Morita Equivalent Blocks of Symmetric Groups

A well-known result of Scopes states that there are only finitely many Morita equivalence classes of $p$-blocks of symmetric groups with a given weight (or defect). In this note we investigate a lower bound on the number of those Morita equivalence classes.


Introduction
Every p-block B of a symmetric group S n is uniquely determined by its weight w ≥ 0 and its core µ (a partition of n − pw, see next Section 3 for details). Scopes [12] showed that there are only finitely many Morita equivalence classes of p-blocks of symmetric groups over the field F p with a given weight (this confirms a special case of Donovan's conjecture). Puig [9] extended her result to blocks defined over Z. More precisely, the number of these so-called Scopes classes is 1 p wp p−1 provided w > 0. If p > 2 and w > 1, then there exist pairs of distinct Scopes classes whose cores are conjugate to one another. Since the corresponding blocks of such pairs are Morita equivalent (even isomorphic), the number of Morita equivalence classes M (p, w) of p-blocks with weight w satisfies M (p, w) ≤ 1 2p for w > 0 (see [10,Corollary 3.10]). In this paper we are interested in the sharpness of this bound. Obviously, there is only one class of blocks with weight (defect) 0 or 1 respectively. The blocks of weight 2 have been investigated by many authors. For instance, Richards [10] obtained a formula for the decomposition matrix of those blocks. Since the decomposition matrix is a Morita invariant over Z (up to permutations of rows and columns), it is possible to count Morita equivalence classes at least when p is small. We have used the computer algebra system GAP [4] to confirm the following conjecture for p ≤ 11. For blocks of weight w = 3 and p ≥ 5, Fayers [3] has shown that all decomposition numbers are 0 or 1. Using the Jantzen-Schaper formula [11] it is therefore possible to compute the decomposition matrices recursively. In this way we obtain M (5, 3) = 147 and M (7, 3) = 3936 using GAP. Note that (1.1) is sharp in these cases as well.
Much less is known about the blocks of weight w > 3. Nevertheless, for small primes the following complementary result seems to hold.
Finally, we remark that our observations should also apply to Hecke algebras.

A Morita invariant
Since the decomposition matrix of a block is usually not available (like in the situation of Conjecture 1.2), it is important to know other Morita invariants. The following approach applies to any finite group G and any prime p. We denote the set of p-regular elements of G by G 0 and the set of irreducible Brauer characters by IBr(G). For χ, ψ ∈ Irr(G) ∪ IBr(G) we are interested in the following p-scalar product The following lemma states that these numbers are preserved under Morita equivalence.
Lemma 2.1. Let B 1 and B 2 be Morita equivalent blocks of (possibly different) finite groups with respect to a complete discrete valuation ring. Let for i = 1, 2. Then there exists a permutation matrix T such that Since B 1 and B 2 are Morita equivalent, there exist permutation matrices S, T such that T Q 1 S = Q 2 . Since T is orthogonal, we conclude that Recall that the height h(χ) ≥ 0 of χ ∈ Irr(B) is defined by χ(1) p = p a−d+h(χ) where |G| p = p a and d is the defect of B. If h(χ) = 0, then the p-adic valuation of [χ, ψ] 0 equals h(ψ) − d for every ψ ∈ Irr(B) (see [5,Theorem 3.24]). Moreover, if h(χ) > 0, then the p-adic valuation of [χ, χ] 0 is larger than −d (see [5,Lemma 3.22(a)]). In this way, M encodes all character heights. Finally, one can show that M determines the elementary divisors of the Cartan matrix of B. However, the next theorem implies that all these invariants do not suffice to distinguish blocks of symmetric groups. We will see in the following that for blocks of symmetric groups the matrix M can be computed from the character table of a local subgroup which only depends on the weight of the block (see Theorem 4.1). At the same time we determine the signs of T in Theorem 2.2 without using the characters explicitly. To do so, we need to introduce a lot of notation.

Notation
We fix the following notation (details can be found in [7]). A partition of n ∈ N 0 is a nonincreasing sequence of positive integers λ = (λ 1 , . . . , λ l ) such that |λ| := l i=1 λ i = n. The number l(λ) := l is called the length of λ. We allow the empty partition () of 0. The set of partitions of n is denoted by P(n).
It is well-known that the conjugacy classes of the symmetric group S n of degree n consist of the elements with a common cycle structure. Hence, we can choose a set of representatives {s λ : λ ∈ P(n)} of those conjugacy classes. Similarly, the irreducible characters of S n are parametrized by P(n) and we will write Let p be a prime. Successively removing all hooks of length p from (the Young diagram of) λ ∈ P(n) yields the p-core λ (p) of λ. The number of removed hooks is called the weight w of λ. Observe that n = |λ (p) | + pw. Moreover, the p-sign of λ is defined by δ p (λ) = (−1) l i where the l i are the leg lengths of the removed hooks. By Nakayama's conjecture, characters χ λ , χ µ ∈ Irr(S n ) lie in the same p-block if and only if λ (p) = µ (p) . Therefore, we may speak of the core and the weight of a block B of S n .
Next we define the p-quotient of λ ∈ P(n). Let l = l(λ) and choose s ≥ 0 such that l + s ≡ 0 (mod p). Define where we omit parts which are zero. Finally, the p-tuple of partitions is called the p-quotient of λ. It turns out that |P (β i )| equals the weight of λ. In general, the set of p-tuples λ = (λ 0 , . . . , λ p−1 ) of partitions such that |λ i | = w is denoted by P p (w). The elements of P p (w) will always be denoted by bold Greek letters. Now we relate the characters of the block B above to the characters of the wreath product G w := C p S w where C p is a (fixed) group of order p. Let Irr(C p ) = {ψ 0 = 1, . . . , ψ p−1 } and λ ∈ P p (w). For i = 0, . . . , p − 1, the linear character ψ ⊗|λ i | := ψ i ⊗ · · · ⊗ ψ i ∈ Irr(C |λ i | p ) has a unique extension to C p S |λ i | (still denoted by ψ ⊗|λ i | ) which acts trivially on S |λ i | . Then we can define The degree of this character is By the hook formula and [6, Lemma 2.1], the map Irr(B) → Irr(G w ), χ λ → ψ λ is a height preserving bijection, i.e., p w (w!) p χ λ (1) p = (n!) p ψ λ (1) p for every χ λ ∈ Irr(B).
In order to obtain information on the decomposition matrix of B we need to label the conjugacy classes of G w . Here we identify C p with Z/pZ to simplify notation. For (x 1 · · · x w , σ) ∈ G w (with x 1 , . . . , x w ∈ Z/pZ, σ ∈ S w ) we define λ ∈ P p (w) as follows: For every cycle (a 1 , . . . , a s ) in σ let s ∈ λ xa 1 +···+xa s . It turns out that two elements of G w are conjugate if and only if they yield the same λ. Let g λ : λ ∈ P p (w) be a set of representatives for the conjugacy classes of G w . The class sizes can be computed with the formula where the row of χ λ corresponds to the row of ψ λ .

Osima's result
We remark that (4.1) does not depend on the labeling of IBr(B), since any permutation of Brauer characters can be realized by S. As in the proof of Lemma 2.1, we have where X denotes the complex conjugate of X. The second orthogonality relation for G w implies that In this way, M can be computed from the character table of G w and some elementary combinatorics for D. Doing so, we are able to distinguish the Scopes classes for p = 3 and w ≤ 15. For p = 5 and w ∈ {4, 5}, some of the Scopes classes yield the same M (see last section). Hence, we obtain only a lower bound on M (5, w) in these cases. For p = 2 we refine our method in the next section in order to deal with larger weights.
5 The case p = 2 in Conjecture 1.2 For p = 2 we notice first that X is integral and (4.2) simplifies to X t X = diag 2 l(µ) |C Sw (s µ )| : µ ∈ P(w) .
In the final paragraph of this section we investigate further properties of M which might be of interest for a potential theoretical proof. We consider the diagonal of M which is also the diagonal of It is easy to see that [χ λ , χ λ ] 0 = [χ µ , χ µ ] 0 if λ 0 = (µ 0 ) . This means that the diagonal entries of M come in pairs. On the other hand, our calculations indicate that [χ λ , χ λ ] 0 > [χ µ , χ µ ] 0 whenever λ 0 > µ 0 > (w/2, 1 w/2 ) where > denotes the lexicographical order. If this turns out to be true, then the permutation matrix T in Lemma 2.1 must be a product of disjoint transpositions. In this case it suffices to find characters χ λ , χ µ ∈ Irr(B 1 ) and χ σ , χ τ ∈ Irr(B 2 ) such that λ 0 = σ 0 = (µ 0 ) = (τ 0 ) and δ p (λ)δ p (µ) = δ p (σ)δ p (τ ). We do not know if this can be done in general.

Implementation
To check Conjecture 1.1 for small primes, we have implemented the following steps in GAP by making use of the hecke package: • determine the cores of all Scopes classes for given p and w up to conjugation of partition, • use the Jantzen-Schaper formula to compute the decomposition matrices of Scopes classes (this is somewhat faster than Richard's formula), • compute the multisets of row sums and column sums for a given decomposition matrix, • partition the set of decompositions matrices according to their row sums and column sums (decomposition matrices in different parts of this partition cannot correspond to Morita equivalent blocks), • use the GAP command TransformingPermutations to distinguish decomposition matrices with the same row sums and column sums.
We have verified our code with the known decomposition matrices of S n with n ≤ 18. The applicability of the procedure above is only limited by the time it takes to run through all Scopes classes, but it can run in parallel. The computation of M (11, 2) = 29,624 took about one day on an Intel Xeon E5520 processor. We could not check Conjecture 1.1 for p = 13, since there are already 372,308 classes to consider (hence ≈ 10 11 comparisons) and the decomposition matrix has size 104 × 90 in each case. Similarly, computing M (11, 3) is hopeless, since this number is expected to be 4,209,504.
On the other hand, our GAP programs for Conjecture 1.2 are mainly limited by the available physical memory. Our procedure here is given as follows: • compute the columns of the character table of G w corresponding to the elements g µ with µ ∈ Γ using a code provided by Thomas Breuer (this uses much less memory than to compute the full character table via CharacterTableWreathSymmetric), • determine the cores of the Scopes classes as above, • compute the p-signs δ p (λ) for χ λ ∈ Irr(B) with respect to the Scopes class just using the definition, • compute M 0 for each Scopes class, • compare the matrices M 0 using TransformingPermutations, • for p = 2 we only consider χ ∈ ∆.
All computations for Conjecture 1.2 take about five days on an Intel Xeon E5520 processor with 128GB memory.
Finally, for the exceptional case (p, w) = (2, 3) in Conjecture 1.2 we have computed the basic algebras of the blocks over F 2 in Magma and checked that these are isomorphic. All codes can be found on the author's homepage.

Concluding remarks
In the following paragraph we indicate an alternative approach to the conjectures stated in the introduction. Let T be the signed permutation matrix from Theorem 2.2 with T M 1 = M 2 T . Suppose that B 1 and B 2 are Morita equivalent. Then we may assume that M := M 1 = M 2 by Lemma 2.1. Since M is real and symmetric, M and T are simultaneously diagonalizable. Since M 2 = M (see (2.1)), all eigenvalues of M are 0 and 1. However, the situation gets more interesting if we restrict to characters of a fixed height. As in Section 5, let p = 2 and w be a 2-power. Then the matrix M 0 in (5.1) seems to have very interesting eigenvalues. For instance, the eigenvalues of 2 12 M 0 for w = 8 are 2 11 , 2 10 , 2 9 , 2 7 · 3, 2 5 · 3 2 , 2 4 · 3 · 5, 2 3 · 5 2 , 5 2 · 7.
If we can show that the eigenvalues of M 0 are pairwise distinct, then M and T can be diagonalized by the same real basis transformation. In this case T has real eigenvalues and we conclude that T is a product of disjoint transpositions (modulo signs). For p = 3 in Conjecture 1.2 it is not always enough to consider only height 0 characters. Even worse for p ≥ 5, Lemma 2.1 cannot tell all Scopes classes apart. For example the 5-blocks of weight 2 with cores 8, 4 2 , 1 4 and 9, 5 2 , 2 3 have decomposition matrices respectively. Since the first matrix has five rows with only one non-zero entry and the second matrix has only four such rows, it is clear that the blocks cannot be Morita equivalent. However, both matrices yield the same matrix M .