Cyclic sieving, necklaces, and branching rules related to Thrall's problem

We show that the cyclic sieving phenomenon of Reiner--Stanton--White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraskiewicz--Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b \hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem.


Introduction
The Lie module L n is the nth degree component of the free Lie algebra over C with m generators, which is naturally a GL(C m )-module. The Lie modules were famously studied by Thrall [Thr42] in the 1940's and have been extensively studied by Brandt [Bra44], Klyachko [Kly74], Kraśkiewicz-Weyman [KW01], Garsia [Gar90], Gessel-Reutenauer [GR93], Reutenauer [Reu93], Sundaram [Sun94], Schocker [Sch03], and many others. Thrall more generally introduced a certain GL(C m )-decomposition ⊕ λ∈Par L λ of the tensor algebra of C m arising from the Poincaré-Birkhoff-Witt theorem, where L (n) = L n . The L λ are sometimes called the higher Lie modules. Thrall's original paper considered the determination of the multiplicity of the irreducible V µ in L λ , which is often referred to as Thrall's problem. This problem is still open 75 years later. See Section 2.6 and [Reu93] for more background on Thrall's problem and [Rei15] for a recent summary of related work. See Section 2 for missing definitions.
Kraśkiewicz-Weyman [KW01] gave a combinatorial solution to Thrall's problem when λ = (n). In particular, they showed the multiplicity of V µ in L (n) is #{T ∈ SYT(µ) : maj(T ) ≡ n 1}, i.e. the number of standard tableaux of shape µ with major index 1 modulo n. Their argument crucially hinges upon the formula ω n is a primitive nth complex root of unity, σ n is an n-cycle in the symmetric group S n , and χ µ is the character of the S n -irreducible indexed by a partition µ of n. The analysis in [KW01] is somewhat indirect. It involves results of Lusztig and Stanley on coinvariant algebras and an intricate though beautiful argument involving ℓ-decomposable partitions.
Equation (1) bears a striking resemblance to the cyclic sieving phenomenon (CSP) of Reiner-Stanton-White, which we now recall.
Definition 1.1. [RSW04] Suppose C n is a cyclic group of order n generated by σ n , W is a finite set on which C n acts, and f (q) ∈ Z ≥0 [q]. We say the triple (W, C n , f (q)) exhibits the cyclic sieving phenomenon (CSP) if for all r ∈ Z, f (ω r n ) = #W σ r n := #{w ∈ W : σ r n · w = w} = χ W (σ r n ), where ω n is a primitive nth root of unity and χ W is the character of W as a C nmodule.
See [Sag11] for an excellent survey and introduction to cyclic sieving. The following cyclic sieving result also due to Reiner-Stanton-White is intimately related to (1). We use W stat (q) to denote w∈W q stat(w) .
Theorem 1.2. [RSW04, Theorem 8.3, Proposition 4.4] Let α n, let W α denote the set of all words of content α, let C n act on W α by rotation, and let maj denote the major index statistic. Then, the triple W α , C n , W maj α (q) exhibits the CSP.
Since the sets W α are precisely the S n -orbits for the natural S n action on length n words, Theorem 1.2 may be thought of as a "universal sieving result" as follows. A very similar observation appeared in [BER11,Prop. 3.1]. Corollary 1.3. Let W be a finite set of length n words closed under the S n -action. Then, the triple W, C n , W maj (q) exhibits the CSP.
In [AS18], the authors introduced a new statistic on words, flex. As an example, flex(221221) = 2 · 3 = 6 since 221221 is the concatenation of 2 copies of the primitive word 221 and 221221 is third in lexicographic order amongst its 3 cyclic rotations. See Definition 2.3 for details. The flex statistic was designed to be "universal" for cyclic rather than symmetric actions on words in the following sense. A corollary of these universal sieving results is the following equidistribution result. A more refined statement appeared in [AS18].
Theorem 1.5. [AS18,Theorem 8.4] Let W n denote the set of length n words, let maj n denote the major index modulo n taking values in {1, . . . , n}, and let cont denote the content of a word. We then have W cont,maj n n (x; q) = W cont,flex n (x; q).
In Section 3, we show that the following well-known result of Kraśkiewicz-Weyman is essentially a corollary of Theorem 1.5. Here χ r is the linear representation of the cyclic group C n given by χ r (σ n ) = ω r n . Theorem 1.6. [KW01] We have where a λ,r := #{Q ∈ SYT(λ) : maj(Q) ≡ n r}.
Klyachko [Kly74,Prop. 1] showed that the Lie modules L n and the induced representations χ 1 ↑ Sn Cn are Schur-Weyl duals. The λ = (n) case of Thrall's problem thus follows from Theorem 1.6 when r = 1. More precisely, Klyachko expressed both the characteristic of χ 1 ↑ Sn Cn and the character of L n as content generating functions on primitive necklaces of length n words. We generalize this observation in Section 3 as follows, which also naturally motivates the flex statistic.
Theorem 1.7. Let NFD n,r denote the set of necklaces of length n words with frequency dividing r, F n,r denote the set of length n words with flex equal to r, and M n,r (x) denote the set of length n words with maj n equal to r. Then ch χ r ↑ Sn Cn = NFD cont n,r (x) = F cont n,r (x) = M cont n,r (x). Our new proof of Kraśkiewicz-Weyman's result reduces the problem of finding a bijective proof of a well-known symmetry result following from Theorem 1.6 to finding a bijective proof of the above equidistribution result, Theorem 1.5; see Corollary 3.5. It also provides a thus far rare example of an instance of cyclic sieving being used to prove other results rather than vice-versa.
In Section 4, we give a new proof of a result of Stembridge [Ste89] which settled a conjecture of Stanley describing the irreducible multiplicities of induced representations χ r ↑ Sn σ for arbitrary σ ∈ S n . The corresponding generalized major index statistics arise very naturally from the combinatorics of orbits and cyclic sieving.
In Section 5, we prove and generalize a result of Schocker [Sch03] concerning the higher Lie modules. Thrall's problem may be reduced to the λ = (a b ) case by the Littlewood-Richardson rule. Bergeron-Bergeron-Garsia [BBG90] identified the Schur-Weyl dual of L (a b ) as a certain induced module χ 1,1 ↑ S ab Ca≀S b where C a ≀ S b is a wreath product; see Section 2.7 for details. Schocker gave a formula for the multiplicity of the irreducible V µ in L (a b ) , though it involves many subtractions and divisions in general. We generalize Schocker's formula to all one-dimensional representations of C a ≀ S b . In our approach, the subtractions and divisions in Schocker's formula arise naturally from the underlying combinatorics using Möbius inversion and Burnside's lemma.
The basic outline of each argument is the same: we obtain an orbit generating function from an explicit basis of a GL(V )-module, we construct an appropriate necklace generating function, we use cyclic sieving to rewrite this generating function using words and descent statistics like the major index, and we finally apply RSK to get a Schur expansion. Transitioning from an orbit generating function to a necklace generating function where we can apply cyclic sieving involves various combinatorial techniques.
In Section 6, we discuss applying aspects of our approach to Thrall's problem in general. The arguments in the preceding sections strongly suggest attacking Thrall's problem by considering all branching rules for the inclusion C a ≀ S b ֒→ S ab rather than considering only one such rule. To that end, consider the irreducible representations S λ of C a ≀ S b , which are indexed by the set of a-tuples λ = (λ (1) , . . . , λ (a) ) of partitions with a r=1 |λ (r) | = b. We first give the following plethystic expression for the corresponding characteristic.
We then identify the analogues of the flex and maj n statistics in this context, which send words to such a-tuples of partitions. We consequently give the following monomial expansion of the corresponding graded Frobenius series. See Section 2.7 and Section 6 for details.
The rest of the paper is organized as follows. In Section 2, we review combinatorial and representation-theoretic background. In particular, we summarize work related to Kraśkiewicz-Weyman's result, Theorem 1.6, in Section 2.5, and we discuss the current status of Thrall's problem in Section 2.6. In Section 3, we present our proof of Kraśkiewicz-Weyman's result, Theorem 1.6, using cyclic sieving. In Section 4, we give an analogous proof of Stembridge's result, Theorem 4.11. In Section 5, we give generalizations of Schocker's result, Theorem 5.11. In Section 6, we define the statistics flex b a and maj b a , prove Theorem 1.8 and Theorem 1.9, and discuss how the approach could be used to find the branching rules for C a ≀ S b ֒→ S ab .

Background
Here we provide background on words, tableaux, Schur-Weyl duality, Kraśkiewicz-Weyman's result, Thrall's problem, and certain wreath products for use in later sections. All representations will be over C. We write [n] := {1, . . . , n}, #S for the cardinality of a set S, and 2.1. Words. We now recall standard combinatorial notions on words and fix some notation. A word w of length n is a sequence w = w 1 w 2 · · · w n of letters The content of a word w, written cont(w), is the sequence α = (α 1 , α 2 , . . .) where α j is the number of j's in w. Such a sequence α is called a (weak) composition of n, written α n. For n ≥ 1 and α n, we write the set of words of length n or content α as The set of all words with letters from Z ≥1 is a monoid under concatenation. A word is primitive if it is not a power of a smaller word. Any non-empty word w may be written uniquely as w = v f for f ≥ 1 with v primitive. The period of w, denoted period(w), is the length of v. The frequency of w, denoted freq(w), is f .
Definition 2.1. An orbit of w ∈ W n under rotation is a necklace, denoted [w]. Note that period(w) = #[w] and freq(w) · period(w) = n. Content, primitivity, period, and frequency are all well-defined on necklaces. For n ≥ 1, we write N n := {necklaces of length n words}.
2.2. Generating Functions. In most triples (W, C n , f (q)) that have been found to exhibit the CSP, f (q) is a statistic generating function on W for some well-known statistic. Given stat : W → Z ≥0 , we write the corresponding generating function as We use natural multivariable analogues of this notation as well. For example, letting where x (α 1 ,...,αm) := x α 1 1 · · · x αm m .
2.3. Tableaux. A partition of n, denoted λ ⊢ n, is a composition of n whose parts weakly decrease. Write Par for the set of all partitions. The Young diagram of λ is the upper-left justified collection of cells with λ i entries in the ith row starting from the top. We may write a partition in exponential form as λ = 1 m 1 2 m 2 · · · ⊢ n where m i is the number of parts of λ of size i. In this case, the number of elements of S n with cycle type λ is n! z λ where z λ := 1 m 1 2 m 2 · · · m 1 !m 2 ! · · · . A semistandard Young tableau of shape λ is a filling of the Young diagram of λ with entries from Z ≥1 which weakly increases along rows and strictly increases along columns. The set of semistandard Young tableaux of shape λ is denoted SSYT(λ). The content of P ∈ SSYT(λ), denoted cont(P ), is the composition whose j-th entry is the number of j's in P . The set of standard Young tableaux of shape λ, denoted SYT(λ), is the subset of SSYT(λ) consisting of tableaux of content (1, . . . , 1) n. The descent set of a tableau Q ∈ SYT(λ), denoted Des(Q), is the set of all i ∈ [n−1] such that i + 1 lies in a lower row of Q than i.
The shape of w under RSK, denoted sh(w), is the common shape of P (w) and Q(w). Two well-known properties of the RSK correspondence are cont(w) = cont(P (w)), Des(w) = Des(Q(w)).
The fact that Des(w) = Des(Q(w)) is originally due to Schützenberger [ We will repeatedly use the RSK correspondence to transition from the monomial to the Schur basis. These arguments all rely on the following result. Proof. Using RSK and (6), we have 2.4. Schur-Weyl Duality. We next summarize a few key points from the representation theory of S n and GL(C m ). See [Ful97] for more. The complex irreducible inequivalent representations of S n are canonically indexed by partitions λ ⊢ n and are called Specht modules, written S λ . The Frobenius characteristic map ch is defined by ch S λ := s λ (x) and is extended additively to all Let E be a finite-dimensional, polynomial representation of GL(V ) and pick a basis Thus, for any S n -module M , we have In light of this, we often leave dependence on m or V implicit.
2.5. Kraśkiewicz-Weyman Symmetric Functions. The symmetric functions appearing in Theorem 1.5 have a wealth of important interpretations. Here we summarize some of these interpretations.
These symmetric functions are intimately related to the irreducible representations of certain cyclic groups.
Definition 2.9. Recall σ n := (1 2 · · · n) ∈ S n and C n := σ n ≤ S n be the cyclic group of order n it generates. Fixing any primitive nth root of unity ω n , write the irreducible characters of C n as χ 1 , . . . , χ n where χ r (σ n ) := ω r n . We sometimes write χ r n if we want to specify the cyclic group C n as well. Theorem 1.6 gives our first interpretation of KW n (x; q), Since the regular representation of C n is ⊕ n r=1 χ r , when q = 1 the right-hand side of (8) is the Frobenius characteristic of the regular representation of S n , denoted CS n . The right-hand side of (8) is hence similar to a graded Frobenius series for CS n and tracks branching rules for the inclusion C n ֒→ S n . By Theorem 1.7, we can also write this series as Now consider the action of σ n on the S n -irreducible S λ . Since σ n n = 1 ∈ S n , the action of σ n on S λ is diagonal with eigenvalues ω k 1 n , ω k 2 n , . . . where ω n is a fixed primitive nth root of unity and 1 ≤ k i ≤ n for each i. Let P λ (q) := q k 1 + q k 2 + · · · be the generating function of the cyclic exponents k 1 , k 2 , . . ., which were studied extensively by Stembridge [Ste89]. Using the right-hand side of (8) and Frobenius reciprocity quickly gives the following. . The cyclic exponent generating function for S n is given by Next, extend the regular representation CS n to an S n × C n -module by letting S n act on the left and C n act on the right. There is a straightforward notion of an S n × C n -Frobenius characteristic map given by sending an irreducible S λ ⊗ χ r to s λ (x)q r where q is an indeterminate satisfying q n = 1. The following now follows easily using the right-hand side of (8).
Corollary 2.11. [KW01] The S n × C n -Frobenius characteristic of the regular representation is It is well-known that the type A n−1 coinvariant algebra R n is a graded S n -module which is isomorphic as an ungraded S n -module to CS n . We may give R n an S n × C nmodule structure by letting C n act on the kth degree component of R n by σ n · f := ω k n f , where ω n is a fixed primitive nth root of unity. Springer and, independently, Kraśkiewicz-Weyman showed that CS n and R n are isomorphic as S n × C n -modules. Consequently, from the right-hand side of (11), we have the following.
The graded Frobenius characteristic of the coinvariant algebra is the modified See also [Rho10, §3] for a nice summary of this connection. We may instead use the right-hand side of (7) as a starting point. From Lemma 2.7, it follows that From Theorem 1.5 and (14), our final interpretation of KW n (x; q) in this subsection is 2.6. Thrall's Problem. We next define the Lie modules L λ and summarize the status of Thrall's problem. See [Reu93] for more details.
as graded GL(V )-representations, where the sum is over all partitions and Sym m (M ) is the mth symmetric power of M [Reu93, Lemma 8.22]. The higher Lie module associated to λ = 1 m 1 2 m 2 · · · is defined to be Thrall's problem is the determination of the multiplicity of V µ in L λ (V ), for instance by counting explicit combinatorial objects. The well-known Littlewood-Richardson rule solves the analogous problem for V µ ⊗ V ν . It follows from (16) and the Littlewood-Richardson rule that, for the purposes of Thrall's problem, we may restrict our attention to the case when λ = (a b ) is a rectangle. Since the single-row case is particularly fundamental. Hall [Hal59, Lemma 11.2.1] introduced what is now called the Hall basis for L n (V ), which, in the m → ∞ limit, is in content-preserving bijection with primitive necklaces NFD n,1 . For each primitive necklace, Hall associates a bracketing of its elements using what is now known as the Lyndon factorization [CFL58]. He gives an explicit, though computationally complex, algorithm to express any bracketing as a linear combination of the bracketings associated to primitive necklaces. Linear independence of these generators follows from a dimension count.
Klyachko consequently observed that the Schur character of L n is the corresponding content generating function NFD cont n,1 (x). Taking symmetric powers, it follows that in the m → ∞ limit, L (a b ) (V ) has a basis indexed by multisets of primitive necklaces and the Schur character is the following content generating function. One formulation of Thrall's problem is hence to find the Schur expansion of the expressions in Lemma 2.13. While we will not have direct need of it, we would be remiss if we did not mention the following beautiful and important result of Gessel and Reutenauer [GR93, (2.1)]. The expansion of ch L λ in terms of Gessel's fundamental quasisymmetric functions is Gessel and Reutenauer gave an elegant bijective proof of (18) in [GR93] involving multisets of primitive necklaces as in Lemma 2.13. Another formulation of Thrall's problem is thus to convert the right-hand side of (18) to the Schur basis.
Since χ r ↑ Sn Cn depends up to isomorphism only on n and gcd(n, r), we also have the following well-known symmetry.
Remark 2.16. A bijective proof of this symmetry is currently unknown.
Thrall's problem is an instance of a plethysm problem as we next describe. See [Sta99, Appendix 2] for more details. Given polynomial representations of general linear groups where V, W, X are finite-dimensional complex vector spaces, the plethysm of their Schur characters is the Schur character of their composite: It is easy to see that ch Remark 2.17. At present, Thrall's problem has only been solved in the following cases: • when λ = (n) has a single part (see Corollary 2.14); 2.7. Wreath Products. The Schur-Weyl duals of the higher Lie modules L λ have also been identified in terms of induced representations of certain wreath products.
Here we summarize this connection as well as some related aspects of the representation theory of wreath products which will be used in Section 6. Our presentation largely mirrors [Ste89].
Definition 2.18. Given a group G, the wreath product of G with S n , denoted G≀S n , is the semidirect product explicitly described as follows. G ≀ S n is the set G n × S n with multiplication given by for all g 1 , . . . g n , h 1 , . . . , h n ∈ G and σ, τ ∈ S n . Furthermore, given α n, set which has a natural inclusion into G ≀ S n . Roughly speaking, G ≀ S n can be considered as the group of n × n "pseudo-permutation" matrices with entries from G. Now suppose U is a G-set and V is an S n -set. There is a natural notion of U ≀ V as a G ≀ S n -set. Explicitly, let U ≀ V be the set U n × V with G ≀ S n -action given by for all g 1 , . . . , g n ∈ G, σ ∈ S n , u 1 , . . . , u n ∈ U, v ∈ V . There is an analogous notion if U is a G-module and V is an S n -module, namely U ≀V := U ⊗n ⊗V with G≀S n -action Since S a acts naturally and faithfully on and noting that the action remains faithful gives an inclusion S a ≀ S b ֒→ S ab . Similarly we have an inclusion C a ≀ S b ֒→ S ab . More concretely, C a ≀ S b acts faithfully on [ab] by permuting the b size-a intervals in [ab] amongst themselves and cyclically rotating each size-a interval independently.
Remark 2.19. The induction product of two symmetric group representations corresponds to the product of their Frobenius characteristics, so that if U is an S amodule and V is an S b -module, then [Sta99, Prop. 7.18.2], In Section 2.6, we considered the plethysm of Schur characters of general linear group representations. The corresponding operation for Frobenius characters of symmetric group representations is less well-known and involves wreath products as follows. Given two symmetric functions f and g = m 1 + m 2 + · · · where the m i are all monomials, their plethysm is given by [Sta99,Def. A2.6] which is well-defined since f is symmetric. Then, if U is an S a -module and V is an When G is a finite group, Specht [Spe32] described the complex inequivalent irreducible representations of G ≀ S n in terms of those for G and S n , the conjugacy classes of G, and wreath products. In the case C a ≀ S b , they are indexed by the following objects.
One consequence of Theorem 2.20 is Another consequence is an explicit description of the one-dimensional representations of C a ≀ S b , which are as follows.
Definition 2.21. Fix integers a, b ≥ 1. Let χ r,1 := χ r a ≀ 1 b and χ r,ǫ := χ r a ≀ ǫ b where r = 1, . . . , a and 1 b and ǫ b are the trivial and sign representations of S b , respectively. When b = 1, ǫ b = 1 b , in which case χ r,1 = χ r,ǫ = χ r a . We sometimes write χ r,1 (a b ) or χ r,ǫ (a b ) if we want to specify the group C a ≀ S b as well. Bergeron-Bergeron-Garsia [BBG90] extended Klyachko's observation by showing that the Schur-Weyl dual of L (a b ) is χ 1,1 ↑ S ab Ca≀S b . We next give a different argument of this fact which is straightforward given the preceding background and which uses a lemma we will require later in Section 6.
Lemma 2.22. We have Proof. By Lemma 2.25 below and the fact that (22) and the r = 1 case of Theorem 1.7, Indeed, the Schur-Weyl duals of general L λ can be expressed very explicitly in terms of induced linear representations as follows. Suppose σ ∈ S n has cycle type λ. Write Z λ for the centralizer of σ in S n . When λ = (a b ), it is straightforward to see that Corollary 2.24 (see [Reu93,Thm. 8.24]). Suppose λ = 1 b 1 2 b 2 · · · k b k ⊢ n. Let χ 1,1 λ denote the linear representation of Z λ ≤ S n given by the (outer) tensor product of the representations χ 1,1 Proof. Using in order (16), multiplicativity of Schur characters under tensor products, Corollary 2.23, (20), Lemma 2.26 and transitivity of induction, the fact that , and the definition of χ 1,1 λ , we have The result will be complete once we prove Lemma 2.26.
Lemma 2.26. Suppose that H 1 , . . . , H k are subgroups of groups G 1 , . . . , G k and that U i is an H i -module for 1 ≤ i ≤ k. Then Proof. Having chosen bases for both sides, there is a natural C-linear map between them. It is easy to check this is also G 1 × · · · × G k -equivariant. The details are omitted.

Cyclic Sieving and Kraśkiewicz-Weyman's Result
In this section, we first build on work of Klyachko to prove Theorem 1.7. We then recover Kraśkiewicz-Weyman's result, Theorem 1.6, and discuss some benefits of our approach.
Klyachko observed in [Kly74, Prop. 1] that E(χ 1 ↑ Sn Cn ), like L (n) , also has a basis indexed by primitive necklaces. Klyachko's argument may be readily generalized to E(χ r ↑ Sn Cn ) as follows. Recall from the introduction that NFD n,r := {N ∈ N n : freq(N ) | r}, In particular, NFD n,n = N n , and NFD n,1 is the set of primitive necklaces of length n.
Theorem 3.1. There is a basis for E(χ r ↑ Sn Cn ) indexed by necklaces of length n words with letters from [m] and with frequency dividing r. Moreover, ch χ r ↑ Sn Cn = NFD cont n,r (x).
Proof. Suppose the underlying vector space V has basis {v 1 , . . . , v m }. By a slight abuse of notation, we may view χ r as the vector space C with the left C n -action σ n · 1 := ω r n . Since χ r ↑ Sn Cn := CS n ⊗ CCn χ r , we have E(χ r ↑ Sn Cn ) = V ⊗n ⊗ CSn CS n ⊗ CCn χ r ∼ = V ⊗n ⊗ CCn χ r (28) where C n acts on V ⊗n on the right by "rotating" the components of simple tensors. A spanning set for V ⊗n ⊗ CCn χ r is given by all v i 1 ⊗· · ·⊗v in ⊗1, which we abbreviate as [i 1 · · · i n ]. Acting by σ −1 n on χ r on the left or on V ⊗n on the right gives the relation This relation shows that [i 1 · · · i n ] is well-defined on the level of necklaces, at least up to nonzero scalar multiplication, which explains our notation. If the word i 1 · · · i n has frequency f and period p, we then find Since ω p n is a primitive n/p = f -th root of unity, the factor f −1 ℓ=0 ω ℓpr n is nonzero if and only if ω pr n = 1, so if and only if f | r. Picking representatives for necklaces with frequency dividing r thus gives a spanning set for E(χ r ↑ Sn Cn ), and it is easy to see it is in fact a basis. Diagonal matrices act on this basis via diag(x 1 , . . . , x n ) · [i 1 · · · i n ] = x cont(i 1 ···in) [i 1 · · · i n ], from which it follows that the Schur character is the content generating function of necklaces of length n words with letters from [m] and with frequency dividing r. Letting m → ∞, (27) follows. Since flex(w) = freq(w) lex(w) = r, we have freq(w) | r, so [w] ∈ NFD n,r . Thus, ι is in fact a map from F n,r to NFD n,r . Since each necklace in NFD n,r contains exactly one word with flex equal to r, ι is a content-preserving bijection. Therefore, NFD cont n,r (x) = F cont n,r (x).
Using Theorem 1.5, we have W cont,flex n (x; q) = W cont,maj n n (x; q), which means F cont n,r (x) = M cont n,r (x).
Remark 3.3. From Theorem 3.1 and Lemma 3.2, the Schur character of χ r ↑ Sn Cn may be described as a content generating function for certain necklaces or for certain words. This proves Theorem 1.7 from the introduction.
We may now present our remarkably direct proof of Kraśkiewicz-Weyman's result, Theorem 1.6, using cyclic sieving. Using universal cyclic sieving on words for S n -orbits and C n -orbits as described in the introduction, Theorem a λ,r s λ (x) q r .
Combining all of these equalities and extracting the coefficient of q r gives the result.
Every step of the preceding proof uses an explicit bijection with the exception of the appeal to cyclic sieving through Theorem 1.5. This suggests the problem of finding a bijective proof of Theorem 1.5.
Problem 3.4. For each n ≥ 1, find an explicit, content-preserving bijection φ : W n → W n such that maj n (w) = flex(φ(w)). Remark 3.6. The most difficult step in our proof of Theorem 1.6 is the universal S n -cyclic sieving result, Corollary 1.3, or equivalently Theorem 1.2. The proof in [RSW04] of Theorem 1.2 perhaps unsurprisingly uses several of the interpretations of the Kraśkiewicz-Weyman symmetric functions from Section 2.5, in particular Theorem 2.12 involving ch Sn×Cn R n . However, both Kraśkiewicz-Weyman's and Springer's original proofs of Theorem 2.12 hinge upon (1). Indeed, Kraśkiewicz-Weyman showed explicitly in [KW01,Prop. 3] that ch Sn×Cn CS n = ch Sn×Cn R n is easily equivalent to (1). Springer's argument proving (1) uses a Molien-style formula, while Kraśkiewicz-Weyman's argument uses a recursion involving ℓ-cores and skew hooks.
One may thus ask about the relationship between (1) and the cyclic sieving result, Theorem 1.2. Using stable principal specializations, one can consider earlier approaches to have been "in the s-basis" and our approach to have been "in the h-basis" in the following sense. Let τ λ be the S n -character of 1↑ Sn S λ , which has ch(1↑ Sn S λ ) = h λ . We have where the first equality is (1), the second is [Sta99,Prop. 7.19.11], the third is Theorem 1.2, and the fourth is [Sta99,Prop. 7.8.3] and [Mac13, Art. 6]. Our approach suggests that, as far as the Kraśkiewicz-Weyman theorem is concerned, the h-basis arises more directly.
In [AS18], the authors proved a refinement of Theorem 1.2. Since earlier approaches to Theorem 1.2 involving representation theory could not readily be adapted to this refinement, the argument instead uses completely different and highly combinatorial techniques. Thus, the arguments in [AS18] and the proof of Theorem 1.6 together give an essentially self-contained proof of Kraśkiewicz-Weyman's result.

Induced Representations of Arbitrary Cyclic Subgroups of S n
We next generalize the discussion in Section 3 to branching rules for general inclusions σ ֒→ S n , recovering a result of Stembridge, Theorem 4.11. Following the outline of the previous section, we express the relevant characters in turn as a certain orbit generating function, Theorem 4.2, a necklace generating function, Lemma 4.6, and a generating function on words, Lemma 4.7. Two variations on the major index, maj ν and maj ν , arise quite naturally from our argument. The CSP Theorem 1.2 again plays a decisive role.
Throughout this section, let σ ∈ S n , let C be the cyclic group generated by σ, and let ℓ := #C be the order of σ. Fixing a primitive ℓ-th root of unity ω ℓ , let χ r : C → C for r = 1, . . . , ℓ be the linear C-module given by χ r (σ) := ω r ℓ . We begin by updating our notation for this setting and generalizing Theorem 3.1.  Proof. The proof of Theorem 3.1 goes through verbatim with the C-action replacing the C n -action.
Our goal is broadly to replace OFD cont C,r (x) with a necklace generating function, apply cyclic sieving to get a major index generating function on words, and then apply RSK to get a Schur expansion. Notation 4.3. For the rest of the section, suppose that σ has disjoint cycle decomposition σ = σ 1 · · · σ k with ν i := |σ i |. Consequently, ℓ = | σ | = lcm(ν 1 , . . . , ν k ). Further, write In Section 3, we considered the C n -orbits of W n , namely necklaces N ∈ N n . The frequency of N is the stabilizer-order of N , i.e. freq(N ) = # Stab Cn (N ). We may group together C n -orbits of W n according to their stabilizer sizes by letting NF n,r := {N ∈ N n : freq(N ) = r} (33) be the set of necklaces of length n words with frequency r. Similarly, NFD n,r consists of C n -orbits of W n whose stabilizer is contained in the common stabilizer of NF n,r .
Analogously, the C ν -orbits of W n can be identified with products of necklaces N 1 × · · · × N k or equivalently with tuples (N 1 , . . . , N k ) where N j ∈ N ν j . Since we may group together C ν -orbits of W n according to their stabilizers as follows.
The elements of NF ν,ρ all have the same stabilizer, and the elements of NFD ν,ρ are precisely those whose stabilizer is contained in the common stabilizer of elements of NF ν,ρ . We write ρ | ν to mean that ρ i | ν i for all i = 1, . . . , r. Note that NF ν,ρ = ∅ if and only if ρ | ν.
Given a group G acting on a set W and a subgroup H of G, each G-orbit of W is partitioned into H-orbits. Consequently, C ν -orbits of W n are unions of C-orbits, which we exploit as follows.
In Section 3, we used cyclic sieving to turn generating functions involving NFD cont n,r (x) into Schur expansions. Thus our next goal is to turn the necklace generating function in Lemma 4.6 into an analogous generating function over NFD cont ν,ρ (x). To accomplish this, one could in principle use Möbius inversion on the lattice of stabilizers of C ν -orbits to convert from NF cont ν,ρ (x) to NFD cont ν,ρ (x). However, the following argument is more direct.
The result follows from Lemma 4.6.
Our next goal is to convert the necklace expansion in Lemma 4.7 into a Schur expansion. Recalling from Section 3 that M n,r := {w ∈ W n : maj n (w) = r}, Lemma 3.2 tells us Interpreting the right-hand side of (34) in terms of words and comparing with the indexing set in Lemma 4.7 motivates the following variations on the major index.
Furthermore, let maj ν : W n → [ℓ] be defined by Consequently, we have maj (n) = maj n . Note that both maj ν and maj ν are functions of Des(w). We may thus define both maj ν and maj ν on Q ∈ SYT(n) using only Des(Q) in the same way. Equivalently, we may set maj ν (Q) := maj ν (w) and maj ν (Q) := maj ν (w) for any w such that Q = Q(w).
Since maj ν (w) depends only on Des(w), we can apply the RSK bijection again through Lemma 2.7 to get W cont,maj ν n a ν λ,r s λ (x)q r .
Remark 4.12. Stembridge showed the equality of the first and third terms in Theorem 4.11 using the skew analogue of (1) and branching rules along Young subgroups of S n . By contrast, W cont,maj ν n (x; q) played a key role in our approach.
Since the isomorphism type of χ r ↑ Sn C , or equivalently the Schur expansion of OFD cont C,r (x), depends only on ν, the cycle type of a generator of C, and gcd(ℓ, r), we have the following generalization of Corollary 2.15.
For use in the next section, we record the Schur expansion of M cont ν,τ (x). The proof is analogous to the last step of the proof of Theorem 4.11 using Lemma 2.7.
Corollary 4.15. Suppose ν = (ν 1 , . . . , ν k ) is the cycle type of some σ ∈ S n , τ ∈ [ν 1 ] × · · · × [ν k ], π ∈ S k , and λ ⊢ n. Then, a ν λ,τ = a π·ν λ,π·τ . Proof. Since reordering does not affect contents, we have NFD cont ν,τ (x) = NFD cont π·ν,π·τ (x). Now apply Corollary 4.14 and equate coefficients of s λ (x). 5. Inducing 1-dimensional Representations from C a ≀ S b to S ab We next apply the approach of Section 3 and Section 4 to prove a generalization of a formula due to Schocker [Sch03] for the Schur expansion of L (a b ) . In particular, we give Schur expansions of the characteristics of Note that ch L (a b ) = ch L 1,1 (a b ) by Corollary 2.23. The argument in Corollary 2.23 and the fact that ch(ǫ b ) = e b (x) immediately yield the following more general result, which also follows from an appropriate modification of Theorem 3.1.
Lemma 5.1. We have Our first goal is to manipulate the necklace generating functions in Lemma 5.1 in such a way that we may apply cyclic sieving. We use Burnside's lemma and a sign-reversing involution to unravel these multiset and subset generating functions, respectively. Proof. Multisets of b necklaces from NFD a,r can be thought of as S b -orbits of lengthb tuples (N 1 , . . . , N b ) of necklaces N i ∈ NFD a,r under the natural S b -action. The tuples (N 1 , . . . , N b ) fixed by an element σ ∈ S b are those tuples which are constant on blocks corresponding to cycles of σ. It follows that if σ has cycle type ν ⊢ b, By Burnside's lemma, we may count S b -orbits of necklaces (N 1 , . . . , N b ) of fixed content by averaging the number of σ-fixed tuples of fixed content over all σ ∈ S b . The result follows by grouping together permutations of a given cycle type.
Consequently, one may replace the combinatorial manipulations in Lemma 5.2 and Lemma 5.3 with symmetric function manipulations. In the next section, we will prove Theorem 1.8, which generalizes the first equalities in (38) and (39).
Remark 5.5. Let ω be the involution on the algebra of symmetric functions defined by ω(s λ (x)) = s λ ′ (x) where λ ′ is the conjugate of λ, obtained by reflecting λ through the line y = −x. One may show in a variety of ways that For instance, we can prove (40) using Theorem 1.6 as follows. Since conjugation Therefore, by Theorem 1.6, letting s = n 2 − r, From the symmetry result Corollary 2.15, it follows that ch χ r ↑ Sn Cn is fixed under ω when n is odd. When n is even, ch χ r ↑ Sn Cn may or may not be fixed. For instance, when r = 1, we find if n/2 is odd, ch L n = ch χ 1 ↑ Cn Sn otherwise.
Here L (2) n is the deformation of L n recently studied by Sundaram [Sun18]. Further standard properties of plethysm together with (38) and (39) give for a odd, and ω ch L r,1 (a b ) = ch L s,1 Consequently, one may obtain the Schur expansion of ch L r,ǫ (a b ) from the Schur expansion of ch L r,1 (a b ) simply by applying the ω map if and only if a is odd. When a is even, these two cases are more fundamentally different.
Next, we convert NFD cont a,r (x ν j 1 , x ν j 2 , . . .) into a linear combination of NF cont k,s (x)'s and then apply Mobius inversion to convert to a linear combination of NFD cont k,s (x)'s. We will need the following variation on the number-theoretic Möbius function µ.
We may now state and generalize Schocker's formula for ch L (a b ) = ch L 1,1 (a b ) .
Remark 5.12. Schocker's approach to [Sch03,Thm. 3.1] uses Jöllenbeck's noncommutative character theory and involved manipulations with Klyachko's idempotents and Ramanujan sums. Much of Schocker's argument generalizes immediately to all r. The argument presented above is comparatively self-contained and direct. Two perhaps mysterious aspects of the formula, the appearance of Möbius functions and the average over S b , arose naturally from Burnside's lemma and a change of basis using Möbius inversion. Our argument uses explicit bijections at each step except for the appeal to Burnside's lemma and the use of Lemma 5.3.

Higher Lie Modules and Branching Rules
The argument in Section 3 solves Thrall's problem for λ = (n) by considering all branching rules for C n ֒→ S n simultaneously and using cyclic sieving and RSK to convert from the monomial to the Schur basis. We now turn to analogous considerations for the higher Lie modules and more generally branching rules for C a ≀S b ֒→ S ab . We give an analogue of the flex statistic and the monomial basis expansion for such branching rules from Section 2.5. We then show how to convert from the monomial to the Schur basis assuming the existence of a certain statistic on words we call mash which interpolates between maj n and the shape under RSK.
We now recall and prove Theorem 1.8 from the introduction, after introducing some notation. Theorem. For all a, b ≥ 1 and λ = (λ (1) , . . . , λ (a) ) ∈ P b a , we have Proof of Theorem 1.8. We have where the first and third isomorphisms use transitivity of induction, the second isomorphism uses Lemma 2.26, and the fourth isomorphism uses Lemma 2.25. Consequently, using (20), (22), and Theorem 3.1, we have Recall from Section 2.3 that given a word w, the shape of w, denoted sh(w), is the common shape of P (w) and Q(w) under RSK.
Theorem. Fix a, b ≥ 1. We have where the S λ are irreducible representations of C a ≀ S b and the q λ are independent indeterminates.
The first equality in Theorem 1.9 now follows from combining (45) and (47) with (44). The second equality in Theorem 1.9 follows similarly.
While Theorem 1.9 determines the monomial expansion of the graded Frobenius series tracking branching rules for C a ≀ S b ֒→ S ab , we are ultimately interested in the corresponding Schur expansion. We next describe how the approach in the preceding sections might be used to find this Schur expansion. The key properties used in the proof of Theorem 1.6 converting from the monomial basis to the Schur basis were that maj n is equidistributed with flex on each W α and maj n (w) depends only on Q(w). In order to apply a similar argument for ch(S λ ↑ S ab Ca≀S b ), we need a statistic as follows.
Problem 6.4. Fix a, b ≥ 1. Find a statistic mash b a : W ab → P b a with the following properties.
(i) For all α ab, maj b a (or equivalently flex b a ) and mash b a are equidistributed on W α . (ii) If v, w ∈ W ab satisfy Q(v) = Q(w), then mash b a (v) = mash b a (w).
Finding such a statistic mash b a would determine the Schur decomposition of ch(S λ ↑ S ab Ca≀S b ) as follows. The result follows by equating coefficients of q λ .
Remark 6.6. When a = 1 and b = n, we may replace λ with λ ⊢ n. Under this identification, maj n 1 (w) = sh(w), which clearly satisfies Properties (i) and (ii). When a = n and b = 1, we may replace λ with an element r ∈ [n]. Under this identification, we may set mash 1 n (w) = maj n (w), which satisfies Properties (i) and (ii). In this sense mash b a interpolates between the major index maj n and the shape under RSK, hence the name.
While maj b a trivially satisfies Property (i), it fails Property (ii) already when a = b = 2, as in the following example.
Remark 6.8. When defining flex b a and maj b a , we somewhat arbitrarily chose the lexicographic order on W a . Any other total order would work just as well. However, maj b a continues to fail Property (ii) using any other total order when a = b = 2 in Example 6.7 since either 14 < 23 or 23 < 14.