Hook-Shape Immanant Characters from Stanley-Stembridge Characters

We consider the Schur-positivity of monomial immanants of Jacobi-Trudi matrices, in particular whether a non-negative coefficient of the trivial Schur function implies non-negative coefficients for other Schur functions in said immanants. We prove that this true for hook-shape Schur functions using combinatorial methods in a representation theory setting. Our main theorem proves that hook-shape immanant characters can be written as finite non-negative integer sums of Stanley-Stembridge characters, and provides an explicit combinatorial formula for these sums. This resolves a special case of a longstanding conjecture of Stanley and Stembridge that posits such a sum exists for all immanant characters. We also provide several simplifications for computing immanant characters, and several corollaries applying the main result to cases where the coefficient of the trivial Schur function in monomial immanants of Jacobi-Trudi matrices is known to be non-negative.


Introduction
A fundamental class of objects in the theory and construction of symmetric functions is Jacobi-Trudi matrices.They are indexed by skew shapes, which are ordered pairs of partitions where the Young diagram of the second is contained in that of the first.Jacobi-Trudi matrices are of particular interest as their determinants are skew-Schur functions.In the case where the skew shape is simply a partition, the determinant is the Schur function indexed by that partition.Schur functions are essential in combinatorics and the representation theory of symmetric groups.
Less studied is the theory of immanants.A virtual character of the symmetric group is a function from the symmetric group to the integers that is constant on conjugacy classes.In particular, characters of representations are virtual characters.The sign character of S n is w → sgn(w), and appears in the definition of a determinant of an n × n matrix M = [m ij ], det(M ) = w∈Sn sgn(w)m 1,w(1) , . . ., m n,w(n) .
Immanants are analogues of determinants in which the sign character is replaced with a virtual character of the symmetric group.When the chosen virtual character is the character of an irreducible representation, the corresponding immanant is called ordinary.This paper is motivated by the study of immanants that use virtual characters In [16, §4], Stembridge defined a virtual character Γ θ µ/ν , where θ ⊢ N is a partition and µ/ν is a skew shape with such that N = |µ/ν|.The character Γ θ µ/ν is defined in detail in Section 2 below.We call Γ θ µ/ν the immanant character, so named as it yields an equivalent formulation of Conjecture 1.1.
Conjecture 1.2.[16,Conj. 4.1 ′ ] Γ θ µ/ν is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup.
Conjectures 1.1 and 1.3 are also stated in [15] using the language of symmetric functions.
Perhaps better known (and more often studied, as in [12,4,7,11,9,2,1], and many others) is the Stanley-Stembridge conjecture [16,15], which is Conjecture 1.2 in the particular case that θ = (N ) (the original Stanley-Stembridge conjecture is a more general statement, but was reduced to this form in [7]).Because of this, we call Γ Conjectures 1.1 and 1.3 are proven assuming the skew shape µ/ν contains no 2 × 2 box in its Young diagram [15, §2].This paper proves the case of Conjecture 1.3 when θ is a hook-shape partition (N − k, 1, . . ., 1) and µ/ν is arbitrary.Our main theorem is the following.
Theorem 1.4 is Corollary 4.3 below, which gives an explicit combinatorial construction of the Stanley-Stembridge character summands.
We apply Theorem A to prove new cases of Conjecture 1.2.
Corollary 1.5.Let θ be a hook partition and µ/ν a skew shape such that µ/ν either: • is pre-abelian, or We now give a brief overview of the contents of this paper.Section 2 gives necessary constructions and definitions for our proofs.Subsection 2.3 in particular contains background material on the connection to Hessenberg functions that is necessary to understand the proof of our main theorem.Section 3 contains computational reductions whose proofs are relegated to Appendix A. These reduce Conjectures 1.1 and 1.3 to a smaller class of skew shapes.Section 4 contains the proof of Theorem 1.4, including an explicit decomposition of Γ θ µ/ν into Stanley-Stembridge characters when θ is a hook partition.This section also contains several corollaries of interest, including the results of Corollary 1.5.

Characters, immanants, and Hessenberg functions
A partition λ of length ℓ =: ℓ(λ) of a positive integer n is a weakly decreasing sequence (λ 1 , . . ., λ ℓ ) of positive integers such that ℓ i=1 λ i = n.If λ is a partition of n we write λ ⊢ n.The Young diagram of λ is a collection of upper-left justified boxes containing λ i boxes in row i.A standard Young tableau is a filling of a Young diagram with distinct integers from [n] that increases along rows and down columns.A semi-standard Young tableau is a filling of those boxes with positive integers weakly increasing along rows and strictly increasing down columns.The content of a semi-standard Young tableau is the list c = (c 1 , . ..) such that the c i is the number of times i appears in the tableau.
Given a partition λ ⊢ n and any sequence of non-negative integers c that sum to n, the Kostka number K λ,c is the number of semi-standard Young tableaux with shape λ and content c.The value K λ,c is unaffected by re-ordering the entries of c or removing zeros.For example, if c = (4, 2, 3, 1) and Note that K θ,c = 3  2 in this case.The following lemma shows that Kostka numbers associated to hook partitions are particularly nice.Lemma 2.2.Let θ = (N − k, 1, . . ., 1) be a partition of N and c a content with r nonzero entries.Then K θ,c = r−1 k .Proof.We may assume that c 1 , . . ., c r are the nonzero entries of c.Consider the r−1 k size-k subsets of {2, . . ., r}.Note that the top-left box in any semi-standard tableaux of shape θ and content c must be 1.Given a semistandard tableau of shape θ and content c, the set of entries in rows 2 through k + 1 determine a unique size-k subset of {2, . . ., r}.This correspondence defines a bijection.□ Given two partitions µ and ν such that ℓ(µ) ⩾ ℓ(ν) and µ i ⩾ ν i for all i, then we say ν < µ and the pair of partitions is called a skew shape, denoted µ/ν.The Young diagram of µ/ν is the diagram of µ with the boxes of ν removed.If µ/ν is a skew shape, its length ℓ(µ/ν) is the largest index i such that ν i ⪇ µ i .If µ ⊢ N µ and ν ⊢ N ν then the size of µ/ν is |µ/ν| := N µ −N ν .Standard and semistandard tableaux of skew shapes are defined with the same conditions on rows and columns as for partitions.
2.1.Characters and symmetric functions.Let S n be the symmetric group on n letters, and C(w) the conjugacy class and Z(w) the centralizer of w in S n .When the particular symmetric group is not clear from context, we will write C n (w) and Z n (w) to denote the conjugacy class and centralizer of w in S n .A virtual character of S n is a function from S n to Z that is constant on conjugacy classes.As conjugacy classes are in bijection with partitions, virtual characters are also functions from the set {λ ⊢ n} of partitions of n to Z.We will denote the conjugacy class of S n associated to λ ⊢ n by C(λ).A virtual character is a character if it arises as the character of a representation of S n .
Example 2.3.The length ℓ(w) of a permutation w ∈ S n is the number of inversions of w.The sign character of S n is defined by sgn(w) = (−1) ℓ (w) .A slightly more complicated example is the character that counts the number of fixed points of each permutation: Both are virtual characters, and also happen to be characters of the sign and natural representations of S n respectively.Symmetric functions (with coefficients in Z) are formal power series in Z[x 1 , . ..] invariant under any permutation of the variables.The symmetric functions form a graded ring over Z denoted by Λ with several important bases.Each of these bases is indexed by partitions of positive integers.The bases used herein are the monomial, homogeneous, and Schur symmetric functions, denoted by {m λ }, {h λ }, and {s λ }, respectively.We also make use of the power-sum symmetric functions {p λ }, which form a basis of Q ⊗ Λ.For more information on the enumerative combinatorics of symmetric function bases see [13,14].
There is a natural inner product on the space of virtual characters.If χ and ψ are virtual characters of S n , the character inner product is the bilinear map on the space of virtual characters given by ⟨χ, ψ⟩ = 1 n! w∈Sn χ(w)ψ(w).
Much like Λ, the space of all virtual characters of symmetric groups can also be given a graded ring structure.The induction product of virtual characters ϕ of S k and ψ of S ℓ is which is itself a character of S k+ℓ [10].
There is also an inner product on the ring of symmetric functions.Let ⟨•, •⟩ : Λ → Q be defined so that the Schur symmetric functions basis is orthonormal.With this inner product the monomial and homogeneous functions form dual bases, so There is an isometric isomorphism between the ring of symmetric functions and the ring of virtual characters of all symmetric groups via the Frobenius characteristic map Frob, which produces a symmetric function from a virtual character χ on S n defined by: The Frobenius characteristic map sends the characters of irreducible representations to the Schur symmetric function basis and the characters of representations defined Algebraic Combinatorics, Vol. 7 #1 (2024) by the action of the symmetric group on cosets of Young subgroups to the homogeneous symmetric function basis.The monomial symmetric functions are mapped to by virtual characters called monomial virtual characters.We fix the notation Frob −1 (s λ ) =: χ λ , Frob −1 (h λ ) =: η λ , and Frob −1 (m λ ) =: ϕ λ .
Since the {χ λ } are characters of irreducible representations, we will call them irreducible characters.Since the {η λ } correspond to induced characters of the trivial character on Young subgroups, we will call them induced trivial characters.As such, if χ is a virtual character of S n , the following are equivalent: (1) χ is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup, (2) Frob(χ) = λ⊢n c λ h λ where every c λ is a non-negative integer (i.e.Frob(χ) is h-positive), and (3) χ = λ⊢n c λ η λ where every c λ is a non-negative integer.
Recall that {h λ } and {m λ } are dual bases in symmetric functions, so {η λ } and {ϕ λ } are dual bases in the space of virtual characters of S n .
For more information on the correspondence between symmetric functions, characters, and representations see [10], and for a very thorough treatment of symmetric functions see [13].

2.2.
Immanants and the immanant character.An immanant is a generalization of the determinant where the sign character is replaced with any virtual character.Definition 2.4.Let M = [m ij ] 1⩽i,j⩽n be an n×n matrix with entries from an algebra over C, and χ : S n → Z a virtual character.The immanant of M with respect to χ is When χ = χ λ is the character of an irreducible representation χ λ [M ] is referred to as an ordinary immanant, and when χ = ϕ λ is a monomial virtual character ϕ λ [M ] is called a monomial immanant.
Computing the associated immanants, we obtain sgn The determinants of H µ/ν are well studied, as by the Jacobi-Trudi identity det(H µ/ν ) = s µ/ν is a skew-Schur function.In particular if ν = ∅ then det(H µ/ν ) = s µ is a Schur function.Skew-Schur functions are known to be Schur positive, a fact that follows from the Littlewood-Richardson rule [13].Recall that Haiman proved Theorem 2.7.[8] Ordinary immanants of Jacobi-Trudi matrices are Schur-positive.
Conjecture 1.1 considers whether monomial immanants of Jacobi-Trudi matrices are Schur positive, as is the case with the determinant and other ordinary immanants.
To study Conjecture 1.1, we introduce a character originally defined by Stembridge [16].
The following is stated in [16] and we include the proof here for the reader's convenience.
Lemma 2.9.[16] Let ϕ be any virtual character.Then the inner product Proof.First, the coefficient of s θ in the Schur expansion of h λ is the Kostka number K θ,λ .Let ϕ be a virtual character.The character inner product of Γ θ µ/ν with ϕ is On the other hand, the immanant of H µ/ν with respect to ϕ is This concludes the proof.□ Recall that η λ and ϕ λ are dual bases, so ⟨Γ θ µ/ν , ϕ λ ⟩ is the coefficient of η λ in Γ θ µ/ν .Thus Lemma 2.9 is the connection between Conjectures 1.1 and 1.2 above.
Theorem 2.7 states that ordinary immanants of Jacobi-Trudi matrices are Schurpositive, so ⟨Γ θ µ/ν , χ λ ⟩ is a non-negative integer for all partitions λ.The irreducible characters form an orthonormal basis of the space of characters of S n , so Γ θ µ/ν is an integer sum of irreducible characters, and in particular, Γ θ µ/ν is indeed a character.The pattern of nonzero entries in a Jacobi-Trudi matrix corresponds to a combinatorial object called a Hessenberg function.

Definition 2.10. A Hessenberg function is a weakly increasing function h : [n] → [n] such that h(i) ⩾ i for all i ∈ [n]. Each such function is denoted by a vector, (h(1), h(2), . . . , h(n)).
The Hessenberg function h determined by the pattern of zeros in H µ/ν is given by Recall that the i-th row and j-th column of H µ/ν is h µi−νj +j−i .Thus in regards to the matrix H µ/ν , h(j) is the row index of the last nonzero entry in the j-th column of There are three nonzero entries in the first two columns, four in the third column, and five in the fourth and fifth columns.So the associated Hessenberg function is h = (3, 3, 4, 5, 5).Lemma 2.12.Let µ/ν be a skew shape.Then w −1 has no negative entries if and only if this concludes the proof.□ In the case of Example 2.11, the set of w ∈ S n such that w −1 has no negative entries is {w ∈ S 5 | w(1), w(2) ⩽ 3 and w(3) ⩽ 4}.Given a Hessenberg function h, the indicator function h : S n → {0, 1} will denote whether or not µ + δ − w −1 (ν + δ) has negative entries.By Lemma 2.12, Example 2.13.Let h = (3, 3, 4, 4).The following diagrams depict each permutation matrix imposed over a diagrams with box (i, j) shaded whenever i ⩽ h(j).
Proof.The Kostka number K (N ), w is 1 whenever w has only non-negative entries and 0 otherwise.By Lemma 2.12 this occurs precisely when w −1 (i) ⩽ h(i) for all i ∈ [n].Since C(w) is closed under inverses, we may ignore the nuance of distinguishing w and w −1 in the sum for Γ h .□ By Lemma 2.14, if µ/ν is a skew shape with corresponding Hessenberg function h, we let Γ The following example demonstrates how to compute a particular Γ h .
Example 2.15.Let h = (3,3,4,4).The following table lists each conjugacy class by the associated partition λ, the permutations w in that conjugacy class such that h(w) = 1, and the value that Γ h takes on that conjugacy class.

Computational simplifications
In this section, we give several simplifications for computing Γ θ µ/ν .The methods of proof are technical representation-theoretic computations and are not used in the other sections of our paper.As such, the proofs are delayed to Appendix A.
Note that the skew shape skew shape µ/ ν is simply the skew shape µ/ν with the empty third row removed.
Proposition 3.1 below asserts that the immanant characters Γ θ µ/ν and Γ θ µ/ ν are equal.In particular, it tells us that to compute Γ θ µ/ν it suffices consider skew shapes without empty (zero) rows in the middle.
Appending empty rows to a skew shape µ/ν allows one to consider Γ θ µ/ν as a character of a symmetric group on more letters than µ/ν has nonzero rows.Proposition 3.2 below confirms that this process does not meaningfully alter the character.
µi/νi in S n .In particular, if Conjecture 1.3 is true for characters Γ θ µ/ν of S ℓ(µ/ν) then it is true in general.By Proposition 3.2, we may always assume that Γ θ µ/ν is a character of S n where n is the number of nonempty rows in the skew diagram.
Proposition 3.4 can be generalized to all immanant characters.For example Proposition 3.5 below allows one to compute the immanant character Γ θ µ/ν of S 4 using immanant characters of the skew shapes (3, 2)/(1) and (2, 1).In general, it allows one to compute Γ θ µ/ν from the immanant characters of the connected components of µ/ν.Proposition 3.5.Let µ/ν be a disconnected skew shape with components µ 0 /ν 0 and Where c θ λσ is the Littlewood-Richardson coefficient.We summarize the consequences of the above computational reductions to Conjectures 1.1 and 1.3 in Corollary 3.6 below.
Corollary 3.6.If Conjectures 1.1 and 1.3 hold for skew shapes µ/ν and n such that µ/ν is connected and n is the length of µ/ν, then they hold in full generality.
Proof.Assume the conjectures hold as in the claim.Proposition 3.2 ensures we may take n to be the length of µ/ν.The immanant character for a disconnected skew shape can be written as the non-negative integral sum of immanant characters of connected components via Proposition 3.5.The induction product of induced trivial characters is itself an induced trivial character, so Conjecture 1.1 follows from Proposition 3.5.The induction product distributes over sums of characters, so Conjecture 1.3 follows from Proposition 3.5 and Proposition 3.4.□

The hook partition case
We aim to prove Conjecture 1.3 when θ is a hook partition, which asserts that Γ θ µ/ν = i∈I Γ (Ni)  Proof.Say that Since λ was arbitrary, Γ θ µ/ν = i∈I Γ h i .By Lemma 2.14 the claim follows.□ Recall from Lemma 2.2 that the Kostka numbers for hook partitions are particularly nice, as if ℓ(c) is the number of nonzero entries in c, and θ is a hook partition of The following allows us to apply Lemma 4.1 in the case where θ is a hook.This is the key combinatorial result of this section.

. . , 1) be a hook partition and µ/ν a skew shape with no empty rows and with associated Hessenberg function
for all w ∈ S n , where for each J ⊂ [n − 1], h J is the Hessenberg function Proof of Theorem 4.2.First we verify that h J is in fact a Hessenberg function.If This contradicts our assumption that µ/ν had no nonzero rows.So we have that h Thus every h J is a Hessenberg function.
Since h J (i) ⩽ h(i) for all subsets J and for all i ∈ [n], h(w) = 0 implies that h J (w) = 0 for all J. Thus it suffices to restrict our attention to those w such that h(w) = 1.
Recall w denotes the sequence µ Consider w ∈ S n such that h(w) = 1 and h J (w) = 0. We will show that J ∩ and so i ∈ Z w .Thus we have the other direction of containment and We next claim that h J (w) = 1 if and only if J ∩ Z w = ∅.We use the presentation such that h J (i) < w(i), and so J ∩ Z w = ∅.On the other hand, if h J (w) = 0 then there exists an i ∈ There are precisely many such subsets J.This concludes the proof.□ We apply Theorem 4.2 to obtain an expansion for hook partition immanant characters in terms of Stanley-Stembridge characters.

. , 1) be a hook partition and µ/ν a skew shape with no empty rows and associated Hessenberg function
where Furthermore, if we collect terms in (2) so that where each h J is a unique Hessenberg function, then Proof.Equation ( 2) follows directly from Lemma 4.1 and Theorem 4.2.By construction c J is the number of Given Lemma 2.9, Corollary 4.3 states that in the Schur expansion of the ψimmanant of a Jacobi-Trudi matrix for any virtual character ψ, the hook partition Schur coefficients are non-negative sums of trivial Schur coefficients in the Schur expansions for ψ-immanants of some collection of Jacobi-Trudi matrices.
As an application of our result, we apply Theorem 4.2 where the Stanley-Stembridge conjecture is already known in order to prove the hook partition version of Conjecture 1.1 in those cases.A Hessenberg function is abelian if h(h(1) + 1)) = n or if h(1) = n.In the abelian case, the Stanley-Stembridge conjecture is known.Theorem 4.5.[9] If h is abelian, then Γ h is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup.Definition 4.6.Let µ/ν be a skew shape and H µ/ν the associated Jacobi-Trudi matrix.Let H ′ µ/ν be the matrix obtained from H µ/ν by replacing all 1-s with 0-s.The pattern of nonzero entries in H ′ µ/ν determines a Hessenberg function h ′ .We say the skew shape µ/ν is pre-abelian if h ′ is an abelian Hessenberg function.
On the other hand, the following skew shapes are pre-abelian, and appear with the corresponding Jacobi-Trudi matrices.
Algebraic Combinatorics, Vol. 7 #1 (2024) (4, 4, 4, 4)/(1) ( In essence, a skew shape is pre-abelian if the sum (2)  Note the Dahlberg result is actually stronger, as the paper proves the result for a much larger collection of Hessenberg functions.The larger collection is not as conducive to applying Corollary 4.3.
Corollary 4.10.Suppose θ is a hook partition.If µ/ν is a skew shape associated to Hessenberg function h such that h(i) − i ⩽ 2 for all i ∈ [n], then Γ θ µ/ν is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup.In other words, under these assumptions Conjecture 1.2 holds.
Proof.Each h J from the decomposition in Corollary 4.3 has the property that h J (i) ⩽ h(i).Apply Theorem 4.9.□ There are several other classes of Hessenberg functions for which the Stanley-Stembridge conjecture is known.Any time the summands from Corollary 4.3 are known to fall exclusively within the known cases, we obtain a partial proof of Conjecture 1.1.In particular, whenever the Jacobi-Trudi matrix does not contain any 1-s at all, the decomposition in Corollary 4.3 will simply be many copies of the original Hessenberg function.This occurs for skew shape µ/ν with associated Hessenberg function h when µ h(i) − ν i + i − h(i) > 0 for all i ∈ [n].For any Hessenberg function h, it is possible to construct a Jacobi-Trudi matrix whose pattern of nonzero entries corresponds to h and that contains no entries that are 1.As such, for any particular Hessenberg function h where the Stanley-Stembridge conjecture is known, there are skew shapes µ/ν for which the decomposition in equation ( 2 On the other hand, the first n − 1 elements of δ n are exactly one greater than the those in δ n−1 .More specifically, under point-wise addition of integer sequences (and considering δ n−1 as a sequence of length n by appending the integer 0), δ n = δ n−1 + (1, 1, . . ., 1).In particular, noting w ′ ((1, 1, . . ., 1)) = (1, 1, . . ., 1), up to appending a 0, Thus Γ n (w) = (Γ n−1 )  Sn Sn−1 (w) for all w ∈ S n−1 .The proposition now follows by linearity of induced characters.The "particular" part of our claim follows by induction on n. □ Proposition A.5 (Proposition 3.3).Let µ 0 /ν 0 and µ 1 /ν 1 be two skew shapes whose Young diagrams have identical connected components.Let θ Proof.This will proceed similarly to the proof of Proposition 3.1 (Proposition A.2). First, let n = ℓ(µ 0 /ν 0 ) = ℓ(µ 1 /ν 1 ), so Γ θ µ 0 /ν 0 and Γ θ µ 1 /ν 1 are characters of S n .A disconnected skew shape µ/ν is naturally associated to a Young subgroup S of S n , defined by the rule that the simple transposition (i, i + 1) ∈ S if µ i+1 ⪈ ν i (i.e. if the i-th and i + 1-th rows of µ/ν are in the same connected component).By Lemma A.2, K θ,µ+δ−w ′ (ν+δ) = 0 whenever w ′ / ∈ S. Let S 0 and S 1 be the Young subgroups corresponding to µ 0 /ν 0 and µ 1 /ν 1 respectively.Let I 0 , I 1 be, respectively, sets orbits of S 0 and S 1 on [n].
Finally let σ ∈ S n so that permuting the rows of µ 0 /ν 0 via σ (i.e.sending row i to row σ(i)) gives µ 1 /ν 1 .We require that σ be order preserving within the row indices of each connected component of µ 0 /ν 0 .

Hook-shape immanant characters
As in the proof of Proposition 3.2 (Proposition A.3), we let δ r = (r − 1, . . ., 1, 0).For any element x = x k x r of S k × S r , where • is concatenation of sequences.Let w ∈ S n be arbitrary.We proceed by evaluating and simplifying the expression By definition, Now Γ λ µ k /ν k × Γ σ µ r /ν r is defined to be zero on all elements not in S k × S r .To shorten notation moving forward, let C kr n (w) := C n (w) ∩ (S k × S r ).Then |Z n (w)| x k xr∈C kr n (w) Γ λ µ k /ν k (x k ) Γ σ µ r /ν r (x r ) .

2. 3 .
Hessenberg functions.If θ = (N ), then all of the Kostka numbers in equation (1) are either 0 or 1 depending on whether or not µ + δ − w(ν + δ) has any negative entries.Thus the character Γ (N ) µ/ν depends only on the pattern of zeros in the Jacobi-Trudi matrix H µ/ν .

Lemma 4 . 1 .
Fix θ and µ/ν and set n = ℓ(µ/ν).If there exists a finite set of Hessenberg functions {h i | i ∈ I} such that for all w ∈ S n , h i (w) = 1 for precisely K θ, w -many i ∈ I, then Conjecture 1.3 holds for the character Γ θ µ/ν .
and let z w = |Z w |.Note n / ∈ Z w .The content w has n − z w many nonzero terms, and thus K θ, w = n−zw−1 k by Lemma 2.2.

=.
|Z n (w)| x k xr∈C kr n (w) 1 |C k (x k )| |C r (x r )| x ′ k ∈C k (x k ) x ′ r ∈Cr(xr) K λ, x ′ k K σ, x ′ rNow we decompose C kr n (w) further.If λ k , λ r are partitions, we write λ k • λ r for the partition constructed by concatenating λ k and λ r and reordering to be decreasing.Let ρ n (w) be the cycle type of w in S n so thatC n (w) = {w ′ ∈ S n | ρ n (w ′ ) = ρ n (w)}.Define ρ k and ρ r similarly.Let C n (λ) denote the conjugacy class of cycle type λ in Algebraic Combinatorics, Vol. 7 #1 (2024) 155 N. R. T. Lesnevich S n .ThenC n (w) ∩ (S k × S r ) = {x k x r ∈ S k × S r | ρ k (x k ) • ρ r (x r ) = ρ n (w)} = λ k •λr=ρn(w) λ k ⊢k, λr⊢r {x k x r ∈ S k × S r | ρ k (x k ) = λ k , ρ r (x r ) = λ r )} = λ k •λr=ρn(w) λ k ⊢k, λr⊢r C k (λ k ) × C r (λ r ) Now Γ λ µ k /ν k • Γ σ µ r /ν r (w) simplifies to |Z n (w)| λ k •λr=ρn(w) λ k ⊢k, λr⊢r x k ∈Cw(λ k ) xr∈Cr(λr) in Theorem 4.2 yields only abelian Hessenberg functions.Corollary 4.8.If θ is a hook partition and µ/ν is pre-abelian, then Γ θ µ/ν is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup.In other words, under these assumptions Conjecture 1.2 holds.Proof.If h is pre-abelian then h J is abelian for all J ⊂ [n − 1].So the Hessenberg functions in the decomposition from Corollary 4.3 are all abelian.Apply Theorem 4.5.□ The following result is due to Dahlberg.Theorem 4.9.[3, Thm.5.4] If h is such that h(i) − i ⩽ 2, then Γ h is the character of a permutation representation of S n whose transitive components are each isomorphic to the action of S n on cosets of a Young subgroup.