Universal characters twisted by roots of unity

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.


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 λ at a set of variables "twisted" (not his term) by a primitive t-th root of unity ζ [27, §7.3].In modern terminology, Littlewood's theorem asserts that s λ evaluated at the variables ζ j x i for 1 ⩽ i ⩽ n and 0 ⩽ j ⩽ t − 1 is zero unless the t-core of λ is empty.Moreover, when it is nonzero, it factors as a product of Schur functions indexed by the elements of the t-quotient of λ, each with the variables x t 1 , . . ., x t n .The Schur functions are characters of the irreducible polynomial representations of the general linear group GL(n, C).Ayyer and Kumari [5] have recently generalised Littlewood's theorem to the characters of the other classical groups O(2n, C), Sp(2n, C) and SO(2n + 1, 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-core(λ) is of the form (a | a + z) in Frobenius notation, where z = −1, 1, 0, for O(2n, C), Sp(2n, C) and SO(2n + 1, C) respectively.Note that these are the same partitions occurring in Littlewood's Schur expansion of the Weyl denominators for types B n , C n and D n [27, p. 238] (see also [30, 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 [23,Theorem 5.8.2] have given an alternate proof based on the Jacobi-Trudi formula (4).This approach was already known to S. P. Albion Farahat, who used it to extend Littlewood's theorem to skew Schur functions s λ/µ where µ is the t-core of λ [12,Theorem 2].The full skew Schur case was then given by Macdonald [30, 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 O(2n, C), Sp(2n, C) and SO(2n + 1, 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 [42,Theorems 7.8.E & 7.9.A].For the universal characters we generalise the notion of "twisting" a set of variables by introducing operators φ t : Λ −→ Λ for each integer t ⩾ 2 which act on the complete homogeneous symmetric functions as (1) φ t h r = h r/t if t divides r, 0 otherwise.
It is not at all hard to show that the image of φ t acting on a symmetric function at the variables x t 1 , . . ., x t n agrees with the result of twisting the variables x 1 , . . ., x n by ζ.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 rs λ,µ (11) associated with a rational representation of GL(n, 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 , . . ., 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.

Preliminaries
2.1.Partitions.A partition λ = (λ 1 , λ 2 , λ 3 , . . . ) is a weakly decreasing sequence of nonnegative integers such that only finitely many of the λ i are nonzero.The nonzero λ i are called parts and the number of parts the length, written l(λ).We say λ is a partition of Two partitions are regarded as the same if they agree up to trailing zeroes, and the set of all partitions is written P.A partition is identified with its Young diagram, which is the left-justified array of squares consisting of λ 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 λ ′ by reflecting the diagram of λ in the main diagonal x = y, so that the conjugate of (6, 4, 3, 2) above is (4,4,3,2,1,1).If λ = λ ′ then λ is called self-conjugate.For a square at coordinate (i, j) where 1 ⩽ i ⩽ l(λ) and 1 ⩽ j ⩽ λ i the hook length is h(i, j) = λ i + λ ′ j − i − j + 1.For example the square (1, 2) in has hook length 8, with its hook shaded.A partition λ is a t-core if it contains no squares of hook length t, the set of which is denoted C t .For a pair of partitions λ, µ we write µ ⊆ λ if the diagram of µ can be drawn inside the diagram of λ, i.e. if µ i ⩽ λ i for all i ⩾ 1.In this case we can form the skew shape λ/µ by removing the digram of µ from that of λ.For example (3, 2, 1, 1) ⊆ (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 × 2 square.A t-ribbon is a ribbon with t boxes.The height of a t-ribbon R, written 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 ht(R) = 3.We say a skew shape is tileable by t-ribbons or t-tileable if there exists a sequence of partitions A sequence D = (ν (0) , . . ., ν (k) ) (not to be confused with the t-quotient of ν below, for which we use the same notation) satisfying ( 2) is called a ribbon decomposition (or border strip decomposition) of λ/µ.We define the height of a ribbon decomposition to be the sum of the heights of the individual ribbons: ht(D) : ).As shown by van Leeuwen [26, Proposition 3.3.1]and Pak [34,Lemma 4.1] (also in [2, §6]), the quantity (−1) ht(D) is the same for every ribbon decomposition of λ/µ.We therefore define the sign of a t-tileable skew shape λ/µ as (3) sgn t (λ/µ) := (−1) ht (D) .
Let rk(λ) be the greatest integer such that rk(λ) ⩾ λ rk(λ) , usually called the Frobenius rank of λ.Equivalently, rk(λ) is the side length of the largest square which fits inside the diagram of λ (the Durfee square).A partition can alternatively be written in Frobenius notation as Any pair of integer sequences For z ∈ Z and an integer sequence of predetermined length a = (a 1 , . . ., a k ) we write a + z := (a 1 + z, . . ., a k + z).Following Ayyer and Kumari [5, Definition 2.9], λ is called z-asymmetric if it is of the form λ = (a | a + z) for some integer sequence a and integer z.Clearly a 0asymmetric partition is self-conjugate.Partitions which are −1-and 1-asymmetric are called orthogonal and symplectic respectively.

S. P. Albion
2.2.Cores and quotients.We now describe the t-core and t-quotient of λ arithmetically following [30, p. 12].There are many equivalent descriptions, see for instance [13,15,16,41].We begin with the beta set of a partition, which is simply the set of n integers where n ⩾ l(λ) is fixed.The number of elements in this set congruent to r modulo t is denoted by m r (λ; n) = m r .An element which falls into residue class r for 0 ⩽ r ⩽ t−1 can be written as ξ (r) mr ⩾ 0. These integers are used to define a partition with parts λ and the ordered sequence (λ (0) , . . ., λ (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 ⩽ k ⩽ m r (λ; n) − 1 and 0 ⩽ r ⩽ t−1 in descending order, say as ξ1 > • • • > ξn .Then t-core(λ) i := ξi −n+i.If t-core(λ) is empty then we say λ 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 tcores 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 β(λ; n).For an example, let λ = (4, 4, 3, 2, 1) so that β(λ; 6) = {9, 8, 6, 4, 2, 0}.Then the bead configuration for λ with t = 3 and n = 6 and the beads labelled by their position is Moving a bead up one space is equivalent to reducing one of the elements of β(λ; n) by t.This is, in turn, equivalent to removing a t-ribbon from λ such that what remains is still a Young diagram (see for instance [30, p. 12]).Pushing all beads to the top will give the bead configuration of t-core(λ), and this is clearly independent of the order in which the beads are pushed.It follows that t-core(λ) is the unique partition obtained by removing t-ribbons (in a valid way) from the diagram of λ 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 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 (λ; n) beads.For our example, this means that β(t-core(λ); 6) = {6, 5, 3, 2, 1, 0}, so that t-core(λ) = (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 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 [32].The t-quotient of a partition has its origin in the star diagrams of Nakayama, Osima, Robinson and Staal [33,38,39], which were shown to be equivalent to Littlewood's t-quotient by Farahat [10].
Let w t (λ; n) be the permutation of β(λ; 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 (λ; n) will be denoted sgn(w t (λ; n)).The permutation w t (λ; 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 (λ; 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 λ has empty t-core if and only if it is ttileable.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.
Proof.The skew shape being t-tileable is equivalent to the diagram of µ being obtainable from the diagram of λ by removing t-ribbons.In other words, we can obtain the bead configuration of µ from that of λ, where both have nt beads, by moving beads upwards.Assume that this is the case.Then m r (λ; nt) = m r (µ; nt) for each 0 ⩽ r ⩽ t − 1, so that the r-th runner has the same number of beads in each diagram.This implies that t-core(λ) = t-core(µ).It also follows that the i-th bead in each runner of λ's bead configuration must lie weakly below the i-th bead in the same runner of µ's bead configuration.Equivalently, r) .The reverse direction is now clear.□ Note that the lemma is also true when the t-quotients of λ and µ are computed using the same integer n of any residue class modulo t.If λ/µ is t-tileable, then we think of λ (0) /µ (0) , . . ., λ (t−1) /µ (t−1) as its t-quotient.When λ/µ is not t-tileable, it is not so clear how to define the t-quotient.

Symmetric functions and universal characters.
Here we discuss some basics of the theory of symmetric functions, following [30].Let Λ denote the ring of symmetric functions in an arbitrary countable set of variables X = (x 1 , x 2 , x 3 , . . .), called an alphabet.Where possible, we write elements of Λ 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 Λ reduce to symmetric polynomials in the variables (x 1 , . . ., x n ).Another common specialisation sets x n+i = x −1 i for 1 ⩽ i ⩽ 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., BC n -symmetric functions).We will later write (x ± 1 , . . ., x ± n ) for this alphabet.Two fundamental algebraic bases for Λ are the complete homogeneous symmetric functions and the elementary symmetric functions, defined for any positive integer r respectively.We further set h 0 = e 0 := 1 and h −r = e −r = 0 for positive r.These admit the generating functions The h r and e r for r ⩾ 1 are algebraically independent over Z and generate Λ.In view of this, we can define a homomorphism ω : Λ −→ Λ by ωh r = e r .It then follows from the relation H z (X)E −z (X) = 1 that ωe r = h r , so that ω is an involution.We also define the power sums by for r ⩾ 1 and p 0 := 1.These satisfy ωp 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 λ/µ is a skew shape and n an integer such that n ⩾ l(λ) we define ( 4) This is independent of n as long as n ⩾ l(λ).If µ ̸ ⊆ λ then we set s λ/µ := 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) Restricting to the µ empty case, we have s (r) = h r and s (1 r ) = e r .Moreover, it is clear that If the set of variables (x 1 , . . ., x n ) is finite then the Schur function for µ = 0 admits another definition as a ratio of alternants (5) s The denominator is the Vandermonde determinant and has the product representation det 1⩽i,j⩽n In this case we also define 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.
For l(λ) ⩽ n, each of the three above universal characters become actual characters of irreducible representations of their associated groups when specialised to (x ± 1 , . . ., x ± n ) (hence the name universal characters).The irreducible polynomial representations of GL(n, 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 [40].Such sequences are equivalent to pairs of partitions λ, µ such that l(λ) + l(µ) ⩽ n.Given such a pair, one defines the associated staircase [λ, µ] by [λ, µ] i := λ i − µ n−i+1 for 1 ⩽ i ⩽ n.The characters of the rational representations of GL(n, C) are then given by s [λ,µ] (x 1 , . . ., x n ) for all staircases with n entries.Note that (6) implies that this object is just a Schur function up to a power of x 1 • • • x n .In [28], Littlewood gave the expansion (10) s For a pair of partitions λ, µ and sets of indeterminates X, Y , this may be used to define the universal character associated to a rational representation of GL(n, C) as Note that the only terms which contribute are those with ν ⊆ λ and ν ′ ⊆ µ.If we let ω X and ω Y denote the involution ω acting on the set of variables in its subscript, then Algebraic Combinatorics, Vol. 6 #6 (2023)

S. P. Albion
As shown by Koike [18], this object has a Jacobi-Trudi-type expression as a block matrix where n ⩾ l(λ) and m ⩾ l(µ).As for the other determinants, this is independent of n and m as long as n ⩾ l(λ) and m ⩾ l(µ).The relation under ω X ω Y implies we have the dual form [18, Definition 2.1] where n ⩾ λ 1 and m ⩾ µ 1 .The definition (11) and the determinants ( 12) and ( 13) 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 ( 11) and ( 10) it is immediate that, for l(λ) + l(µ) ⩽ n, We will always take X = Y in rs λ,µ (X; Y ), which we write as rs λ,µ in the rest of the paper.In particular we note that rs λ,µ = rs µ,λ .
As mentioned in the introduction, the notion of twisting the set of variables x 1 , . . ., x n by a primitive t-th root of unity ζ is replaced by the operator φ t (1), which has been considered by Macdonald [30,p. 91] and, for t = 2, by Baik and Rains [6, p. 25].Let X t := (x t 1 , x t 2 , x t 3 . . . ) and denote by ψ t the homomorphism Since ψ t H z (X) = H z t (X t ), both φ t and ψ t act on the h r in the same way, i.e. the diagram commutes, where the arrow labelled X t is the substitution map.This implies the claim of the introduction that the action of φ t is equivalent to twisting the alphabet X by a primitive t-th root of unity ζ.If one wishes to think about this as a map Λ X −→ Λ X where X is some concrete alphabet, then substitute each x ∈ X by its set of t-th roots x 1/t , ζx 1/t , . . ., ζ 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 [23, §5.8] φ t e r = (−1) r(t−1)/t e r/t if t divides r, 0 otherwise.
Therefore ω and φ 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.

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 φ t .The first of these is the action of the map on the skew Schur functions.
Theorem 3.1.We have that φ t s λ/µ = 0 unless λ/µ is tileable by t-ribbons, in which case For µ empty, this result is due to Littlewood [27, p. 131], who proves it by direct manipulation of the ratio of alternants (5).By (14) with X = (x 1 , . . ., x n ), one can recover Littlewood's result by simply evaluating the right-hand side of the equation at (x t 1 , . . ., x t n ).He also states the µ 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 [30, p. 91].Curiously, Prasad recently rediscovered the µ empty case independently, with a proof identical to Littlewood's, but in a more representation-theoretic context [36].A version of the result for Schur's P -and Q-functions has been given by Mizukawa [31,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 [43] 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.
Our next result is the same factorisation for the symplectic character.
Finally, we can claim a similar factorisation for so λ .

S. P. Albion
Theorem 3.4.Let λ be a partition of length at most nt.Then φ t so λ = 0 unless t-core(λ) is self-conjugate, in which case φ t so λ = (−1) ε so λ;nt sgn(w t (λ; nt)) where (1) Some remarks are in order.Firstly, the three signs sgn(w t (λ; nt))(−1) ε • λ;nt are actually independent of n as long as nt ⩾ l(λ), 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 → 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(λ) 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.
We remark that the second relation does not hold with so λ replaced by sp λ or o λ as written above since ωso − λ = so − λ ′ .
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].
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 λ be a partition of length at most nt.If t-core(λ) is orthogonal, then Next, we show that the sign of a tileable skew shape can be expressed in terms of the signs of the permutations w t (λ; n).Lemma 4.5.For λ/µ t-tileable and any integer n such that n ⩾ l(λ), sgn t (λ/µ) = sgn(w t (λ; n)) sgn(w t (µ; n)).
Proof.Since λ/µ is t-tileable, it has a ribbon decomposition D = (ν (0) , . . ., ν (k) ) where ν (0) = µ and ν (k) = λ.Also, ν (k−1) can be obtained from λ by moving one bead at some position upward one space.By our characterisation of the inversions in the permutation w t (λ; n), we see that moving a bead at position ℓ up one space changes the sign by (−1) b k where b k is the number of beads at positions between ℓ − t and ℓ.In other words, sgn(w t (λ We also have the following useful relationship between the sign of λ/µ and λ ′ /µ ′ .

S. P. Albion
Proof.To prove the claim of the lemma we will proceed by induction on |λ/µ|.If |λ/µ| = 0 then λ = µ and the equation is trivial.Now fix µ and assume the result holds for λ/µ being t-tileable.Adding a t-ribbon to λ/µ moves one of the beads, say at position b, in the bead configuration for λ down a single space.The change in the number of inversions in w t (λ; nt) is the number of beads b ′ such that b < b ′ < b + 1.
A consequence of Lemma 4.1 is that w t (λ ′ ; 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 λ/µ changes some element of the t-quotient of λ 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 λ be an orthogonal or symplectic t-core whose diagram is contained in an nt × nt square.Then sgn(w t (λ; nt)) sgn(w t (λ ′ ; nt)) = (−1) rk(λ) .
Proof.Assume that λ is a non-empty, orthogonal t-core (if λ 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 λ ′ with nt beads can be obtained from the bead configuration of λ with nt beads by reducing the labels by 1 modulo t.For example if λ = (12, 7, 5, 3, 2, 2, 1, 1, 1, 1, 1) then λ ′ = (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 = ±1 and Lemma 4.1.When passing from λ to λ ′ , 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 λ.Then the sign change is sgn(w t (λ; nt)) = sgn(w t (λ ′ ; nt))(−1) n 2 (t−1)+b .Since λ is an orthogonal t-core, n 2 (t − 1) + b ≡ rk(λ) (mod 2) by ( 16) and Lemma 4.3 with z = −1. □ The next lemma proves the claim made after Theorems 3.2-3.4that the signs occurring in those factorisations are independent of n.
Proof.Assume that nt ⩾ l(λ).Incrementing n by one adds a row of beads to the top of the bead configuration of λ, and so m r (λ; (n + 1)t) = m r (λ; nt) + 1.In the inversion count, the rth bead in the new first row sees Algebraic Combinatorics, Vol. 6 #6 (2023) other beads.Summing over k = 0, . . ., t − 1 we see that sgn(w t (λ; (n + 1)t)) = sgn(w t (λ; nt))(−1) ⌊t/2⌋ r=1 (m2r−1(λ;nt)+1) .Now assume that λ has an orthogonal t-core.Then by Lemma 4.3 with z = −1 the above has the same parity as A short calculation shows that where the last equality uses (16).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 λ, µ be partitions of length at most nt and assume that for 0 ⩽ r, s ⩽ t − 1 we have for some k, ℓ such that 1 ⩽ k ⩽ m r (λ; nt) and 1 ⩽ ℓ ⩽ m s (µ; nt).Alternatively, if r + s + z ≡ 0 (mod t) then Proof.We first write by the definition of the t-quotient.The second claim is analogous.□

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 [30, 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.

S. P. Albion
5.1.Proof of Theorem 3.1.Let n be a nonnegative integer and µ ⊆ λ be a pair of partitions such that l(λ) ⩽ nt.Consider the Jacobi-Trudi determinant Before applying the map φ t , we rearrange the rows and columns of this determinant by the permutations w t (λ; nt) and w t (µ; nt) respectively.By Lemma 4.5 this introduces a sign of sgn t (λ/µ).The rows and columns are now arranged in such a way that the residue classes of λ i − i and µ 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 φ t since φ t h λi−µj −i+j vanishes unless λ i − i ≡ µ j − j (mod t).Therefore, φ t s λ/µ has a block-diagonal structure, with each block having size m r (λ; nt) × m r (µ; nt) for 0 ⩽ r ⩽ t − 1.We conclude that φ t s λ/µ = 0 unless m r (λ; nt) = m r (µ; nt) for all 0 ⩽ r ⩽ 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 for some k and ℓ with 1 ⩽ k, ℓ ⩽ n.Note that the rows and columns are in the desired order (i.e. in each n × 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 (λ; nt) = m r (µ; nt) for all 0 ⩽ r ⩽ t − 1, then r) for any r such that 0 ⩽ r ⩽ t − 1 this expression will give zero, from which we conclude, by Lemma 2.1, that φ t s λ/µ = 0 unless λ/µ 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 λ = (12, 12, . We therefore have that 4-core(λ) = (4, 1, 1), which is clearly orthogonal, and λ (0) , λ (1) , λ (2) , λ (3) = (2, 2), ( Now choose n = 3, so that nt = 12 ⩾ l(λ).Using the definition of o λ as a Jacobi-Trudi-type determinant (7) we immediately see that , where we write • 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 (λ; 12) and w 4 (0; 12), respectively.In this case, the first permutation is odd and the second even, so we are Algebraic Combinatorics, Vol. 6 #6 (2023) left with The top-left 3 × 3 minor and central 3 × 3 minor occupying rows 6-8 and columns 7-9 are clearly equal to o (2,2) and so − (3,2,1) , respectively.One way to isolate the copy of so − (3,2,1) is to push it so that it is the bottom-right 3 × 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 Our goal is to show that this final unidentified determinant is equal to rs (4,1),( . 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 rs (4,1),(2,1,1) .Thus, Note that (−1) ε o λ;12 = 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 o λ , sp λ and so λ .5.3.Proof of Theorem 3.2.Let λ be a partition such that l(λ) ⩽ nt and consider the definition (7) We permute the rows and columns by w t (λ; nt) and w t (0; nt) respectively, which introduces a sign of (19) (−1) ( n+1 2 )( t 2 ) sgn(w t (λ; nt)).
The modular behaviour of the indices of each row is now known.There are three possibilities for the entries of φ t o λ : both h's may survive, one h may survive, or the entry is necessarily zero.For both to survive, we see that h λi−i+j and h λi−i−j are nonzero under φ t if and only if λ i − i ≡ −j ≡ 0 (mod t) or, if t is even, λ i − i ≡ −j ≡ t/2 (mod t).In the first instance, by Lemma 4.9, , where 1 ⩽ k ⩽ m t/2 (λ; nt) and 1 ⩽ ℓ ⩽ n.Again, all other entries in these rows and columns are necessarily zero under φ t .Given a row corresponding to the residue class r where 1 ⩽ r ⩽ ⌊(t − 1)/2⌋, 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, The set of indices of the complete homogeneous symmetric functions in such a row are If we look at the complementary row corresponding to t−r, then a similar computation shows that the indices are We have now identified the entries which do not necessarily vanish under φ 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 (λ; nt) rows and n columns, which picks up a sign of (−1) We then group the rows and columns corresponding to the residue classes r and t − r together with 0 ⩽ r ⩽ ⌊(t − 1)/2⌋ increasing.The determinant is now blockdiagonal and the blocks have dimension m 0 (λ; nt) × n, (m r (λ; nt) + m t−r (λ; nt)) × 2n for 1 ⩽ r ⩽ ⌊(t − 1)/2⌋ and, if t is even, m t/2 (λ; nt) × n.Since the determinant of a block-diagonal matrix vanishes if one of the blocks is not a square, we can therefore conclude that φ t o λ vanishes unless the conditions (15) with z = −1 hold in Lemma 4.3, i.e. unless t-core(λ) is orthogonal.In this case the top-left n × n minor is equal to o λ (0) and if t is even the bottom-right minor corresponds to so − λ (t/2) .Note that in this case the grouping of the 2n × 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 ⩽ r ⩽ ⌊(t − 1)/2⌋ these final minors are of the form where we write m r = m r (λ; nt).Clearing the negatives in this minor produces the sign (−1) mr(λ;nt)+n .If m r (λ; nt) = m t−r (λ; nt) = n then we are done.If m r (λ; nt) > n then we need to move the columns n+1 to m r (λ; nt) so they are the first m r (λ; nt)−n columns, and then reverse the order.This gives a sign of (−1) n(mr(λ;nt)−n)+( mr (λ;nt)−n 2 ) = (−1) ( mr (λ;nt) 2 If m r (λ; nt) < n then we need to push the m t−r (λ; nt) − n missing rows past the n − m r rows to their right and then reverse again, giving the same sign In each of the three cases the resulting determinant is equal to rs λ (r) ,λ (t−r) .Collecting all of the above determinant manipulations, the value of ε o λ;nt is To see that this agrees with the sign of Ayyer and Kumari, we use ( 16) 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 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 sp λ (8).However, it will be more insightful to begin with the e Jacobi-Trudi-type formula ( 9) , where we assume that n is an integer such that nt ⩾ λ 1 .We further assume that nt ⩾ l(λ), since, in the end, our sign will be independent of n.The values of φ t h r and φ 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 o λ ′ ((7) with λ → λ ′ ), 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 λ ′ is simply the reverse of the t-quotient of λ.We can therefore already claim that φ t sp λ vanishes unless t-core(λ) is symplectic, in which case + nrk(t-core(λ)) t even, 0 t odd.

S. P. Albion
All that remains now is to show that this sign agrees with that of Theorem 3.
As before we apply the permutations w t (λ; nt) and w t (0; nt) to the rows and columns of this determinant, introducing the sign (19).Unlike before, there is only one case in which both h's may survive.If t is odd and λ i − i ≡ −j ≡ (t − 1)/2 (mod t) then we have where 1 ⩽ k ⩽ m (t−1)/2 (λ; nt) and 1 ⩽ ℓ ⩽ n.These entries lie in the rows 1 + m r (λ; 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 ⩽ r ⩽ ⌊(t − 2)/2⌋.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 Again, a similar computation holds for the row corresponding to t − r − 1, and the sets of indices agree with (20) but with t − r → t − r − 1 in (20b).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 (λ;nt) (t−3)/2 r=0 mr(λ;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 sizes (m r (λ; nt) + m t−r−1 (λ; nt)) × 2n for 0 ⩽ r ⩽ ⌊(t − 2)/2⌋ plus one of size m (t−1)/2 (λ; nt) × n if t is odd.Thus the determinant vanishes unless (15) holds with z = 0, i.e. unless t-core(λ) is self-conjugate.Accounting for the sign of (−1) ( mr (λ;nt) 2 )+( n2 ) from reordering the columns in the copies of rs λ (r) ,λ (t−r−1) , the exponent ε so λ;nt has the value By (18) this has the same parity as ε so λ;nt = t−1 r=⌊(t+1)/2⌋ m r (λ; nt) + 1 2 + 0 t even, nrk(t-core(λ)) t odd.alphabet X, we let εX be the alphabet with all variables negated.The complete homogeneous supersymmetric function used in [22] may be defined as 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 − εY .= so (λ1+1,...,λn+1) so − (λ2,...,λn+1) .To get back to the results of Ayyer and Behrend one simply evaluates both sides of each equation at the alphabet (x ± 1 , . . ., x ± n ).The precise forms present in [3, Equations ( 18)-( 21)] then follow from (6). (2) As identities for Laurent polynomials the pairs of identities ( 21) and ( 22) 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 rs λ,µ .

Following
Koike and Terada we define the universal characters for O(2n, C) and Sp(2n, C) as the symmetric functions [20, Definition 2.1.1]

Lemma 4 . 1 .
Let λ be a partition of length at most nt such that λ 1 ⩽ mt.Then the bead configuration for β(λ ′ ; mt) can be obtained from the bead configuration for β(λ; nt) with n + m rows by rotating the picture by 180 • and then interchanging beads and spaces.