Almost all wreath product character values are divisible by given primes

For a finite group $G$ with integer-valued character table and a prime $p$, we show that almost every entry in the character table of $G \wr S_N$ is divisible by $p$ as $N \to \infty$. This result generalizes the work of Peluse and Soundararajan on the character table of $S_N$.


Introduction
Let S N be the symmetric group on N letters.The complex irreducible characters of S N were calculated by Frobenius in 1900; in particular, Frobenius showed that the characters are integer-valued [Fro00].In 2019, Alex Miller investigated the distribution of the parity of entries of the character table of S N .He made the remarkable conjecture that for any prime p and exponent ≥ 1, the proportion of entries of the character table of S N divisible by p (and later p for ≥ 1) tends to 1 as N → ∞ [Mil19a;Mil19b].This conjecture was recently proved by Peluse and Soundararajan in the case = 1 in [PS22].
This leaves the question of investigating the distribution of residues modulo p for more general finite groups with integer-valued character tables.A natural infinite family of such is the wreath product G S N as N → ∞.When G is a fixed group with integer-valued character table, it is known that the characters of G S N are also integer-valued [Jam06,Corollary 4.4.11].These families include the Weyl group of type B N , when G = Z/2Z, and wreath products S M S N of symmetric groups.
Our main result is a generalization of Peluse and Soundararajan's theorem: Theorem (see Theorem 3.8 below).Let G be a group with integer-valued character table and let G S N be the wreath product of G with S N .For all primes p, the proportion of entries in the character table of G S N which are divisible by p tends to 1 as N → ∞.
The proof relies on the combinatorics of the representations of G S N .If G has k conjugacy classes, then conjugacy classes and representations of G S N are both naturally labelled by kmultipartitions of N .One of the key inputs is characterizing when two elements of G S N have 2. Preliminaries 2.1.Representation Theory of the Wreath Product.Let G be a finite group and let S N be the symmetric group on N letters.
Definition 2.1.The wreath product of G with S N , denoted G S N , is the group of N × N permutation matrices with nonzero entries in G.
We begin by recalling the representation theory of G S N .The representation theory of wreath products was first studied in Specht's dissertation [Spe32], anticipated by Young's work on the case G = Z/2Z [You30]; see also [Zel81;Jam06] for more modern treatments.If we take the representation theory of G as input data and let N vary, the representation theory has structural similarities to the representation theory of S N , the case when G = 1.While representations of the symmetric group are labelled by partitions of N , representations of the wreath product are labelled by multipartitions: Suppose that G has k conjugacy classes.Then k-multipartitions of N label the conjugacy classes of G S N .We will not need to use the specific form of this bijection in this paper; it is used in the proofs of character formulas in Propositions 2.4 and 2.10, which we omit.
Proposition 2.3 ([Jam06], Theorem 4.2.8).If G has k conjugacy classes, then the conjugacy classes of G S N are indexed by k-multipartitions of N .Given x ∈ G S N , the multipartition λ corresponding to x is formed as follows: for each cycle in x of length , if the product of the nonzero entries in that cycle is in the ith conjugacy class of G, then add to λ i .
One can check the assignment of a conjugacy class to a multipartition is well-defined by checking under conjugation by S N and by diagonal matrices G N ⊆ G S N .Conjugating an element of G S N by S N does not change the set of cycle products at all.If (g 1 , . . ., g N ) ∈ G N and (12 • • • N ) is an N -cycle, the conjugate of (12 • • • N )(g 1 , g 2 , . . ., g N ) by (g, 1, . . ., 1) is (12 • • • N )(g 1 g −1 , g 2 , . . ., gg N ); these two elements have conjugate cycle products g N • • • g 2 g 1 and g(g N • • • g 2 g 1 )g −1 .The general case of conjugation by G N reduces to the above case.
To find the complex irreducible representations of G S N , we need the complex irreducible representations of G as input; call the irreducible G-representations V 1 , . . ., V k .
where S λ i is the Specht module for S N corresponding to λ i .
Character values of wreath products can be calculated using a modified version of the Murnaghan-Nakayama rule for the symmetric group.Let χ λ be the character of V λ and χ λ µ be the value of χ λ on the conjugacy class corresponding to µ.Then χ λ µ is calculated by decomposing the of Young diagrams of λ i for all i using rimhooks: Definition 2.6.For k-multipartitions λ and µ, a rimhook decomposition of λ by µ is obtained by repeatedly removing rimhooks in λ with parts of µ in a fixed ordering such that after all rimhooks have been taken, there are no boxes of λ left.All the possible ways to take rimhooks of λ with parts of µ is the set RHD(λ, µ).
The Murnaghan-Nakayama rule can be modified for wreath products as follows: Proposition 2.7 ([Jam06], Theorem 4.4.10).Let λ and µ be k-multipartitions of N .Let χ 1 , χ 2 , . . ., χ k be the irreducible characters of G.For ρ ∈ RHD(λ, µ), let ψ(ρ) be defined by where c h is the conjugacy class of G associated to h.Then where ht(ρ) is the height of the rimhook decomposition.
The permutation module characters of wreath products form another basis for the space of class functions of G S N that is easier to work with.Definition 2.8.Let λ = (λ 1 , . . ., λ k ) be a k-multipartition of N and let a i = |λ i |.For each λ i , let S λ i be the Young subgroup of S a i corresponding to λ i and let G λ i = G S λ i .Then the permutation module M λ for G S N is defined by There is a character formula for M λ using row decompositions instead of rimhook decompositions.It is as follows: Definition 2.9.Let λ and µ be k-multipartitions of N .A row decomposition of λ by µ is a function ρ : {rows of µ} → {rows of λ} such that if r is a row of λ, then the rows in ρ −1 (r) have the same total length as r.The set of all row decompositions of λ by µ is denoted RD(λ, µ).
We will think of row decompositions of λ by µ as a tiling of the Young diagrams of λ by rows, where rows of µ are placed in a fixed ordering.Proposition 2.10.Let λ and µ be k-multipartitions of N .Let χ 1 , χ 2 , . . ., χ k be the irreducible characters of G.For ρ ∈ RD(λ, µ), let α(ρ) be defined by where c r is the conjugacy class of G associated to r.Then the character for permutation module M λ at µ is The proof follows from the character formula for induced representations.
We now describe the change-of-basis between irreducible and permutation characters.
Definition 2.11.The dominance order on k-multipartitions is defined by λ η if and only if λ i dominates η i for all i.
Lemma 2.12.The matrix of multiplicities [M λ : V η ] of the irreducible representations of G S N in permutation modules is unimodular and upper-triangular with respect to dominance order.
Proof.Recall the Kostka numbers K β,γ for β, γ partitions of N are defined by where S β is the Young subgroup corresponding to β and V γ is the Specht module corresponding to γ.Note that our notation for M β and V γ agrees with that of wreath products G S N when G = 1.The Kostka numbers satisfy K β,β = 1 and K β,γ > 0 if and only if β γ in dominance order [Mac98, p. I.6].
We claim that (1) where we make M λ i ⊗ V ⊗a i i a representation of G a i by having S a i act diagonally and G a i naturally on V ⊗a i i .Then (1) follows from multilinearity of the tensor product and linearity of induction.Now since the matrix of Kostka numbers is unimodular and upper-triangular with respect to dominance order, the same is true of the matrix {c(λ, µ)} λ,µ .
2.2.Asymptotics of Partitions.We recall a form of the Hardy-Ramanujan asymptotic for the number of partitions of N , denoted p(N ).
for sufficiently large N .
Let p k (N ) denote the number of k-multipartitions of N .
for sufficiently large N .
This formula also appears in [Mur13].We provide an elementary inductive proof.
Proof.We proceed by induction on k.The base case k = 1 is Proposition 2.13.For δ > 0, let δ = 4 5 δ.By inductive hypothesis, there exists a constant B such that if By considering the size of the first partition in a k-multipartition, it follows that We break up the sum for p k (N ) into distinct parts: let We now consider D 1 and D 3 .Note that for a ∈ [0, B), we have p(a)p k−1 (N − a) ≤ p(B)p k−1 (N ), and for a ∈ (N − B, N ], we have p(a) Combining (2), (3), and (4), we have that for sufficiently large N .
Proof.Pick 0 < ε < δ and 1 ≤ i ≤ k.The number of k-multipartitions of N where By Claim 2.14, the rate at which (5) approaches infinity is significantly slower than the rate at which p k (N ) approaches infinity.Since δ > ε, we can conclude that the number of k-multipartitions λ such that for all i tends to 1 as N → ∞.

Character Table Column
Congruences.Corollary 3.3 below, which we call "the mashing rule," gives a criterion for mod p congruence of two columns of the character table of G S N in terms of k-multipartitions.
In this section, we must assume that G has integer-valued character table.By [Ser77, §13.1], the group G has integer-valued character table if and only if σ ∈ G is conjugate to σ j whenever j is prime to the order of σ.Definition 3.1.Let ∼ p be the equivalence relation on k-multipartitions generated by the following: µ ∼ p ν if there is j such that µ i = ν i for i = j, and ν j is formed by replacing one part of size mp in µ j with p parts of size m in ν j .Lemma 3.2.Let p be a prime and G be a group with integer-valued character table.Let Proof.It suffices to show M λ µ ≡ M λ ν (mod p) if there exists j such that µ i = ν i for all i = j, µ j = (ξ, mp) for some ξ, and ν j = (ξ, m p ).We break RD(λ, ν) into two cases.In case one, we consider the row decompositions of ν where m p is tiled in the same row of λ.In case two we consider the row decompositions when m p is not tiled in the same row.Recalling our formula for characters of permutation modules in Proposition 2.10, let so that M λ ν = β + γ.Case one will show β ≡ M λ µ (mod p).Case two shows γ ≡ 0 (mod p).Together, these two congruences imply M λ µ ≡ M λ ν (mod p).In both cases, we break into subcases based on the ways to tile µ i for i = j and ξ.In case one, we have compatible tilings for µ and ν, and in case two, we have additional tilings for ν.
In case one, assume we have tiled all rows of µ i for all i = j and we have tiled ξ.We now have one row remaining.There is only one way to tile the last row for both µ and ν: put the remaining pieces into the remaining row.Let these row decompositions be denoted ρ µ and ρ ν respectively.
For ρ µ , say that we place the final row r of size mp in the partition λ q .The associated cycle product is c j because mp comes from µ j .Then mp contributes χ q (c j ) to the product α(ρ µ ).Then for ρ ν , the p rows of size m are placed into λ q .The conjugacy class of G associated with the p rows of size m is again c j , so m p contributes a factor of χ q (c j ) p to α(ρ ν ).
Summing over all the tilings in case one, we find M λ µ ≡ β (mod p).In case two, assume we have tiled all rows of µ i for i = j and ξ, after which there are t > 1 remaining unfilled rows of the Young diagrams of λ.If T ⊆ RD(λ, ν) is the set of row decompositions extending our given tiling by µ i for i = j and ξ, then we will show ρ∈T α(ρ) ≡ 0 (mod p).
Then γ is the sum over all such T of ρ∈T α(ρ), from which it will follow γ ≡ 0 mod p.
Call the lengths of the remaining rows (m 1 , m 2 , . . ., m t ).Since the elements of T are in bijection with choices of placements of p cycles of length m into these rows, (8) |T | = p 1 , 2 , . . ., t .
Let ρ ∈ T .Note that all pieces of m p come from µ j , and thus have cycle product c j , while all other cycles in µ are in the same place in T .Thus α(ρ) = α(ρ ) for all ρ, ρ ∈ T .Hence ρ∈T α(ρ) is a sum of |T | identical terms.Then ρ∈T α(ρ) ≡ 0 (mod p) because |T | is divisible by p.
Case one has shown that M λ µ ≡ β (mod p), and case two has shown that γ ≡ 0 (mod p).
Proof.The set of irreducible characters and the set of characters of permutation modules form bases for the space of class functions on G S N .Since the change of basis matrix between these two bases is unimodular and upper-triangular, as stated in Lemma 2.12, χ λ can be expressed as an integral linear combination of M η for all k-multipartitions λ.It follows from Lemma 3.2 that µ ∼ p ν implies χ λ µ ≡ χ λ ν (mod p).Peluse and Soundararajan proved the following estimate of the number of t-cores when t is slightly larger than the typical longest cycle in a random conjugacy class: Proposition 3.5 ([PS22], Proposition 1).Let L be a positive integer, and let A be a real number with 1 ≤ A ≤ log L/ log log L. Additionally suppose that t is a positive integer with Then the number of partitions λ of L which are not t-cores is at most , independent of t satisfying (9).
Complementing the estimate in Proposition 3.5, Peluse and Soundararajan also estimated how many columns of the character table are congruent to a column corresponding to a partition with a large first part: Proof.The number of irreducible representations of D N of the form Res B N D N V λ,µ for λ = µ equals 1 2 (p 2 (N ) − p(N/2)) when N is even, and 1 2 p 2 (N ) when N is odd.The number of irreducible representations appearing as a summand of Res B N D N V λ,λ is 2p(N/2) when N is even and 0 when N is odd.By Claim 2.14, we have p 2 (N ) p(N/2) for large enough N , so the proportion of irreducibles of the form Res B N D N V λ,µ goes to 1 as N → ∞.Since D N ⊆ B N is of index two, at least half of the conjugacy classes in B N intersect D N .Since most entries in the character table of B N are divisible by p, the same is true when we restrict to the columns which intersect D N , since they are at least half of the columns.Hence, the proportion of entries in the character table of D N which are divisible by p goes to 1 as N → ∞.
Proposition 2.4 ([Jam06], Theorem 4.4.3).If G has k conjugacy classes, then the irreducible representations of G S N are in bijection with k-multipartitions of N .For λ = (λ 1 , . . ., λ k ) a k-multipartition of N , let a i = |λ i | and G a = G S a .Then the irreducible representation of G S N corresponding to λ is

Figure 2 .
Figure 2. All valid row decompositions of ( , ) by (31, 21).The numbers in the boxes indicate the order in which parts of µ are placed into rows of λ, with fixed right-to-left placement.

Figure 3 .
Figure 3. Example of three conjugacy classes which are congruent mod 3 in Z/2Z S N (note that m = 2 in the first cycle type and m = 1 in the second).
Proposition 3.6 ([PS22], Proposition 2).Let p ≤ (log L) (log log L) 2 be a prime.Starting with a partition µ of L, we repeatedly replace every occurrence of p parts of the same size m by one part of size mp until we arrive at a partition μ where no part appears more than p − 1 times.Then the largest part of μ exceeds √ most O p(L) exp −L 1 15p partitions µ.
[PS22]oof of Main Theorem.Using Corollary 3.3, the existence of one zero in the character table implies many more entries are divisible by p.We proceed, following Peluse and Soundararajan in[PS22], by using Proposition 2.7 to show sufficiently many entries of the character table are zero.