Generalised Cartan invariants of symmetric groups

K\"ulshammer, Olsson, and Robinson developed an l-analogue of modular representation theory of symmetric groups where l is not necessarily a prime. They gave a conjectural combinatorial description for invariant factors of the Cartan matrix in this context. We confirm their conjecture by proving a more precise blockwise conjecture due to Bessenrodt and Hill.


Introduction
Fundamental theory of representations of a finite group G over an algebraically closed field of characteristic ℓ > 0 was developed by Brauer. An essential feature of ℓ-modular representation theory is the construction of two sets of class functions defined on the elements of G of order prime to ℓ, namely, the irreducible Brauer characters and the projective indecomposable characters (see e.g. [16,Chapter 2]). These sets are dual to each other with respect to the usual scalar product. Further, there is a natural partition of each of these sets (as well as the set of ordinary irreducible characters of G) into disjoint subsets that correspond to the ℓ-blocks of G. For the symmetric group S n , Külshammer, Olsson, and Robinson [14] generalised character-theoretic aspects of Brauer's theory to the case when ℓ is not necessarily a prime and developed an analogue of block theory in this case. We begin by reviewing some of their definitions.
For any finite group G, denote by Irr(G) the set of ordinary irreducible characters of G and by C(G) the abelian group Z[Irr(G)] of virtual characters of G. Let ℓ, n ∈ N. An element g ∈ S n is called ℓ-singular if the decomposition of g into disjoint cycles includes at least one cycle of length divisible by ℓ. Define P(S n ) = {ξ ∈ C(G) | ξ(g) = 0 for all ℓ-singular g ∈ S n }.
Let {φ t } t∈T be a Z-basis of P(S n ), indexed by a finite set T . The ℓ-modular Cartan matrix of S n is the T × T -matrix Cart ℓ (n) = ( φ t , φ t ′ ) t,t ′ ∈T , where ·, · is the usual scalar product of class functions. In this paper we are only concerned with the invariant factors of Cart ℓ (n). They do not depend on the choice of the basis. (If ℓ is prime, then projective indecomposable characters defined with respect to ℓ form a basis of P(S n ).) The set Irr(S n ) is parameterised by the partitions of n in a standard way, and we write s λ for the irreducible character corresponding to a partition λ. If λ = (λ 1 , . . . , λ t ) is a partition (so that λ 1 ≥ · · · ≥ λ t > 0), we write |λ| = i λ i and l(λ) = t.
For r ∈ Z ≥0 , define If ℓ ∈ N and ℓ = i p ri i is the prime factorisation of ℓ, set [3,Definition 3.5]). Let a, b ∈ Z ≥0 . Write a ⋆b for the sequence a, . . . , a with b entries. Define k(b, a) to be the number of tuples (λ (1) , . . . , λ (b) ) of partitions such that b i=1 |λ (i) | = a. If R ⊂ R ′ are rings and A and B are R ′ -valued a × b-matrices, then A and B are said to be equivalent over R if there exist U ∈ GL a (R) and V ∈ GL b (R) such that B = U AV . (If the ring R is not specified, it is assumed to be Z.) The main aim of this paper is to prove the following result, conjectured by Bessenrodt and Hill (see [3,Conjecture 5.3]).
Using results of Hill [10], Bessenrodt and Hill [3] proved that Theorem 1.1 is implied by Theorem 3.15. Their reduction relies on the translation of the problem to Hecke algebras H n (q) where q is an ℓ-th root of unity (see Remark 1.4) and on results that relate the Grothendieck groups of finitely generated projective H n (q)modules to the basic representation of the affine Kac-Moody algebra of type A (1) ℓ−1 (see [2], [9,Theorem 14.2] and [13,Chapter 9]). In Section 3 we give a more direct and elementary proof of the reduction of Theorem 1.1 to Theorem 3.15 that uses only character theory of symmetric groups and wreath products. Our proof relies on an isometry constructed by Rouquier [18] between the block C(S n , ρ) of Theorem 1.1 and the "principal ℓ-block" of the wreath product S ℓ ≀ S w and on a result concerning class functions on wreath products proved in [8].
Intermediate results proved in Section 3 show that certain matrices studied by Hill in [10] may be interpreted naturally in terms of scalar products of class functions on G≀S w , where G is a finite group. These matrices are related to the inner product ·, · ℓ defined by Macdonald on the space of symmetric functions (see Remark 3.16). The results of this paper determine the invariant factors of these matrices (see Corollary 3.18). Theorem 3.15 is proved in Sections 4 and 5. In Section 4 we use Brauer's characterisation of characters to reduce Theorem 3.15 to the problem of finding the invariant factors of a certain matrix Y with rows and columns indexed only by the partitions λ such that all parts λ i are powers of a fixed prime p (cf. the definitions 4 ANTON EVSEEV before Theorem 4.9). Finally, in Section 5, we establish the invariant factors of Y by a direct combinatorial argument and thereby complete the proof of Theorem 1.1.
Remark 1.6. An important Z-grading on the Iwahori-Hecke algebras H n (q) (and, more generally, on cyclotomic Hecke algebras of type A) was discovered by Brundan and Kleshchev [6]. Recently, Ando, Suzuki, and Yamada [1] and Tsuchioka [20] have obtained formulae for determinants of graded Cartan matrices and proposed conjectures concerning their invariant factors. In particular, [20,Conjecture 7.17] generalises the statement of Theorem 1.1 (in the case when ℓ is a prime power).

Notation and preliminaries
In this section we introduce some general notation and review standard results that are used in the paper, in particular, those related to class functions on symmetric groups. Throughout, Z ≥0 and N denote the sets of nonnegative and positive integers respectively.
Matrices. Let T and Q be sets. If A is a T × Q-matrix, that is, a matrix with rows indexed by T and columns indexed by Q, we write A tq for the (t, q)-entry of A. In Section 3, A n tq denotes the n-th power of A tq (on the other hand, (A n ) tq is the (t, q)-entry of A n ). All matrices considered will have only finitely many non-zero entries in each row and each column, so matrix multiplication is unambiguously defined even for infinite matrices. By diag{(a t ) t∈T } we denote the diagonal T × Tmatrix with (t, t)-entry equal to a t for each t. We write A tr for the transpose of a matrix A. The identity T × T -matrix is denoted by I T .
Let R ⊂ R ′ be rings. As usual, GL T (R) denotes the group of invertible R-valued T × T -matrices A such that A −1 is R-valued. Two R ′ -valued T × Q-matrices A and B are said to be row equivalent over R if there exists U ∈ GL T (R) such that B = U A. The row space of A over R is the R-span of the rows of A as elements of (R ′ ) Q , the free R ′ -module of vectors indexed by Q.
Tuples and partitions. Let T be a set and w ∈ Z ≥0 . We define I(T ) to be the set of maps j : T → Z ≥0 such that j(t) = 0 for all but finitely many t ∈ T . Further, I w (T ) is the set of j ∈ I(T ) such that t j(t) = w.
Class functions on symmetric groups. Let Λ = ⊕ w≥0 C(S w ). For any finite group G write CF(G) for the set of Q-valued class functions on G. Then Λ Q = Q⊗ Z Λ may be identified with ⊕ w≥0 CF(S w ). The scalar product ·, · on Λ Q is defined via the standard scalar product on CF(S w ) in such a way that the components CF(S w ) are orthogonal.
By a graded basis of Λ Q we mean a Q-basis u = (u λ ) λ∈Par such that (u λ ) λ∈Par(w) is a basis of CF(S w ) for every w. If u = (u λ ) and v = (v λ ) are graded bases of Λ Q , we say that (u, v) is a dual pair if u λ , v µ = δ λµ for all λ, µ ∈ Par, where δ λµ is the Kronecker delta.
If G, H are finite groups and φ ∈ CF(G), ψ ∈ CF(H), then the outer tensor product φ ⊗ ψ ∈ CF(G × H) is defined by (φ ⊗ ψ)(g, h) = φ(g)ψ(h). If w = w 1 +· · ·+w n (w i ≥ 0), then the direct product of the symmetric groups S w1 , . . . , S wn is viewed as a subgroup of S w (known as a Young subgroup) in the usual way. An element f ∈ Λ Q is graded if f ∈ CF(S w ) for some w. In this case we write deg(f ) = w. If f and f ′ are graded elements of Λ Q of degrees d and w respectively, then their product is defined by With this product, Λ Q becomes a (graded) Q-algebra. The symbol Π, applied to elements of Λ Q , means this product. When applied to sets or groups, Π represents the usual direct product.
By A ×w we mean the direct product of w copies of a set or a group A. If φ is a class function on a group G, we write φ ⊗w = φ ⊗ · · · ⊗ φ ∈ CF(G ×w ). If U ≤ V are abelian groups, then V ⊗w is the tensor product (over Z) of w copies of V , and U ⊗w is viewed as a subgroup of V ⊗w in the obvious way.
While we find it convenient to use notation usually reserved for symmetric functions, the elements just defined are to be viewed as class functions on symmetric groups, and our arguments are essentially character-theoretic. One may identify Λ with the ring of symmetric functions via the isomorphism of [15, §I.7]. With this identification, the elements p λ , s λ and h λ are the same as those defined in [15, §I.2-3].

Scalar products of class functions on wreath products
We begin this section by summarising some notation and results concerning class functions on wreath products; for more detail, see [8, §2.3 and §4.1]. Let G be a finite group and w ∈ Z ≥0 . The wreath product G ≀ S w consists of the tuples (x 1 , . . . , x w ; σ) with x i ∈ G and σ ∈ S w . The group operation is defined by where we use the standard left action of S w on [1, w]. If w = 0, then G ≀ S w is the trivial group.
By a cycle in S w we understand either a non-identity cyclic permutation in S w or a 1-cycle (i) for some i ∈ [1, w]. Whenever (i) is to be viewed as an element of S w , it is interpreted as the identity element. The support of (i) is defined as {i}, while the support of a non-identity cycle σ is the set of points in [1, w] moved by σ. By o(σ) we mean the order of a cycle σ, with the order of (i) defined to be 1. A tuple σ 1 , . . . , σ n is called a complete system of cycles in S w if these cycles have disjoint supports and i o(σ i ) = w.
Whenever σ is a cycle in S w and x ∈ G, we set where x appears in an entry belonging to the support of σ (say, the first such entry). There is a unique equivalence relation on G ≀ S w satisfying the following rule: if σ 1 , . . . , σ n is a complete system of cycles, two elements of the form (u 1 , . . . , u w ; τ ) and y σ1 (x 1 ) · · · y σn (x n ) are equivalent if and only if τ = σ 1 · · · σ n and x j = u t u σ −1 . Each equivalence class contains exactly one element of the form y σ1 (x 1 ) · · · y σn (x n ) with σ 1 , . . . , σ n being a complete system, and the equivalence class of such an element has size |G| w−n . By [12,Theorem 4.2.8], any two equivalent elements of G ≀ S w are G ≀ S w -conjugate (even G ×wconjugate). By the same theorem, if σ 1 , . . . , σ n is a complete system, two elements y σ1 (x 1 ) · · · y σn (x n ) and y σ1 (u 1 ) · · · y σn (u n ) are G ≀ S w -conjugate if and only if there is a permutation τ of [1, n] In the case when φ is a character afforded by a QG-module, φ ⊗w is afforded by a corresponding Q(G ≀ S w )-module: see [12,Lemma 4.3.9]. Consider a tuple where φ i ∈ CF(G) and each f i is a graded element of Λ Q . Let w i = deg(f i ) and suppose that w = i w i . Then we define

Here, Inf
G≀Sw i Sw i f i is the inflation of f i , sending every g ∈ G ≀ S wi to f i (gG ×wi ), and · is the inner tensor product: (f · f ′ )(g) = f (g)f ′ (g) for all g. In the important special case when Ξ = ((φ, f )) with f ∈ CF(S w ), we have Let T be a finite set and φ : T → CF(G). For every λ ∈ PMap w (T ) define ζ If T is a subset of CF(G) and φ is the identity map, we will write ζ λ instead of ζ (φ) λ . These definitions are motivated, in part, by the fact that (3.4) Irr(G ≀ S w ) = {ζ λ | λ ∈ PMap w (Irr(G))} and the characters ζ λ are distinct for different λ ∈ PMap w (Irr(G)) (see [12,Theorem 4.3.34]).
For every λ ∈ Par(w) and χ ∈ CF(G ≀ S w ) define ω λ (χ) ∈ CF(G ×l(λ) ) by where n = l(λ) and σ 1 , . . . , σ n form a complete system of cycles in S w with o(σ i ) = λ i for each i. We will view ω λ (χ) as an element of CF(G) ⊗l(λ) . Let X be a finitely generated subgroup of the abelian group CF(G). The subgroup X ≀ S w of CF(G ≀ S w ) is defined to be the Z-span of the class functions ζ Ξ over all tuples Ξ as in (3.1) such that φ i ∈ X and f i ∈ Λ for all i. A subgroup U of a free abelian group V is said to be pure in V if for every v ∈ V such that nv ∈ U for some n ∈ Z − {0} we have v ∈ U . . Let X be a pure subgroup of C(G). Then X ≀ S w is precisely the set of all ξ ∈ C(G ≀ S w ) such that ω λ (ξ) ∈ X ⊗l(λ) for all λ ∈ Par(w).
If T is a finite set, let I w (T ) denote the set of all maps j : T → Z ≥0 such that t∈T j(t) = w. Lemma 3.2. Let X be a finitely generated subgroup of the abelian group CF(G). Let B be a Z-basis of X . Then the class functions ζ λ , λ ∈ PMap w (B), form a Z-basis of X ≀ S w .
Further, by Theorem 3.1, ω µ (ξ) ∈ C pri (S ℓ ) ⊗n for all µ. Let X = C pri (S ℓ )∩P(S ℓ ). Since both C pri (S ℓ ) and P(S ℓ ) are pure in C(S ℓ ), one easily sees that Since F preserves scalar products and maps P(S ℓw+e , ρ) onto X ≀ S w , we have proved the following result.
After conjugation by the diagonal matrix with the (i, i)-entry equal to (−1) i , this becomes the classical Cartan matrix of type A ℓ−1 . As is well known, the invariant factors of this matrix are ℓ, 1, 1, . . . , 1 (with 1 appearing ℓ − 2 times). This observation and Proposition 3.4 suggest the following general problem: given a finite set T and a map φ : q∈T . In the case when |T | = 1, the answer is given by Theorem 3.15, which is proved in Sections 4 and 5, and by Corollary 3.17. The rest of this section is devoted to an unsurprising reduction of the general problem to the case |T | = 1 (see Corollary 3.18).
where ν runs through PMap(T × Q) and t, q run through T, Q respectively.
Note that the summand indexed by ν in the above formula is zero unless |λ| The preceding definition is motivated by the following result. Lemma 3.6. Let φ : T → CF(G) and ψ : Q → CF(G) be arbitrary maps, where T, Q are finite sets and G is a finite group. Let A = ( φ(t), ψ(q) ) t∈T, q∈Q . Then for every w ≥ 0 and λ ∈ PMap w (T ), µ ∈ PMap w (Q), we have First, we prove a simpler lemma.
Lemma 3.7. Let G be a finite group and φ, ψ ∈ CF(G). If λ, µ ∈ Par(w), then Proof. The proof is similar to that of [8,Lemma 7.2]. Observe that ζ (φ,p λ ) vanishes outside the preimage in G≀S w of the conjugacy class of S w consisting of the elements of cycle type λ. A similar statement holds for ζ (ψ,pµ) , so the lemma holds if λ = µ. Assume that λ = µ and fix a complete system of cycles σ 1 , . . . , σ n with orders λ 1 , . . . , λ n in S w , where n = l(λ). With respect to the equivalence relation on G≀S w described above, the equivalence class of an element of the form y σ1 (x 1 ) · · · y σn (x n ) contains exactly |G| w−n elements, which are all conjugate to y σ1 (x 1 ) · · · y σn (x n ). Also, σ = σ 1 · · · σ n has w!/z λ conjugates in S w . Therefore, Proof of Lemma 3.6. One may parameterise the double t S |λ(t)|q S |µ(q)| -cosets in S w by the maps j ∈ I w (T × Q) such that Here, as usual, the double coset containing g ∈ S w corresponds to the map j defined by S |λ(t)| ∩ g S |µ(q)| ≃ S j(t,q) , where S |λ(t)| and S |µ(q)| are the appropriate direct factors of the two Young subgroups being considered. Using the definition of ζ (3.2)) and applying the Mackey formula, we see that ζ (3.8) and the summands are Fix a map j ∈ I w (T × Q) satisfying (3.8). For every q ∈ Q, where the sum is over all ν ∈ PMap |µ(q)| (T ) such that ν(t) = j(t, q) for all t. Indeed, s µ(q) , t p ν(t) is the value of the character s µ(q) on an element of cycle type t ν(t). Similarly, for every t ∈ T , where the sum is over all η ∈ PMap |λ(t)| (T ) such that |η(q)| = j(t, q) for all q. After using (3.10) and substituting (3.11) and (3.12), Eq. (3.9) becomes Here η and ν run through the set of elements of PMap w (T × Q) such that |η(t, q)| = j(t, q) = |ν(t, q)| for all t, q, and the second equality holds by Lemma 3.7. Summing over all j satisfying (3.8), we obtain where ν now runs through the elements of PMap w (T × Q) such that q |ν(t, q)| = |λ(t)| for all t and t |ν(t, q)| = |µ(q)| for all q. Moreover, this formula remains true if we sum over all ν ∈ PMap(T × Q), as the extra summands are all equal to 0. Comparing with Definition 3.5, we deduce the result. In the remainder of this section, T, Q, Z denote arbitrary finite sets. Let M be a Proof. Due to the duality conditions, we have Substituting (3.13) and (3.14) into (3.7), one obtains the result after a straightforward calculation.
Remark 3.10. The remaining proofs of this section (except for those of Lemmas 3.12 and 3.13) use essentially the same arguments as those presented in [10,Sections 3,4,6] and [3, Section 3], applied to a slightly more general situation.
Let A be a T × Q-matrix, where T, Q are finite, and let n ∈ Z ≥0 . Denote by T a Q-vector space with basis T . The n-th symmetric power Sym n ( T ) has a basis that consists of the monomials t∈T t i(t) where i runs through I n (T ). It is easy to see that, with respect to this basis and the analogous basis of Sym n ( Q ), the matrix Sym n (A) of the n-th symmetric power of the operator A : T → Q may be described as follows: where the sum is over all f ∈ I n (T × Q) such that q f (t, q) = i(t) for all t and t f (t, q) = j(q) for all q. Here, i ∈ I n (T ), j ∈ I n (Q) are arbitrary, and whenever the product AB of matrices is defined.
Proof. We begin with the case when u = p and v =p. Note that, if (λ i ) i is a tuple of partitions and α = i λ i , then p α = i p λ i (cf. [15, §I.2]). Also, recall that p λ = z −1 λ p λ for all λ and that (p,p) is a dual pair. Using these facts and applying Definition 3.5, we obtain , where the sum is over all ν ∈ PMap(T × Q) such that q ν(t, q) = λ(t) for all t and t ν(t, q) = µ(q) for all q.
In particular, A ≀ (p,p) = 0 unless t λ(t) = q µ(q). So we have a blockdiagonal decomposition of A ≀ (p,p), with blocks indexed by maps j ∈ I(N): the block of j is the intersection of the rows indexed by the maps λ ∈ PMap(T ) such that t m d (λ(t)) = j(d) for all d ∈ N and the columns indexed by the maps µ ∈ PMap(Q) such that q m d (µ(q)) = j(d) for all d.
If E is any finite set and α ∈ PMap(E), define α = ( α d ) d∈N ∈ d∈N I(E) by α d (e) = m d (α(e)) for all d ∈ N, e ∈ E (cf. [10,Notation 3.2]). Fix j ∈ I(N), and let C (j) be the corresponding block of A ≀ (p,p). The map λ → λ is a bijection from the set of rows of C (j) onto d∈N I j(d) (T ). Similarly, µ → µ is a bijection from the set of columns of C (j) onto d∈N I j(d) (Q).
Consider a row λ and a column µ of the block j. Let ν ∈ PMap(T × Q), and write i (d) (t, q) = ν d (t, q) for all d ∈ N, t ∈ T , q ∈ Q. Observe that ν satisfies the conditions stated after Eq.
Substituting this into (3.17), we obtain where i (d) runs through the elements of I j(d) (T × Q) satisfying the above conditions (for each d ∈ N). Comparing this with (3.15), we see that after the identifications λ → λ and µ → µ the block C (j) becomes equal to d∈N Sym j(d) (A).
Due to (3.16), we deduce that Now consider the general case and let M = M (u, p). Using Lemma 3.9 and Eq. (3.18), we obtain Proof. Let G be the cyclic group of order |T |, and let φ : T → Irr(G) be an arbitrary bijection. For each q ∈ Q set ψ(q) = t A tq φ(q), so that A = ( φ(t), ψ(q) ) q,t . By Lemma 3.6, the entries of A ≀w (s, s) are of the form ζ , so all entries of A ≀w (s, s) are integers. Since A ≀ (s, s) is block-diagonal with blocks A ≀w (s, s), the result follows. Proof. Let G be the cyclic group of order |T | and φ : T → Irr(G) a bijection. Let w ∈ Z ≥0 . As we observed above (see (3.4)), the functions ζ   Let ℓ ∈ Z. Applying Definition 3.5 to the 1×1-matrix (ℓ), set X (u,v) ℓ,w = (ℓ) ≀w (u, v) for any graded bases u and v of Λ Q . That is, X (u,v) ℓ,w is the Par(w) × Par(w)-matrix given by In particular, ℓ,w = diag{(ℓ l(λ) ) λ∈Par(w) }. By Lemma 3.9, Therefore, the determinant of X (s,s) ℓ,w is a power of ℓ (cf. [10, Section 6]). In Sections 4 and 5 we will prove the following key result. Here, c p,r (λ) are the integers defined by (1.2). ℓ,w (as {s λ } λ∈Par(w) is a Z-basis of C(S w )). Theorem 3.15, stated in terms of the form ·, · p r , was proved by Hill for r ≤ p and conjectured to hold in general: see [10, Theorem 1.3]. Our proof uses a different approach to that of Hill. In fact, the arguments of Section 5 become much simpler if r is large (more precisely, if p r > w): see Remark 5.2.

Corollary 3.18. Suppose that a T × T -matrix
A is equivalent to diag{(a t ) t∈T } for some a t ∈ Z ≥0 . Then A ≀w (s, s) is equivalent to the diagonal matrix with diagonal entries t∈T ϑ λ(t) (a t ), λ ∈ PMap w (T ).
Proof. Due to Proposition 3.14, we may assume that A = diag{(a t ) t∈T }. As A tq = 0 whenever t = q, Eq. The result now follows from Corollary 3.17, as invariant factors are well-behaved with respect to tensor products of matrices.
Theorem 1.1 may be deduced as follows. Consider A = ( β i , β j ) 0≤i,j≤ℓ−2 , a Cartan matrix of the principal ℓ-block of S ℓ (see (3.6)), so that A has invariant factors ℓ, 1, . . . , 1. By Proposition 3.4 and Lemma 3.6, the matrix Cart ℓ (S ℓw+e , ρ) is equivalent to A ≀w (s, s). Note that ϑ λ (1) = 1 for all λ ∈ Par. Hence, by Corollary 3.18, the matrix A ≀w (s, s) is equivalent to is equivalent to the diagonal matrix described in Theorem 1.1. Thus, it remains only to prove Theorem 3.15.

Reduction to p-power partitions
From now on, we fix a prime p and r ∈ Z ≥0 . Also, we adopt the convention that diagonal matrices are denoted by lower-case letters. If x = (x tq ) t,q is a diagonal matrix, we will write x t for x tt . Let w ≥ 0. Define the diagonal Par(w) × Par(w)matrix a = a (w) by a λ = p rl(λ) , so that a = X It is well known that M (h, s; w) ∈ GL Par(w) (Z) (see [12,Eq. 2.3.7]), so X ′ is equivalent to X (s,s) p r ,w . (In fact, it is the matrix X ′ rather than X (s,s) p r ,w that is considered in [10].) Let λ, µ ∈ Par(w). Define M λµ to be the set of all maps f : Since h λ is the permutation character corresponding to the Young subgroup i S λi , we obtain Remark 4.1. We may work over Z (p) rather than Z when proving Theorem 3.15. Indeed, since det(X ′ ) is a power of p, any diagonal matrix with p-power diagonal entries which is equivalent to X ′ over Z (p) must be equivalent to X ′ (and hence to X) over Z. Thus, we may replace M in the formula X ′ = M aM −1 by any matrix L which is row equivalent to M over Z (p) , that is, such that the rows of L span C (p) (S w ) (in the sense that is made precise below). In this section we will use Brauer's characterisation of characters to find an especially nice matrix L that satisfies this property; in particular, L is block-diagonal with respect to a certain partition of the set Par(w). This will considerably simplify the problem.
Let G be a finite group and h ∈ G be a p ′ -element (that is, an element of order prime to p). Define a class function χ G,h = χ h ∈ CF(G; Q) as follows: Here, as usual, g p ′ denotes the p ′ -part of g (that is, g p ′ is a p ′ -element and g = g p g p ′ = g p ′ g p for some p-element g p ∈ G). Further, if P is a subgroup of C G (h), define a map Res G,h P : CF(G; Q) → CF(P ; Q) by Res G,h P (ξ)(x) = ξ(hx) for all ξ ∈ CF(G, Q), x ∈ P. The following standard lemma, which is a consequence of Brauer's characterisation of characters, is key to the arguments of this section. (It will be used only in the case G = S w .) Proof. Part (i) follows immediately from [11,Lemma 8.19].
We note that the hypothesis of part (ii) for a given p ′ -element h does not depend of the choice of P because all Sylow p-subgroups of C G (h) are conjugate. By Brauer's characterisation of characters (see [11,Theorem 8.4]), the conclusion of (ii) will follow once we show that Res G E (ξ) ∈ C (p) (E) for all elementary subgroups E of G. For every such E we have E = Q × P where P is a p-group and Q is a p ′ -group. Let Q be a set of representatives of the conjugacy classes of Q. Then Res G E (ξ) = q∈Q (χ Q,q ⊗ Res G,q P (ξ)).
By the hypothesis, Res G,q P (ξ) ∈ C (p) (P ) for all q ∈ Q. Also, χ Q,q ∈ C (p) (Q) by (i). The result follows.

Definition 4.3. Let T be a finite set. Let R be an integral domain with field of fractions K. Denote by K T the vector space of row vectors
Let T = ⊔ i T i be a set partition of T and A be a finite T × T -matrix with entries in K. Let V ⊂ K T be the row space of A over R. We say that A splits over R with respect to the given set partition of T if π Ti (V ) ⊂ V for all i.
We use this definition in the case T = Par(w) as follows. Let N p ′ be the set of all natural numbers that are prime to p. Denote by Par ′ (w) the set of all partitions ν ∈ Par(w) such that ν i ∈ N p ′ for all i (such ν are called p-class regular in [14]). Let ν ∈ Par ′ (w). Recall that g ν ∈ S w is a fixed element of cycle type ν. For each ν ∈ Par ′ (w) define Par(w, ν) to be the set of all λ ∈ Par(w) such that n≥0 m jp n (λ)p n = m j (ν) for all j ∈ N p ′ . This leads to the set partition Par(w, ν).
Note that an element g ∈ S w has cycle type belonging to Par(w, ν) if and only if g p ′ has cycle type ν. We will identify Q Par(w) with CF(S w ) via With this identification, C(S w ) is the row space of the character table M (s,p). The row space of M = M (h,p; w) also equals C(S w ) since M (h, s; w) ∈ GL Par(w) (Z).
Let ξ ∈ C(S w ) be the character corresponding to a row of M . Then, for every ν ∈ Par ′ (w), the class function χ gν · ξ corresponds to π Par(w,ν) (ξ). However, by Lemma 4.2(i), we have χ gν · ξ ∈ C (p) (S w ). This means that M splits over Z (p) with respect to the set partition (4.2). Since M is rational-valued, we deduce the following more precise result using standard ring theory. We will use the following general result on split matrices. Lemma 4.5. Let R be an integral domain with field of fractions K. Suppose that T = ⊔ i T i , where T is a finite set. Let A be a T × T -matrix with entries in K that splits over R with respect to this set partition. Suppose that A is lower-triangular with respect to some total order on T and that A tt = 0 for all t ∈ T . Define the T × T -matrix A by Then A is row equivalent to A over R.
Proof. Let < be the given total order on T , and write A t for row t of A. It is enough to prove the following: if t ∈ T i and j = i, then Indeed, applying (4.4) repeatedly, one easily obtains A from A by elementary row operations defined over R. To prove (4.4), note that, as A splits, we have π Tj (A t ) = u∈T α u A u for some α u ∈ R. Since A is lower-triangular with non-zero diagonal entries, it follows that α u = 0 for all u ≥ t, so (4.4) holds.
Let M be the block-diagonal "truncation" of M defined as in the statement of Lemma 4.5 with respect to the set partition Par(w) = ν∈Par ′ (w) Par(w, ν). It is well known (and easy to see from the definition) that M is lower-triangular with respect to the lexicographic order on Par(w) and that the diagonal entries of M are non-zero. Hence, applying Proposition 4.4 and Lemma 4.5, we obtain the following result. LetM =M (w) be the Par(w)×Pow(w)-submatrix of M (w) . The following result is an immediate consequence of Lemma 4.6, due to the block-diagonal structure of M (note that Pow(w) = Par(w, (1 w ))).
Lemma 4.7. The row spaces ofM (w) and N (w) over Z (p) are the same.
Define a map ι : Par → j∈N p ′ Pow, λ → (λ j ) j∈N p ′ , by the identity m p n (λ j ) = m jp n (λ) for all j ∈ N p ′ , n ≥ 0. Let w ≥ 0 and ν ∈ Par ′ (w). Then ι restricts to a bijection from Par(w, ν) onto j∈N p ′ Pow(m j (ν)), also denoted by ι. Let so that L(ν) is a square matrix with rows and columns indexed by j Pow(m j (ν)). Define a Par(w) × Par(w)-matrix L by so that L is block-diagonal with respect to the set partition (4.2). Proof. Let γ ∈ Par(w) and consider the class function ξ γ ∈ CF(S w ) corresponding to row γ of L (via the identification (4.3)). For each η ∈ Par ′ (w) let P η be a Sylow p-subgroup of C Sw (g η ). We will verify that ξ γ satisfies the hypothesis of Lemma 4.2(ii), i.e. that (4.6) Res Sw,gη Pη ξ γ ∈ C (p) (P η ) for all η.
First, it follows from (4.5) that Res Sw,gη Pη ξ γ = 0 if γ / ∈ Par ′ (w, η). Let ν ∈ Par ′ (w) be such that γ ∈ Par(w, ν). Consider an arbitrary partition µ ∈ Par(w, ν). Write ι(γ) = (γ j ) j∈N p ′ and ι(µ) = (µ j ) j∈N p ′ . We have For each j, let G j denote the usual complement to the base subgroup in the wreath product C j ≀ S mj (ν) , so that G ≃ S mj (ν) ; we view G j as a subgroup of S w . Let P j be a Sylow p-subgroup of G j . Since the index of j G j in C Sw (g ν ) is prime to p, we may assume that P ν = j P j . Consider an element x ∈ P defined by x = j x j where x j ∈ P j , viewed as an element of S mj (ν) , has cycle type µ j . Let g ν = j g ν,j be the decomposition of g ν with respect to the direct product (4.7), so that g ν,j has cycle type (j ⋆mj (ν) ) as an element of S w . Since p ∤ j and x j is a p-element, it is easy to see that the cycle type of g ν,j x j ∈ C j ≀ S mj (w) as an element of S w is obtained from µ j by multiplying each part by j. It follows that j (g ν,j x j ) ∈ S w has cycle type µ. Therefore, by (4.5), where h γ j is viewed as a character of G j thanks to the identification of that group with S mj (ν) . It follows that Thus, (4.6) holds in all cases. By Lemma 4.2(ii), we have ξ γ ∈ C (p) (S w ). That is, row γ of L belongs to the row space of M over Z (p) . Since both L and M are rational-valued, the same holds over Z (p) . So the row space of L over Z (p) is contained in that of M . However, it is easy to see that det(M ) = det(L). Indeed, one obtains explicit expressions for det(M ) and det(L) using the definition of L and the fact that the matrices M and N are lower-triangular. The lemma follows. N (w) ). In Section 5 we will prove the following result. Assuming this, we can deduce Theorem 3.15 as follows. By Lemma 4.8, X ′ = M aM −1 is equivalent to X ′′ = LaL −1 over Z (p) . Recall that L is block-diagonal with respect to the set partition Par(w) = ν∈Par ′ (w) Par(w, ν). Since a is diagonal, the matrix LaL −1 is block-diagonal with respect to the same set partition.
We have Pow(w) = κ∈K Pow κ (w). In the sequel, "blocks" of a Pow(w) × Pow(w)matrix are understood to be ones corresponding to this partition of Pow(w). In particular, a Pow(w) × Pow(w)-matrix Z is said to be block-diagonal if Z λµ = 0 wheneverλ =μ. Further, Z is block-scalar if Z λµ = αλδ λµ for all λ, µ ∈ Pow(w), where (α κ ) κ∈K is a tuple of rational numbers.
Remark 5.2. In the case when p r > w, we have Pow κ (w) = {κ} for all κ ∈ K, and the proof below becomes much simpler (in particular, see Remark 5.5). The reader may find it helpful to consider the case p r > w in the first instance. Roughly speaking, the proof in the general case is obtained by applying the (trivial) proof for the case r = 0 "within blocks" and the proof for the case p r > w "between blocks".
Define diagonal Pow(w) × Pow(w)-matrices x <r , x ≥r , y <r , y ≥r ,ỹ as follows: for all λ ∈ Pow(w), It is easy to verify that x = x <r x ≥r and (5.5) y = y <r y ≥rỹ . (5.6) Define a Pow(w) × Pow(w)-matrix C as follows: so that b <r λ = p rl(λ <r ) for all λ. Note that b <r , x <r and y <r are block-scalar, and hence these matrices commute with C.
We have Since S ∈ GL Pow(w) (Z (p) ), the matrix is equivalent to Y ′ , and hence to Y , over Z (p) .
Remark 5.3. If we remove U and V from the product on the right-hand side of (5.15) and simplify the resulting expression, we are left with b <r x <r (y <r ) −1 . An easy calculation shows that v p (b <r λ x <r λ (y <r λ ) −1 ) = c p,r (λ) for all λ ∈ Pow(w) (see (5.29) below). Hence, to prove Theorem 4.9, it is enough to show that removing U and V from the product (5.15) does not change the invariant factors. Lemma 5.6 gives general sufficient conditions for this to be true for products of this kind. The fact that these conditions hold in our case is established at the end of the paper using Lemma 5.4, which gives detailed information on the entries of U .
For each λ ∈ Pow, define Note that Proof. We begin with (ii). Consider the block Pow κ for a fixed κ ∈ K. Write N (κ), A(κ), U (κ) for the Pow κ × Pow κ -submatrices of N, A, U respectively. We have that is to say, N λµ = x κ N (|λ ≥r |) λ ≥r µ ≥r for all λ, µ ∈ Pow κ . Indeed, every f ∈ M λµ satisfies λ f (i) = µ i for all i ∈ [1, l(µ)] such that µ i < p r . Hence, such a map f is determined by an element of M λ ≥r ,µ ≥r together with a permutation of the set {i | µ i = p t } for each t ∈ [0, r − 1] (and the correspondence is bijective).
Using (5.20) and the definition of U (cf. (5.8) and (5.13)), we obtain In order to prove (i) and (iii), we first need to establish a decomposition of N as a product, which may be informally described as follows. If λ, µ ∈ Pow(w), then an element of M λµ may be viewed as a way to aggregate the parts µ j into "lumps" and to associate bijectively some i ∈ [1, l(λ)] with each lump in such a way that λ i is the sum of the parts µ j in the lump. This process may be split into two stages: first, aggregate the parts µ j ≥ p r that are supposed to go to the same lump, without touching the parts µ j < p r ; then, aggregate the parts µ j < p r with each other and with the lumps obtained in the first stage to obtain the desired element of M λµ . This leads to a decomposition of N as a product of two matrices. The following construction makes this argument precise.
Note that λ → (λ <r , λ ≥r ) is a bijection from Pow(w) onto P. For every Pow(w) × Pow(w)-matrix Z, we write Z ⋆ for the Pow(w) × P-matrix obtained from Z by 24 ANTON EVSEEV relabelling the set of columns via this bijection; and Z ⋆⋆ denotes the P × P-matrix obtained by relabelling both rows and columns. Let ι : P → Pow(w) be the inverse of our bijection.
Due to (5.23), we deduce that v p (U λµ ) > v p ((x <r λ ) −1 ) + e µ + f λ − f µ = −e λ + e µ + f λ − f µ = k λ − k µ . Remark 5.5. In the special case when p r > w, the proof of Lemma 5.4 is much simpler and may be sketched as follows. First, parts (i) and (ii) are obvious. (Note that in the given case C = I Pow(w) , and so U = x −1 N : see (5.8) and (5.13).) Secondly, part (iii) follows from part (i) together with the facts that U λµ = 0 if M λµ = ∅ and k λ < k µ if M λµ = ∅ andλ =μ. The latter inequality can easily be proved by reducing to the case when µ is obtained from λ by replacing one part p j (for some j > 0) with p parts of size p j−1 .
Lemma 5.6. Let R be a discrete valuation ring with field of fractions K and valuation v : K → Z∪{∞}. Let I be a finite set. Suppose that s, t, u, P, Q ∈ GL I (K) and s, t, u are diagonal. Set ρ i = v(s i ) + v(t i ) + v(u i ) for all i ∈ I. Suppose that there exist tuples (α i ) i∈I and (β i ) i∈I of rational numbers such that for all i, j ∈ I the following hold: Then sP tQu is equivalent to stu over R.
Proof. Let π be a uniformising element of R. For d ∈ N, consider the simple extension K ′ of K generated by a d-th root of π, and let R ′ be the integral closure of R in K ′ . Then R ′ is a discrete valuation ring (see e.g. [19,Chapter 1,Proposition 17]). If we view all the matrices in the lemma as ones with entries in K ′ rather than K, then all valuations are multiplied by d. Thus, choosing an appropriate d, we may assume that α i and β i are integers for all i.
Similarly, Q ′ ∈ Γ by (iii). So P ′ Q ′ ∈ Γ. Fix a total order ≤ on I such that i ≤ j implies ρ i ≤ ρ j for all i, j ∈ I. Using standard Gaussian elimination, one can decompose any element of Γ as a product of a lower-triangular and an upper-triangular matrix (with respect to this order) such that both matrices belong to Γ. In particular, P ′ Q ′ = JH for some lower-triangular J ∈ Γ and upper-triangular H ∈ Γ. Let J ′ = st (1) J(st (1) ) −1 and H ′ = (t (2) u) −1 Ht (2) u. Then Now J ′ is lower-triangular and v(J ′ ii − 1) > 0 for all i ∈ I because J has the same properties. Further, if i > j are elements of I, then ρ i ≥ ρ j , and hence v( Hence, J ′ ∈ Γ ≤ GL I (R). By a similar argument, it follows from (v) that H ′ ∈ GL I (R). Therefore, Z is equivalent to stu over R.
We are now in a position to complete the proof of Theorem 4.9. We will apply Lemma 5.6 to the product Y ′′ = x <r U b <r (x <r ) −1 (y <r ) −1 V x <r (see (5.15)), with α λ = k λ /2 and β λ = −k λ /2 for all λ ∈ Pow(w). We check the conditions of the lemma one by one.
So condition (ii) holds.