OF THE PLETHYSTIC MURNAGHAN–NAKAYAMA RULE USING LOEHR’S LABELLED ABACUS

. The plethystic Murnaghan–Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form p r ◦ h m as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Nicholas A. Loehr. “Abacus proofs of Schur function identities”. In:


Introduction
In 2010, Loehr [7] introduced a labelled abacus as a combinatorial model for antisymmetric polynomials a β .By considering appropriate moves of labelled beads and their collisions, he proved standard formulas for decompositions of products of Schur polynomials with other symmetric polynomials, namely Pieri's rule, Young's rule, the Murnaghan-Nakayama rule and the Littlewood-Richardson rule, as well as the equivalence of the combinatorial and the algebraic definitions of Schur polynomials and a formula for inverse Kostka numbers.We follow the slogan 'when beads bump, objects cancel' from [7] and enrich this collection of results by proving the plethystic Murnaghan-Nakayama rule.
See §2.1 for definitions of sgn r and r-decomposable skew partitions, which are the skew partitions for which sgn r is non-vanishing.
More precisely, we prove the formula in Theorem 1.1 for symmetric polynomials in N variables where N ≥ |µ| + rm.This is equivalent to Theorem 1.1 as both sides of the formula have degree |µ| + rm.One can easily extend the result to a decomposition of s µ (p ρ • h ν ) as a sum of Schur functions for any non-empty partitions ρ and ν by iterating Theorem 1.1 and using the formula p ρ • h ν = i,j p ρ i • h ν j .
The plethystic Murnaghan-Nakayama rule is a generalisation of the usual Murnaghan-Nakayama rule, which can be obtained from Theorem 1.1 by letting m = 1, that is by replacing the plethysm p r • h m with p r .By letting r = 1 instead, one obtains Young's rule which describes the decomposition of s µ h m as a sum of Schur functions.
While a description of the plethysm p r •h m = p r •s (m) is known and follows from Theorem 1.1 after letting µ = ø, in general, it is a difficult problem to decompose a plethysm as a sum of Schur functions; see [11, Problem 9], which asks for a decomposition of plethysms of the form s (a) •s (b) .Plethysms play an important role not only in the study of symmetric functions but also in the representation theory of symmetric groups and general linear groups; see [10,Chapter 7: Appendix 2].The connection of plethysms and representation theory was used, for instance, in [1] to find the maximal constituents of plethysms of Schur functions using the highest weight vectors.
The plethystic Murnaghan-Nakayama rule appeared first in [3, p.29], where it was proved using Muir's rule.Since then, it has been proved using several different methods: in [4, Proposition 4.3] characters of symmetric groups are used, [12] uses James' (unlabelled) abacus and induction on m and [2, Corollary 3.8] uses vertex operators.In comparison to these proofs, our elementary proof using the labelled abaci arises naturally by 'merging' the proofs from [7] of the Murnaghan-Nakayama rule and Young's rule.
Since the publication of the original paper introducing the labelled abaci, Loehr has used it to prove the Cauchy product identities in [6], and together with Wills they introduced abacus-tournaments to study Hall-Littlewood polynomials in [8].

Partitions.
A partition λ = (λ 1 , λ 2 , . . ., λ l ) is a non-increasing sequence of positive integers.The size of a partition λ, denoted by |λ|, equals l i=1 λ i .We call the number of elements of λ the length of λ and denote it by ℓ(λ).We use the convention that for i > ℓ(λ) we have λ i = 0, and we allow ourselves to attach extra zeros to a partition without changing it.We write Par ≤N for the set of partitions of length at most N , Par(n) for the set of partitions of size n and Par ≤N (n) for the intersection of these two sets.
The Young diagram of a partition λ is Y λ = (i, j) ∈ N 2 : i ≤ ℓ(λ), j ≤ λ i and we refer to its elements as boxes.We write µ ⊆ λ whenever Y µ ⊆ Y λ .A skew partition λ/µ is a pair of partitions µ ⊆ λ and its Young diagram is Y λ/µ = Y λ \ Y µ .We define the top of a skew partition λ/µ, denoted as t(λ/µ), to be 0 if λ = µ, and the least i such that λ i = µ i otherwise.We similarly define the bottom b(λ/µ) of a skew partition by replacing the word 'least' with 'greatest'.
Let r be a positive integer.An r-border strip is a skew partition λ/µ consisting of r edge-adjacent boxes such that for all (i, j) ∈ Y λ/µ we have (i + 1, j + 1) / ∈ Y λ/µ .It follows from the definition that for any partition λ and a non-negative integer t, there is at most one r-border strip λ/µ with t(λ/µ) = t.A skew partition λ/µ is r-decomposable if there are partitions such that the skew partition γ (i+1) /γ (i) is an r-border strip for all 0 ≤ i ≤ d − 1 and t(γ (1) /γ (0) ) ≥ t(γ (2) /γ (1) ).If such a decomposition exists, it is unique as there is a unique choice for γ (d−1) as λ/γ (d−1) is an r-border strip with t(λ/γ (d−1) ) = t(λ/µ) and an inductive argument then applies.Examples of Young diagrams of r-border strips and an r-decomposable partition are in Figure 1.  1) .The bottoms of these 5-border strips are 5, 5 and 3, respectively, while their tops are 2, 2 and 1, respectively.As the tops are in non-increasing order, the skew partition λ/µ is 5-decomposable.

Symmetric polynomials.
A composition of a non-negative integer m is a sequence β = (β 1 , β 2 , . . ., β N ) of non-negative integers such that N i=1 β i = m.The length of a composition is the number of its elements and we write Com N (m) for the set of compositions of m of length N .
We now introduce the required elements of the ring of symmetric polynomials in N variables, called Λ N , as defined, for instance, in [9, §I.2].For a positive integer m, the complete homogeneous symmetric polynomial h m (x 1 , x 2 , . . ., x N ) is defined as β∈Com N (m) x β , where x β is the monomial x rβ , where rβ = (rβ 1 , rβ 2 , . . ., rβ N ).One can define a plethysm f • g for any elements f and g of Λ N by extending the map To define the final ingredient, Schur polynomials, we introduce the antisymmetric polynomials a β : for a positive integer N and a composition β of length N we let a β = det(x Given a partition λ of length at most N , we now define the Schur polynomial where While compared to other definitions such as [10, Definition 7.10.1], it is not immediately obvious that Schur polynomials are polynomials, we use this definition as one requires antisymmetric polynomials to use the labelled abaci.
Example 2.1.Let N = 3.Then 2.3.Labelled abacus.Most of our terminology and notation for labelled abaci comes from [7].A labelled abacus with N beads is a sequence w = (w 0 , w 1 , w 2 , . . . ) indexed from 0 with precisely N non-zero entries, which are 1, 2, . . ., N .For 1 ≤ B ≤ N , we let w −1 (B) to be the index i such that for the indices i such that w i is non-zero and define the support supp(w) to be {ι i (w) : Finally, the sign sgn(w) is the sign of the permutation σ w ∈ S N given by σ w (B) = w ι B (w) .
One should imagine that a labelled abacus w consists of a single runner with positions 0, 1, 2, . . ., where the position i is empty if w i = 0, and is occupied by a bead labelled by w i otherwise.The value w −1 (B) is the position of bead B, the support is the set of the non-empty positions and ι t (w) is the t-th largest occupied position.The permutation of beads in w, starting from beads ordered in decreasing order, is then σ w .With this in mind, we introduce the following intuitive terminology.
Fix positive integer r and B = C ≤ N and write y = w −1 (B) and z = w −1 (C) for the positions of beads B and C, respectively.A labelled abacus w ′ is obtained from w by swapping beads B and If that is the case, a labelled abacus w ′ is obtained from w by r-moving bead B if w ′ y = 0, w ′ y+r = B and w ′ i = w i otherwise.If bead B in not r-mobile, we say that it r-collides with bead w y+r .Similarly, bead B is left-r-mobile if y ≥ r and w y−r = 0.If that is the case, a labelled abacus w ′ is obtained from w by r-moving bead B leftwards if w is obtained from w ′ by r-moving bead B. Finally, for t ≤ N , the t-th rightmost bead of w is σ w (t).
It is easy to see how the sign changes when performing the above operations.To state the formula, we define the number of beads between positions Proof.
The importance of labelled abaci comes from the simple identity which holds for any λ ∈ Par ≤N ; see [7, p.1359].Note that the identity is just the expansion of a λ+δ(N ) = det(x

Proof of the plethystic Murnaghan-Nakayama rule
The following is an immediate consequence of a well-known result connecting moves on an (unlabelled) abacus and removals of border strips.Lemma 3.1.Let r, t and N be positive integers such that t ≤ N .For λ ∈ Par ≤N and w ∈ Abc N (λ) the following holds: (i) There is a bijection θ between left-r-mobile beads of w and r-border strips of the form λ/µ given by mapping bead B to an r-border strip λ/µ, where µ is the shape of the labelled abacus obtained from w by r-moving bead B leftwards.(ii) If λ/µ is an r-border strip with top t, then θ −1 (λ/µ) is the t-th rightmost bead of w. (iii) With µ as in (ii), the number of beads between positions ι t (w) and ι t (w) − r equals b(λ/µ) − t(λ/µ).(iv) Continuing with the same µ, there is a bijection φ : Abc N (λ) → Abc N (µ) given by r-moving the t-th rightmost bead leftwards.
Proof.Part (i) (without labels) is [5, Lemma 2.7.13].Now suppose that w ′ is obtained from w by r-moving bead B leftwards.If j is the least index in which sh(w) and sh(w ′ ) differ, then bead B is the j-th rightmost bead of w.
Similarly, if j is the largest such index, then bead B is the j-th rightmost bead of w ′ .Thus we deduce (ii) and (iii).Finally, (iv) follows from (i) and (ii).
We can rephrase this result to obtain a characterisation of r-decomposable partitions.In the statement, one should bear in mind that in (ii) and (iii) the r-moves are made consecutively, and thus a bead may r-move multiple times.
Corollary 3.4.Let r and N be positive integers and let λ, µ ∈ Par ≤N .The following are equivalent: (i) λ/µ is an r-decomposable skew partition.(ii) There are labelled abaci w ∈ Abc N (µ) and w ′ ∈ Abc N (λ) such that w ′ is obtained from w by a series of r-moves of beads from positions i 1 , i 2 , . . ., i m , where is obtained from w by a series of r-moves of beads from positions i 1 , i 2 , . . ., i m , where Moreover, if (i)-(iii) hold true, then the choice of w and the series of rmoves in (iii) is unique, |λ| = |µ| + mr where m is the number of r moves in (ii) and (iii) and sgn(w ′ ) = sgn r (λ/µ) sgn(w).
We are now ready to prove the main theorem.
Proof of Theorem 1.1.Fix a partition µ and positive integers r, m and N such that N ≥ |µ| + rm.Using the definition of Schur polynomials by antisymmetric polynomials in (2), our desired equality (in N variables) becomes Using (3) and the definition of the plethysm p r • h m , we can expand the left-hand side as where the weight wt r (w, β) equals wt(w)x rβ and the sign sgn(w, β) is just sgn(w).Given a labelled abacus w ∈ Abc N (µ) and a composition β ∈ Com N (m), we consider a process on w in which we read w from left and every time we see bead B with β B ≥ 1 we attempt to r-move it, provided that we have not already r-moved it β B -times.
The process always terminates as when we look at position i, the beads which are yet to be r-moved (that is beads C such that α C ≥ 1) lie on positions greater or equal to i.We write I and J for the set of pairs (w, β) which are unsuccessful and successful, respectively.For (w, β) ∈ I, write B for the label of the non-r-mobile bead which terminated the process and C for the label of the bead that bead B r-collided with.We define a labelled abacus w ′ to be obtained from w by swapping beads B and C. We also define a sequence β ′ of length N by We then define ǫ(w, β) as (w ′ , β ′ ).For (w, β) ∈ J, we define a labelled abacus ψ(w, β), which is the labelled abacus v at the end of the process.See Figure 4 for an example.3.As observed, we have (w, (0, 2, 0, 0, 1, 0)), (w, (0, 2, 1, 0, 0, 0)) ∈ J and (w, (0, 2, 0, 1, 0, 0)) ∈ I.The diagrams above display the images of maps ǫ and ψ applied to these three pairs.We claim that (4) follows once we establish the following two statements: (i) The map ǫ is a weight-preserving involution on I, which reverses the sign.(ii) The map ψ is a weight-preserving bijection from J to λ Abc N (λ), where the union is taken over partitions λ ∈ Par(|µ| + rm) such that λ/µ is an r-decomposable skew partition.Moreover, for any (w, β) ∈ J we have sgn(ψ(w, β)) = sgn r (λ/µ) sgn(w, β), where λ is the shape of ψ(w, β).
We now prove this claim.We can split the final sum of (5), which equals the left-hand side of (4), as The first sum is zero from (i), while the second sum can be rewritten, using (ii), as sgn(w) wt(w), which is the right-hand side of ( 4) by ( 3), as required.
Thus it remains to show (i) and (ii).For (i), let (w, β) ∈ I and write (w ′ , β ′ ) for ǫ(w, β).We keep the notations B and C for the labels of the beads such that the process terminated with non-r-mobile bead B which rcollided with bead C. Clearly, w ′ lies in Abc N (µ).We now check that β lies in Com N (m).Since the beads only r-move, we have r | w −1 (C) − w −1 (B); thus β ′ is an integral sequence.We also see that bead C has not r-moved during the process; hence w −1 (C) > w −1 (B) and we must have checked whether bead B is r-mobile at least w −1 (C) − w −1 (B) /r -times.The latter statement implies that β B ≥ w −1 (C) − w −1 (B) /r, and in turn the entries of β ′ are non-negative.From (6), clearly, β ′ and β have the same sum of entries, and hence β ′ ∈ Com N (m).

Lemma 2 . 3 .
Let w be a labelled abacus with N beads.Fix B ≤ N and write y = w −1 (B) for the position of bead B. (i) If C ≤ N and C = B and w ′ is obtained from w by swapping beads B and C, then sgn(w ′ ) = − sgn(w).(ii) If bead B is r-mobile for some chosen positive integer r and w ′ is obtained from w by r-moving bead B, then sgn(w ′ ) = (−1) u sgn(w), where u is the number of beads between y and y + r.
for a diagrammatic example of two such sequences.Lemma 3.3.Let r, m and N be positive integers and λ, µ ∈ Par ≤N .