A generalisation of bar-core partitions

When p and q are coprime odd integers no less than 3, Olsson proved that the q -bar-core of a p -bar-core is again a p -bar-core. We establish a generalisation of this theorem: that the p -bar-weight of the q -bar-core of a bar partition λ is at most the p -bar-weight of λ . We go on to study the set of bar partitions for which equality holds and show that it is a union of orbits for an action of a Coxeter group of type ˜ C ( p − 1) / 2 × ˜ C ( q − 1) / 2 . We also provide an algorithm for constructing a bar partition in this set with a given p -bar-core and q -bar-core.


Introduction
The study of the set of core partitions, those partitions whose Young diagrams are without s-hooks for some natural number s, has revealed a great deal about the representation theory of the symmetric groups, since two irreducible characters of S n are in the same s-block if and only if the partitions labelling them have the same s-core. The motivation behind the study of bar partitions is their correspondence to projective, or spin representations of the symmetric group [5]. The irreducible spin representations of S n corresponding to bar partitions λ and µ lie in the same p-block if and only if they have the same p-bar-core.
The purpose of this paper is to establish analogues for bar partitions, i.e. partitions with distinct parts (or simply finite subsets of N), of the results in Fayers' 'A generalisation of core partitions' [4], which includes a generalisation of Olsson's result that shows the s-core of a t-core is again a t-core [9]. Although the definitions differ, the ideas in [4] translate very well to the notion of p-bar-cores, and this paper includes a generalisation of a further result established by Olsson in [9]: that the q-bar-core of a p-bar-core is again a p-bar-core.
We will begin with a few definitions, which will seem very familiar to those acquainted with the representation theory of the symmetric group, that have been adapted to suit our purposes. Using some basic results, we consider an action of W p , the Weyl group of typeC (p−1)/2 , on the set of bar partitions P, and discover some interesting symmetry. We then consider the problem of constructing the smallest bar partition with a given p-bar-core and q-bar-core, for coprime odd p, q 3. We finish by investigating the orbits of the Yin and Yang partitions [1] under the group action of W p × W q .

Definitions
A bar partition λ ∈ P is a decreasing sequence of distinct positive integers (often referred to as a 2-regular or strict partition). For odd integers p 3, the p-runner abacus [2] has p infinite vertical runners numbered from left to right (p+1) /2, (p+3) /2, . . . , p − 1, 0, 1, . . . , (p−1) /2, with the positions on runner i labelled with the integers with p residue i, increasing down the runner, so that position x + 1 appears directly to the right of position x (for x ≡ (p−1) /2 mod p). We obtain a visual representation of λ on the p-runner abacus by placing a bead on position x for each x ∈ λ and each integer x < 0 such that −x ∈ λ; position 0 remains empty.
This differs from the way that Olsson, for example, represents bar partitions and bar-cores, but this p-runner abacus will be more useful for our purposes.
Example 2.1. The bar partition (9,8,7,5,3) has the following bead configuration on the 5-runner abacus (we indicate the zero position with a white bead).
Let A(λ) denote the set containing all integers that label positions occupied by beads in the bead configuration for λ ∈ P on the p-runner abacus: x ∈ A(λ) ⇔ x ∈ λ, x > 0, or −x ∈ λ, x < 0.
Note that this is independent from the choice of p. For odd integers p 3, removing a p-bar from λ ∈ P means either (i) removing x ∈ λ such that 0 x − p ∈ λ, and replacing x with x − p if x = p; or (ii) removing two parts x, p − x ∈ λ (where 0 < x < p). (p must be odd because of the incompatible possibility that a bar partition could have a 2p-bar but not a p-bar, e.g. p = 4 and the partition (6,2).) In terms of the abacus, removing a p-bar from λ corresponds to moving a bead at position x to position x − p (replacing x ∈ A(λ) with x − p), then moving the bead at position p − x to position −x (replacing p − x ∈ A(λ) with −x).
When it is not possible to remove any p-bars from λ, i.e. when x − p ∈ A(λ) for all x ∈ A(λ), we say that λ is a p-bar-core, and we denote the set of p-bar-cores by C p . Since removing a p-bar always corresponds to moving beads up on their runners to unoccupied positions, we have reached the bead configuration of a p-bar-core when all beads are moved up their runners as far as possible. The order in which these moves are made is irrelevant; we always end up at the same bead configuration. Hence we may define the p-bar-core of a bar partition λ, which we denote by λ p . The number of p-bars which can be successively removed from λ is the p-bar-weight of λ, and we denote this quantity by wt p (λ); denoting by |µ| the sum of the parts of the bar partition µ, The number of bead moves needed to reach the bead configuration for λ p from the bead configuration for λ is equal to twice the p-bar-weight of λ, because removing a Algebraic Combinatorics, Vol. 5 #4 (2022) p-bar corresponds to two moves of the beads. The p-bar-weight of λ is therefore equal to half the number of pairs (x, a) ∈ A(λ) × N such that x − ap ∈ A(λ).
Example 2.2. The 5-bar-core of the bar partition from Example 2.1 is (9,8,7,5, 3) 5 = (4, 3), and has the following bead configuration on the 5-runner abacus: We are now equipped with tools analogous to those James used in his seminal book on the representation theory of the symmetric group via the combinatorics of (not necessarily strict) partitions [6]. For the benefit of readers unfamiliar with James' work, we outline the theory of rim-hooks and cores here.
We may visually represent a partition α = (α 1 , α 2 , . . . , α r ), i.e. a decreasing sequence of (not necessarily distinct) positive integers α 1 α 2 · · · α r , by its Young diagram [α], which has α i nodes in the i th row, for each i ∈ {1, . . . , r}, with each row starting in the first column. The hook-length h i,j of the (i, j)-node, in the i th row and j th column of [α], is found by adding the number of (k, j)-nodes with k i to the number of (i, l)-nodes with l > j. We refer to the (i, j)-nodes with (i + 1, j + 1) ∈ [α] as the rim of [α]. The h i,j pairwise adjacent nodes along the rim of [α] from the lowest node in the j th column, i.e. the (k, j)-node with k maximal, to the (i, α i )-node are collectively called a rim h i,j -hook. Whenever a diagram [α] has an (i, j)-node with hook-length s := h i,j , we may remove a rim s-hook from [α] to obtain the Young diagram of a different partition. If instead [α] has no nodes with hook-length s, then we say that the partition α is an s-core. Example 2.3. Below is the Young diagram [(4, 4, 2, 1)], which has just one 5-hook. The (1, 2)-node is highlighted with a •, the (1, 2)-hook is highlighted in red, and the removable nodes of the corresponding rim 5-hook are highlighted with ×'s.
Adopting the convention that α i = 0 for each i greater than some fixed r ∈ N, the strictly decreasing sequence of integers α 1 − 1 + k > α 2 − 2 + k > . . . , for some k ∈ Z, is called a beta-set for the partition α = (α 1 , α 2 , . . . ), and is denoted by B α k . James' s-abacus has s runners extending infinitely in both directions, with the leftmost runner labelled by multiples of s, and the position directly to the right of i labelled by i+1. A bead configuration is associated with a partition α via the beta-set B α := B α 0 by placing a bead at the position labelled by α i − i for each i ∈ N. Removing a rim s-hook from [α] then corresponds to removing an element x ∈ B α such that x−s ∈ B α , and replacing x with x−s. Thus we obtain the bead configuration for an s-core by moving the beads in the configuration for α on the s-abacus up their runners as far as possible. Since the order in which we move the beads is irrelevant, Algebraic Combinatorics, Vol. 5 #4 (2022)

D. Yates
there is only one s-core which can be obtained from a partition α by removing rim s-hooks, and we denote the s-core of α byα s . The number of moves needed to reach the bead configuration ofα s from the configuration of α, or equivalently, the number of rim s-hooks which can be removed from the diagram [α], is the s-weight of α; we denote this quantity by wt s (α).
The s-quotient of α is the s-tuple of partitions corresponding to the bead configuration of each runner of the s-abacus as s separate 1-abaci. Each partition α is uniquely determined by its s-coreα s and its s-quotient. Removing rim s-hooks has a strong connection with the modular representation theory of the symmetric group, as the two ordinary irreducible representations corresponding to the partitions α and β belong to the same s-block of s-modular irreducible constituents if and only ifα s =β s . This important result was first conjectured by Nakayama, and should be referred to as the Brauer-Robinson Theorem after those who first proved it in 1947. Now that we have seen how the combinatorics of bar partitions is related to James' work, we introduce a useful way to encode bar partitions, just as the s-core and squotient encode partitions [6].
Define the p-set [3] of a bar partition λ to be the set {∆ i mod p λ|i ≡ 0, 1, . . . , p − 1}, where ∆ i mod p λ is the smallest integer x ≡ i modulo p such that x ∈ A(λ p ). Since x ∈ A(λ p ) ⇔ −x ∈ A(λ p ), for any bar partition λ and k ≡ 0 (mod p) we have ∆ k mod p λ + ∆ −k mod p λ = p, so all of the elements in the p-set of any bar partition sum to p(p−1) /2.
(We will drop the 'mod p' in our notation for both the p-set and p-quotient when it is clear which p we are referring to.) The parts of the bar partition λ (0 mod p) are the elements of the set { x /p|x ∈ λ, x ∈ pZ}. For j ≡ 0 (mod p), the i th part of the (not necessarily strict) partition λ (j mod p) is equal to the number of empty spaces above the i th lowest bead on runner j in the bead configuration for λ on the p-runner abacus.
It follows from the definition of the p-runner abacus that for each j ≡ 0 (mod p), the partition λ (−j mod p) is equal to (λ (j mod p) ) , the conjugate of the partition λ (j mod p) , the parts of which are the lengths of the columns in the Young diagram for λ (j mod p) .

A level q group action on bar partitions
Now we will consider an action of W p , the affine Coxeter group of typeC(p−1) /2 , with generators δ 0 , . . . , δ(p−1) /2 and relations For coprime odd integers p, q 3, we define a level q action of W p on Z [3]: For the rest of this paper, we will assume that p and q are coprime odd integers no less than 3.
Lemma 4.1. The above defines a group action of W p on Z, and this can be extended to an action on P.
Proof. We must have p > 3 for the fourth and fifth relations of the generators of W p to hold for the group action. The first relation δ 2 i = 1 is clear for all i. Moreover, the generators δ i , δ j commute when 0 i < j − 1 (p−3) /2 because they act on distinct congruence classes of integers modulo p. For the third relation, when 1 i (p−5) /2 and x ∈ Z we have Algebraic Combinatorics, Vol. 5 #4 (2022) For the fourth and fifth relations, assuming p > 3, If X is a subset of Z\{0} that is bounded above, and its complement in Z is bounded below, then it is easy to see that the same is true for aX := {ax|x ∈ X}, for any a ∈ W p . Moreover, when x ∈ X ⇔ −x ∈ X, for all x ∈ Z\{0}, then the set aX also satisfies this rule. Hence, this action can be extended to an action on bar partitions λ by defining δ i λ to be the bar partition with A(δ i λ) = δ i A(λ). We obtain the bead configuration of the bar partition (13, 6, 5, 2) = (δ 0 δ 2 )(9, 8, 7, 5, 3) by first subtracting q := 3 from each element in A( (9,8,7,5,3)) that is congruent to 4 (mod 5), and adding 3 to each element congruent to 1 (mod 5), then subtracting 6 from each element congruent to 3 (mod 5), and adding 6 to each element congruent to 2 (mod 5). We thus obtain the set

D. Yates
Notice that both bar partitions have the same q-bar-core; this will always be the case, as the level q action defined on the generators of W p always corresponds to adding or removing q-bars. On the 3-runner abacus, the action of δ 0 δ 2 ∈ W 5 has the following effect on the bead configuration of (9,8,7,5,3).

−→
We now give some invariants of the level q action of W p which we will later use to give an explicit criterion for when two bar partitions lie in the same orbit under the level q action, which we refer to as a level q orbit. Lemma 4.3. Suppose λ ∈ P and a ∈ W p , and define aλ using the level q action. ( Proof. The relations occurring in all four parts are transitive, so we need only prove them in the case where a is simply a generator δ i of W p . (1) An element x ∈ A(λ) is fixed by the level q action of δ 0 if and only if .
Thus the action of δ 0 on A(λ) corresponds to removing 2q-bars from or adding 2q-bars to λ. .
Thus the action of δ i on A(λ) corresponds to removing q-bars from or adding q-bars to λ. Moreover, since there are only finitely many x ∈ A(λ) such that at least one of (3) This follows from (2) and Lemma 3.1(1) (taking c = 1): the bead configuration for δ i λ on the p-runner abacus is the same as that of λ but with the runners reordered, so the p-bar-weights of the two bar partitions are equal.
We suppose the contrary.
When i = 0, we may assume that j ≡ ±q ≡ −k (mod p), as otherwise j = k and , so we must have i > 0, and we may also assume that j ≡ ±iq or ±(i + 1)q, and j ≡ k ± q (mod p). But then Next we will give a criterion for when two bar partitions lie in the same level q orbit; to this end, we will first establish a condition for two p-bar-cores to lie in the same level q orbit.
Proposition 4.4. Suppose λ, µ ∈ C p , and that the multisets are equal. Then λ q = µ q , and λ and µ lie in the same level q orbit (of p-bar-cores).
Proof. The fact that λ and µ have the same q-bar-core is established by Fayers' [3, Proposition 4.1], and it follows from the definition of the level q action of W p on the set of p-bar-cores that this action preserves the q-bar-core of a bar partition as by definition δ i does not change the multiset of residues modulo q of the elements of the p-set. Therefore each orbit of the level q action on C p can contain at most one q-bar-core.
In the same paper [3], Fayers proves the following result: Suppose O is a level q orbit. Let ν be an element of O for which the sum Then ν is a q-bar-core. This ν is uniquely defined as each level q orbit contains no more than one q-bar-core. Thus letting ν be the q-bar-core of both λ and µ, it must be contained in both the level q orbit containing λ and the level q orbit containing µ; so these orbits coincide.
For the more general result, we define the q-weighted p-quotient of λ ∈ P with p-set {∆ 0 λ, . . . , ∆ p−1 λ} and p-quotient Q p (λ) = (λ (0) , . . . , λ (p−1) ) to be the multiset Proof. Firstly suppose that λ and µ lie in the same level q orbit; we may assume that otherwise, For the other direction, suppose that λ and µ share the q-weighted p-quotient Q q p (λ) = Q q p (µ). By the definition of the p-set, and since all components of the pquotient of a p-bar-core are equal to the empty bar partition, the p-bar-cores of λ and µ must have the same q-weighted p-quotient Q q p (λ p ) = Q q p (µ p ). Thus, by Proposition 4.4 we may find a, b ∈ W p such that a(λ p ) = b(µ p ) = σ, where σ is the q-bar-core of both λ p and µ p . Then by Lemma 4.3(4) we have (aλ) p = (bµ) p = σ, so using Lemma 4.3(1) we see that σ is the p-bar-core and the q-bar-core of both aλ and bµ; in particular, aλ and bµ have the same p-set. Moreover, by our assumption and the proof of the only 'only if' part of the proposition above both aλ and bµ have q-weighted p-quotient Q q p (λ). From the proof of [3, Proposition 4.1] and the fact that σ ∈ C p ∩ C q it follows that for each k ∈ {0, . . . , q − 1}, the elements ∆ j σ in the p-set (of σ, aλ and bµ) that are congruent to k modulo q form an arithmetic progression with common difference q. By the first paragraph of this proof we can therefore apply the level q action to aλ and arbitrarily reorder the elements (aλ) (j) ∈ Q p (aλ) such that j ≡ k (mod q), for each k, without affecting the q-weighted p-quotient.

Generalised bar-cores
Now that we have covered all of the necessary definitions and basic results relating to the action of W p , we arrive at the first of our main results. The following proposition is a generalisation of a theorem by Olsson [9,Theorem 4] which states that the q-barcore of a p-bar-core is again a p-bar-core, or in the notation used above, Proof. We use induction on wt q (λ), with the trivial case being that λ is a q-bar-core.
Assuming that this is not the case, we may find a removable q-bar: y ∈ λ such that y − q ∈ A(λ). We will describe how to remove q-bars from λ to obtain a new partition with the same q-bar-core as λ, with q-bar-weight strictly less than wt q (λ), and with p-bar-weight no more than wt p (λ).
Algebraic Combinatorics, Vol. 5 #4 (2022) Let y be any part of λ such that y − q ∈ A(λ). For any x ∈ A(λ) congruent to y modulo p such that x − q ∈ A(λ), replace x with x − q, then replace q − x ∈ A(λ) with −x. We keep repeating this process until there are no more such x (the process will terminate because λ has finitely many removable q-bars), then we name our new bar partition ν. Since each action corresponds to removing a q-bar from λ, and since we have removed at least one q-bar (replacing y with y − q, and q − y with −y, in A(λ)), we have ν q = λ q and wt q (ν) < wt q (λ).
We remarked earlier that the p-bar-weight of a bar partition λ is equal to half the number of pairs (x, a) ∈ A(λ) × N such that x − ap ∈ A(λ). We will call such a pair (x, a) a p-bar-weight pair for λ. It follows from our construction of ν that for any x ≡ y, y − q, q − y, −y (mod p) and a ∈ N, (x, a) is a p-bar-weight pair for ν if and only if it is a p-bar-weight pair for λ. We will consider the remaining possibilities for the residue of y modulo p and show that in each case ν has no more p-bar-weight pairs than λ, hence wt p (ν) wt p (λ).
First suppose that y ≡ q (mod p), so that we obtain A(ν) by repeatedly replacing each x ∈ A(λ) such that x ≡ q (mod p) and q − x ∈ A(λ) with x − q, then replacing q − x with −x, until there are no more such x. Since in this case y − q ≡ 0 ≡ q − y (mod p), we may compare the p-bar-weights of λ and ν by counting how many of the three pairs (x, a), (x − q, a), (x − 2q, a) are p-bar-weight pairs for each of the two bar partitions when x ≡ q (mod p) and a ∈ N. We will do this by considering each of the four possibilities for the size of X : , so the number of p-bar-weight pairs for λ, and for ν, amongst the three pairs (x, a), (x − q, a), and a) is not a p-bar-weight pair for ν. If only one of the three pairs is a p-bar-weight pair for ν, it must be (x − q, a) as necessarily If (x − q, a) and (x − 2q, a) are both p-bar-weight pairs for ν, then we must have so λ also has two p-bar-weight pairs out of the three. 2q, a) can be p-bar-weight pairs for ν as , a), (x − 2q, a) are p-bar-weight pairs for ν.
Next suppose that y ≡ 0 (mod p), so that we obtain A(ν) by repeatedly replacing each x ∈ A(λ) such that p|x and q − x ∈ A(λ) with x − q, then replacing q − x with −x, until there are no much such x. Since y ≡ −y (mod p), we can apply the same argument as above, when y ≡ q (mod p), and conclude that ν has no more p-barweight pairs than λ amongst (x + q, a), (x, a), (x − q, a), and thus wt p (ν) wt p (λ).

D. Yates
Finally, suppose that y ≡ q, 0 (mod p), so that y ≡ −y and y − q ≡ q − y. In this case, we need only consider replacing all pairs x, q − x ∈ A(λ) such that x ≡ y (mod p) with x − q, −x, so we are in a simpler situation; wt p (ν) wt p (λ) since for any x ≡ y (mod p) and a ∈ N, ν has no more p-bar-weight pairs than λ amongst (x, a) and (x − q, a).
Hence ν has no more p-bar-weight pairs than λ, and therefore has p-bar-weight no more than the p-bar-weight of λ. The result follows by induction.
From this purely combinatorial result we obtain an interesting algebraic corollary.
Corollary 5.2. For any µ ∈ P, if w is the weight of the p-block containing [µ], a spin representation of the symmetric group S r (r ∈ N), and [λ] is a spin representation of S r+iq corresponding to λ ∈ P obtained by adding q-bars to µ, for any i ∈ N, then [λ] belongs to a p-block of weight w.
In particular, if [µ] belongs to a p-block of weight w > 0, then [λ] belongs to a block of positive weight.
Next we will consider the set C p,q containing all bar partitions λ which satisfy wt p (λ) = wt p (λ q ). Proof. Say that (a, b, c) is a bad triple for λ if a, b, c satisfy the conditions above. When (a, b, c) is bad, either a > c or b + c − a > b; either way, since a ≡ c (mod q) we find that λ has a removable q-bar and is thus not a q-bar-core. Hence the proposition is true when λ ∈ C q . Now we assume λ is not a q-bar-core, and choose y ∈ λ such that y − q ∈ A(λ). We define a new bar partition ν as in the proof of Proposition 5.1: by repeatedly replacing pairs x, q − x ∈ A(λ) with x − q and −x, respectively, when x ≡ y (mod p).
By induction it suffices to show that either: (1) wt p (ν) = wt p (λ), and there is a bad triple for ν iff. there is a bad triple for λ; or (2) wt p (ν) < wt p (λ), and there is a bad triple for λ.
Suppose first that there are no pairs x, q − x satisfying x, q − x ∈ A(λ) and x ≡ y (mod p).
Algebraic Combinatorics, Vol. 5 #4 (2022) We first assume that y ≡ q (mod p), and let x ≡ q (mod p). Then there are eight different possibilities for the intersection of A(λ) and {x, x − q, x − 2q}: However, the last four possibilities are all excluded by our assumption that there are no pairs x, q − x satisfying (1), so we find that ν = δ 0 λ. Therefore wt p (ν) = wt p (λ) by If y ≡ 0 (mod p) we are in an identical situation to the above: ν = δ 0 λ. When y ≡ q, 0 (mod p), we have a similar situation: since there are no pairs x, q − x satisfying (1), we have ν = δ i λ, where (i + 1)q ≡ y (mod p). Hence wt p (ν) = wt p (λ), Finally, we assume that there is a pair x, q − x satisfying (1), so that (y, x, y − q) is a bad triple for λ. We argue that wt p (ν) < wt p (λ), as in the proof of Proposition 5.1: If y ≡ q (mod p) and we let z := max{x, y}, l := |x−y| /p, then exactly one of (z, l), is a p-bar-weight pair for λ, then (z − q, l) is a p-bar-weight pair for ν but neither of (z, l), (z − 2q, l) are; and if (z − 2q, l) is not a p-bar-weight pair for λ, then none of (z, l), (z − q, l), (z − 2q, l) are p-bar-weight pairs for ν.
If instead we have y ≡ 0 (mod p), and again let z := max{x, y} and l := |x−y| /p, then exactly one of (z, l), (z − q, l) is a p-bar-weight pair for λ. Now if (z + q, l) is a p-bar-weight pair for λ, then (z, l) is a p-bar-weight pair for ν but neither of (z + q, l), (z − q, l) are; and if (z + q, l) is not a p-bar-weight pair for λ, then none of (z + q, l), (z, l), (z − q, l) are p-bar-weight pairs for ν.
If y ≡ q, 0 (mod p) then we define z and l as above so that exactly one of (z, l), (z − q, l) is a p-bar-weight pair for λ and neither is a p-bar-weight pair for ν.
Thus it follows from the proof of Proposition 5.1 that there are less p-bar-weight pairs for ν then there are p-bar-weight pairs for λ.
Proof. The condition in Proposition 5.3 is symmetric in p and q.
While the last result may seem surprising given the definition of C p,q , this symmetry is the motivation behind the study of this set. Furthermore, the next result shows that C p,q is closed under the level q action of W p .
Proposition 5.5. For any λ ∈ P and a ∈ W p , if λ ∈ C p,q , then aλ ∈ C p,q .
Interchanging p and q and appealing to Corollary 5.4, we see that C p,q is also a union of orbits for the level p action of W q . The actions of W p and W q clearly commute because the action of W p on an integer does not change its residue modulo Algebraic Combinatorics, Vol. 5 #4 (2022) D. Yates q, and the action of W q does not change its residue modulo p. Hence C p,q is a union of orbits for the action of W p × W q . We will look at these orbits in more detail, first by considering just the level q action of W p .
Proposition 5.6. Suppose λ ∈ P, and let O be the orbit containing λ under the level q action of W p . Then the following are equivalent: Proof. Since C q ⊂ C p,q , Proposition 5.5 shows that if O contains a q-bar-core, then λ ∈ C p,q . Hence the second statement implies the first. Trivially the third statement implies the second, so it remains to show that the first implies the third. So suppose that λ ∈ C p,q , and we can assume that λ is not a q-bar-core or the third statement is trivial. Thus we may find a pair y, q − y ∈ A(λ). By the proof of Proposition 5.3, there are no pairs x, q − x ∈ A(λ) with either x or q − x ≡ y (mod p), and if we take i ∈ {0, . . . , (p−1) /2} such that iq ≡ y (mod p), then the bar partition ν = δ i λ satisfies ν q = λ q and wt q (ν) < wt q (λ). Since wt p (ν) = wt p (δ i λ) = wt p (λ), ν is also in C p,q , and by induction the orbit containing ν contains ν q .
Corollary 5.7. Let O be an orbit of W p × W q consisting of bar partitions in C p,q . Then O contains exactly one bar partition that is both a p-bar-core and a q-bar-core.
Proof. Let λ be a bar partition in O. Then by Proposition 5.6, λ q ∈ O, and by the same result with p and q interchanged, the bar partition ν = (λ q ) p lies in O. Obviously ν is a p-bar-core, and by Proposition 5.1, it is also a q-bar-core. Now suppose that there is another bar partition in O that is both a p-bar-core and a q-bar-core. We can write this as baν, with a ∈ W p and b ∈ W q . Since wt q (δ j λ) = wt q (λ) for any j ∈ {0, . . . , (q−1) /2} (by interchanging p and q in the proof of Proposition 5.5), it follows that wt q (aν) = wt q (baν) = 0, hence it follows from (δ i λ) q = λ q (for any i ∈ {0, . . . , (p−1) /2}) that Similarly bν = ν, and thus baν = ν.
Remark. From Proposition 5.6 and Corollary 5.7, we see that two bar partitions λ, µ ∈ C p,q lie in the same orbit of W p × W q if and only if the p-bar-cores of (λ q ) and (ν q ) are equal. However, it does not seem to be easy to tell when two arbitrary bar partitions lie in the same orbit.
Proof. If we can remove a pq-bar from λ to obtain a new bar partition ν, then we can also remove q successive p-bars, or p successive q-bars, to obtain ν from λ. Thus ν q = λ q and wt p (ν) wt p (λ) − q, so we have It follows from Proposition 5.1 that (λ q ) p and (λ p ) q are both p-bar-cores and qbar-cores, and by Proposition 5.6 they both lie in the same orbit as λ under the action of W p × W q . Hence the result follows from Corollary 5.7.

The sum of a p-bar-core and a q-bar-core
In the present section we will give a constructive method for determining the bar partition in C p,q with a given p-bar-core µ and q-bar-core σ such that µ q = σ p . The resulting bar partition can be interpreted as the 'sum' of µ and σ. Proposition 6.1. Suppose µ ∈ C p and σ ∈ C q , and that µ q = σ p . Then there is a unique bar partition λ ∈ C p,q with λ p = µ and λ q = σ. Moreover, and λ is the unique smallest bar partition with p-bar-core µ and q-bar-core σ.
Proof. Let τ = µ q , and consider the action of W p × W q on P. By Proposition 5.6 we can find a ∈ W p and b ∈ W q such that aτ = µ and bτ = σ, and we let λ = aσ, so that λ ∈ C p,q (as it lies in the same orbit as the q-bar-core σ). Then we have λ q = σ q = σ, and by the proof of Proposition 5.5, we have Moreover, we have Now suppose ν is a bar partition distinct from λ with ν p = µ and ν q = σ, and let a, b be as above. Then we have (a −1 ν) q = ν q = σ, but a −1 ν = a −1 λ = σ, so |a −1 ν| > |σ|. Hence, again using the proof of Proposition 5.5, we have which means that ν ∈ C p,q . Furthermore, we see that wt p (ν) > wt p (λ), so |ν| > |λ|. Hence λ is the unique smallest bar partition with p-bar-core µ and q-bar-core σ.
Proof. Suppose λ ∈ C p,q and let µ = λ p , σ = λ q . Then by Proposition 6.1, λ is the unique smallest bar partition in C p,q with p-bar-core µ and q-bar-core σ, and therefore the only one whose parts sum to N .
Conversely, suppose λ ∈ C p,q . Then we can find integers a, b, c such that a ≡ b  (mod p), a ≡ c (mod q), a, b + c − a ∈ A(λ) and b, c ∈ A(λ) (by Proposition 5.3).
We first assume that a, b, c, b + c − a, −a, −b, −c and a − b − c are distinct integers. Define a new bar partition ν by its bead configuration on the p-runner abacus, Then ν can be obtained from λ by removing a |b − a|-bar and adding a |b − a|-bar, so |ν| = |λ| and since p|(a − b), we have ν p = λ p . Alternatively, we can obtain ν from λ Algebraic Combinatorics, Vol. 5 #4 (2022) 681 D. Yates by removing a |c − a|-bar and adding a |c − a|-bar, so it follows from the divisibility of a − c by q that ν q = λ q . Hence λ is not the only partition with p-bar-core µ and q-bar-core σ whose parts sum to N .
If instead the integers a, b, c and b + c − a are not distinct, i.e. if a = b + c − a or b = c, then they must all be congruent modulo pq, and λ therefore cannot be a pq-bar-core. By adding and removing the same number of pq-bars, we can obtain a new partition ν from λ with ν pq = λ pq , |ν| = |λ|, and ν = λ. Then it is easy to see that ν p = µ and ν q = σ, as removing a pq-bar is the same as removing p q-bars or q p-bars.
Now we may assume that a, b, c, b + c − a are distinct but , a contradiction; so we need to consider six separate cases (or three, up to symmetry): , then we may define A(ν) = {−a, c, a − c} ∪ A(λ)\{a, −c, c − a} so that we can obtain ν from λ by removing and adding |a|-bars, or by removing and adding |c − a|-bars, and thus ν has the same size, p-bar-core, and q-bar-core as λ, since p|a and q|(a − c), but is distinct from λ. If instead a − c ∈ A(λ), defining A(ν) = {−a, c − a, 2a − c} ∪ A(λ)\{a, a − c, c − 2a}, we also have |ν| = |λ|, and we can obtain ν from λ by adding and removing |a|-bars, or by adding and removing |c − a|-bars.
Proof. When σ p = Υ p,q , the non-zero elements of the shared p-set of Υ p,q and σ belong to two congruence classes modulo p, so by Lemmas 3.1(1) and 6.3, Q p (σ) consists of a q-bar-core σ (0 mod p) and at most two distinct q-cores. Moreover, since (σ (j mod p) ) = σ (−j mod p) for each j ≡ 0 (mod p), the p-quotient of σ consists of σ (0 mod p) (the parts of σ which are multiples of p, divided by p) and either p − 1 other empty bar partitions (when σ = Υ p,q ), or (p−1) /2 copies each of two conjugate partitions. When σ p = Υ q,p , all of the non-zero elements in the p-set of σ are congruent modulo p, so again by Lemmas 3.1(1) and 6.3, the q-quotient Q q (σ) simply consists of a q-bar-core σ (0 mod p) and p − 1 copies of a self-conjugate q-core.
The above lemma means that the construction of µ σ becomes even more straightforward when µ and σ are contained in the Υ-orbit. Proposition 7.3. Suppose µ ∈ C p and σ ∈ C q are such that µ q = σ p = Υ p,q . Then µ σ is the bar partition λ with λ p = µ, λ (0 mod p) = σ (0 mod p) , and otherwise.
Moreover, µ σ is also the bar partition with q-bar-core σ and the same q-quotient as µ.

D. Yates
Proof. There are (p+1) /2 elements in the p-set of both Υ p,q = σ p and σ that are divisible by q, and the other (p−1) /2 elements are congruent to p modulo q. Hence, it follows from Proposition 6.4 that µ σ = λ. The elements ∆ 1 mod q σ, . . . , ∆ p−1 mod q σ of the q-set of σ are all congruent modulo p, so the p-quotients of the two bar partitions σ and µ σ are exactly the same.
(2) Suppose λ ∈ P and a ∈ W p . Then (aλ) (0 mod q) = a(λ (0 mod q) ), where a acts at level q on P and at level 1 on C p .
Then µ ∈ C p and σ ∈ C q by Proposition 7.7, so we can define Ψ(X, α, β) to be the bar partition µ σ, which is contained in the orbit of Υ p,q under the action of W p × W q by Proposition 4.6.
Then since µ σ shares a q-quotient with µ and has the same p-quotient as σ up to reordering, we find that Φ(µ σ) = (X, α, β). Hence Φ and Ψ are mutual inverses, and thus bijections.