A Note on Cartan Matrices for Symmetric Groups

Christine Bessenrodt

Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg

D-39016 Magdeburg, Germany Email: bessen@mathematik.uni-magdeburg.de

Jørn B. Olsson

Matematisk Afdeling, University of Copenhagen Copenhagen, Denmark Email: olsson@math.ku.dk

January 15, 2002

Abstract Using generating functions a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.

1 Introduction

In general, by a theorem of R. Brauer ([4], Theorem IV.3.11), the elementary divisors of the p-Cartan matrix of a finite group G, p a prime, are known to be the orders of the p-defect groups of the p-regular conjugacy classes of G. In case of the symmetric group S_{n} explicit calculations were made in [7]. These included formulae for the multiplicity of any given power of p as an elementary divisor in the p-Cartan matrix of S_{n} and in the Cartan matrix of any p-block of S_{n}. From this information it should in principle be possible to compute the determinants of these Cartan matrices, but the calculations for specific blocks appear to be rather complicated.

In this paper we show that these determinants may be computed by very simple formulae using generating functions, see Theorems 3.3 and 3.4. A special rôle is played by the generating function T (q) = ∑_{n≥0} t(n)q^{n}, where t(n) is defined as the number of divisors of the integer n.

2000 Mathematics Subject Classification. Primary 20C30 Secondary 05A15

Our calculations work also when p is composite. As explained in Section 3 this may have applications for a recently developed e-modular theory for symmetric groups.

2 The total length function

In this section we prove some results on the sums of the lengths of partitions and this is applied in Section 3 to Cartan matrices.

As is easily seen the generating function T (q) for the number of divisors of n may be expressed as follows:

T (q) = ∑_{i≥1}(q^{i} + q^{2i} + ...) = ∑_{i≥1} \frac{q^{i}}{1 − q^{i}} (1).

We let p(n) denote the number of partitions λ ⊢ n of the integer n and P (q) = ∑_{n≥0} p(n)q^{n} the corresponding generating function. When λ is a partition of n then l(λ) is the length (number of parts) of λ. We consider the integers l(n) = ∑_{λ⊢n} l(λ), the total length for n. Using the conjugacy map on partitions (see e.g. [1], [5]), l(n) is also the sum of the first parts of all λ ⊢ n.

It is convenient to use the exponential notation for partitions. Write λ = (1^{a_{1}(λ)}, 2^{a_{2}(λ)}, ...), where a_{i}(λ) is the multiplicity of i as a part in the partition λ. Thus l(λ) = ∑_{i≥1} a_{i}(λ). The following result is known (see [9]), but we include a proof, as we need to generalize it.

Proposition 2.1 Let L(q) = ∑_{n≥0} l(n)q^{n} be the generating function for l(n).

Then

L(q) = P (q)T (q) .

Proof. Using the well-known formula P (q) = ∏_{i≥1} \frac{1}{1−q^{i}} we see that P (q)(1 − q^{i}) is the generating function for the number of partitions of n with no part equal to i, and thus P (q)(1 − q^{i})q^{ia} counts the number of partitions with a_{i}(λ) = a. Thus the generating function for the numbers ∑_{λ⊢n} a_{i}(λ) may be expressed as

P (q)(1 − q^{i}) ∑_{a≥1} aq^{ia} = P (q)(q^{i} + q^{2i} + ...) = P (q)q^{i}/(1 − q^{i}) .

Therefore from (1) we obtain L(q) = ∑_{i≥1} P (q)q^{i}/(1 − q^{i}) = P (q)T (q), as de-

sired. □

Let e ∈ ℕ, e ≥ 2. We may divide the divisors of an integer n into two disjoint sets according to whether e divides the divisor or not. For the generating function T (q) this has as a consequence

T (q) = T (q^{e}) + T_{e}(q) (2),

where T_{e}(q) is the generating function for the number of divisors of n, which are not divisible by e. By iterating this, we obtain

T (q) = ∑_{j≥0} T_{e}(q^{e^{j}}) (3).

Clearly

T_{e}(q) = ∑_{e∤i} \frac{q^{i}}{1 − q^{i}} (4).

A partition λ of n is called e-class regular if a_{i}(λ) = 0, whenever e|i. In that case we write λ ⊢_{e} n. When p is a prime, the p-class regular partitions of n give the cycle types of the conjugacy classes of p-regular elements in S_{n}. The generating function P_{e}(q) for the number p_{e}(n) of e-class regular partitions of n is (see [5])

P_{e}(q) = P (q)/P (q^{e}) (5).

We refine the definition of l(n) above to l_{e}(n) = ∑_{λ⊢_{e}n} l(λ). Then we have

Proposition 2.2 Let L_{e}(q) = ∑_{n≥0} l_{e}(n)q^{n} be the generating function for l_{e}(n).

Then

L_{e}(q) = P_{e}(q)T_{e}(q) .

Proof. We modify the proof of Proposition 2.1 to see that for an integer i the generating function for the numbers ∑_{λ⊢_{e}n} a_{i}(λ) is P_{e}(q)q^{i}/(1 − q^{i}) if e ∤ i

and is 0 otherwise. Then the result follows from (4) above. □

Combining the propositions above with (2) and (5) we may deduce the following interesting identity.

Corollary 2.3 For all e ∈ ℕ, e ≥ 2 we have

L(q) = P_{e}(q)L(q^{e}) + P (q^{e})L_{e}(q) .

3 Cartan matrices

We keep the notations of the previous section.

The structure of centralizers of elements in S_{n} is well-known. Let the element x be contained in a conjugacy class labelled by the partition λ ⊢ n. Thus x contains a_{i}(λ) disjoint cycles of length i for each i and the centralizer C_{S_{n}}(x) is factored as a direct product of wreath products ℤ_{i} ≀ S_{a_{i}(λ)}.

Let p be a prime. When λ ⊢_{p} n is p-class regular, i.e., a_{i}(λ) = 0 whenever p | i, then the p-defect group of the conjugacy class labelled by λ is isomorphic to the p-Sylow subgroup of the direct product ∏ S_{a_{i}(λ)}. The p-Sylow subgroup of the symmetric group S_{a} has order p^{d_{p}(a)}, where

d_{p}(a) = ∑_{j≥1} [\frac{a}{p^{j}} ] .

Here [...] signifies “integral part of”. Thus the p-defect of the class of λ is given by

d_{p}(λ) = ∑_{i,j≥1} [\frac{a_{i}(λ)}{p^{j}}] (6).

Then by a result of Brauer, the determinant of the p-Cartan matrix C_{p}(n) of S_{n} is p^{c_{p}(n)}, where

c_{p}(n) = ∑_{λ⊢_{p}n} d_{p}(λ) (7).

The formula (6) makes sense also when p is not a prime. For e ∈ ℕ, e ≥ 2, and λ e-class regular we define the e-defect of λ by the formula

d_{e}(λ) = ∑_{i,j≥1} [\frac{a_{i}(λ)}{e^{j}}] (8).

As shown by the work [3], [6] it may be reasonable to consider an e-modular theory for characters of symmetric groups. A better understanding of this may contribute to the positive solution of a conjecture of Mathas (for further details on this see [3]). Also the theory provides a nice application of the theory of π-blocks. Mathas’ conjecture about the Cartan matrices for Hecke algebras of type A at an e-th root of unity is settled affirmatively if it can be proved that the determinant of a suitably defined e-Cartan matrix C_{e}(n) of S_{n} is a power of e. This Cartan matrix is in the non-prime case not unique. Choose a ℤ-basis (a “basic set”) for the ℤ-space of the restrictions of generalized characters of S_{n} to the e-regular classes. The matrix D_{e}(n) of coefficients expressing the restrictions of the irreducible characters of S_{n} to the e-regular classes as linear combinations of the characters in the basic set is considered as an e-analogue of the decomposition matrix and then as usual C_{e}(n) = D_{e}(n)^{t}D_{e}(n) is an e-analogue of the Cartan matrix. We may make the conjecture even more explicit:

Conjecture. In analogy with (7) above, the determinant of C_{e}(n) equals e^{c_{e}(n)}, where

c_{e}(n) = ∑_{λ⊢_{e}n} d_{e}(λ) (9).

It should be remarked that when e = p is a prime, then the elementary divisors of C_{p}(n) are exactly the p^{d_{p}(λ)}’s but this is false in general.

We proceed to compute the generating function C_{e}(q) for the c_{e}(n)’s. In the case of a prime this gives a very simple formula for the determinant of the p-Cartan matrix in terms of the numbers of p-regular partitions of the non-negative integers n − pv, v ≥ 1.

A partition λ ⊢_{e} n is said to be of e-class defect 0, if d_{e}(λ) = 0. This is the case, if and only if a_{i}(λ) ≤ e − 1 for all i (i.e., λ is also e-regular in the usual sense). The number of such partitions is denoted d_{e}^{0}(n). The corresponding generating function is

D_{e}^{0}(q) = \frac{P (q)P (q^{e^{2}})}{P (q^{e})^{2}} = \frac{P_{e}(q)}{P_{e}(q^{e})} (10)

(see e.g. Lemma (2.3) in [7]).

Proposition 3.1 The following relation holds:

c_{e}(n) = ∑_{v≥1} (c_{e}(v) + l_{e}(v)) d_{e}^{0}(n − ev) .

Proof. Let λ ⊢_{e} n. Write for each i ≥ 1 a_{i}(λ) = m_{i}(λ)e + r_{i}(λ), where 0 ≤ r_{i}(λ) ≤ e − 1. Let μ = (i^{m_{i}(λ)}) and ρ = (i^{r_{i}(λ)}). Clearly μ is e-class regular and ρ is of e-class defect 0. Moreover |μ|e + |ρ| = n and λ is uniquely determined by μ and ρ. Now since a_{i}(μ) = m_{i}(λ) for all i we have

d_{e}(λ) = ∑_{i,j≥1} [\frac{a_{i}(λ)}{e^{j}}] = ∑_{i,j≥1} [\frac{m_{i}(λ)e}{e^{j}}] = ∑_{i,j≥1} [\frac{a_{i}(μ)}{e^{j}}]+∑_{i≥1} a_{i}(μ) = d_{e}(μ)+l(μ),

which is independent of ρ. Thus sorting the partitions λ ⊢_{e} n according to the size v of μ we obtain

\begin{aligned}c_{e}(n) & = & ∑_{λ⊢_{e}n} d_{e}(λ) = ∑_{v≥1} ( ∑_{μ⊢_{e}v}(d_{e}(μ) + l(μ))) d_{e}^{0}(n − ev) \\ & = & ∑_{v≥1} (c_{e}(v) + l_{e}(v)) d_{e}^{0}(n − ev) , \end{aligned}

as desired. □

In terms of generating functions, Proposition 3.1 may be formulated as

Corollary 3.2 Let C_{e}(q) = ∑_{n≥0} c_{e}(n)q^{n} be the generating function for c_{e}(n).

Then

C_{e}(q) = (C_{e}(q^{e}) + L_{e}(q^{e})) D_{e}^{0}(q) .

We may now prove the first main result.

Theorem 3.3 We have

C_{e}(q) = P_{e}(q)T (q^{e}) .

Proof. Define C_{e}^{∗}(q) = C_{e}(q)/P_{e}(q). We need to show C_{e}^{∗}(q) = T (q^{e}). By Corollary 3.2 and (10) we have

C_{e}(q) = (C_{e}(q^{e}) + L_{e}(q^{e})) D_{e}^{0}(q) = (C_{e}(q^{e}) + L_{e}(q^{e})) P_{e}(q)/P_{e}(q^{e}) .

Then Proposition 2.2 implies

C_{e}^{∗}(q) = C_{e}^{∗}(q^{e}) + T_{e}(q^{e}) .

Iterating this and using (3) we obtain

C_{e}^{∗}(q) = ∑_{j≥1} T_{e}(q^{e^{j}}) = T (q^{e}) . □

Example. The 3-Cartan matrix of S_{12} is a square 36 × 36-matrix with elementary divisors 243, 81, 27, 9^{6}, 3^{13}, 1^{14}, where the exponents are the multiplicities. Thus the determinant is 3^{37}. By Theorem 3.3 the exponent 37 may be calculated as c_{3}(12) = p_{3}(9) · t(1) + p_{3}(6) · t(2) + p_{3}(3) · t(3) + p_{3}(0) · t(4) = 16 · 1 + 7 · 2 + 2 · 2 + 1 · 3.

Next we consider the “block version” of Theorem 3.3. It is well-known that the p-Cartan matrix of a finite group, p a prime, may be arranged in a diagonal block form if the characters are arranged according to the p-blocks of G. In case of the symmetric groups the ordinary irreducible and p-modular irreducible characters are distributed into p-blocks according to the so-called “Nakayama Conjecture” (see [5], 6.1.21). Two irreducible characters are in the same p-block if and only if the partitions labelling them have the same p-core. Thus each p-block B of S_{n} has an associated combinatorial invariant w = w_{B}, called the weight of B ([5], Section 6.2).

In [8] and [7] it was proved that the elementary divisors of the Cartan matrix C_{B} of a p-block B of S_{n} depend only on the weight w_{B}. Thus the same is true for the determinant of C_{B}. In [6] this was generalized to suitably defined e-blocks of S_{n}. In this case two irreducible characters are defined to be in the same e-block if the partitions labelling them have the same e-core and the weight of an e-block is defined correspondingly. Then the elementary divisors and the determinant of the Cartan matrix of an e-block depend only on the weight w of the block. Thus assuming that the conjecture mentioned above is true, this determinant should be a power of e, say e^{\overline{c}_{e}(w)}. Using Theorem 3.3 we may compute the generating function for these numbers. From the above it is clear that the number of e-blocks of weight w in S_{n} is equal to the number \overline{d}_{e}(n − ew) of e-core partitions of n − ew, “e-blocks of defect 0”. It is known (see e.g. [7]) that the generating function for the numbers \overline{d}_{e}(n) is

\overline{D}_{e}(q) = \frac{P (q)}{P (q^{e})^{e}} (11).

Theorem 3.4 Let \overline{C}_{e}(q) be the generating function for \overline{c}_{e}(n). Then

\overline{C}_{e}(q) = P (q)^{e−1}T (q) .

Proof. We decompose c_{e}(n) according to the e-blocks. Since there are \overline{d}_{e}(n − we) e-blocks of weight w in S_{n}, we see that

c_{e}(n) = ∑_{w≥0} \overline{c}_{e}(w) \overline{d}_{e}(n − ew) .

For the generating functions we then obtain using (11) that

C_{e}(q) = \overline{C}_{e}(q^{e})\overline{D}_{e}(q) = P (q)\frac{\overline{C}_{e}(q^{e})}{P (q^{e})^{e}} .

On the other hand Theorem 3.3 and (5) imply

C_{e}(q) = P_{e}(q)T (q^{e}) = P (q)\frac{T (q^{e})}{P (q^{e})} .

Thus

\frac{\overline{C}_{e}(q^{e})}{P (q^{e})^{e}} = \frac{T (q^{e})}{P (q^{e})} .

Replacing q^{e} by q, the theorem now follows. □

Remarks. (a) The coefficient k(e − 1, w) of q^{w} in P (q)^{e−1} is in fact the size of the Cartan matrix of an e-block of weight w.

(b) For e = 2, we obtain for the generating function \overline{C}_{2}(q) the very simple formula

\overline{C}_{2}(q) = P (q)T (q) = L(q) .

Example. The Cartan matrix of the principal 3-block of S_{12} (weight 4) is a square 20 × 20-matrix with elementary divisors 243, 81, 27, 9^{4}, 3^{7}, 1^{6}, where the exponents are the multiplicities. Thus the determinant is 3^{27}. By Theorem 3.4 the exponent 27 may be calculated as \overline{c}_{3}(4) = k(2, 3) · t(1) + k(2, 2) · t(2) + k(2, 1) · t(3) + k(2, 0) · t(4) = 10 · 1 + 5 · 2 + 2 · 2 + 1 · 3.

Report on recent developments. After this paper was submitted for publication there have been significant developments, which we briefly describe here. Brundan and Kleshchev have published a preprint [2] where the conjecture of Mathas mentioned above is proved. Indeed, let H_{n} be the Iwahori-Hecke algebra associated to S_{n} at a primitive e-th root of unity. Then the determinant of the Cartan matrix of a block of e-weight w of H_{n} is e^{\overline{c}_{e}(w)}. It is shown in [3] that an e-analogue C_{e}(n) of the Cartan matrix for S_{n} has the same determinant as the Cartan matrix of H_{n}. As described in Section 7 of [6] this implies that the conjecture mentioned above is true. Thus for an arbitrary e ≥ 2 we have det(C_{e}(n)) = e^{c_{e}(n)}. Correspondingly it is also true that the determinant of the Cartan matrix of an e-block of weight w in S_{n} is e^{\overline{c}_{e}(w)}.

Acknowledgment. The authors are grateful to the Danish Natural Science Research Council for the support of their collaboration.

References

[1] G.E. Andrews, The theory of partitions, Addison-Wesley 1976 (Reprinted Cambridge University Press 1998).

[2] J. Brundan, A. Kleshchev, Cartan determinants and Shapovalov forms, preprint 2001.

[3] S. Donkin, Representations of Hecke algebras and characters of symmetric groups, preprint 2001, to appear in Studies in Memory of Issai Schur, edited by A. Joseph, A. Melnikov, R. Rentschler, Progress in Mathematics.

[4] W. Feit, The representation theory of finite groups, North-Holland, 1982.

[5] G. James, A. Kerber, The representation theory of the symmetric group, Addison-Wesley 1981.

[6] B. Külshammer, J.B. Olsson, G.R. Robinson, Generalized blocks for symmetric groups, preprint 2002.

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[9] Sequence A006128. In N.J.A. Sloane’s On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/∼njas/sequences/