of the quotient is unchanged upon replacing $n\mapsto n+1$,
so the product in the evaluation is independent of $n$. It is in principle
possible to carry out our proof technique below under the assumption that
$l(\lambda)$ is bounded by an arbitrary integer, say $k$, where $k$ is not
necessarily a multiple of $t$. In this case the evaluation is of course the
same, however the sign will be expressed differently and the $t$-quotients in
the evaluations will be a cyclic permutation of the ones presented. Since the
proof is simplest when this $k$ is a multiple of $t$, we stick to this case.

To obtain the theorems of Ayyer and Kumari one evaluates the right-hand side
of each identity at the set of variables $(x_{1}^{\pm t},\dots,x_{n}^{\pm
t})$. Using (2.5) and the definition of $\mathrm{rs}_{\lambda,\mu}$ it follows
that in this case the rational universal characters occurring in each
evaluation agree with the Schur functions $s_{\mu^{(k)}_{i}}(x_{1}^{\pm
t},\dots,x_{n}^{\pm t})$ in the notation of [5].

As we have already seen the maps $\omega$ and $\varphi_{t}$ do not commute in
general. However, when the symmetric function is one of
$s_{\lambda/\mu},\mathrm{o}_{\lambda},\mathrm{sp}_{\lambda},\mathrm{so}_{\lambda}$,
the maps commute up to a sign which can be computed.

######  Proposition 3.5.

We have the relationship

$\omega\varphi_{t}s_{\lambda/\mu}=(-1)^{(t-1)(\lvert\lambda^{(0)}/\mu^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}/\mu^{(t-1)}\rvert)}\varphi_{t}\omega
s_{\lambda/\mu}.$

Moreover,

$\omega\varphi_{t}\bullet_{\lambda}=(-1)^{(t-1)(\lvert\lambda^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}\rvert)}\varphi_{t}\omega\bullet_{\lambda}$

for $\bullet\in\\{\mathrm{o},\mathrm{sp},\mathrm{so}\\}$.

##  4\. Auxiliary results

The purpose of this section is to collect all the small facts about beta sets
and the signs (2.2) which we need to prove our main results. To begin, we
relate the bead configurations of a partition and its conjugate.

######  Lemma 4.1.

Let $\lambda$ be a partition of length at most $nt$ such that
$\lambda_{1}\leqslant mt$. Then the bead configuration for
$\beta(\lambda^{\prime};mt)$ can be obtained from the bead configuration for
$\beta(\lambda;nt)$ with $n+m$ rows by rotating the picture by $180^{\circ}$
and then interchanging beads and spaces.

###### Proof.

This is a consequence of the fact [31, p. 3] that for $l(\lambda)\leqslant n$
and $\lambda_{1}\leqslant m$,

$\\{0,1,\dots,m+n-1\\}=\\{\lambda_{i}+n-i:1\leqslant i\leqslant
n\\}\sqcup\\{m+n-1-(\lambda_{j}^{\prime}+m-j):1\leqslant j\leqslant m\\},$

where $\sqcup$ denotes a disjoint union. ∎

This lemma immediately implies the following relationship between the $t$-core
and $t$-quotient of $\lambda$ and $\lambda^{\prime}$.

######  Corollary 4.2.

For a partition $\lambda$ we have
$t\textup{-core}(\lambda^{\prime})=t\textup{-core}(\lambda)^{\prime}$ and the
$t$-quotient of $\lambda^{\prime}$ is
$\big{(}(\lambda^{(t-1)})^{\prime},\dots,(\lambda^{(0)})^{\prime}\big{)}$.

The next pair of lemmas are due to Ayyer and Kumari, the first of which
characterises partitions with $z$-asymmetric $t$-cores in terms of their beta
sets [5, Lemma 3.6].

######  Lemma 4.3.

For a partition $\lambda$ of length at most $nt$, $t\textup{-core}(\lambda)$
is of the form $(a\mid a+z)$ for some integer $-1\leqslant z\leqslant t-1$ if
and only if

(4.1a) $\displaystyle m_{r}(\lambda,nt)+m_{t-r-z-1}(\lambda,nt)$
$\displaystyle=2n\quad\text{for $0\leqslant r\leqslant t-z-1$},$ (4.1b)
$\displaystyle m_{r}(\lambda,nt)$ $\displaystyle=n\quad\text{for $t-z\leqslant
r\leqslant t-1$},$

where the indices of the $m_{r}$ are taken modulo $t$.

The second lemma of Ayyer and Kumari we need is [5, Lemma 3.13], which is used
later on to simplify signs.

######  Lemma 4.4.

Let $\lambda$ be a partition of length at most $nt$. If
$t\textup{-core}(\lambda)$ is orthogonal, then

(4.2)
$\displaystyle\mathrm{rk}(t\textup{-core}(\lambda))=\sum_{r=1}^{\lfloor(t-1)/2\rfloor}\lvert
m_{r}(\lambda;nt)-n\rvert$
$\displaystyle=\sum_{r=\lfloor(t+2)/2\rfloor}^{t-1}\lvert
m_{r}(\lambda;nt)-n\rvert$ If $t\textup{-core}(\lambda)$ is symplectic, then
(4.3)
$\displaystyle\mathrm{rk}(t\textup{-core}(\lambda))=\sum_{r=0}^{\lfloor(t-3)/2\rfloor}\lvert
m_{r}(\lambda;nt)-n\rvert$ $\displaystyle=\sum_{r=\lfloor
t/2\rfloor}^{t-2}\lvert m_{r}(\lambda;nt)-n\rvert$ If
$t\textup{-core}(\lambda)$ is self-conjugate, then (4.4)
$\displaystyle\mathrm{rk}(t\textup{-core}(\lambda))=\sum_{r=0}^{\lfloor(t-2)/2\rfloor}\lvert
m_{r}(\lambda;nt)-n\rvert$
$\displaystyle=\sum_{r=\lfloor(t+1)/2\rfloor}^{t-1}\lvert
m_{r}(\lambda;nt)-n\rvert$

Next, we show that the sign of a tileable skew shape can be expressed in terms
of the signs of the permutations $w_{t}(\lambda;n)$.

######  Lemma 4.5.

For $\lambda/\mu$ $t$-tileable and any integer $n$ such that $n\geqslant
l(\lambda)$,

$\operatorname{sgn}_{t}(\lambda/\mu)=\operatorname{sgn}(w_{t}(\lambda;n))\operatorname{sgn}(w_{t}(\mu;n)).$

###### Proof.

Since $\lambda/\mu$ is $t$-tileable, it has a ribbon decomposition
$D=(\nu^{(0)},\dots,\nu^{(k)})$ where $\nu^{(0)}=\mu$ and $\nu^{(k)}=\lambda$.
Also, $\nu^{(k-1)}$ can be obtained from $\lambda$ by moving one bead at some
position upward one space. By our characterisation of the inversions in the
permutation $w_{t}(\lambda;n)$, we see that moving a bead at position $\ell$
up one space changes the sign by $(-1)^{b_{k}}$ where $b_{k}$ is the number of
beads at positions between $\ell-t$ and $\ell$. In other words,
$\operatorname{sgn}(w_{t}(\lambda;n))=(-1)^{b_{k}}\operatorname{sgn}(w_{t}(\nu^{(k-1)};n))$.
Moreover, $b_{k}=\mathrm{ht}(\nu^{(k)}/\nu^{(k-1)})$, so that

$\operatorname{sgn}(w_{t}(\lambda;n))\operatorname{sgn}(w_{t}(\mu;n))=(-1)^{\sum_{i=1}^{k}b_{i}}=(-1)^{\mathrm{ht}(D)}=\operatorname{sgn}_{t}(\lambda/\mu).\qed$

We also have the following useful relationship between the sign of
$\lambda/\mu$ and $\lambda^{\prime}/\mu^{\prime}$.

######  Lemma 4.6.

For $\lambda/\mu$ $t$-tileable,

$\operatorname{sgn}_{t}(\lambda/\mu)\operatorname{sgn}_{t}(\lambda^{\prime}/\mu^{\prime})=(-1)^{(t-1)(\lvert\lambda^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}\rvert-\lvert\mu^{(0)}\rvert-\cdots-\lvert\mu^{(t-1)}\rvert)}.$

###### Proof.

To prove the claim of the lemma we will proceed by induction on
$\lvert\lambda/\mu\rvert$. If $\lvert\lambda/\mu\rvert=0$ then $\lambda=\mu$
and the equation is trivial. Now fix $\mu$ and assume the result holds for
$\lambda/\mu$ being $t$-tileable. Adding a $t$-ribbon to $\lambda/\mu$ moves
one of the beads, say at position $b$, in the bead configuration for $\lambda$
down a single space. The change in the number of inversions in
$w_{t}(\lambda;nt)$ is the number of beads $b^{\prime}$ such that
$b<b^{\prime}<b+1$. A consequence of Lemma 4.1 is that
$w_{t}(\lambda^{\prime};mt)$ will change by the number of empty spaces between
$b$ and $b+1$. There are $t-1$ spaces and beads between $b$ and $b+1$, so the
left-hand side changes by $(-1)^{t-1}$ when adding a $t$-ribbon. But adding a
$t$-ribbon to $\lambda/\mu$ changes some element of the $t$-quotient of
$\lambda$ by a single box, also corresponding to a change in sign of
$(-1)^{t-1}$. ∎

There is another sign relation between orthogonal and symplectic $t$-cores,
but this time using the permutations $w_{t}$.

######  Lemma 4.7.

Let $\lambda$ be an orthogonal or symplectic $t$-core whose diagram is
contained in an $nt\times nt$ square. Then

$\operatorname{sgn}(w_{t}(\lambda;nt))\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))=(-1)^{\mathrm{rk}(\lambda)}.$

###### Proof.

Assume that $\lambda$ is a non-empty, orthogonal $t$-core (if $\lambda$ is
empty the result is trivial) and fix $n$ so that the condition of the theorem
holds. The key observation is that for an orthogonal $t$-core, the bead
configuration of $\lambda^{\prime}$ with $nt$ beads can be obtained from the
bead configuration of $\lambda$ with $nt$ beads by reducing the labels by $1$
modulo $t$. For example if $\lambda=(12,7,5,3,2,2,1,1,1,1,1)$ then
$\lambda^{\prime}=(11,6,4,3,3,2,2,1,1,1,1,1)$ and their bead configurations
for $t=6$ and $n=2$ are

and

respectively, where we have suppressed the labels. This is a consequence of
Lemma 4.3 with $z=\pm 1$ and Lemma 4.1. When passing from $\lambda$ to
$\lambda^{\prime}$, the inversions contributed by the beads in the first
runner are removed and replaced by additional inversions associated to the
remaining beads in the first $n$ rows. Modulo two, this is equivalent to each
bead in the zeroth runner now seeing all of the beads in the same row twice,
plus all other beads in the other runners once. Let $b$ be the number of beads
in the first $n$ rows of the runners from $1$ to $t-1$ in the bead
configuration of $\lambda$. Then the sign change is

$\operatorname{sgn}(w_{t}(\lambda;nt))=\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))(-1)^{n^{2}(t-1)+b}.$

Since $\lambda$ is an orthogonal $t$-core,
$n^{2}(t-1)+b\equiv\mathrm{rk}(\lambda)\ (\mathrm{mod}\ 2)$ by (4.2) and Lemma
4.3 with $z=-1$. ∎

The next lemma proves the claim made after Theorems 3.2–3.4 that the signs
occurring in those factorisations are independent of $n$.

######  Lemma 4.8.

The signs
$(-1)^{\varepsilon_{\lambda;nt}^{\bullet}}\operatorname{sgn}(w_{t}(\lambda;nt))$
for $\bullet\in\\{\mathrm{o},\mathrm{sp},\mathrm{so}\\}$ are independent of
$n$ as long as $nt\geqslant l(\lambda)$.

###### Proof.

Assume that $nt\geqslant l(\lambda)$. Incrementing $n$ by one adds a row of
beads to the top of the bead configuration of $\lambda$, and so
$m_{r}(\lambda;(n+1)t)=m_{r}(\lambda;nt)+1$. In the inversion count, the $r$th
bead in the new first row sees

$\sum_{k=r+1}^{t-1}(m_{k}(\lambda;nt)+1)$

other beads. Summing over $k=0,\dots,t-1$ we see that

$\operatorname{sgn}(w_{t}(\lambda;(n+1)t))=\operatorname{sgn}(w_{t}(\lambda;nt))(-1)^{\sum_{r=1}^{\lfloor
t/2\rfloor}(m_{2r-1}(\lambda;nt)+1)}.$

Now assume that $\lambda$ has an orthogonal $t$-core. Then by Lemma 4.3 with
$z=-1$ the above has the same parity as

$\sum_{r=1}^{\lfloor
t/2\rfloor}(m_{2r-1}(\lambda;nt)+1)\equiv\begin{cases}\frac{(n+1)t}{2}&\text{$t$
even,}\\\\[5.69054pt]
\frac{t-1}{2}+\sum_{r=1}^{(t-1)/2}m_{r}(\lambda;nt)&\text{$t$
odd},\end{cases}\ (\mathrm{mod}\ 2).$

A short calculation shows that

$\displaystyle\varepsilon_{\lambda;(n+1)t}^{\mathrm{o}}$
$\displaystyle=\varepsilon_{\lambda;nt}^{\mathrm{o}}+\sum_{r=1}^{\lfloor(t-1)/2\rfloor}m_{r}(\lambda;nt)+\begin{cases}n+\frac{t}{2}+\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$
even},\\\\[2.84526pt] \frac{t-1}{2}&\text{$t$ odd},\end{cases}$
$\displaystyle\equiv\varepsilon_{\lambda;nt}^{\mathrm{o}}+\begin{cases}\frac{(n+1)t}{2}&\text{$t$
even},\\\\[2.84526pt]
\frac{t-1}{2}+\sum_{r=1}^{(t-1)/2}m_{r}(\lambda;nt)&\text{$t$ odd}\end{cases}\
(\mathrm{mod}\ 2).$

where the last equality uses (4.2). The remaining two cases follow similarly.
∎

We conclude this section with a small lemma relating the indices in the
Jacobi–Trudi determinants with partition quotients.

######  Lemma 4.9.

Let $\lambda,\mu$ be partitions of length at most $nt$ and assume that for
$0\leqslant r,s\leqslant t-1$ we have $\lambda_{i}-i\equiv r\ (\mathrm{mod}\
t)$, $\mu_{j}-j\equiv s\ (\mathrm{mod}\ t)$ for $1\leqslant i,j\leqslant nt$.
If $r-s+z\equiv 0\ (\mathrm{mod}\ t)$ for some $z\in\mathbb{Z}$, then

$\frac{\lambda_{i}-\mu_{j}+j-i+z}{t}=\lambda^{(r)}_{k}-\mu^{(s)}_{\ell}-k+\ell+m_{r}(\lambda;nt)-m_{s}(\mu;nt)+(r-s+z)/t,$

for some $k,\ell$ such that $1\leqslant k\leqslant m_{r}(\lambda;nt)$ and
$1\leqslant\ell\leqslant m_{s}(\mu;nt)$. Alternatively, if $r+s+z\equiv 0\
(\mathrm{mod}\ t)$ then

$\frac{\lambda_{i}+\mu_{j}-i-j+z}{t}=\lambda^{(r)}_{k}+\mu^{(s)}_{\ell}-k-\ell-2n+1+m_{r}(\lambda;nt)+m_{s}(\mu;nt)+(r+s+z)/t$

for some $k,\ell$ such that $1\leqslant k\leqslant m_{r}(\lambda;nt)$ and
$1\leqslant\ell\leqslant m_{s}(\mu;nt)$.

###### Proof.

We first write $\lambda_{i}+nt-i=\xi^{(r)}_{k}t+r$ and
$\mu_{j}+nt-j=\pi_{\ell}^{(s)}t+s$ for $1\leqslant k\leqslant
m_{r}(\lambda;nt)$ and $1\leqslant\ell\leqslant m_{s}(\mu;nt)$. Then

$\displaystyle\frac{\lambda_{i}-\mu_{j}-i+j+z}{t}$
$\displaystyle=\xi^{(r)}_{k}+\pi_{\ell}^{(s)}+(r-s+z)/t$
$\displaystyle=\lambda^{(r)}_{k}+\mu^{(s)}_{\ell}-k+\ell+m_{r}(\lambda;nt)-m_{s}(\mu;nt)+(r-s+z)/t,$

by the definition of the $t$-quotient. The second claim is analogous. ∎

##  5\. Proofs of theorems

In this section we provide proofs of Theorems 3.2, 3.3 and 3.4. Since our
proof strategy follows that of Macdonald’s proof of the skew Schur case [31,
p. 91] (Theorem 3.1 above), we reproduce this proof in detail as preparation
for what remains. We also give a detailed example in the orthogonal case in
Section 5.2 to further elucidate the structure of the remaining proofs.

###  5.1. Proof of Theorem 3.1

Let $n$ be a nonnegative integer and $\mu\subseteq\lambda$ be a pair of
partitions such that $l(\lambda)\leqslant nt$. Consider the Jacobi–Trudi
determinant

$s_{\lambda/\mu}=\det_{1\leqslant i,j\leqslant
nt}(h_{\lambda_{i}-\mu_{j}-i+j}).$

Before applying the map $\varphi_{t}$, we rearrange the rows and columns of
this determinant by the permutations $w_{t}(\lambda;nt)$ and $w_{t}(\mu;nt)$
respectively. By Lemma 4.5 this introduces a sign of
$\operatorname{sgn}_{t}(\lambda/\mu)$. The rows and columns are now arranged
in such a way that the residue classes of $\lambda_{i}-i$ and $\mu_{j}-j$ are
grouped in ascending order, and the values within each class are decreasing.
From this vantage point it is easy to apply the map $\varphi_{t}$ since
$\varphi_{t}h_{\lambda_{i}-\mu_{j}-i+j}$ vanishes unless
$\lambda_{i}-i\equiv\mu_{j}-j\ (\mathrm{mod}\ t)$. Therefore,
$\varphi_{t}s_{\lambda/\mu}$ has a block-diagonal structure, with each block
having size $m_{r}(\lambda;nt)\times m_{r}(\mu;nt)$ for $0\leqslant r\leqslant
t-1$. We conclude that $\varphi_{t}s_{\lambda/\mu}=0$ unless
$m_{r}(\lambda;nt)=m_{r}(\mu;nt)$ for all $0\leqslant r\leqslant t-1$.
Assuming this is the case, then the entries of the of the minor corresponding
to the residue class $r$ are given by Lemma 4.9, and are

$h_{(\lambda_{i}-\mu_{j}-i+j)/t}=h_{\lambda^{(r)}_{k}-\mu^{(r)}_{\ell}-k+\ell}$

for some $k$ and $\ell$ with $1\leqslant k,\ell\leqslant n$. Note that the
rows and columns are in the desired order (i.e., in each $n\times n$ minor the
indices increase from $1$ to $n$) thanks to the permutations we applied at the
beginning of the proof. We have therefore shown that if
$m_{r}(\lambda;nt)=m_{r}(\mu;nt)$ for all $0\leqslant r\leqslant t-1$, then

$\varphi_{t}s_{\lambda/\mu}=\operatorname{sgn}_{t}(\lambda/\mu)\prod_{r=0}^{t-1}s_{\lambda^{(r)}/\mu^{(r)}}.$

Now, if $\mu^{(r)}\not\subseteq\lambda^{(r)}$ for any $r$ such that
$0\leqslant r\leqslant t-1$ this expression will give zero, from which we
conclude, by Lemma 2.1, that $\varphi_{t}s_{\lambda/\mu}=0$ unless
$\lambda/\mu$ is $t$-tileable.

###  5.2. An example

The structure of the remaining proofs is best outlined through a detailed
example. To this end, let $t=4$ and $\lambda=(12,12,12,8,8,8,7,7,3,3,2)$. We
therefore have that $4\text{-core}(\lambda)=(4,1,1)$, which is clearly
orthogonal, and

$\big{(}\lambda^{(0)},\lambda^{(1)},\lambda^{(2)},\lambda^{(3)}\big{)}=\big{(}(2,2),(4,1),(3,2,1),(2,1,1)\big{)}.$

Now choose $n=3$, so that $nt=12\geqslant l(\lambda)$. Using the definition of
$\mathrm{o}_{\lambda}$ as a Jacobi–Trudi-type determinant (2.6) we immediately
see that

$\varphi_{4}\mathrm{o}_{\lambda}=\setcounter{MaxMatrixCols}{12}\begin{vmatrix}h_{3}&\cdot&-h_{2}&\cdot&h_{4}&\cdot&-h_{1}&\cdot&h_{5}&\cdot&-1&\cdot\\\
\cdot&h_{3}-h_{2}&\cdot&\cdot&\cdot&h_{4}-h_{1}&\cdot&\cdot&\cdot&h_{5}-1&\cdot&\cdot\\\
-h_{2}&\cdot&h_{3}&\cdot&-h_{1}&\cdot&h_{4}&\cdot&-1&\cdot&h_{5}&\cdot\\\
\cdot&\cdot&\cdot&h_{2}-1&\cdot&\cdot&\cdot&h_{3}&\cdot&\cdot&\cdot&h_{4}\\\
h_{1}&\cdot&-1&\cdot&h_{2}&\cdot&\cdot&\cdot&h_{3}&\cdot&\cdot&\cdot\\\
\cdot&h_{1}-1&\cdot&\cdot&\cdot&h_{2}&\cdot&\cdot&\cdot&h_{3}&\cdot&\cdot\\\
\cdot&\cdot&\cdot&h_{1}&\cdot&\cdot&\cdot&h_{2}&\cdot&\cdot&\cdot&h_{3}\\\
1&\cdot&\cdot&\cdot&h_{1}&\cdot&\cdot&\cdot&h_{2}&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&h_{1}&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot&\cdot&h_{1}&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1\\\
\end{vmatrix},$

where we write $\cdot$ in place of $0$ to avoid clutter. The next step is to
permute the rows and columns of the matrix according to the permutations
$w_{4}(\lambda;12)$ and $w_{4}(0;12)$, respectively. In this case, the first
permutation is odd and the second even, so we are left with

$\varphi_{4}\mathrm{o}_{\lambda}=-\setcounter{MaxMatrixCols}{12}\begin{vmatrix}h_{2}-1&h_{3}&h_{4}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
h_{1}&h_{2}&h_{3}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&1&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&h_{3}&h_{4}&h_{5}&\cdot&\cdot&\cdot&-h_{2}&-h_{1}&-1\\\
\cdot&\cdot&\cdot&\cdot&1&h_{1}&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&h_{3}-h_{2}&h_{4}-h_{1}&h_{5}-1&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&h_{1}-1&h_{2}&h_{3}&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&h_{1}&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&-h_{2}&-h_{1}&-1&\cdot&\cdot&\cdot&h_{3}&h_{4}&h_{5}\\\
\cdot&\cdot&\cdot&-1&\cdot&\cdot&\cdot&\cdot&\cdot&h_{1}&h_{2}&h_{3}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1&h_{1}&h_{2}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&1\end{vmatrix}.$

The top-left $3\times 3$ minor and central $3\times 3$ minor occupying rows
6–8 and columns 7–9 are clearly equal to $\mathrm{o}_{(2,2)}$ and
$\mathrm{so}_{(3,2,1)}^{-}$, respectively. One way to isolate the copy of
$\mathrm{so}_{(3,2,1)}^{-}$ is to push it so that it is the bottom-right
$3\times 3$ submatrix, while preserving the order of the other rows and
columns. In this case such a procedure will introduce a sign of $-1$. Putting
this together, we have shown that

$\varphi_{4}\mathrm{o}_{\lambda}=\mathrm{o}_{(2,2)}\mathrm{so}_{(3,2,1)}^{-}\begin{vmatrix}h_{3}&h_{4}&h_{5}&-h_{2}&-h_{1}&-1\\\
\cdot&1&h_{1}&\cdot&\cdot&\cdot\\\ -h_{2}&-h_{1}&-1&h_{3}&h_{4}&h_{5}\\\
-1&\cdot&\cdot&h_{1}&h_{2}&h_{3}\\\ \cdot&\cdot&\cdot&1&h_{1}&h_{2}\\\
\cdot&\cdot&\cdot&\cdot&\cdot&1\\\ \end{vmatrix}.$

Our goal is to show that this final unidentified determinant is equal to
$\mathrm{rs}_{(4,1),(2,1,1)}$. Clearly the extra signs can be cleared by
multiplying the first two rows and first three columns by $-1$ each,
generating an overall sign of $-1$. Then one need only push the first column
past the second and third, which does not change the sign, and the resulting
determinant is precisely a copy of $\mathrm{rs}_{(4,1),(2,1,1)}$. Thus,

$\varphi_{4}\mathrm{o}_{\lambda}=-\mathrm{o}_{(2,2)}\mathrm{so}_{(3,2,1)}^{-}\mathrm{rs}_{(4,1),(2,1,1)}.$

Note that $(-1)^{\varepsilon_{\lambda;12}^{\mathrm{o}}}=1$ so the overall sign
clearly agrees with Theorem 3.2. In the next sections we show that, with a
little extra work, this argument also works in general for the universal
characters $\mathrm{o}_{\lambda},\mathrm{sp}_{\lambda}$ and
$\mathrm{so}_{\lambda}$.

###  5.3. Proof of Theorem 3.2

Let $\lambda$ be a partition such that $l(\lambda)\leqslant nt$ and consider
the definition (2.6) of $\mathrm{o}_{\lambda}$

$\mathrm{o}_{\lambda}=\det_{1\leqslant i,j\leqslant
nt}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j}\big{)}.$

We permute the rows and columns by $w_{t}(\lambda;nt)$ and $w_{t}(0;nt)$
respectively, which introduces a sign of

(5.1)
$(-1)^{\binom{n+1}{2}\binom{t}{2}}\operatorname{sgn}(w_{t}(\lambda;nt)).$

The modular behaviour of the indices of each row is now known. There are three
possibilities for the entries of $\varphi_{t}\mathrm{o}_{\lambda}$: both $h$’s
may survive, one $h$ may survive, or the entry is necessarily zero. For both
to survive, we see that $h_{\lambda_{i}-i+j}$ and $h_{\lambda_{i}-i-j}$ are
nonzero under $\varphi_{t}$ if and only if $\lambda_{i}-i\equiv-j\equiv 0\
(\mathrm{mod}\ t)$ or, if $t$ is even, $\lambda_{i}-i\equiv-j\equiv t/2\
(\mathrm{mod}\ t)$. In the first instance, by Lemma 4.9,

$\varphi_{t}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j}\big{)}=h_{\lambda^{(0)}_{k}-k+\ell+m_{0}(\lambda;nt)-n}-h_{\lambda^{(0)}_{k}-k-\ell+m_{0}(\lambda;nt)-n},$

where $1\leqslant k\leqslant m_{0}(\lambda;nt)$ and $1\leqslant\ell\leqslant
n$. Moreover, all other entries in the first $m_{0}(\lambda;nt)$ rows and $n$
columns are zero. If $t$ is even then we also find a submatrix of size
$m_{t/2}(\lambda;nt)\times n$ in the rows
$1+\sum_{r=0}^{(t-2)/2}m_{r}(\lambda;nt)$ to
$\sum_{r=0}^{t/2}m_{r}(\lambda;nt)$ and columns $1+nt/2$ to $n(t+2)/2$. The
entries of this submatrix are

$\varphi_{t}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j}\big{)}=h_{\lambda^{(t/2)}_{k}-k+\ell+m_{t/2}(\lambda;nt)-n}-h_{\lambda^{(t/2)}_{k}-k-\ell+m_{t/2}(\lambda;nt)-n+1},$

where $1\leqslant k\leqslant m_{t/2}(\lambda;nt)$ and $1\leqslant\ell\leqslant
n$. Again, all other entries in these rows and columns are necessarily zero
under $\varphi_{t}$. Given a row corresponding to the residue class $r$ where
$1\leqslant r\leqslant\lfloor(t-1)/2\rfloor$, there are two possibilities for
the entry to potentially survive: the column corresponds to the residue class
$r$ or $t-r$. Again, by Lemma 4.9,

$\varphi_{t}\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j}\big{)}=\begin{cases}h_{\lambda^{(r)}_{k}-k+\ell+m_{r}(\lambda;nt)-n}&\text{if
$j\equiv-r\ (\mathrm{mod}\ t)$},\\\
-h_{\lambda^{(r)}_{k}-k-\ell+m_{r}(\lambda;nt)-n+1}&\text{if $j\equiv r\
(\mathrm{mod}\ t)$}.\\\ \end{cases}$

The set of indices of the complete homogeneous symmetric functions in such a
row are

(5.2a) $\displaystyle\big{\\{}\lambda^{(r)}_{k}-k-\ell+m_{r}(\lambda,nt)+1\mid
1\leqslant\ell\leqslant 2n\big{\\}}$
$\displaystyle\,=\big{\\{}\lambda^{(r)}_{k}-k+\ell\mid 1\leqslant\ell\leqslant
m_{r}(\lambda,nt)\big{\\}}\sqcup\big{\\{}\lambda^{(r)}_{k}-k-\ell+1\mid
1\leqslant\ell\leqslant m_{t-r}(\lambda,nt)\big{\\}}.$ If we look at the
complementary row corresponding to $t-r$, then a similar computation shows
that the indices are (5.2b)
$\displaystyle\big{\\{}\lambda^{(t-r)}_{k}-k-\ell+m_{t-r}+1\mid
1\leqslant\ell\leqslant 2n\big{\\}}$
$\displaystyle\,=\big{\\{}\lambda^{(t-r)}_{k}-k+\ell\mid
1\leqslant\ell\leqslant
m_{t-r}(\lambda,nt)\big{\\}}\sqcup\big{\\{}\lambda^{(t-r)}_{k}-k-\ell+1\mid
1\leqslant\ell\leqslant m_{r}(\lambda,nt)\big{\\}}.$

We have now identified the entries which do not necessarily vanish under
$\varphi_{t}$. These can be rearranged into a block-diagonal matrix. If $t$ is
even, we move the submatrix corresponding to $t/2$ to the bottom-right
$m_{t-1}(\lambda;nt)$ rows and $n$ columns, which picks up a sign of

$(-1)^{m_{t/2}(\lambda;nt)\sum_{r=(t+2)/2}^{t-1}m_{r}(\lambda;nt)+n^{2}(t-2)/2}.$

We then group the rows and columns corresponding to the residue classes $r$
and $t-r$ together with $0\leqslant r\leqslant\lfloor(t-1)/2\rfloor$
increasing. The determinant is now block-diagonal and the blocks have
dimension $m_{0}(\lambda;nt)\times n$,
$(m_{r}(\lambda;nt)+m_{t-r}(\lambda;nt))\times 2n$ for $1\leqslant
r\leqslant\lfloor(t-1)/2\rfloor$ and, if $t$ is even,
$m_{t/2}(\lambda;nt)\times n$. Since the determinant of a block-diagonal
matrix vanishes if one of the blocks is not a square, we can therefore
conclude that $\varphi_{t}\mathrm{o}_{\lambda}$ vanishes unless the conditions
(4.1) with $z=-1$ hold in Lemma 4.3, i.e., unless $t\textup{-core}(\lambda)$
is orthogonal. In this case the top-left $n\times n$ minor is equal to
$\mathrm{o}_{\lambda^{(0)}}$ and if $t$ is even the bottom-right minor
corresponds to $\mathrm{so}_{\lambda^{(t/2)}}^{-}$. Note that in this case the
grouping of the $2n\times 2n$ minors does not change the sign of the
determinant since each row and column is pushed past an even number of rows or
columns. For each $1\leqslant r\leqslant\lfloor(t-1)/2\rfloor$ these final
minors are of the form

$\begin{pmatrix}h_{\lambda^{(r)}_{1}+m_{r}-n}&\cdots&h_{\lambda^{(r)}_{1}+m_{r}-1}&-h_{\lambda^{(r)}_{1}+m_{r}-n-1}&\cdots&-h_{\lambda^{(r)}_{1}+m_{r}-2n}\\\
\vdots&&\vdots&\vdots&&\vdots\\\
h_{\lambda^{(r)}_{m_{r}}+1-n}&\cdots&h_{\lambda^{(r)}_{m_{r}}}&-h_{\lambda^{(r)}_{m_{r}}-n}&\cdots&-h_{\lambda^{(r)}_{m_{r}}-2n+1}\\\
-h_{\lambda^{(t-r)}_{1}+m_{t-r}-n-1}&\cdots&-h_{\lambda^{(t-r)}_{1}+m_{t-r}-2n}&h_{\lambda^{(t-r)}_{1}+m_{t-r}-n}&\cdots&h_{\lambda^{(t-r)}_{1}+m_{t-r}-1}\\\
\vdots&&\vdots&\vdots&&\vdots\\\
-h_{\lambda^{(t-r)}_{m_{t-r}}-n}&\cdots&-h_{\lambda^{(t-r)}_{m_{t-r}}-2n+1}&h_{\lambda^{(t-r)}_{m_{t-r}}+1-n}&\cdots&h_{\lambda^{(t-r)}_{m_{t-r}}}\end{pmatrix},$

where we write $m_{r}=m_{r}(\lambda;nt)$. Clearing the negatives in this minor
produces the sign $(-1)^{m_{r}(\lambda;nt)+n}$. If
$m_{r}(\lambda;nt)=m_{t-r}(\lambda;nt)=n$ then we are done. If
$m_{r}(\lambda;nt)>n$ then we need to move the columns $n+1$ to
$m_{r}(\lambda;nt)$ so they are the first $m_{r}(\lambda;nt)-n$ columns, and
then reverse the order. This gives a sign of

$(-1)^{n(m_{r}(\lambda;nt)-n)+\binom{m_{r}(\lambda;nt)-n}{2}}=(-1)^{\binom{m_{r}(\lambda;nt)}{2}-\binom{n}{2}}.$

If $m_{r}(\lambda;nt)<n$ then we need to push the $m_{t-r}(\lambda;nt)-n$
missing rows past the $n-m_{r}$ rows to their right and then reverse again,
giving the same sign

$(-1)^{(m_{t-r}(\lambda;nt)-n)m_{r}(\lambda;nt)+\binom{m_{t-r}(\lambda;nt)-n}{2}}=(-1)^{\binom{m_{r}(\lambda;nt)}{2}-\binom{n}{2}},$

since $m_{t-r}(\lambda;nt)-n=n-m_{r}(\lambda;nt)$. In each of the three cases
the resulting determinant is equal to
$\mathrm{rs}_{\lambda^{(r)},\lambda^{(t-r)}}$. Collecting all of the above
determinant manipulations, the value of
$\varepsilon^{\mathrm{o}}_{\lambda;nt}$ is

$\frac{(n+1)nt(t-1)}{4}+\sum_{r=1}^{\lfloor(t-1)/2\rfloor}\bigg{(}\binom{m_{r}+1}{2}+\binom{n+1}{2}\bigg{)}\\\
+\begin{cases}n\sum_{r=1}^{(t-2)/2}m_{t-r}+\frac{n^{2}(t-2)}{2}&\text{$t$
even},\\\ 0&\text{$t$ odd}.\end{cases}$

To see that this agrees with the sign of Ayyer and Kumari, we use (4.2)
together with the fact that for odd $t$, $(t+1)(t-1)n(n+1)/4$ is even and for
even $t$, $n(n+1)(t^{2}-2)/4$ has the same parity as $n(n+1)/2$. The above
exponent therefore has the same parity as

$\varepsilon^{\mathrm{o}}_{\lambda;nt}=\sum_{r=\lfloor(t+2)/2\rfloor}^{t-1}\binom{m_{r}(\lambda;nt)+1}{2}+\mathrm{rk}(t\textup{-core}(\lambda))+\begin{cases}\binom{n+1}{2}+n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$
even},\\\ 0&\text{$t$ odd}.\end{cases}$

This completes the proof.

###  5.4. Proof of Theorem 3.3

It is of course possible to prove Theorem 3.3 by direct manipulation of the
$h$ Jacobi–Trudi-type formula for $\mathrm{sp}_{\lambda}$ (2.7). However, it
will be more insightful to begin with the $e$ Jacobi–Trudi-type formula (2.8)

$\mathrm{sp}_{\lambda}=\det_{1\leqslant i,j\leqslant
nt}\big{(}e_{\lambda_{i}^{\prime}-i+j}-e_{\lambda_{i}^{\prime}-i-j}\big{)},$

where we assume that $n$ is an integer such that $nt\geqslant\lambda_{1}$. We
further assume that $nt\geqslant l(\lambda)$, since, in the end, our sign will
be independent of $n$. The values of $\varphi_{t}h_{r}$ and $\varphi_{t}e_{r}$
differ by a sign of $(-1)^{(t-1)r/t}$, and the indices of the $e$’s in this
formula are the same as the $h$’s in the formula for
$\mathrm{o}_{\lambda^{\prime}}$ ((2.6) with $\lambda\mapsto\lambda^{\prime}$),
so we can simply replace each $h$ by a signed $e$ in the previous proof.
Moreover, by Corollary 4.2, we know that the $t$-quotient of
$\lambda^{\prime}$ is simply the reverse of the $t$-quotient of $\lambda$. We
can therefore already claim that $\varphi_{t}\mathrm{sp}_{\lambda}$ vanishes
unless $t\textup{-core}(\lambda)$ is symplectic, in which case

$\varphi_{t}\mathrm{sp}_{\lambda}=(-1)^{\delta}\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))\mathrm{sp}_{\lambda^{(t-1)}}\prod_{i=0}^{\lfloor(t-3)/2\rfloor}\mathrm{rs}_{\lambda^{(i)},\lambda^{(t-i-2)}}\times\begin{cases}\mathrm{so}_{\lambda^{((t-2)/2)}}&\text{$t$
even},\\\ 1&\text{$t$ odd},\end{cases}$

where

$\delta=(t-1)\sum_{r=0}^{t-1}\lvert\lambda^{(r)}\rvert+\sum_{r=\lfloor(t+1)/2\rfloor}^{t-1}\binom{m_{r}(\lambda^{\prime};nt)}{2}\\\
+\mathrm{rk}(t\textup{-core}(\lambda))+\begin{cases}\binom{n+1}{2}+n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$
even},\\\ 0&\text{$t$ odd}.\end{cases}$

All that remains now is to show that this sign agrees with that of Theorem
3.3. By a combination of Lemmas 4.6 and 4.7 we may replace
$\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))$ by
$\operatorname{sgn}(w_{t}(\lambda;nt))$, which cancels
$\mathrm{rk}(t\textup{-core}(\lambda))+(t-1)\sum_{r=0}^{t-1}\lvert\lambda^{(r)}\rvert$
in $\delta$. If we call this new exponent $\delta^{\prime}$, then we also have
by Lemma 4.1 that $m_{r}(\lambda^{\prime};nt)=m_{r-1}(\lambda;nt)$ for
$\lfloor(t+1)/2\rfloor\leqslant r\leqslant t-1$, which implies
$\delta^{\prime}=\varepsilon_{\lambda;nt}^{\mathrm{sp}}$.

###  5.5. Proof of Theorem 3.4

The final proof closely follows the first. Let $\lambda$ be a partition of
length at most $nt$ and consider

$\mathrm{so}_{\lambda}=\det_{1\leqslant i,j\leqslant
nt}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+1}\big{)}.$

As before we apply the permutations $w_{t}(\lambda;nt)$ and $w_{t}(0;nt)$ to
the rows and columns of this determinant, introducing the sign (5.1). Unlike
before, there is only one case in which both $h$’s may survive. If $t$ is odd
and $\lambda_{i}-i\equiv-j\equiv(t-1)/2\ (\mathrm{mod}\ t)$ then we have

$\varphi_{t}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+1}\big{)}\\\\[2.84526pt]
=h_{\lambda^{((t-1)/2)}_{k}-k+\ell+m_{(t-1)/2}(\lambda;nt)-n}+h_{\lambda^{((t-1)/2)}_{k}-k-\ell+1+m_{(t-1)/2}(\lambda;nt)-n}.$

where $1\leqslant k\leqslant m_{(t-1)/2}(\lambda;nt)$ and
$1\leqslant\ell\leqslant n$. These entries lie in the rows
$1+\sum_{r=0}^{(t-3)/2}m_{r}(\lambda;nt)$ to
$\sum_{r=0}^{(t-1)/2}m_{r}(\lambda;nt)$ and columns $1+n(t-1)/2$ to
$n(t+1)/2$, and outside of their intersection, all other entries in these rows
and columns are zero. Now consider a row corresponding to the residue class
$r$ for $0\leqslant r\leqslant\lfloor(t-2)/2\rfloor$. Then the column must
fall into the residue class $r$ or $t-r-1$ in order for the entry to not
necessarily vanish. In this case we now have

$\varphi_{t}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+1}\big{)}=\begin{cases}h_{\lambda^{(r)}_{k}-k+\ell_{1}+m_{r}(\lambda;nt)-n}&\text{if
$j\equiv-r\ (\mathrm{mod}\ t)$},\\\
h_{\lambda^{(r)}_{k}-k-\ell_{1}+m_{r}(\lambda;nt)-n+1}&\text{if $j\equiv r+1\
(\mathrm{mod}\ t)$}.\\\ \end{cases}$

Again, a similar computation holds for the row corresponding to $t-r-1$, and
the sets of indices agree with (5.2) but with $t-r\mapsto t-r-1$ in (5.2b). If
$t$ is odd we move the central submatrix corresponding to $(t-1)/2$ to the
top-left, picking up a sign of

$(-1)^{m_{(t-1)/2}(\lambda;nt)\sum_{r=0}^{(t-3)/2}m_{r}(\lambda;nt)+n^{2}(t-1)/2}.$

The grouping and rearrangement of the remaining minors is the same as in the
first proof above. We only remark that the result is the determinant of a
block-diagonal matrix with blocks of dimensions
$(m_{r}(\lambda;nt)+m_{t-r-1}(\lambda;nt))\times 2n$ for $0\leqslant
r\leqslant\lfloor(t-2)/2\rfloor$ plus one of size
$m_{(t-1)/2}(\lambda;nt)\times n$ if $t$ is odd. Thus the determinant vanishes
unless (4.1) holds with $z=0$, i.e., unless $t\textup{-core}(\lambda)$ is
self-conjugate. Accounting for the sign of
$(-1)^{\binom{m_{r}(\lambda;nt)}{2}+\binom{n}{2}}$ from reordering the columns
in the copies of $\mathrm{rs}_{\lambda^{(r)},\lambda^{(t-r-1)}}$, the exponent
$\varepsilon^{\mathrm{so}}_{\lambda;nt}$ has the value

$\frac{(n+1)nt(t-1)}{4}+\sum_{r=0}^{\lfloor(t-2)/2\rfloor}\bigg{(}\binom{m_{r}}{2}+\binom{n}{2}\bigg{)}+\begin{cases}0&\text{$t$
even},\\\ n\sum_{r=0}^{(t-3)/2}m_{r}+n^{2}(t-1)/2&\text{$t$ odd}.\end{cases}$

By (4.4) this has the same parity as

$\varepsilon^{\mathrm{so}}_{\lambda;nt}=\sum_{r=\lfloor(t+1)/2\rfloor}^{t-1}\binom{m_{r}(\lambda;nt)+1}{2}+\begin{cases}0&\text{$t$
even},\\\ n\mathrm{rk}(t\textup{-core}(\lambda))&\text{$t$ odd}.\end{cases}$

###  5.6. Proof of Proposition 3.5

To close out this section, we sketch the proof of Proposition 3.5. In the
Schur case, by Corollary 4.2 and the fact that $\lambda/\mu$ is $t$-tileable
if and only if $\lambda^{\prime}/\mu^{\prime}$ is, we already have
$\omega\varphi_{t}s_{\lambda/\mu}=\pm\varphi_{t}\omega s_{\lambda/\mu}$. The
precise difference in sign is then provided by Lemma 4.6.

In the orthogonal case, again using Corollary 4.2, we have

$\displaystyle\omega\varphi_{t}\mathrm{o}_{\lambda}$
$\displaystyle=(-1)^{\varepsilon^{\mathrm{o}}_{\lambda;nt}}\operatorname{sgn}(w_{t}(\lambda;nt))\mathrm{sp}_{(\lambda^{(0)})^{\prime}}\prod_{i=1}^{\lfloor(t-1)/2\rfloor}\\!\mathrm{rs}_{(\lambda^{(i)})^{\prime},(\lambda^{(t-i)})^{\prime}}\\!\times\\!\begin{cases}\mathrm{so}_{(\lambda^{(t/2)})^{\prime}}^{-}&\text{$t$
even},\\\ 1&\text{$t$ odd},\end{cases}$
$\displaystyle=(-1)^{\varepsilon^{\mathrm{o}}_{\lambda;nt}+\varepsilon^{\mathrm{sp}}_{\lambda^{\prime};nt}}\operatorname{sgn}(w_{t}(\lambda;nt))\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))\varphi_{t}\mathrm{sp}_{\lambda^{\prime}},$

where $n$ should be large enough so that $\lambda$ is contained in an
$nt\times nt$ box. Combining Lemmas 4.5–4.7 shows that, in this case,

$\operatorname{sgn}(w_{t}(\lambda;nt))\operatorname{sgn}(w_{t}(\lambda^{\prime};nt))=(-1)^{\mathrm{rk}(t\textup{-core}(\lambda))+(t-1)(\lvert\lambda^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}\rvert)}.$

A sign computation as in Section 5.4 above shows that the total sign agrees
with the proposition, and is equal to
$(-1)^{(t-1)(\lvert\lambda^{(0)}\rvert+\cdots+\lvert\lambda^{(t-1)}\rvert)}$.
The case of $\mathrm{so}_{\lambda}$ is nearly identical, and so it is omitted.

##  6\. Other factorisations

###  6.1. Littlewood-type factorisations

In [27, §7.3], Littlewood proves a factorisation slightly more general than
the one contained in Theorem 3.1 for $\mu$ empty; see also [5, Theorem 2.7].
Here, and below, we let $\lambda^{(r)}=\lambda^{(k)}$ if $k\equiv r\
(\mathrm{mod}\ t)$.

######  Theorem 6.1.

Let $\lambda$ be a partition of length at most $nt+1$ and
$X=(x_{1},\dots,x_{n})$ a set of variables. Then for another variable $q$,

$s_{\lambda}(X,\zeta X,\dots,\zeta^{t-1}X,q)=0$

unless $t\textup{-core}(\lambda)=(c)$ for some $0\leqslant c\leqslant t-1$, in
which case

$s_{\lambda}(X,\zeta
X,\dots,\zeta^{t-1}X,q)=\operatorname{sgn}_{t}(\lambda/(c))q^{c}s_{\lambda^{(c-1)}}(X^{t},q^{t})\prod_{\begin{subarray}{c}r=0\\\
r\neq c-1\end{subarray}}^{t-1}s_{\lambda^{(r)}}(X^{t}).$

This theorem can also be placed in our framework, however in a somewhat less
elegant manner than our other results. The operator
$\varphi_{t}^{q}:\Lambda\longrightarrow\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}[q]$
which gives the above may be defined by

$\varphi_{t}^{q}h_{at+b}:=q^{b}\sum_{k\geqslant
0}q^{kt}h_{a-k}=\sum_{k\geqslant 0}q^{k}\varphi_{t}h_{at+b-k}.$

Note that the sums are finite since $h_{r}$ vanishes for negative $r$, and
that for $q=0$ this reduces to the operator $\varphi_{t}$ from the earlier
sections. Alternatively, since

$h_{r}(X,q)=\sum_{k\geqslant 0}q^{k}h_{r-k}(X),$

the image of $\varphi_{t}^{q}$ acting on any symmetric function $f$ is the
same as the image of $\varphi_{t}$ acting on $f(X,q)$, where $\varphi_{t}$
acts only on the $X$ variables. Using $\varphi_{t}^{q}$, Littlewood’s above
theorem may be phrased as follows. After the statement we provide a short
proof which relies only on Theorem 3.1.

######  Proposition 6.2.

We have that $\varphi_{t}^{q}s_{\lambda}=0$ unless
$t\textup{-core}(\lambda)=(c)$ for some $c$ such that $0\leqslant c\leqslant
t-1$, in which case

$\varphi_{t}^{q}s_{\lambda}=\operatorname{sgn}_{t}(\lambda/(c))q^{c}\prod_{\begin{subarray}{c}r=0\\\
r\neq c-1\end{subarray}}^{t-1}s_{\lambda^{(r)}}\sum_{k\geqslant
0}q^{kt}s_{\lambda^{(c-1)}/(k)}.$

###### Proof.

The first observation is that

$\varphi_{t}^{q}s_{\lambda}=\sum_{k\geqslant
0}q^{k}\varphi_{t}s_{\lambda/(k)},$

which is a simple consequence of the branching rule for Schur functions [31,
p. 72]. In the case that $l(t\textup{-core}(\lambda))>1$ then each term in the
sum on the right-hand side vanishes by Theorem 3.1 as the $t$-cores of the
inner and outer shape can never be equal. Now assume
$t\textup{-core}(\lambda)=(c)$ for some $0\leqslant c\leqslant t-1$, which is
a complete set of $t$-cores with length one. Then the nonzero terms in the sum
on the right-hand side are those for which $k$ is of the form $\ell t+c$ with
$\ell\geqslant 0$ and $\ell t+c\leqslant\lambda_{1}$. Therefore

$\varphi_{t}^{q}s_{\lambda}=q^{c}\sum_{\ell\geqslant 0}q^{\ell
t}\varphi_{t}s_{\lambda/(\ell t+c)}=q^{c}\sum_{\ell\geqslant
0}\operatorname{sgn}_{t}(\lambda/(\ell t+c))q^{\ell
t}s_{\lambda^{(c-1)}/(\ell)}\prod_{\begin{subarray}{c}r=0\\\ r\neq
c-1\end{subarray}}^{t-1}s_{\lambda^{(r)}},$

again by Theorem 3.1. By our convention we always compute the $t$-quotient
using a beta set with number of elements a multiple of $t$. This means that
the single row $(\ell t+c)$ has one non-empty element in its $t$-quotient,
$\lambda^{(c-1)}$. Moreover, since the partitions $(\ell t+c)$ all differ by a
ribbon of height zero, the sign of each term in the sum is the same and equal
to $\operatorname{sgn}_{t}(\lambda/(c))$. Putting all of this together, we
arrive at

$\varphi_{t}^{q}s_{\lambda}=\operatorname{sgn}_{t}(\lambda/(c))q^{c}\prod_{\begin{subarray}{c}r=0\\\
r\neq c-1\end{subarray}}^{t-1}s_{\lambda^{(r)}}\sum_{\ell\geqslant 0}q^{\ell
t}s_{\lambda^{(c-1)}/(\ell)}.\qed$

We do not see, at this stage, whether it is possible to extend the previous
result to skew Schur functions. If we expand $\varphi_{t}^{q}s_{\lambda/\mu}$
as in the proof above, we find that

$\varphi_{t}^{q}s_{\lambda/\mu}=\sum_{\nu\succ\mu}q^{\lvert\nu\rvert-\lvert\mu\rvert}\varphi_{t}s_{\lambda/\nu},$

where $\nu\succ\mu$ means that $\nu/\mu$ is a _horizontal strip_ , i.e.,
$\nu\supseteq\mu$ and $\nu/\mu$ contains at most one box in each column of its
Young diagram. Of course, this implies that $\varphi_{t}^{q}s_{\lambda/\mu}=0$
if there does not exist a $\nu$ such that $\nu\succ\mu$ and $\lambda/\nu$ is
$t$-tileable. However, the sum may vanish even if such a $\nu$ exists. For
example,

$\displaystyle\varphi_{2}^{q}s_{(4,4)/(1)}$
$\displaystyle=q\varphi_{t}\big{(}s_{(4,4)/(2)}+s_{(4,4)/(1,1)}\big{)}+q^{3}\varphi_{t}\big{(}s_{(4,4)/(4)}+s_{(4,4)/(3,1)}\big{)}$
$\displaystyle=q(s_{(2)}s_{(2)/(1)}-s_{(2)}s_{(2)/(1)})+q^{3}(s_{(2)}-s_{(2)})$
$\displaystyle=0.$

In a similar direction Pfannerer [36, Theorem 4.4] has shown that, if
$\lambda$ has empty $t$-core and $m=\ell t+k$ is any integer, then the Schur
function $s_{\lambda}(1,\zeta,\dots,\zeta^{m-1})$ always factors as a product
of Schur functions with variables all one indexed by the $t$-quotient of
$\lambda$. When $m$ is a multiple of $t$ this becomes a special case of
Littlewood’s theorem (Theorem 3.1 with $\mu$ empty) noted by Reiner, Stanton
and White [38, Theorem 4.3]. Pfannerer’s result has subsequently been
generalised by Kumari [22, Theorem 2.2], in addition to analogues of Theorem
6.1 for other classical group characters. It is an open problem to see how
these factorisations fit into our story.

###  6.2. Factorisations of supersymmetric Schur functions

Recently, Kumari has given a version of Theorem 3.1 for the so-called _skew
hook Schur functions_ (or _supersymmetric skew Schur functions_) [21, Theorem
3.2]. For two independent sets of variables (_alphabets_), we denote their
plethystic difference by $X-Y$; see, e.g., [14, 23] for the necessary
background on plethystic notation. We also note that for an alphabet $X$, we
let $\varepsilon X$ be the alphabet with all variables negated. The _complete
homogeneous supersymmetric function_ used in [21] may be defined as

$\sum_{j=0}^{r}h_{j}(X)e_{r-j}(Y)=h_{r}[X-\varepsilon Y].$

The hook Schur function is then the Jacobi–Trudi determinant of these
functions, so that

$s_{\lambda/\mu}[X-\varepsilon Y]=\det_{1\leqslant i,j\leqslant
n}\big{(}h_{\lambda_{i}-\mu_{j}-i+j}[X-\varepsilon Y]\big{)}.$

From this, it follows readily that Kumari’s factorisation for the hook Schur
functions is contained in Theorem 3.1 above at the alphabet $X-\varepsilon Y$.

###  6.3. Factorisations of $\mathrm{rs}_{\lambda,\mu}$

To close, we point out that the universal character
$\mathrm{rs}_{\lambda,\mu}$ can be used to lift some factorisation results,
discovered by Ciucu and Krattenthaler [8, Theorems 3.1–3.2] and subsequently
generalised by Ayyer and Behrend [3, Theorems 1–2], to the universal character
level. In the next result we write
$\lambda+1^{n}=(\lambda_{1}+1,\dots,\lambda_{n}+1)$ where $n\geqslant
l(\lambda)$.

######  Theorem 6.3.

For $\lambda$ a partition of length at most $n$, there holds

(6.1a) $\displaystyle\mathrm{rs}_{\lambda,\lambda}$
$\displaystyle=\mathrm{so}_{\lambda}\mathrm{so}_{\lambda}^{-},$ and (6.1b)
$\displaystyle\mathrm{rs}_{\lambda+1^{n},\lambda}$
$\displaystyle=\mathrm{o}_{\lambda+1^{n}}\mathrm{sp}_{\lambda}.$

Moreover, for $\lambda$ a partition of length at most $n+1$,

(6.2a)
$\displaystyle\mathrm{rs}_{(\lambda_{1},\dots,\lambda_{n}),(\lambda_{2},\dots,\lambda_{n+1})}+$
$\displaystyle\mathrm{rs}_{(\lambda_{1}-1,\dots,\lambda_{n+1}-1),(\lambda_{2}+1,\dots,\lambda_{n}+1)}$
$\displaystyle\qquad\qquad\quad=\mathrm{sp}_{(\lambda_{1},\dots,\lambda_{n})}\mathrm{o}_{(\lambda_{2},\dots,\lambda_{n+1})},$
and (6.2b)
$\displaystyle\mathrm{rs}_{(\lambda_{1}+1,\dots,\lambda_{n}+1),(\lambda_{2},\dots,\lambda_{n+1})}+$
$\displaystyle\mathrm{rs}_{(\lambda_{1},\dots,\lambda_{n+1}),(\lambda_{2}+1,\dots,\lambda_{n}+1)}$
$\displaystyle\qquad\qquad\quad=\mathrm{so}_{(\lambda_{1}+1,\dots,\lambda_{n}+1)}\mathrm{so}_{(\lambda_{2},\dots,\lambda_{n+1})}^{-}.$

To get back to the results of Ayyer and Behrend one simply evaluates both
sides of each equation at the alphabet $(x_{1}^{\pm},\dots,x_{n}^{\pm})$. The
precise forms present in [3, Equations (18)–(21)] then follow from
(2.5).222The factor of $(1+\delta_{0,\lambda_{n+1}})$ in [3, Equation (20)] is
not present in our generalisation (6.2a) since the second character vanishes
if $\lambda_{n+1}=0$. As identities for Laurent polynomials the pairs of
identities (6.1) and (6.2) admit uniform statements. However no such uniform
statement will exist for the above generalisation, since this requires
characters indexed by half-partitions, which cannot be handled by the
universal characters. Ayyer and Fischer [4] have also given skew analogues of
the non-universal case of Theorem 6.3. Jacobi–Trudi formulae for the
symplectic and orthogonal characters have recently been derived in [1, 17],
and so there are candidates for the universal characters for those objects.
However, the main obstacle in lifting Ayyer and Fischer’s results to the
universal level is the lack of a skew analogue of $\mathrm{rs}_{\lambda,\mu}$.

###### Proof of Theorem 6.3.

First up is (6.1a), which is the simplest of the four. In the determinant

$\mathrm{rs}_{\lambda,\lambda}=\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j}\big{)}_{1\leqslant i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i,j\leqslant n\end{subarray}}\\\\[5.69054pt]
\big{(}h_{\lambda_{i}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i,j\leqslant n\end{subarray}}&\big{(}h_{\lambda_{i}-i+j}\big{)}_{1\leqslant
i,j\leqslant n}\end{pmatrix},$

add the blocks on the right to the blocks on the left, and then subtract the
blocks on the top from the blocks on the bottom, giving

$\displaystyle\mathrm{rs}_{\lambda,\lambda}$ $\displaystyle=\det_{1\leqslant
i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+1}\big{)}_{1\leqslant
i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i,j\leqslant n\end{subarray}}\\\\[5.69054pt]
0&\big{(}h_{\lambda_{i}-i+j}-h_{\lambda_{i}-i-j+1}\big{)}_{1\leqslant
i,j\leqslant n}\end{pmatrix}$
$\displaystyle=\mathrm{so}_{\lambda}\mathrm{so}_{\lambda}^{-}.$

For the second identity (6.1b),

$\mathrm{rs}_{\lambda+1^{n},\lambda}=\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j+1}\big{)}_{1\leqslant i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+2}\big{)}_{\begin{subarray}{c}1\leqslant
i,j\leqslant n\end{subarray}}\\\\[5.69054pt] \big{(}h_{\lambda-
i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant i,j\leqslant
n\end{subarray}}&\big{(}h_{\lambda_{i}-i+j}\big{)}_{1\leqslant i,j\leqslant
n}\end{pmatrix},$

and we add columns $1,\dots,n-1$ to the columns $n+2,\dots,2n$ and then
subtract the bottom two blocks from the top two, resulting in

$\displaystyle\mathrm{rs}_{\lambda+1^{n},\lambda}$
$\displaystyle\quad=\frac{1}{2}\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j+1}-h_{\lambda_{i}-i-j+1}\big{)}_{1\leqslant
i,j\leqslant n}&0\\\ \big{(}h_{\lambda-
i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant i,j\leqslant
n\end{subarray}}&\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+2}\big{)}_{1\leqslant
i,j\leqslant n}\end{pmatrix}$
$\displaystyle\quad=\mathrm{o}_{\lambda+1^{n}}\mathrm{sp}_{\lambda}.$

In the third identity, we consider the second determinant in the sum in (6.2a)

$\displaystyle\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j-1}\big{)}_{1\leqslant i,j\leqslant
n+1}&\big{(}h_{\lambda_{i}-i-j}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n+1\\\ 1\leqslant j\leqslant n-1\end{subarray}}\\\\[8.53581pt]
\big{(}h_{\lambda_{i+1}-i-j+2}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n-1\\\ 1\leqslant j\leqslant
n+1\end{subarray}}&\big{(}h_{\lambda_{i+1}-i+j+1}\big{)}_{1\leqslant
i,j\leqslant n-1}\end{pmatrix}.$

Push the first column so it becomes the $(n+1)$-st, and then push the
$(n+1)$-st row to the final row, which picks up a minus sign. The resulting
determinant differs from that of
$\mathrm{rs}_{(\lambda_{1},\dots,\lambda_{n}),(\lambda_{2},\dots,\lambda_{n+1})}$
in only the last row, so we can take the sum of the two, giving

$\displaystyle\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j}\big{)}_{1\leqslant i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n\end{subarray}}\\\\[5.69054pt]
\big{(}h_{\lambda_{i+1}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n-1\\\ 1\leqslant j\leqslant
n\end{subarray}}&\big{(}h_{\lambda_{i+1}-i+j}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n-1\\\ 1\leqslant j\leqslant n\end{subarray}}\\\\[8.53581pt]
(h_{\lambda_{n+1}-n-j+1}-h_{\lambda_{n+1}-n+j-1})_{1\leqslant j\leqslant
n}&(h_{\lambda_{n+1}-n+j}-h_{\lambda_{n+1}-n-j})_{1\leqslant j\leqslant
n}\end{pmatrix}.$

In this new determinant, add columns $n+1,\dots,2n-1$ to columns $2,\dots,n$,
and then subtract rows $2,\dots,n$ from rows $n+1,\dots,2n-1$, which gives

$\displaystyle\frac{1}{2}\det_{1\leqslant i,j\leqslant
2n}\begin{pmatrix}\big{(}h_{\lambda_{i}-i+j}+h_{\lambda_{i}-i-j+2}\big{)}_{1\leqslant
i,j\leqslant
n}&\big{(}h_{\lambda_{i}-i-j+1}\big{)}_{\begin{subarray}{c}1\leqslant
i\leqslant n\end{subarray}}\\\\[5.69054pt]
0&\big{(}h_{\lambda_{i+1}-i+j}-h_{\lambda_{i+1}-i-j}\big{)}_{1\leqslant
i,j\leqslant n}\\\\[8.53581pt] \end{pmatrix},$

which equals
$\mathrm{sp}_{(\lambda_{1},\dots,\lambda_{n})}\mathrm{o}_{(\lambda_{2},\dots,\lambda_{n+1})}$.
The final factorisation (6.2b) follows almost identically. ∎

### Acknowledgements

I would like to thank Christian Krattenthaler for many useful comments,
corrections and suggestions. This work also benefitted from discussions with
Ilse Fischer, Hans Höngesberg, Josef Küstner, Florian Schreier-Aigner and Ole
Warnaar.

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