On the second largest eigenvalue of some Cayley graphs of the symmetric group

Let Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_n$$\end{document} and An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{n}$$\end{document} denote the symmetric and alternating group on the set {1,…,n},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,\ldots ,n\},$$\end{document} respectively. In this paper we are interested in the second largest eigenvalue λ2(Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )$$\end{document} of the Cayley graph Γ=Cay(G,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma =\mathrm{Cay}(G,H)$$\end{document} over G=Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=S_{n}$$\end{document} or An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{n}$$\end{document} for certain connecting sets H. Let 1<k≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<k\le n$$\end{document} and denote the set of all k-cycles in Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n}$$\end{document} by C(n, k). For H=C(n,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,n)$$\end{document} we prove that λ2(Γ)=(n-2)!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )=(n-2)!$$\end{document} (when n is even) and λ2(Γ)=2(n-3)!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )=2(n-3)!$$\end{document} (when n is odd). Further, for H=C(n,n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,n-1)$$\end{document} we have λ2(Γ)=3(n-3)(n-5)!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )=3(n-3)(n-5)!$$\end{document} (when n is even) and λ2(Γ)=2(n-2)(n-5)!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )=2(n-2)(n-5) !$$\end{document} (when n is odd). The case H=C(n,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,3)$$\end{document} has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let 1≤r<k<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le r<k<n$$\end{document} and let C(n,k;r)⊆C(n,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(n,k;r) \subseteq C(n,k)$$\end{document} be set of all k-cycles in Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{n}$$\end{document} which move all the points in the set {1,2,…,r}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2,\ldots ,r\}.$$\end{document} That is to say, g=(i1,i2,…,ik)(ik+1)⋯(in)∈C(n,k;r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=(i_{1},i_{2},\ldots ,i_{k})(i_{k+1})\dots (i_{n})\in C(n,k;r)$$\end{document} if and only if {1,2,…,r}⊂{i1,i2,…,ik}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2,\ldots ,r\}\subset \{i_{1},i_{2},\ldots ,i_{k}\}.$$\end{document} Our main result concerns λ2(Γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )$$\end{document}, where Γ=Cay(G,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma =\mathrm{Cay}(G,H)$$\end{document} with H=C(n,k;r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,k;r)$$\end{document} with 1≤r<k<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le r<k<n$$\end{document} when G=Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=S_{n}$$\end{document} if k is even and G=An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=A_{n}$$\end{document} if k is odd. Here we observe that λ2(Γ)≥(k-2)!n-rk-r1n-r((k-1)(n-k)-(k-r-1)(k-r)n-r-1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{2}(\Gamma )\ge (k-2)! {n-r \atopwithdelims ()k-r} \frac{1}{n-r} \big ((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big ). \end{aligned}$$\end{document}We prove that this bound is attained in the special case k=r+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=r+1$$\end{document} , giving λ2(Γ)=r!(n-r-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}(\Gamma )=r!(n-r-1)$$\end{document}. The cases with H=C(n,3;1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,3;1)$$\end{document} and H=C(n,3;2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=C(n,3;2)$$\end{document} were considered earlier in [6].


Introduction
Let be a finite undirected graph and let A denote its adjacency matrix. The eigenvalues of this matrix, together with their multiplicities, are an important invariant of the graph. Since A is symmetric, all eigenvalues are real. For a regular graph of degree d, it is well-known that the largest eigenvalue of A is λ 1 ( ) = d. The second largest eigenvalue, denoted by λ 2 ( ), plays an important role in many theoretical and practical applications of graph theory, from geometry to computer science. The literature on the second largest eigenvalue is extensive, overviews and further references can be found in [1,3,5,12,13,16]. For regular graphs, the spectral gap λ 1 ( ) − λ 2 ( ) is known as the algebraic connectivity of .
In this paper we are concerned with the second largest eigenvalue of certain Cayley graphs Cay(G, H ) where G is a symmetric or alternating group and where H a connecting set in G. (In particular, we have H = H −1 = {h −1 | h ∈ H }, the identity element of G does not belong to H , and H is not contained in any proper subgroup of G.) For the definition of Cayley graphs see Sect. 2.4. References to recent work on the second largest eigenvalue of Cayley graphs over symmetric groups can be found in [6]. Even for symmetric groups, an explicit computation of it appears to be one of the most challenging problem in the area.
An essential feature of the Cayley graphs in these two theorems is that the connecting set H is invariant under conjugation by G. In fact, C(n, k) is a conjugacy class of S n . If k = n or n − 1, the above results are deduced from some analysis of the group characters in a relatively straightforward way. For arbitrary k, computing the second largest eigenvalue for Cay (G, C(n, k)) is an open problem. For k = 3 this is solved in [6,Theorem 3.4], for k = 2 see [10, Lemma 3].
Notation: All groups considered here are finite. Further, all modules and representations are over the field C of complex numbers. By CG we denote the group algebra of the group G over C. If H ⊂ G, then |H | is the number of elements of H and H + is the sum of all h ∈ H , as an element of CG. A sentence such as 'L is a G-module' means that L is a CG-module. If L is a CG-module and X is a subgroup of G, we write L| X for the restriction of L to X . The trivial G-module and its character are denoted by 1 G . We assume that the reader is familiar with general notions and elementary facts of finite group theory, including the representation theory of S n and A n .

Preliminaries
We collect several well-known facts and prerequisites for the paper. All graphs are finite, undirected and without multiple edges.

Graphs and eigenvalues
Let be a graph on the vertex set V ; we put m = |V |.
The group of all automorphisms of is denoted by Aut( ).
The adjacency matrix A = (a u,v ) of is given by a u,v = 1 if u ∼ v and a u,v = 0 otherwise. Then A determines . The eigenvalues of are, by definition, the eigenvalues of A. These are real since A is symmetric.

Equitable partitions
Let be a graph with vertex set V . Let P be a partition As usual, a second partition P is a refinement of P if every class of P is some union of classes of P .
For any graph examples of equitable partitions arise from the orbits of a group of automorphisms of , see Lemma 2.3. For this reason, equitable partitions are also called generalized orbits. For incidence graphs of geometries also the term tactical decomposition is used, see Dembowski's book [4].
Let CV be the vector space with basis V over C. Then A can be viewed as a linear transformation of CV , and then Av = x∈N (v) x for every v ∈ V . Let (·, ·) be the standard bilinear form on CV defined by (u, v) = δ u,v (the Kronecker delta) Let CP be the vector space with basis w 1 , . . . , w k . In this notation we have:

Lemma 2.2 Let be a graph with vertex set V and adjacency matrix A, and let P be a partition
To be in CP, the right-hand side must be constant on every V j , that is, . . , k} and every v, v ∈ V j . The converse and the second assertion of the lemma follows from (1) as well. Finally, if is regular of degree d then A(w 1 + · · · + w k ) = A v∈V v = d v∈V v = d(w 1 + · · · + w k ), so d is an eigenvalue of A on CP.
Let V = V 1 ∪ · · · ∪ V l be a refinement P of P and let as above w 1 , . . . , w l be a basis of CP . Then every w j ( j = 1, . . . , k) is a linear combination of w 1 , . . . , w l , so CP ⊂ CP , whence the additional statement.

Lemma 2.3
Let be a graph and G a group of automorphisms of . Let V 1 , . . . , V k be the orbits of G on the vertex set V of .

Eigenvalue inequalities
We denote the distinct eigenvalues of a graph by If is regular of degree d, then λ 1 = d by Theorem 2.1. The second largest eigenvalue, λ 2 , plays a distinguished role in graph theory as it provides bounds for the isoperimetric constant of , see also graph expanders, the Kahzdan constant and Cheeger inequality in [11]. From Theorem 2.1 and Lemma 2.2, we immediately have Lemma 2.4 Let P be an equitable partition of the regular graph and let β 2 be the second largest eigenvalue of B(P). Then λ 2 ≥ β 2 .
Upper and lower bounds on eigenvalues of a symmetric matrix over the reals can be obtained from a theorem of Hermann Weyl [18], see also [3, Theorem 2.8.1] and [2]. and We note in particular, for i, j ∈ {1, 2} we have We refer to (4) as the Weyl inequality. For adjacency matrices, we have several applications. First let γ 1 ≥ γ 2 ≥ γ 3 ≥ · · · ≥ γ m be all eigenvalues of , with repetitions. If is regular of degree d and connected, then we have d = γ 1 = λ 1 > λ 2 = γ 2 by Theorem 2.1, for other eigenvalues repetitions may appear.
Let v be a vertex of and let be obtained from by deleting all edges incident with v. Denote the adjacency matrix of by A and let γ 1 ≥ γ 2 ≥ · · · ≥ γ m be all eigenvalues of , with repetitions. Next let be the 'star' at v with d rays, where d is the degree of v, and with m − d − 1 isolated vertices. Denote its adjacency matrix by S. The eigenvalues of the star are , by a simple computation. Then A = A + S and from Theorem 2.5, we obtain the following Corollary 2. 6 Let v be a vertex of the connected graph and let be obtained by deleting all edges incident with v. Let λ 1 > λ 2 and λ 1 > λ 2 be the largest and second largest eigenvalues of and , respectively. Then Accounting for the isolated vertex v, the spectrum of the graph [V \ {v}] induced on V \ {v} is obtained from the spectrum of by removing one eigenvalue λ = 0. The first part of the corollary is therefore an instance of the well-known interlacing theorem, see [3, Theorem 2.5.1]. However, the second part of the corollary appears to be new.

Cayley graphs
Let G be a group with identity element 1 or 1 G and let H be a subset of G. Then H is a connecting set Thus, such a generalization is not essential. However, for the study of eigenvalues of Cay(G 1 , H ) by means of representation theory one sometimes prefers to deal with the adjacency matrix of (G, H ) if the representation theory of G is simpler than that of G 1 . In Sect. 6, we observe this for G = S n and G 1 = A n .
There is no harm in identifying g ∈ G with its matrix in the regular representation of G, of size |G| × |G|. A key fact for adjacency matrices of Cayley graphs then is that h∈H h, the sum over h ∈ H of the matrices just defined, is the adjacency matrix of Cay(G, H ). Thus A(G, H ) = H + under this identification. (Recall that we define H + as an element of the group algebra of G over C.) This is well known and explained, for instance, in [17, p. 384]. In other words, A is the image of H + in the regular representation of the group algebra. Note that A is a linear transformation of the vector space CV = CG, the C-span of G.

Some general results
In this section we discuss representation theoretical aspects of eigenvalue problems for Cayley graphs. Let G be a finite group, H ⊂ G a connecting subset, and put H + = h∈H as an element in the group algebra CG of G over C. If ρ is the regular representation of G then A(G, H ) = ρ(H + ). If we identify g with ρ(g) then A(G, H ) = H + as discussed in Sect. 2.4. Thus, if φ is any representation of G then φ(H + ) is meaningful, and the eigenvalues of φ(H + ) are eigenvalues of . We think that this convention makes the exposition more transparent.
By standard results, ρ is a direct sum of all irreducible representations φ of G, each occurring with multiplicity dim φ. It follows that the set of all eigenvalues of Cay(G, H ) (disregarding the multiplicities) is the union, over the irreducible representations φ of G, of the eigenvalues of φ(H + ). See [11, Proposition 7.1] or elsewhere.
If φ 0 is the trivial representation of G then, obviously, φ 0 (H + ) = |H |. Conversely, if |H | is an eigenvalue of φ(H + ), where φ is an irreducible representation of G, then φ = φ 0 by Theorem 2.1 (as φ 0 occurs in ρ with multiplicity 1; here we use that H is a connecting set).
(iii) Let V 1 = G 1 be the vertices of the Cayley graph Cay(G 1 , H 1 ). As L is an irreducible G 1 -module, L is an irreducible constituent of the regular G 1 -module CV 1 = CG 1 , and hence the eigenvalue of H + 1 on L occurs as an eigenvalue of H + (2) If G = S n , G 1 = S n−1 , then every irreducible G-module M is multiplicity free as a G 1 -module (that is, every irreducible constituent of M| S n−1 occurs exactly once). This follows from the branching rule expressed in terms of Young diagrams (we cannot get the same diagram by removing distinct single boxes from a given Young diagram. This remains true for alternating groups.) In addition, if C is a conjugacy class of S n then C ∩ S n−1 is either empty or a single conjugacy class of G 1 , as two permutations x, y ∈ G 1 are conjugate in S n if and only if they are conjugate in G 1 . In the following we view Sym(1, . . . , k) where T is a trivial F-module of dimension 2. By Schur's lemma, H + acts scalarly on L 1 , L 2 , and hence H + has at most four distinct eigenvalues on M. Clearly, if r = 1 then L 0 = 0 and so H + has at most three distinct eigenvalues.

Symmetric group: the largest conjugacy classes
For basic definitions concerning the characters of symmetric and alternating groups see [8,9]. Let G = S n and let 1 ≤ k ≤ n. We denote the set of all k-cycles in G by C(n, k). Since two permutations are conjugate to each other in G if and only if they have the same cycle type we have |C(n, k)| = |G : C G (g)| when g is any k-cycle and C G (g) denotes the centralizer of g in G. In particular, |C(n, n)| = (n − 1)! and |C(n, n − 1)| = n(n − 2)!. By the same argument, the enumeration of permutations by cycle-type shows that all other conjugacy class sizes in G are smaller if n > 4. It is well-known that the smallest subgroup of S n containing C(n, k) is S n if k is even and A n if k is odd.
In this section, we compute the second largest eigenvalue of H + when H = C(n, n) or H = C(n, n −1). For this we first quote a well-known result for the cycles of length n in representations of S n [8,Lemma 21.4]. Below let χ = χ μ denote an irreducible character of G labeled by the Young diagram μ.
(2) In this case χ , the maximum of the ratio χ m (h)/χ m (1) is attained for m = n − 2 if n is even, and for n = 3 is n is odd. This implies the result.
We expect that the behavior of χ [n−1,1] (h) is not typical when h is a k-cycle with k = n, n − 1. Namely, for k < n − 1 we expect that the second largest eigenvalue is attained for χ = χ [n−1,1] .
By Lemma 2.7 we have:

The natural S n -module and equitable partitions
Let G = S n and let K be the stabilizer of some point in {1, . . . , n}. Then the set G/K of cosets of K in G and {1, . . . , n} are isomorphic as G-sets. The associated permutation module over C is the natural module for G, denoted M = 1 G K . This is probably the most important S n -module. Moreover, we expect that in many cases the second largest eigenvalue of a Cayley graph over S n occurs on M (although we have no means to justify this).
For 1 ≤ r < k ≤ n let C(n, k; r ) ⊆ C(n, k) be the set of all kcycles in S n which move all the points in the set {1, 2, . . . , r }. That is to say, Proof This holds when k = n since C(n, n; r ) = C(n, n) for all r and so it suffices to show that X ⊇ Alt n for 1 ≤ r < k < n. Since for every i ∈ {2, . . . , n} there is some h ∈ C(n, k; r ) with h(i) = 1 it follows that X is transitive on {1, 2, . . . , n}. If i < n we may assume additionally that h(n) = n since k < n. Hence, the stabilizer of n in X is transitive on {1, 2, . . . , n − 1} and so X is doubly transitive. The result follows if k = 2 or = 3, since a double transitive group containing a 2-or 3-cycle contains Alt n . For k ≥ 4, consider the elements h = (1, 2, . .
is a 3-cycle and hence the result follows.
Theorem 5.2 Let G = S n and H = C(n, k; r ) for 2 ≤ r < k < n. Then the eigenvalues of H + on the natural module M are n−r − 1 and μ 4 = −(k − 2)! n−r k−r . In the remainder of this section let k, G and H be as in the theorem and put = Cay(G, H ). We exhibit two equitable partitions of so that their associated eigenvalues belong to {μ 1 , μ 2 , μ 3 , μ 4 }. The proof is completed at the end of the section.
Below a b for integers a ≥ b > 0 denotes the number of choices of b elements from a set of a elements. For uniformity, we need to make the cases with b = 0 and b = −1 meaningful: we define a 0 to be 1, and a −1 to be 0. Lemma 5.3 Let 1 ≤ r < k < n. Then P 1 is an equitable partition of with matrix The eigenvalues of B 1 are μ 1 = (k − 1)! n−r k−r , Proof Put F = Sym(1, . . . , r ) × Sym(r + 1, . . . , n − 1) and let F × K act on V as in Lemma 2.8. Then F × K stabilizes each V i and acts transitively on it. Hence P 1 is equitable by Lemma 2.3. 1) and v 3 = (n, n − 1) we have (i = 1): This gives the matrix above after some further transformations. We have confirmed the eigenvalues using a symbolic calculator such as Wolfram|Alpha [19].
If we take x = n − r − 1, y = 1, z = r − 1 then the equality holds. In addition, the equality fails for any other choice of {x, y, z} ∈ {1, n − r − 1, r − 1}. Whence the result. Proof It is well known that there is a basis of L such that all matrices g ∈ G are orthogonal, that is, if g L is the matrix of g on L then g T L = (g L ) −1 = (g −1 ) L , where (g L ) T is the transpose of g L . Then g L + g −1 L = g L + (g L ) T , whence the claim. Lemma 6.4 Let L be an irreducible S n -module and λ an eigenvalue of H + on L. Then λ ≤ |H | and λ = |H | implies dim L = 1.
Proof If H is connecting then the lemma follows from Theorem 2.1. Indeed, λ = |H | if L is the trivial module, and the trivial module has multiplicity 1 in CS n . If H is not a connecting set then H is a connecting set of A n (Lemma 5.1), and the same argument is valid if L is an A n -module. So |H | is an eigenvalue of H + on L if and only if the restriction of L to A n contains the trivial A n -module. This is well known to imply that dim L = 1.