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Comptes Rendus
Mathématique
Brendon Rhoades
Plethysm and a character embedding problem of Miller
Volume 360 (2022), p. 1113-1116
https://doi.org/10.5802/crmath.363
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1113-1116
https://doi.org/10.5802/crmath.363
Combinatorics / Combinatoire
Plethysm and a character embedding problem
of Miller
Brendon Rhoades∗, a
a Department of Mathematics, University of California, San Diego La Jolla, CA, 92093,
USA
E-mail: bprhoades@ucsd.edu
Abstract. We use a plethystic formula of Littlewood to answer a question of Miller on embeddings of sym-
metric group characters. We also reprove a result of Miller on character congruences.
Funding. B. Rhoades was partially supported by NSF Grant DMS-1953781.
Manuscript received 29 March 2022, revised 4 April 2022, accepted 5 April 2022.
Given d ≥ 1 and a partition λ = (1m1 2m2 3m3 · · · ) of a positive integer n, let d (λ) be the
partition of d 2 ·n given by d (λ) := (d dm1 (2d)dm2 (3d)dm3 · · · ). The Young diagram of d (λ) is
obtained from that of λ by subdividing every box into a d ×d grid, as suggested by the notation.
Let Sn be the symmetric group on n letters. For a partition λ` n, let V λ be the corresponding
Sn-irreducible with character χλ : Sn → C. For d ≥ 1, define a new class function d (χλ) on Sn
whose value on permutations of cycle type µ` n is given by
d
(
χλ
)
µ
:=χd (λ)
d (µ)
. (1)
Thus, the values of the class function d (χλ) on Sn are embedded inside the character table
of the larger symmetric group Sd 2·n . A. Miller conjectured [4] that the class functions d (χλ)
are genuine characters of (rather than merely class functions on) Sn . We prove that this is so
in Theorem 1 using plethysm of symmetric functions.
In the arguments that follow, we use standard material on symmetric functions; for
details see [3]. For µ ` n, let mi (µ) be the multiplicity of i as a part of µ and zµ :=
1m1(µ)2m2(µ) · · ·m1(µ)!m2(µ)! · · · be the size of the centralizer of a permutation w ∈ Sn of cy-
cle type µ.
Let Λ = ⊕n≥0Λn be the ring of symmetric functions in an infinite variable set (x1, x2, . . . ).
Bases of Λ are indexed by partitions; we use the Schur basis {sλ} and power sum basis {pλ}. The
basis pλ is multiplicative: if λ = (λ1,λ2, . . . ) then pλ = pλ1 pλ2 · · · . The transition matrix from the
Schur to the power sum basis encodes the character table of Sn ; for λ` n we have
sλ =
∑
µ`n
χλµ
zµ
pµ.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1114 Brendon Rhoades
Let 〈−,−〉 be the Hall inner product on Λ with respect to which the Schur basis {sλ} is
orthonormal. The power sums are orthogonal with respect to this inner product. We have
〈pλ, pµ〉 = zλ ·δλ,µ where δ is the Kronecker delta.
Write R =⊕n≥0 Rn where Rn is the space of class functions ϕ : Sn →C. The characteristic map
chn : Rn → Λn is given by chn(ϕ) = 1n!
∑
w ∈Sn ϕ(w) ·pcyc(w) where cyc(w) ` n is the cycle type of
w ∈ Sn . The map ch = ⊕n≥0 chn is a linear isomorphism R → Λ. The space R has an induction
product given byϕ◦ψ := IndSn+mSn ×Sm (ϕ⊗ψ) for allϕ ∈ Rn andψ ∈ Rm . Under this product, the map
ch : R →Λ becomes a ring isomorphism. We record two properties of ch.
• We have ch(χλ) = sλ, so that ch sends the irreducible character basis of R to the Schur
basis ofΛ.
• If ϕ : Sn →C is any class function and µ` n, then〈
ch(ϕ), pµ
〉= value of ϕ on a permutation of cycle type µ. (2)
Let ψd : Λ → Λ be the map ψd : F (x1, x2, . . . ) 7→ F (xd1 , xd2 , . . . ) which replaces each variable
xi with its d th power xdi . The symmetric function ψ
d (F ) is the plethysm pd [F ] of F into the
power sum pd . Let φd : Λ→ Λ be the adjoint of ψd characterized by 〈ψd (F ),G〉 = 〈F,φd (G)〉 for
all F,G ∈ Λ. In this note we apply the operators ψd and φd to character theory; see [6] for an
application to the cyclic sieving phenomenon of enumerative combinatorics.
Theorem 1. Let d ≥ 1 and λ ` n. Consider the chain of subgroups ∆(Sn) ⊆ Sdn ⊆ Sdn where
Sdn = Sn × ·· ·×Sn is the d-fold self-product of Sn and ∆(Sn) is the diagonal {(w, . . . , w) : w ∈ Sn}
in Sdn . Then 
d (χλ) is the character of the ∆(Sn) ∼= Sn module
ResSdn
∆(Sn )
(
V λ ◦ · · · ◦V λ
)
(3)
obtained by restricting the d-fold induction product V λ ◦ · · · ◦V λ = IndSdn
Sdn
(V λ⊗·· ·⊗V λ) to ∆(Sn).
Proof. Let λ,µ` n be two partitions and let d ≥ 1. By (2) we have the class function value
χ
d (λ)
d (µ)
=
〈
sd (λ), pd (µ)
〉
=
〈
sd (λ),ψ
d
(
pdµ
)〉
=
〈
φd
(
sd (λ)
)
, pdµ
〉
. (4)
Littlewood [2, p. 340] proved (see also [1, Equation 13]) that for any partition ν` dm, with empty
d-core, the image φd (sν) is given by
φd (sν) = εd (ν) · sν(1) · · · sν(d) (5)
where εd (ν) is the d-sign of ν and (ν
(1), . . . ,ν(d)) is the d-quotient of ν. We refer the reader to [1, 2]
for definitions. In our context we have εd (d (λ)) = +1 (since d (λ) admits a d-ribbon tiling
with only horizontal ribbons) and the d-quotient of d (λ) is the constant d-tuple (λ, . . . ,λ).
Equation (5) reads
φd
(
sd (λ)
)= sdλ . (6)
Plugging (6) into (4) gives
χ
d (λ)
d (µ)
=
〈
φd
(
sd (λ)
)
, pdµ
〉
=
〈
sdλ , p
d
µ
〉
(7)
which (thanks to (2)) agrees with the trace of (w, . . . , w) ∈∆(Sn) on V λ ◦· · ·◦V λ for w ∈ Sn of cycle
type µ. 
Ifλ= (λ1,λ2, . . . ) is a partition, let d ·λ= (dλ1,dλ2, . . . ) be the partition obtained by multiplying
every part of λ by d . The argument proving Theorem 1 applies to show that for λ ` n, the class
function χd ·λ : Sn → C given by (χd ·λ)µ := χd ·λd ·µ is a genuine character (although its module does
not have such a nice description). It may be interesting to find other ways to discover characters
of Sn embedded inside characters of larger symmetric groups.
Brendon Rhoades 1115
In closing, we use plethysm to give a quick proof of a character congruence result of Miller [5,
Thm. 1]. Miller gave an interesting combinatorial proof of the following theorem by introducing
objects called “cascades”.
Theorem 2. (Miller) Let d ≥ 1. For any partitions λ` n and µ` dn, we have
χ
d (λ)
d ·µ ≡ 0 mod d !. (8)
Furthermore, suppose λ,ν` n with d - n. Then
χ
d (λ)
d 2·ν = 0. (9)
Proof. Arguing as in the proof of Theorem 1, we have
χ
d (λ)
d ·µ =
〈
sd (λ), pd ·µ
〉= 〈sd (λ),ψd (pµ)〉= 〈φd (sd (λ)) , pµ〉= 〈sdλ , pµ〉 (10)
where the last equality used Equation (6). We have sλ =
∑
ρ`n
χλρ
zρ
pρ so that
χ
d (λ)
d ·µ =
〈
sdλ , pµ
〉
=
〈( ∑
ρ`n
χλρ
zρ
pρ
)d
, pµ
〉
. (11)
We expand far right of (11) using the orthogonality of the p’s to obtain〈( ∑
ρ`n
χλρ
zρ
pρ
)d
, pµ
〉
= ∑
(µ(1), ...,µ(d))
zµ
zµ(1) · · ·zµ(d)
×χλµ(1) · · ·χλµ(d) (12)
where the sum is over all d-tuples (µ(1), . . . , µ(d)) of partitions of n whose multiset of parts equals
µ. In particular, (12) is zero unless every part of µ is ≤ n; we assume this going forward. We want
to show that (12) is divisible by d !. To show this, we examine what happens when some of the
entries in a tuple (µ(1), . . . , µ(d)) coincide.
Fix a d-tuple (µ(1), . . . , µ(d)) of partitions of n whose multiset of parts is µ. The ratio of z’s in the
corresponding term on the RHS of (12) is a product of multinomial coefficients
zµ
zµ(1) · · ·zµ(d)
=
(
m1(µ)
m1
(
µ(1)
)
, . . . , m1
(
µ(d)
)) · · · ( mn(µ)
mn
(
µ(1)
)
, . . . , mn
(
µ(d)
)). (13)
Let σ = (σ1, . . . , σr ) ` d be the partition of d obtained by writing the entry multiplicities in the
d-tuple (µ(1), . . . , µ(d)) in weakly decreasing order. For example, if n = 3, d = 5, and our d-tuple of
partitions of n is (µ(1), . . . , µ(5)) = ((2,1), (3), (1,1,1), (3), (2,1)), then σ = (2,2,1). Each multinomial
coefficient in (13) for which mi (µ) > 0 is divisible by σ1! · · ·σr !. Since each part of µ is ≤ n, at least
one mi (µ) > 0 and the whole product (13) of multinomial coefficients is divisible by σ1! · · ·σr !.
Thus, the sum of the terms in (12) indexed by rearrangements of (µ(1), . . . , µ(d)) is divisible by( d
σ1, ...,σr
)·σ1! · · ·σr ! = d !, so that (12) itself is divisible by d !. This proves the first part of the theorem.
For the second part of the theorem, let λ,ν` n where d - n. Arguing as above, we have
χ
d (λ)
d 2·ν =
〈( ∑
ρ`n
χλρ
zρ
pρ
)d
, pd ·ν
〉
. (14)
Since d - n, each partition ρ ` n appearing in the first argument of the inner product in (14) has
at least one part not divisible by d . Since the p’s are an orthogonal basis ofΛ, we see that (14) = 0,
proving the second part of the theorem. 
Acknowledgements
B. Rhoades is grateful to Alex Miller for many helpful conversations, as well as comments on a
draft of this manuscript.
1116 Brendon Rhoades
References
[1] A. Lascoux, B. Lecrec, J.-Y. Thibon, “Ribbon tableaux, Hall–Littlewood symmetric functions, quantum affine algebras,
and unipotent varieties”, J. Math. Phys. 38 (1997), no. 2, p. 1041-1068.
[2] D. E. Littlewood, “Modular representations of symmetric groups”, Proc. R. Soc. Lond., Ser. A 209 (1951), p. 333-353.
[3] I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, Oxford
University Press, 1995.
[4] A. R. Miller, “Personal communication”, 2021.
[5] ——— , “Congruences in character tables of symmetric groups”, https://arxiv.org/abs/1908.03741v1, 2019.
[6] B. Rhoades, “Hall–Littlewood polynomials and fixed point enumeration”, Discrete Math. 310 (2010), no. 4, p. 869-876.


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