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DEFECT ZERO p−BLOCKS FOR FINITE SIMPLE GROUPS
Andrew Granville and Ken Ono
Transactions of the American Mathematical Society, 348, 1, 1996, pages 331-347.
Abstract. We classify those finite simple groups whose Brauer graph (or decom- position matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p−blocks remained unclassi- fied were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (n´ee the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjec- ture of Garvan, that establishes certain multiplicative congruences when 5 ≤ p ≤ 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p−blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p−modularly irreducible representations of Sn, for almost all n.
- Introduction
An ordinary representation of a group G of degree n is a group homomorphism from G to Gln(C), the group of invertible n × n matrices with complex coefficients. Such a representation may be viewed as a homomorphism from G to the group of isomorphisms of an n−dimensional complex vector space V to itself. An irreducible representation is an ordinary representation which does not have a non-trivial stable subspace; and, in a finite group G, the equivalence classes of such representations are in 1-1 correspondance with the conjugacy classes of G. In the symmetric group Sn, a conjugacy class is the set of permutations with a given cycle structure, and so they are in a natural 1-1 correspondance with the set of partitions of n (a partition of n is a non-increasing sequence of positive integers whose sum is n). Thus the number of irreducible representations of Sn equals p(n), the number of partitions
of n; which Hardy and Ramanujan showed is ∼ (^4) n^1 √ 3 eπ
2 n/ (^3).
The first author is a Presidential Faculty Fellow and an Alfred P. Sloan Research Fellow. His research is supported in part by the National Science Foundation.
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2 ANDREW GRANVILLE AND KEN ONO
Young [10,19] described a natural correspondence between partitions of n and irreducible representations of Sn: Given a partition [λ] = λ 1 + λ 2 + · · · + λk of n (where λ 1 ≥ λ 2 ≥ · · · ≥ λk), Young constructed a set of matrices for the represen- tation of Sn attached to [λ] (which we also denote by [λ]) by examining the action of Sn on Young tableaux, combinatorial objects constructed from the Ferrers-Young diagram of [λ]. The Ferrers-Young diagram of a partition [λ] of n is an array of nodes with λk nodes in the kth^ row. We assign numbers to the rows and columns, and coordinates to the nodes, just as we do for a matrix. The (i, j) hook is the set of nodes directly below, together with the set of nodes directly to the right of, the (i, j) node, as well as the (i, j) node itself (that is, the nodes (i, k) with k ≥ j together with the nodes (k, j) with k ≥ i). The hook number, denoted by H(i, j), is the total number of nodes on the (i, j) hook. A t-core partition of n is a partition of n in which none of the hook numbers are divisible by t.
Example. The Ferrers-Young diagram of the partition 4 + 3 + 1 of 8 is
1 2 3 4 1 • (1,1) • (1,2) • (1,3) • (1,4) 2 • (2,1) • (2,2) • (2,3) 3 • (3,1)
The hooks at (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), have hook numbers 6 , 4 , 3 , 1 , 4 , 2 , 1 , 1 , respectively. Therefore the partition 4 + 3 + 1 of 8 is a t-core partition for t = 5 and for all t ≥ 7.
The Young tableaux of [λ] are given by the n! different ways of assigning the numbers 1 through n to the nodes of the Ferrers-Young diagram, each node getting a different number. A standard tableaux is one where the numbers are increasing as one goes right or down (that is, the (i, j) entry is less than or equal to the (I, J) entry whenever i ≤ I and j ≤ J). Young showed how to formulate the properties of a given representation [λ] of Sn, in terms of certain d-by-d matrices of rational numbers, one for each element of Sn, which he constructed from the combinatorial properties of the set of standard tableaux (here d is the number of standard tableaux for [λ]). See Appendix I for further details. It turns out that the hooks of [λ] are intimately connected with the structure of the associated representation. In fact the degree d of the representation (which equals the number of standard tableaux) is given by the Frame-Thrall-Robinson hook formula [10,6.1.19]:
(1) d =
n! ∏ i,j H(i, j)^
Although Young’s matrices for the representation [λ] of Sn have rational en- tries, there is a change of basis under which all of the matrices have integer entries [19, 12.13]; hence we may reduce these entries modulo a given prime p to obtain the corresponding p-modular representation [¯λ]. Under this reduction, the charac- teristic 0 representations of Sn form equivalence classes, known as p-blocks. Let ρ 1 , ρ 2 ,... ρp(n) denote the ordinary irreducible representations of Sn. The Brauer graph is constructed by associating a node to each such representation, and then connecting two nodes i and j by an edge if and only if the reductions of ρi and
4 ANDREW GRANVILLE AND KEN ONO
The generating function for ct(n) is given [6, 11] by:
∑^ ∞
n=
ct(n)qn^ =
∏^ ∞
n=
(1 − qtn)t (1 − qn)
If p = 2 then Jacobi’s identity gives
∏^ ∞
n=
(1 − q^2 n)^2 1 − qn^
∑^ ∞
n=
c 2 (n)qn^ =
∑^ ∞
n=
q
n^2 +n (^2).
This confirms what we had above, replacing n by 2x 1 or − 2 x 1 − 1, as n is even or odd. We shall see in section 3 that c 3 (n) =
d| 3 n+
(d 3
where
3
is the Legendre symbol; and is therefore 0 if and only if there exists a prime p ≡ 2 (mod 3) such the exact power of p dividing 3n + 1 is odd. By elementary sieve theory we thus see that there are N/
log N integers n ≤ N for which c 3 (n) is non-zero. Therefore An has no defect zero 3-blocks for almost all n. Garvan, Olsson, Stanton and many others have speculated that cp(n) > 0 for all integers n ≥ 0, whenever prime p ≥ 5. From Proposition 1 this then implies that every symmetric group Sn and every alternating group An has a p−block with defect zero, for each prime p ≥ 5. In fact, it has even been conjectured that ct(n) > 0 for all integers t > 3, the so-called t-core partition conjecture. However since ckt(n) ≥ ct(n) whenever k is a positive integer, it follows that the t-core partition conjecture may be deduced by showing that ct(n) > 0 for all integers n ≥ 0, for all primes t ≥ 5, as well as for t = 4, 6 and 9. The result was proved for p = 5 and p = 7 by Erdmann and Michler [4] using ‘abaci’; and from exact formulae for c 5 (n) and c 7 (n) in [6]. The second author [17,18] proved the result for 4 ≤ t ≤ 11, using the theories of quadratic and modular forms. In this paper we complete the proof of the t-core partition conjecture:
Theorem 1. Every non-negative integer n has at least one t-core partition, pro- vided t ≥ 4.
Corollary 1. For any positive integer n and any prime p ≥ 5 , the symmetric group Sn and the alternating group An have a p−block with defect 0.
We are thus able to complete the classification of defect zero p−blocks in finite simple groups (using the main classification theorem, [8] and [14,23]):
Corollary 2. Every finite simple group G has a p-block of defect 0, for every prime p, except in the following special cases:
- G has no 2-block of defect 0 if it is isomorphic to M 12 , M 22 , M 24 , J 2 , HS, Suz, Ru, C 1 , C 3 , BM , or An where n 6 = 2m^2 + m nor 2 m^2 + m + 2 for any integer m.
- G has no 3-block of defect 0 if it is isomorphic to Suz, C 3 , or An with 3 n+1 = m^2 r where r is squarefree and divisible by some prime q ≡ 2 mod 3.
See Appendix II for a brief description of how these groups arise. In section 4 we investigate congruence properties of cp(n), the number of p−blocks with defect 0 for Sn. In addition to verifying certain multiplicative congruences for cp(n) where 5 ≤ p ≤ 23, conjectured in [7], we prove:
MODULAR REPRESENTATIONS 5
Theorem 2. For any prime p and positive integer m, in almost all symmetric groups Sn, the number of p−blocks with defect 0 is a multiple of m. If p is any prime, then the number of p−modularly irreducible representations of Sn is almost always a multiple of p.
By contrast, p(n) mod m (where p(n) is the number of irreducible representa- tions of Sn) is believed to follow no simple patterns, except in the special arithmetic progressions found by Ramanujan [2].
Acknowledgements: The authors thank Professors Gert Almkvist, Christine Bessen- rodt, Nigel Boston, Jon Carlson, Frank Garvan, Dennis Stanton and especially Jorn Olsson for helpful discussions in the preparation of this paper.
- The t−core conjecture for t ≥ 17 In this section we prove the conjecture for t ≥ 17. The idea behind the proof is to use the freedom available in choosing so many variables in (2) to reduce the problem to an application of Lagrange’s Theorem, that every non-negative integer is representable as a sum of four squares. To begin with though we show that all ‘small’ n can be represented as in (2).
Lemma 1. Every integer n ≤ t^2 / 4 may be represented by (2) with each xi = − 1 , 0 or 1.
Proof. For any fixed integer k ≤ t/2, we let xt−k = xt−(k−1) = · · · = xt− 1 = −1, and for a given I ⊂ { 0 , 1 , 2 ,... , t − k − 1 } of size k, we let xi = 1 if i ∈ I, and xi = 0 otherwise. Then the integer given by (2) is
(k+ 2
i∈I i. We claim that, for any given k ≤ m + 1, and integer r ∈
[(
k 2
, km −
(k 2
)]
, there is
a subset I of { 0 , 1 , 2 ,... , m} with
i∈I i^ =^ r. We prove this by induction on^ r: it is certainly true for r =
(k 2
by taking I = { 0 , 1 , 2 ,... , k − 1 }. Assume that it is true for r − 1, that is we have
j∈J j^ =^ r^ −^ 1. Now select the largest^ j^ ∈^ J^ with^ j^ ≤^ m for which j + 1 6 ∈ J. Evidently such a j exists (unless J = {m + 1 − k,... , m}, in which case r > km −
(k 2
∑ ), so let^ I^ be the set^ J^ with^ j^ replaced by^ j^ + 1, and then i∈I i^ = 1 +^
j∈J j^ =^ r. Thus above we see that every integer in [( k + 1 2
k 2
k + 1 2
k 2
)]
= [k^2 , k(t − k)]
is so represented by (2). Taking the union of these intervals for 0 ≤ k ≤ t/2 gives the result.
We will prove
Proposition 2. Any integer n ≥ 3 t+9 may be represented by (2), provided t ≥ 17.
Now, since 3t + 9 < t^2 /4 for t ≥ 15, Lemma 1 combined with Proposition 2 implies:
Theorem 3. Every non-negative integer has at least one t−core partition, provided t ≥ 17 ; that is, any integer n ≥ 0 may be represented by (2) once t ≥ 17.
Proof of Proposition 2. For any integer n ≥ 3 t + 9, let n 0 be the least residue of n (mod 3). If n 0 = 1 then let x 0 = 1 and xt− 1 = −1; if n 0 = 2 then let x 1 = 1 and
MODULAR REPRESENTATIONS 7
t = 4, 6 , 9 and any prime ≥ 5, since any integer ≥ 4 must be divisible by one of these numbers. Given Theorem 3, it only remains to prove the conjecture for t = 4, 5 , 6 , 7 , 9 , 11 and 13. All these cases have been handled in previous work except for t = 13. We briefly summarize the techniques that have been used: For t = 4 one can use (2) to show that c 4 (n) is the number of integer solutions x, y, z of 8n + 5 = x^2 + 2y^2 + 2z^2 (see [17]); and this is always > 0 by the work of Gauss. For t = 6 one can factor the generating function (3) into a product of two formal power series: The first is the generating function for the number of representations of an integer as the sum of three triangular numbers, and the second has all non- negative coefficients with leading term 1. Thus c 6 (n) > 0 since Gauss’s Eureka theorem asserts that every non-negative integer can be represented as the sum of three triangular numbers. Similarly for t = 9 we factor the generating function (3) into a product of two formal power series, the first of which is a power of an Eisenstein series, that we prove has all positive coeffcients, and the second of which has all non-negative coefficients with leading term 1. This forces all the coefficients of the resulting product to be positive. For prime values of t we can prove the result by explicitly computing the gen- erating function (3) as the sum of an Eisenstein series and a cusp form. This was done for t = 11 in [17], and we shall do it for t = 13 here. We begin by recalling various definitions and facts from the theory of modular forms: Let H be the upper half of the complex plane and let SL 2 (Z) act on it by linear fractional transformations. If N is a positive integer, then let Γ 0 (N ) denote the subgroup of SL 2 (Z) defined by
Γ 0 (N ) :=
a b c d
| ad − bc = 1, and c ≡ 0 mod N
Given a positive integer k and a Dirichlet character χ mod N, we say that a mer- morphic function on H is a modular form of weight k with character χ if
f
az + b cz + d
= χ(d)(cz + d)kf (z)
for all z ∈ H and all
a b c d
∈ Γ 0 (N ).
If f (z) is holomorphic on H as well as at the cusps (that is, at the rational z), then we say that f (z) is a holomorphic modular form of type (k, χ) and level N. The set of all such forms is denoted Mk(N, χ) and is a finite dimensional C−vector space. The subspace of Mk(N, χ) consisting of those modular forms which also vanish at the cusps, the cusp forms, is denoted by Sk(N, χ). Every modular form f (z) in Mk(N, χ) admits a Fourier expansion at infinity of the form
f (z) =
∑^ ∞
n=
a(n)qn
where q := e^2 πiz^. The Hecke operators Tp are natural linear transformations which preserve Mk(N, χ) and Sk(N, χ). For every prime p the image of the modular form
8 ANDREW GRANVILLE AND KEN ONO
f (z) is defined by
(5) f (z) | Tp :=
∑^ ∞
n=
(a(pn) + χ(p)pk−^1 a(n/p))qn.
We say that f (z) is an eigenform with respect to the Hecke operator Tp if there exists a complex number λp satisfying
f (z) | Tp = λpf (z).
One can construct modular forms, out of forms of a lower level, since if f (z) ∈ Sk(N, χ) then both f (z) and f (mz) belong to Sk(mN, χ), for any positive integer m. For any given N we can form the vector space generated by all modular forms in Sk(N, χ) obtained in this way from all Sk(M, χ), where M and the conductor of χ both divide N. This is the subspace of Sk(N, χ) of oldforms, denoted by Sold k (N, χ). The orthogonal complement of Sold k (N, χ) in Sk(N, χ) is Snew k (N, χ). It turns out that S knew (N, χ) has a basis of newforms, which are defined to be elements of this space which are also eigenforms of all of the Hecke operators Tp. Throughout we shall assume that a newform f (z) is normalized so that its Fourier expansion is of the form
f (z) = q +
∑^ ∞
n=
a(n)qn
(that is, we divide out to make the leading coefficient 1). With this normalization we obtain, for every prime p,
(6) f (z) | Tp = a(p)f (z).
Since newforms are eigenforms of the Hecke operators, the Fourier coefficients a(n) possess nice multiplicative properties. Specifically the coefficients satisfy
a(mn) = a(m)a(n) if gcd(m, n) = 1,
and
(7) a(pr^ ) = a(p)a(pr−^1 ) − χ(p)pk−^1 a(pr−^2 ).
For more details on the theory of modular forms see [13,15]. Deligne’s Theorem, implies that if f (z) = q +
n=2 a(n)q
n (^) is a newform of type
(k, χ) of level N , then for every prime p which does not divide N we have
| a(p) |≤ 2 p
k− 1 (^2).
In [15, 4.6.17] Miyake shows that we get the better upper bound | a(p) |≤ p
k− 1 2 when p does divide N. This allows us to obtain the following upper bound for |a(n)|:
10 ANDREW GRANVILLE AND KEN ONO
In the special case p = 3 we replace z by 3z so that the quotient of eta-functions is an Eisenstein series; in fact a modular form belonging to M 1 (9,
n
), with Fourier expansion
∑^ ∞
n=
c 3 (n)q^3 n+1^ =
η^3 (9z) η(3z)
∑^ ∞
n=
σ(n)qn
where σ(n) =
0 if n ≡ 0 mod 3 ∑ d|n
(d 3
if n ≡ 1 , 2 mod 3.
Thus c 3 (n) = σ(3n + 1) as noted in the introduction. We will need a lower bound for σp(n) for p ≥ 5:
Lemma 3. If p is a prime ≥ 5 , and σp(n) is the divisor function defined by
σp(n) =
d|n
χ(n/d)d
p− 2 3 ,
where χ(d) =
(d p
is the usual Legendre symbol, then
σp(n) ≥ n
p− 23 ∏
q|n, q prime
q
p− 3 2
Proof. Since σp(n) is a multiplicative function, it suffices to prove the result for
n = qk, a prime power. Writing Q = q
p− 3 (^2) we either have σp(qk) = Qk^ (if χ(p) = 0) or else
1 Qk^
σp(qk) = 1 ±
Q
Q^2
Q^3
Q^4
(±1)k Qk^
Q
Q
and the result follows.
We restrict our attention here to lower bounds for cp(n) once p ≥ 11:
Theorem 4. There are more than 2 α 5 pn
p− 2 3 p-blocks with defect zero, once n is sufficiently large, provided prime p ≥ 11.
Proof. By (9) we see that cp(n) = αp(σp(N ) +
i cifi(N^ )) where^ N^ =^ n^ +^
p^2 − 1 24 and fi(N ) is the Fourier coefficient of qN^ in the Fourier expansion of fi(z). By Lemma 2, we know that ∣ ∣ ∣ ∣ ∣
i
cifi(N )
i
|ci|
N
p− 3 (^4) (1 +
2)Ω(N^ ),
since each fi(z) has weight k = (p − 1)/2. Now suppose τ satisfies 1 +
2 = 2τ^. Then (1 +
2)Ω(N^ )^ = 2τ^ Ω(N^ )^ ≤ N τ^ and so |
i cifi(N^ )| ≤^ (
i |ci|)^ N^
p− 3 4 +τ^. On
the other hand, since p− 2 3 ≥ 4 when p ≥ 11,
σp(N ) ≥ N
p− 3 2
q|N, q prime
q
p− 23
≥ N
p− 3 2
q prime
q^4
π^4
N
p− 3 (^2).
MODULAR REPRESENTATIONS 11
Therefore, since N ≥ n and 90/π^4 − 1 / 2 > 2 /5, we have
cp(n) ≥ αpN
p− 23
π^4
i |ci| N
p− 3 4 −τ
2 αp 5
n
p− 2 3 ,
once N
p− 3 4 −τ^ ≥ 2
i |ci|.^
As mentioned above, Almkvist [1] has now determined the value of αp explicitly for all primes p. This gives us some hope of finding an explicit upper bound for each ci in the proof above, which would lead to an explicit version of Theorem 4. However we do not yet know how to do this, and have to work hard to even completely solve the case p = 13. What we will do is to fill out the steps of the above proof explicitly, using Maple, so as to determine the actual values of the ci.
Theorem 5. Every non-negative integer has at least one 13 -core partition. Actu- ally n has more than (n/10)^5 such partitions.
Proof. By (9) we have
∑^ ∞
n=
c 13 (n)qn+7^ = η^13 (13z) η(z)
= α 13 E 13 (z) + f (z)
where f (z) ∈ Snew 6 (13,
13
). This space of cusp forms has dimension 6, and Garvan [6] proved that it has the basis,
bi(z) :=
η^13 (13z) η(z)
η^2 (z) η^2 (13z)
) 7 −i for 1 ≤ i ≤ 6.
Now it is well known that Eisenstein series lie in the orthogonal complement to the cusp forms, in the space of modular forms, and so E 13 (z) and the bi(z) are linearly independent. Therefore in order to write f (z) = α 13
i γibi(z) above, and to determine α 13 , we equate the first seven terms of the Fourier expansions of both sides of (10) and solve the resulting linear equations. First note that
bi(z) = qi^
n≥ 1
(1 − qn)^13 −^2 i(1 − q^13 n)^2 i−^1 ≡ qi
(^7) ∏−i
n=
(1 − qn)^13 −^2 i^ mod q^8 ,
and so the first few Fourier coefficients are easily determined. We can compute E 13 (z) from the definition given:
E 13 (z) = q + 31q^2 + 244q^3 + 993q^4 + 3124q^5 + 7564q^6 + 16806q^7 +...
So we solve the matrix equation
1 1 0 0 0 0 0 31 − 11 1 0 0 0 0 244 44 − 9 1 0 0 0 993 − 55 27 − 7 1 0 0 3124 − 110 − 12 14 − 5 1 0 7564 374 − 90 7 5 − 3 1 16806 − 143 135 − 49 10 0 − 1
α 13
γ 1 γ 2 γ 3 γ 4 γ 5 γ 6
MODULAR REPRESENTATIONS 13
3548220835392 Rρ equals
1326473734176 + 870096491748ρ + 94767875376ρ^2 + 7911872133ρ^3 +... · · · + 918595056ρ^4 − 5815887 ρ^5 317901315600 + 321298585860ρ − 3005304012 ρ^2 + 13222526309ρ^3 −... · · · − 101172348 ρ^4 + 95885041ρ^5 − 132164787024 − 199061787876 ρ − 3238095204 ρ^2 − 6279291529 ρ^3 −... · · · − 8785908 ρ^4 − 40796693 ρ^5 24210604656 + 56495681340ρ + 893004468ρ^2 + 1563549235ρ^3 +... · · · + 5006820ρ^4 + 9362375ρ^5 − 7406385780 ρ − 131628545 ρ^3 − 483709 ρ^5 − 1946375244 ρ − 90540651 ρ^3 − 694383 ρ^5
Substituting this into (11), we find that 33463 η
(^13) (13z) η(z) =^ E^13 (z)−
ρ cρfρ(z), where
1182740278464 cρ = 11387025509088 − 18832940453556 ρ + 544407924080ρ^2 − 645774441961 ρ^3 + 3806036272ρ^4 − 4396980293 ρ^5.
Using a floating point routine to evaluate cρ for each ρ we get the values
≈ − 2. 373 ± 5. 33 i, − 5. 156 ± 12. 901 i, 8. 029 ± 26. 472 i,
so that 2
ρ |cρ|^ <^ 190. From the proof of Theorem 4, we need (n + 7)^5 /^2 −ln(1+
√ 2)/ ln 2 (^) ≥ 190 which is
true for n ≥ 65. The result may be verified for n ≤ 64 by explicit computation.
By combining Theorems 3 and 5, we obtain Theorem 1.
- Congruence properties for the number of defect zero p−blocks There has been much interest in congruence properties of p(n) since Ramanujan first conjectured the residue of p(n) modulo powers of 5,7, and 11, for certain values of n. It is now believed that, besides these very special congruences, there are no other moduli m for which p(n) behaves in a predictable way modulo m. However for cp(n) we will find many congruence properties which follow from the theory of modular forms as developed by Deligne, Serre and Sturm. Considering ct(n) where t is a power of 5, 7, or 11, we obtain congruences modulo powers of 5,7, and 11 which are equivalent to the Ramanujan congruences for p(n).
Proposition 3. If ct(n) is the number of t−core partitions of n, then for all k ≥ 1 and every integer n we have
c 5 k (5kn + δ 5 ,k) ≡ 0 mod 5k c 7 k (7kn + δ 7 ,k) ≡ 0 mod 7[k/2]+ and c 11 k (11kn + δ 11 ,k) ≡ 0 mod 11k
where δp,k := 1/ 24 mod pk.
Proof. By Euler’s generating function for p(n) and (3), we find that
∑^ ∞
n=
c`k (n)qn^ =
n=
p(n)qn
n=
(1 − q`
k (^) n )`
k
14 ANDREW GRANVILLE AND KEN ONO
If we let
n=1(1^ −^ q
lk^ n)lk = 1 + ∑∞ n=1 a`,k(n)q
`k^ n, then we obtain
ck (kn + δ,k) = p(kn + δl,k) +
∑^ n
i=
p(kn −ki + δ,k)a,k(i).
The result then follows immediately from Ramanujan’s partition congruences (see [2]), which state that p(km + δ,k) ≡ 0 mod `K^ for any integer m (where K = k if p = 5 or 11, and K = [k/2] + 1 if p = 7).
In [7] Garvan proves various congruences for cp(n) and conjectures [7,5.5] that for 5 ≤ p ≤ 23 we have
(12) cp(prn − δp) + δpcp(pr − δp)cp(pn − δp) + rp−^2 cp((pn/r) − δp) ≡ 0 mod p
for all primes r and any non-negative integer n where δp = p
(^2) − 1
- This was proved in [7] for p = 5, 7 , and 11, and also follows easily from Proposition 3 for these p. To prove Garvan’s conjecture for primes 13 ≤ p ≤ 23, we show that the modular form η
p(pz) η(z) |^ Tp^ is congruent, modulo^ p, to a scalar multiple of the unique weight p − 1 normalized eigenform with respect to SL 2 (Z). These are defined as follows: The canonical Eisenstein series of weights 4 and 6 have Fourier expansions
E 4 (z) = 1 + 240
∑^ ∞
n=
σ 3 (n)qn^ and E 6 (z) = 1 − 504
∑^ ∞
n=
σ 5 (n)qn,
respectively, where σk(n) :=
d|n d
k. The only normalized cusp form of weight
12, with respect to SL 2 (Z), is f 12 (z) := ∆(z) = η^24 (z). The only normalized cusp form of weight k (for k = 16, 18 or 22) with respect to SL 2 (Z), that is also an eigenform of the Hecke operators, is f 16 (z) = E 4 (z)∆(z), f 18 (z) = E 6 (z)∆(z) or f 22 (z) = E 4 (6)E 6 (z)∆(z), respectively.
Proposition 4. If 13 ≤ p ≤ 23 is prime, then
ηp(pz) η(z)
| Tp ≡ 24 fp− 1 (z) mod p.
Proof. Sturm [22] proved that two modular forms with integer coefficients, both of weight k with respect to Γ 0 (N ), are congruent modulo an integer m if the alleged
congruence holds for the first 12 k
p|N
1 − (^) p^12
p(pz) η(z) |^ Tp^ and^
ηp(z) η(pz) are weight^
p− 1 2 modular forms on Γ^0 (p), and so their product has weight p − 1. We checked (in Maple) that the coefficients of qm^ of this modular form and of 24fp− 1 (z) are congruent modulo p, for each
m ≤ p
(^3) −p (^2) −p+ 12 p. Thus, by Sturm’s theorem the two modular forms are congruent modulo p. But this implies the result since the Fourier expansion of η
p(z) η(pz) satisfies the congruence ηp(z) η(pz)
∏^ ∞
n=
(1 − qn)p (1 − qpn)
≡ 1 mod p,
because (1 − Q)p^ ≡ 1 − Qp^ mod p.
As a corollary we deduce Garvan’s conjecture [7]:
16 ANDREW GRANVILLE AND KEN ONO
- R. Brauer, Representations of finite groups, Lect. on Modern Math. 1 (1963), Wiley, New York, 133-175.
- K. Erdmann and G. Michler, Blocks for symmetric groups and their covering groups and quadratic forms preprint.
- P. Fong and B. Srinivasan, The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 121-191.
- F. Garvan, D. Kim and D. Stanton, Cranks and t−cores, Invent. Math. 101 (1990), 1-17.
- F. Garvan, Some congruence properties for partitions that are t−cores, Proc. London Math. Soc. 66 (1993), 449-478.
- D. Gorenstein, Finite simple groups: An introduction to their classification, Plenum Press, New York and London, 1982.
- I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976.
- G. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, Reading, 1979.
- A. Klyachko, Modular forms and representations of symmetric groups, integral lattices and finite linear groups, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 116 (1982).
- B. K¨ulshammer, Landau’s theorem for p−blocks of p−solvable groups, J. reine angew. Math. 404 (1990), 171-188.
- N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, New York,
- G. Michler, A finite simple group of Lie type has p-blocks with different defects if p 6 = 2, J. Algebra 104 (1986), 220-230.
- T. Miyake, Modular forms, Springer-Verlag, New York, 1989.
- J. Olsson, On the p−blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), 188-213.
- K. Ono, On the positivity of the number of t−core partitions, Acta Arithmetica 66 (1994), 221-228.
- , A note on the number of t−core partitions, The Rocky Mtn. J. of Math, to appear.
- G. de Robinson, Representation theory of the symmetric group, Edinburgh Univ. Press, 1961.
- G. Robinson, The number of p−blocks with a given defect group, J. Algebra 84 (1983), 493-
- J.-P. Serre, Divisibilite des coefficients des formes modulaires de poids entier, C.R. Acad. Sci. Paris A 279 (1974), 679-682.
- J. Sturm, On the congruence of modular forms, Springer Lect. Notes in Math. 1240 (1984), Springer Verlag, New York, 275-280.
- W. Willems, Blocks of defect zero in finite simple groups of Lie type, J. Algebra 113 (1988), 511-522.
Appendix I. Young’s matrices The general construction of Young’s matrices, corresponding to a given repre- sentation, is rather complicated to describe (see [19,2.1] for details). However, every permutation σ in Sn can be expressed as a product of transpositions of the form σr := (r, r + 1) (where 1 ≤ r ≤ n − 1), say σ = σr 1 σr 2... σrm. But then the matrix M corresponding to σ is given by M = Mr 1 Mr 2... Mrm , where Mr is the matrix corresponding to σr. Therefore we only need to construct the matrices corresponding to transpositions σr := (r, r + 1). This matrix has dimension d, the number of standard tableaux for partition [λ]. We order the standard tableaux T 1 , T 2 ,... Td in (essentially) lexicographic order: If row i is the first row for which the standard tableaux T ′^ and T ′′^ differ, and j is the first such entry in the ith row, then we write T ′^ < T ′′^ if the (i, j) entry in T ′^ is smaller than the (i, j) entry in T ′′. We then construct Mr , the matrix representing the transposition (r, r + 1), as follows:
- If r and r + 1 are on the same row of Ti then we place the number 1 in the (i, i) entry of Mr.
MODULAR REPRESENTATIONS 17
- If r and r + 1 are on the same column of Ti then we place the number −1 in the (i, i) entry of Mr.
- Otherwise r and r + 1 are on neither the same row nor the same column of Ti, and if we swap them around then we will get another standard tableaux, call it Tj. If i < j, and r and r + 1 are entries (u, v) and (U, V ) of Ti, respectively, then we must have u < U and v > V because of the lexicographic ordering of the tableaux. We then place the numbers −ρ, 1 − ρ^2 , 1, and ρ in the (i, i), (i, j), (j, i), and (j, j) entries of Mr , respectively, where 1/ρ := (U − u) + (v − V ).
- Zeros are placed everywhere else in Mr.
Appendix II. Simple groups Here we briefly describe the finite simple groups which are mentioned in Corol- lary 2. For a long time, the only known sporadic simple groups were the Mathieu groups: M 11 , M 12 , M 22 , M 23 , and M 24. These groups are highly transitive permu- tation groups where the subscript denotes the number of letters in the defining permutations. Many other sporadic simple groups are obtained by examining the Leech lattice, a 24-dimensional lattice which is defined in terms of the Mathieu group M 24. One can obtain J 2 , HS, Suz, C1 and C3 by examining the automorphism group of the Leech lattice. In some cases these groups are realized as automorphism groups of the Leech lattice which stabilize certain low dimensional sublattices, and in other cases they are realized as the full automorphism group of the Leech lattice with an enlarged ring of definition. The monster group M is the largest of the sporadic simple groups. Several of the sporadic groups are non-abelian composition factors for the centralizer of an element in M. The Baby Monster BM is constructed in this manner. The only remaining sporadic simple group occuring in Corollary 2 is Ru, the Rudvalis group. This group is realized as a 28 dimensional matrix group over the finite field with 2 elements. For more on the sporadic simple groups see [8].
Department of Mathematics, The University of Georgia, Athens, Georgia 30602 E-mail address: [email protected]
Department of Mathematics,The University of Illinois, Urbana, Illinois 61801 Current address: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 E-mail address: [email protected]
